Problems and Solutions in Mathematical Finance Volume 2: Equity Derivatives [1 ed.] 1119965829, 978-1-119-96582-4, 978-1-119-96610-4, 978-1-119-96611-1, 978-1-119-19219-0

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Problems and Solutions in Mathematical Finance

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Problems and Solutions in Mathematical Finance Volume 2: Equity Derivatives ´ Eric Chin, Dian Nel and Sverrir Olafsson

This edition first published 2017 © 2017 John Wiley & Sons, Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley publishes in a variety of print and electronic formats and by print-on-demand. Some material included with standard print versions of this book may not be included in e-books or in print-on-demand. If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com. For more information about Wiley products, visit www.wiley.com. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom. If professional advice or other expert assistance is required, the services of a competent professional should be sought. A catalogue record for this book is available from the Library of Congress.

A catalogue record for this book is available from the British Library. ISBN 978-1-119-96582-4 (hardback) ISBN 978-1-119-96610-4 (ebk) ISBN 978-1-119-96611-1 (ebk) ISBN 978-1-119-19219-0 (obk) Cover design: Cylinder Cover image: © Attitude/Shutterstock Set in 10/12pt Times by Aptara Inc., New Delhi, India Printed in Great Britain by TJ International Ltd, Padstow, Cornwall, UK

“Blue dye is derived from the indigo plant and surpassed its parental colour” Xunzi, An Exhortation to Learning

Contents Preface

ix

About the Authors

xi

1

Basic Equity Derivatives Theory 1.1 Introduction 1.2 Problems and Solutions 1.2.1 Forward and Futures Contracts 1.2.2 Options Theory 1.2.3 Hedging Strategies

1 1 8 8 15 27

2

European Options 2.1 Introduction 2.2 Problems and Solutions 2.2.1 Basic Properties 2.2.2 Black–Scholes Model 2.2.3 Tree-Based Methods 2.2.4 The Greeks

63 63 74 74 89 190 218

3

American Options 3.1 Introduction 3.2 Problems and Solutions 3.2.1 Basic Properties 3.2.2 Time-Independent Options 3.2.3 Time-Dependent Options

267 267 271 271 292 305

4

Barrier Options 4.1 Introduction 4.2 Problems and Solutions 4.2.1 Probabilistic Approach 4.2.2 Reflection Principle Approach 4.2.3 Further Barrier-Style Options

351 351 357 357 386 408

viii

Contents

5

Asian Options 5.1 Introduction 5.2 Problems and Solutions 5.2.1 Discrete Sampling 5.2.2 Continuous Sampling

439 439 443 443 480

6

Exotic Options 6.1 Introduction 6.2 Problems and Solutions 6.2.1 Path-Independent Options 6.2.2 Path-Dependent Options

531 531 532 532 586

7

Volatility Models 7.1 Introduction 7.2 Problems and Solutions 7.2.1 Historical and Implied Volatility 7.2.2 Local Volatility 7.2.3 Stochastic Volatility 7.2.4 Volatility Derivatives

647 647 652 652 685 710 769

A Mathematics Formulae

787

B Probability Theory Formulae

797

C Differential Equations Formulae

813

Bibliography

821

Notation

825

Index

829

Preface Mathematical finance is a highly challenging and technical discipline. Its fundamentals and applications are best understood by combining a theoretically solid approach with extensive exercises in solving practical problems. That is the philosophy behind all four volumes in this series on mathematical finance. This second of four volumes in the series Problems and Solutions in Mathematical Finance is devoted to the discussion of equity derivatives. In the first volume we developed the probabilistic and stochastic methods required for the successful study of advanced mathematical finance, in particular different types of pricing models. The techniques applied in this volume assume good knowledge of the topics covered in Volume 1. As we believe that good working knowledge of mathematical finance is best acquired through the solution of practical problems, all the volumes in this series are built up in a way that allows readers to continuously test their knowledge as they work through the texts. This second volume starts with the analysis of basic derivatives, such as forwards and futures, swaps and options. The approach is bottom up, starting with the analysis of simple contracts and then moving on to more advanced instruments. All the major classes of options are introduced and extensively studied, starting with plain European and American options. The text then moves on to cover more complex contracts such as barrier, Asian and exotic options. In each option class, different types of options are considered, including time-independent and time-dependent options, or non-path-dependent and path-dependent options. Stochastic financial models frequently require the fixing of different parameters. Some can be extracted directly from market data, others need to be fixed by means of numerical methods or optimisation techniques. Depending on the context, this is done in different ways. In the riskneutral world, the drift parameter for the geometric Brownian motion (Black–Scholes model) is extracted from the bond market (i.e., the returns on risk-free debt). The volatility parameter, in contrast, is generally determined from market prices, as the so-called implied volatility. However, if a stochastic process is to be fitted to known price data, other methods need to be consulted, such as maximum-likelihood estimation. This method is applied to a number of stochastic processes in the chapter on volatility models. In all option models, volatility presents one of the most important quantities that determine the price and the risk of derivatives contracts. For this reason, considerable effort is put into their discussion in terms of concepts, such as implied, local and stochastic volatilities, as well as the important volatility surfaces. At the end of this volume, readers will be equipped with all the major tools required for the modelling and the pricing of a whole range of different derivatives contracts. They will

x

Preface

therefore be ready to tackle new techniques and challenges discussed in the next two volumes, including interest-rate modelling in Volume 3 and foreign exchange/commodity derivatives in Volume 4. As in the first volume, we have the following note to the student/reader: Please try hard to solve the problems on your own before you look at the solutions!

About the Authors Eric Chin is a quantitative analyst at an investment bank in the City of London where he is involved in providing guidance on price testing methodologies and their implementation, formulating model calibration and model appropriateness on commodity and credit products. Prior to joining the banking industry he worked as a senior researcher at British Telecom investigating radio spectrum trading and risk management within the telecommunications sector. He holds an MSc in Applied Statistics and an MSc in Mathematical Finance both from University of Oxford. He also holds a PhD in Mathematics from University of Dundee. Dian Nel has more than 10 years of experience in the commodities sector. He currently works in the City of London where he specialises in oil and gas markets. He holds a BEng in Electrical and Electronic Engineering from Stellenbosch University and an MSc in Mathematical Finance from Christ Church, Oxford University. He is a Chartered Engineer registered with the Engineering Council UK. ´ Sverrir Olafsson is Professor of Financial Mathematics at Reykjavik University; a Visiting Professor at Queen Mary University, London and a director of Riskcon Ltd, a UK based risk management consultancy. Previously he was a Chief Researcher at BT Research and held academic positions at The Mathematical Departments of Kings College, London; UMIST ´ Manchester and The University of Southampton. Dr Olafsson is the author of over 95 refereed academic papers and has been a key note speaker at numerous international conferences and seminars. He is on the editorial board of three international journals. He has provided an extensive consultancy on financial risk management and given numerous specialist seminars to finance specialists. In the last five years his main teaching has been MSc courses on Risk Management, Fixed Income, and Mathematical Finance. He has an MSc and PhD in mathematical physics from the Universities of T¨ubingen and Karlsruhe respectively.

1 Basic Equity Derivatives Theory In finance, an equity derivative belongs to a class of derivative instruments whose underlying asset is a stock or stock index. Hence, the value of an equity derivative is a function of the value of the stock or index. With a growing interest in the stock markets of the world, and the prevalence of employee stock options as a form of compensation, equity derivatives continue to expand with new product structures continuously being offered. In this chapter, we introduce the concept of equity derivatives with emphasis on forwards, futures, option contracts and also different types of hedging strategies.

1.1 INTRODUCTION Among the many equity derivatives that are actively traded in the market, options and futures are by far the most commonly traded financial instruments. The following is the basic vocabulary of different types of derivatives contracts: s Option A contract that gives the holder the right but not the obligation to buy or sell an asset for a fixed price (strike/exercise price) at or before a fixed expiry date. s Call Option A contract that gives the holder the right to buy an asset for a fixed price (strike/exercise price) at or before a fixed expiry date. s Put Option A contract that gives the holder the right to sell an asset for a fixed price (strike/exercise price) at or before a fixed expiry date. s Payoff Difference between the market price and the strike price depending on derivative type. s Intrinsic Value The payoff that would be received/paid if the option was exercised when the underlying asset is at its current level. s Time Value Value that the option is above its intrinsic value. The relationship can be written as Option Price = Intrinsic Value + Time Value. s Forward/Futures A contract that obligates the buyer and seller to trade an underlying, usually a commodity or stock price index, at some specified time in the future. The difference between a forward and a futures contract is that forwards are over-the-counter (OTC) products which are customised agreements between two counterparties. In contrast, futures are standardised contracts traded on an official exchange and are marked to market on a daily basis. Hence, futures contracts do not carry any credit risk (the risk that a party will not meet its contractual obligations). s Swap An OTC contract in which two counterparties exchange cash flows. s Stock Index Option A contract that gives the holder the right but not the obligation to buy or sell a specific amount of a particular stock index for an agreed fixed price at or before

2

1.1 INTRODUCTION

s

s s s s

s s

a fixed expiry date. As it is not feasible to deliver an actual stock index, this contract is usually settled in cash. Stock Index Futures A contract that obligates the buyer and seller to trade a quantity of a specific stock index on an official exchange at a price agreed between two parties with delivery on a specified future date. Like the stock index option, this contract is usually settled in cash. Strike/Exercise Price Fixed price at which the owner of an option can buy (for a call option) or sell (for a put option) the underlying asset. Expiry Date/Exercise Date The last date on which the option contract is still valid. After this date, the option contract becomes worthless. Delivery Date The last date by which the underlying commodity or stock price index (usually cash payment based on the underlying stock price index) for a forward/futures contract must be delivered to fulfil the requirements of the contract. Discounting Multiplying an amount by a discount factor to compute its present value (discounted value). It is the opposite of compounding, where interest is added to an amount so that the added interest also earns interest from then on. If we assume the risk-free interest rate 𝑟 is a constant and continuously compounding, then the present value at time 𝑡 of a certain payoff 𝑀 at time 𝑇 , for 𝑡 < 𝑇 , is 𝑀𝑒−𝑟(𝑇 −𝑡) . Hedge An investment position intended to reduce the risk from adverse price movements in an asset. A hedge can be constructed using a combination of stocks and derivative products such as options and forwards. Contingent Claim A claim that depends on a particular event such as an option payoff, which depends on a stock price at some future date.

Within the context of option contracts we subdivide them into option style or option family, which denotes the class into which the type of option contract falls, usually defined by the dates on which the option may be exercised. These include: s European Option An option that can only be exercised on the expiry date. s American Option An option that can be exercised any time before the expiry date. s Bermudan Option An option that can only be exercised on predetermined dates. Hence, this option is intermediate between a European option and an American option. Unless otherwise stated, all the options discussed in this chapter are considered to be European.

Option Trading In option trading, the transaction involves two parties: a buyer and a seller. s The buyer of an option is said to take a long position in the option, whilst the seller is said to take a short position in the option. s The buyer or owner of a call (put) option has the right to buy (sell) an asset at a specified price by paying a premium to the seller or writer of the option, who will assume the obligation to sell (buy) the asset should the owner of the option choose to exercise (enforce) the contract.

1.1 INTRODUCTION

3

s The payoff of a call option at expiry time 𝑇 is defined as Ψ(𝑆𝑇 ) = max{𝑆𝑇 − 𝐾, 0} where 𝑆𝑇 is the price of the underlying asset at expiry time 𝑇 and 𝐾 is the strike price. If 𝑆𝑇 > 𝐾 at expiry, then the buyer of the call option should exercise the option by paying a lower amount 𝐾 to obtain an asset worth 𝑆𝑇 . However, if 𝑆𝑇 ≤ 𝐾 then the buyer of the call option should not exercise the option because it would not make any financial sense to pay a higher amount 𝐾 to obtain an asset which is of a lower value 𝑆𝑇 . Here, the option expires worthless. In general, the profit earned by the buyer of the call option is Υ(𝑆𝑇 ) = max{𝑆𝑇 − 𝐾, 0} − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) where 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is the premium paid at time 𝑡 < 𝑇 (written on the underlying asset 𝑆𝑡 ) in order to enter into a call option contract. Neglecting the premium for buying an option, a call option is said to be in-the-money (ITM) if the buyer profits when the option is exercised (𝑆𝑇 > 𝐾). In contrast, a call option is said to be out-of-the-money (OTM) if the buyer loses when the option is exercised (𝑆𝑇 < 𝐾). Finally, a call option is said be to at-the-money (ATM) if the buyer neither loses nor profits when the option is exercised (𝑆𝑇 = 𝐾). Figure 1.1 illustrates the concepts we have discussed. Payoff/Profit

Payoff Profit

+ ( − (

;

;

)

)

Figure 1.1

Long call option payoff and profit diagram.

s The payoff of a put option at expiry time 𝑇 is defined as Ψ(𝑆𝑇 ) = max{𝐾 − 𝑆𝑇 , 0} where 𝑆𝑇 is the price of the underlying asset at expiry time 𝑇 and 𝐾 is the strike price. If 𝐾 > 𝑆𝑇 at expiry, then the buyer of the put option should exercise the option by selling the asset worth 𝑆𝑇 for a higher amount 𝐾. However, if 𝐾 ≤ 𝑆𝑇 then the buyer of the put

4

1.1 INTRODUCTION

option should not exercise the option because it would not make any financial sense to sell the asset worth 𝑆𝑇 for a lower amount 𝐾. Here, the option expires worthless. In general, the profit earned by the buyer of the put option is Υ(𝑆𝑇 ) = max{𝐾 − 𝑆𝑇 , 0} − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) where 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is the premium paid at time 𝑡 < 𝑇 (written on the underlying asset 𝑆𝑡 ) in order to enter into a put option contract. Neglecting the premium for buying an option, a put option is said to be ITM if the buyer profits when the option is exercised (𝐾 > 𝑆𝑇 ). In contrast, a put option is said to be OTM if the buyer loses when the option is exercised (𝐾 < 𝑆𝑇 ). Finally, a put option is said to be ATM if the buyer neither loses nor profits when the option is exercised (𝑆𝑇 = 𝐾). Figure 1.2 illustrates the concepts we have discussed. Payoff/Profit

Payoff − (

;

)

− (

;

)

− (

;

) Profit

Figure 1.2

Long put option payoff and profit diagram.

Forward Contract In a forward contract, the transaction is executed between two parties: a buyer and a seller. s The buyer of the underlying commodity or stock index is referred to as the long side whilst the seller is known as the short side. s The contractual obligation to buy the asset at the agreed price on a specified future date is known as the long position. A long position profits when the price of an asset rises. s The contractual obligation to sell the asset at the agreed price on a specified future date is known as the short position. A short position profits when the price of an asset falls. s For a long position, the payoff of a forward contract at the delivery time 𝑇 is Π𝑇 = 𝑆𝑇 − 𝐹 (𝑡, 𝑇 ) where 𝑆𝑇 is the spot price (or market price) at the delivery time 𝑇 and 𝐹 (𝑡, 𝑇 ) is the forward price initiated at time 𝑡 < 𝑇 to be delivered at time 𝑇 .

1.1 INTRODUCTION

5

s For a short position, the payoff of a forward contract at the delivery time 𝑇 is Π𝑇 = 𝐹 (𝑡, 𝑇 ) − 𝑆𝑇 where 𝑆𝑇 is the spot price (or market price) at the delivery time 𝑇 and 𝐹 (𝑡, 𝑇 ) is the forward price initiated at time 𝑡 < 𝑇 to be delivered at time 𝑇 . s Since there is no upfront payment to enter into a forward contract, the profit at delivery time 𝑇 is the same as the payoff of a forward contract at time 𝑇 . Figure 1.3 illustrates the concepts we have discussed. Payoff Long Forward (

)

(

− (

)

)

Short Forward

Figure 1.3

Long and short forward payoffs diagram.

Futures Contract Similar to a forward contract, a futures contract is also an agreement between two parties in which the buyer agrees to buy an underlying asset from the seller. The delivery of the asset occurs at a specified future date, where the price is determined at the time of initiation of the contract. As in the case of a forward contract, it costs nothing to enter into a futures contract. However, the differences between futures and forwards are as follows: s In a futures contract, the terms and conditions are standardised where trading takes place on a formal exchange with deep liquidity. s There is no default risk when trading futures contracts, since the exchange acts as a counterparty guaranteeing delivery and payment by use of a clearing house. s The clearing house protects itself from default by requiring its counterparties to settle profits and losses or mark to market their positions on a daily basis. s An investor can hedge his/her future position by engaging in an opposite transaction before the delivery date of the contract. In the futures market, margin is a performance guarantee. It is money deposited with the clearing house by both the buyer and the seller. There is no loan involved and hence, no interest is

6

1.1 INTRODUCTION

charged. To safeguard the clearing house, the exchange requires buyers/sellers to post margin (i.e., deposit funds) and settle their accounts on a daily basis. Prior to trading, the trader must post margin with their broker who in return will post margin with the clearing house. s Initial Margin Money that must be deposited in order to initiate a futures position. s Maintenance Margin Minimum margin amount that must be maintained; when the margin falls below this amount it must be brought back up to its initial level. Margin calculations are based on the daily settlement price, the average of the prices for trades during the closing period set by the exchange. s Variation Margin Money that must be deposited to bring it back to the initial margin amount. If the account margin is more than the initial margin, the investor can withdraw the funds for new positions. s Settlement Price Known also as the closing price for a stock. The settlement price is the price at which a derivatives contract settles once a given trading day has ended. The settlement price is used to calculate the margin at the end of each trading day. s Marking-to-Market Process of adding gains to or subtracting losses from the margin account daily, based on the change in the settlement prices from one day to the next. Termination of a futures position can be achieved by: s An offsetting trade (known as a back-to-back trade), entering into an opposite position in the same contract. s Payment of cash at expiration for a cash-settlement contract. s Delivery of the asset at expiration. s Exchange of physicals. Stock Split (Divide) Effect When a company issues a stock split (e.g., doubling the number of shares), the price is adjusted so as to keep the net value of all the stock the same as before the split. Stock Dividend Effect When dividends are paid during the life of an option contract they will inadvertently affect the price of the stock or asset. Here, the direction of the stock price will be determined based on the choice of the company whether it pays dividends to its shareholders or reinvests the money back in the business. Since we may regard dividends as a cash return to the shareholders, the reinvestment of the cash back into the business could create more profit and, depending on market sentiment, lead to an increase in stock price. Conversely, paying dividends to the shareholders will effectively reduce the stock price by the amount of the dividend payment, and as a result will affect the premium prices of options as well as futures and forwards. Hedging Strategies In the following we discuss how an investor can use options to design investment strategies with specific views on the stock price behaviour in the future. s Protective This hedging strategy is designed to insure an investor’s asset position (long buy or short sell).

1.1 INTRODUCTION

7

An investor who owns an asset and wishes to be protected from falling asset values can insure his asset by buying a put option written on the same asset. This combination of owning an asset and purchasing a put option on that asset is called a protective put. In contrast, an investor shorting an asset who will experience a loss if the asset price rises in value can insure his position by purchasing a call option written on the same asset. Such a combination of selling an asset and purchasing a call option on that asset is called a protective call. s Covered This hedging strategy involves the investor writing an option whilst holding an opposite position on the asset. The motivation for doing so is to generate additional income by receiving premiums from option buyers, and this strategy is akin to selling insurance. When the writer of an option has no position in the underlying asset, this form of option writing is known as naked writing. In a covered call, the investor would hold a long position on an asset and sell a call option written on the same asset. In a covered put, the investor would short sell an asset and sell a put option written on the same asset. s Collar This hedging strategy uses a combination of protective strategy and selling options to collar the value of an asset position within a specific range. By using a protective strategy, the investor can insure his asset position (long or short) whilst reducing the cost of insurance by selling an option. In a purchased collar, the strategy consists of a protective put and selling a call option whilst in a written collar, the strategy consists of a protective call and selling a put option. s Synthetic Forward A synthetic forward consists of a long call, 𝐶(𝑆 , 𝑡; 𝐾, 𝑇 ) and a short 𝑡 put, 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) written on the same asset 𝑆𝑡 at time 𝑡 with the same expiration date 𝑇 > 𝑡 and strike price 𝐾. At expiry time 𝑇 , the payoff is 𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 𝑆𝑇 − 𝐾 and, assuming a constant risk-free interest rate 𝑟 and by discounting the payoff back to time 𝑡, we have 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 − 𝐾𝑒−𝑟(𝑇 −𝑡) . The above equation is known as the put–call parity, tying the relationship between options and forward markets together. s Bull Spread An investor who enters a bull spread expects the stock price to rise and wishes to exploit this. For a bull call spread, it is composed of Bull Call Spread = 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) which consists of buying a call at time 𝑡 with strike price 𝐾1 and expiry 𝑇 and selling a call at time 𝑡 with strike price 𝐾2 , 𝐾2 > 𝐾1 and same expiry 𝑇 . For a bull put spread, it is composed of Bull Put Spread = 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) which consists of buying a put at time 𝑡 with strike price 𝐾1 and expiry 𝑇 and selling a put at time 𝑡 with strike price 𝐾2 , 𝐾2 > 𝐾1 and same expiry 𝑇 .

8

1.2.1 Forward and Futures Contracts

s Bear Spread The strategy behind the bear spread is the opposite of a bull spread. Here, the investor who enters a bear spread expects the stock price to fall. For a bear call spread, it is composed of Bear Call Spread = 𝐶(𝑆𝑡 .𝑡; 𝐾2 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) which consists of selling a call at time 𝑡 with strike price 𝐾1 and expiry 𝑇 and buying a call at time 𝑡 with strike price 𝐾2 , 𝐾2 > 𝐾1 and same expiry 𝑇 . For a bear put spread, it is composed of Bear Put Spread = 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) which consists of selling a put at time 𝑡 with strike price 𝐾1 and expiry 𝑇 and buying a put at time 𝑡 with strike price 𝐾2 , 𝐾2 > 𝐾1 and same expiry 𝑇 . s Butterfly Spread The investor who enters a butterfly spread expects that the stock price will not change significantly. It is a neutral strategy combining bull and bear spreads. s Straddle This strategy is used if an investor believes that a stock price will move significantly, but is unsure in which direction. Here such a strategy depends on the volatility of the stock price rather than the direction of the stock price changes. For a long straddle, it is composed of Long Straddle = 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) which consists of buying a call and a put option at time 𝑡 with the same strike price 𝐾 and expiry 𝑇 . For a short straddle, it is composed of Short Straddle = −𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) which consists of selling a call and a put option at time 𝑡 with the same strike price 𝐾 and expiry 𝑇 . s Strangle The strangle hedging strategy is a variation of the straddle with the key difference that the options have different strike prices but expire at the same time. s Strip/Strap The strip and strap strategies are modifications of the straddle, principally used in volatile market conditions. However, unlike a straddle which has an unbiased outlook on the stock price movement, investors who use a strip (strap) strategy would exploit on downward (upward) movement of the stock price.

1.2 PROBLEMS AND SOLUTIONS 1.2.1

Forward and Futures Contracts

1. Consider an investor entering into a forward contract on a stock with spot price $10 and delivery date 6 months from now. The forward price is $12.50. Draw the payoff diagrams for both the long and short forward position of the contract. Solution: See Figure 1.4.

1.2.1 Forward and Futures Contracts

9

Payoff Long Forward $12.50

Spot Price

$12.50

−$12.50

Short Forward

Figure 1.4

Long and short forward payoff diagram.

2. In terms of credit risk, is a forward contract riskier than a futures contract? Explain. Solution: Given that forward contracts are traded OTC between two parties and futures contracts are traded on exchanges which require margin accounts, forward contracts are riskier than futures contracts. 3. Suppose ABC company shares are trading at $25 and pay no dividends and that the riskfree interest rate is 5% per annum. The forward price for delivery in 1 year’s time is $28. Draw the payoff and profit diagrams for a long position for this contract. Solution: As there is no cost involved in entering into a forward contract, the payoff and profit diagrams coincide (see Figure 1.5). Payoff/Profit Long Forward

$28

Spot Price

−$28

Figure 1.5

Long forward payoff and profit diagram.

10

1.2.1 Forward and Futures Contracts

4. Consider a stock currently worth $100 per share with the risk-free interest rate 2% per annum. The futures price for a 1-year contract is worth $104. Show that there exists an arbitrage opportunity by entering into a short position in this futures contract. Solution: At current time 𝑡 = 0, a speculator can borrow $100 from the bank, buy the stock and short a futures contract. At delivery time 𝑇 = 1 year, the outstanding loan is now worth 100𝑒0.02×1 = $102.02. By delivering the stock to the long contract holder and receiving $104, the speculator can make a riskless profit of $104 − $102.02 = $1.98. 5. Let the current stock price be $75 with the risk-free interest rate 2.5% per annum. Assume the futures price for a 1-year contract is worth $74. Show that there exists an arbitrage opportunity by entering into a long position in this futures contract. Solution: At current time 𝑡 = 0, a speculator can short sell the stock, invest the proceeds in a bank account at the risk-free rate and then long a futures contract. At time 𝑇 = 1 year, the amount of money in the bank will grow to 75𝑒0.025×1 = $76.89. After paying for the futures contract which is priced at $74, the speculator can then return the stock to its owner. Thus, the speculator can make a riskless profit of $76.89 − $74 = $2.89. 6. An investor holds a long position in a stock index futures contract with a delivery date 3 months from now. The value of the contract is $250 times the level of the index at the start of the contract, and each index point movement represents a gain or a loss of $250 per contract. The futures contract at the start of the contract is valued at $250,000, and the initial margin deposit is $15,000 with a maintenance margin of $13,750 per contract. Table 1.1 shows the stock index movement over a 4-day period. Table 1.1 Day 1 2 3 4

Daily closing stock index. Closing Stock Index 1002 994 998 997

Calculate the initial stock index at the start of the contract. By setting up a table, calculate the daily marking-to-market, margin balance and the variation margin over a 4-day period. Solution: Since the futures contract is valued at $250,000 at the start of the contract, the = 1000. initial stock index is 250,000 250 Table 1.2 displays the daily marking-to-market, margin balance and the variation margin in order to maintain the maintenance margin. On Day 0, the initial balance is the initial margin requirement of $15,000 while on Day 1, as the change in the stock index is increased by 2 points, the margin balance is increased by $250 × 2 = $500. On Day 2, the margin balance is $13,500 which is below the maintenance margin level of $13,750. Therefore, a deposit of $1,500 is needed to

1.2.1 Forward and Futures Contracts Table 1.2

11

Daily movements of stock index.

Day

Required Deposit

Closing Stock Index

Daily Change

Marking-toMarket

Margin Balance

Variation Margin

0 1 2 3 4

$15,000 0 0 $1,500 0

1000 1002 994 998 997

0 +2 −8 +4 −1

0 $500 −$2,000 $1,000 −$250

$15,000 $15,500 $13,500 $16,000 $15,750

0 0 $1,500 0 0

bring the margin back to the margin requirement of $15,000. Hence, the variation margin is $1,500 occurring on Day 2. 7. An investor wishes to enter into 10 stock index futures contracts where the value of a contract is $250 times the level of the index at the start of the contract and each index point movement represents a gain or a loss of $250 per contract. The stock index at the start of the contract is 1,000 points and the initial margin deposit is 10% of the total futures contract value. Let the continuously compounded interest rate be 5% which can be earned on the margin balance and the maintenance margin be 85% of the initial margin deposit. Suppose the investor position is marked on a weekly basis. What does the maximum stock index need to be in order for the investor to receive a margin call on week 1. Solution: At the start of the contract the total futures contract value is $250 × 1,000 × 10 = 10 = $250,000. The mainte$2,500,000 and the initial margin deposit is $2,500,000 × 100 nance margin is therefore $250,000 ×

85 100

= $187,500. To describe the movement of the stock index for week 1, see Table 1.3.

Table 1.3 Week 0 1

Movement of stock index on week 1. Closing Stock Index

Weekly Change

Marking-toMarket

Margin Balance

Variation Margin

1000 𝑥

0 𝑥 − 1000

0 $2,500 ×(𝑥 − 1000)

$250,000 $250,000 + $2,500 ×(𝑥 − 1000)

0 $187,500

Thus, in order to invoke a margin call we can set 2500(𝑥 − 1000) + 250,000 = 187,500 𝑥 = 975. Therefore, if the stock index were to fall to values below 975 points then a margin call will be issued on week 1. 8. Let 𝑆𝑡 denote the price of a stock with a dividend payment 𝛿 ≥ 0 at time 𝑡. What is the price of the stock immediately after the dividend payment?

12

1.2.1 Forward and Futures Contracts

Solution: Let 𝑆𝑡+ denote the price of the stock immediately after the dividend payment. Therefore, 𝑆𝑡+ = 𝑆𝑡 − 𝛿.

9. Consider the price of a futures contract 𝐹 (𝑡, 𝑇 ) with delivery time 𝑇 on a stock with price 𝑆𝑡 at time 𝑡 (𝑡 < 𝑇 ). Suppose the stock does not pay any dividends. Show that under the no-arbitrage condition the futures contract price is 𝐹 (𝑡, 𝑇 ) = 𝑆𝑡 𝑒𝑟(𝑇 −𝑡) where 𝑟 is the risk-free interest rate. Solution: We prove this result via contradiction. If 𝐹 (𝑡, 𝑇 ) > 𝑆𝑡 𝑒𝑟(𝑇 −𝑡) then at time 𝑡 an investor can short the futures contract worth 𝐹 (𝑡, 𝑇 ) and then borrow an amount 𝑆𝑡 from the bank to buy the asset. By time 𝑇 the bank loan will amount to 𝑆𝑡 𝑒𝑟(𝑇 −𝑡) . Since 𝐹 (𝑡, 𝑇 ) > 𝑆𝑡 𝑒𝑟(𝑇 −𝑡) then using the money received at delivery time 𝑇 , the investor can pay off the loan, deliver the asset and make a risk-free profit 𝐹 (𝑡, 𝑇 ) − 𝑆𝑡 𝑒𝑟(𝑇 −𝑡) > 0. In contrast, if 𝐹 (𝑡, 𝑇 ) < 𝑆𝑡 𝑒𝑟(𝑇 −𝑡) then at time 𝑡 an investor can long the futures contract, short sell the stock valued at 𝑆𝑡 and then put the money in the bank. By time 𝑇 the money in the bank will grow to 𝑆𝑡 𝑒𝑟(𝑇 −𝑡) and after returning the stock (from the futures contract) the investor will make a risk-free profit 𝑆𝑡 𝑒𝑟(𝑇 −𝑡) − 𝐹 (𝑡, 𝑇 ) > 0. Therefore, under the no-arbitrage condition we must have 𝐹 (𝑡, 𝑇 ) = 𝑆𝑡 𝑒𝑟(𝑇 −𝑡) . 10. Consider the price of a futures contract 𝐹 (𝑡, 𝑇 ) with delivery time 𝑇 on a stock with price 𝑆𝑡 at time 𝑡 (𝑡 < 𝑇 ). Throughout the life of the futures contract the stock pays discrete dividends 𝛿𝑖 , 𝑖 = 1, 2, … , 𝑛 where 𝑡 < 𝑡1 < 𝑡2 < ⋯ < 𝑡𝑛 < 𝑇 . Show that under the no-arbitrage condition the futures contract price is 𝐹 (𝑡, 𝑇 ) = 𝑆𝑡 𝑒𝑟(𝑇 −𝑡) −

𝑛 ∑ 𝑖=1

𝛿𝑖 𝑒𝑟(𝑇 −𝑡𝑖 )

where 𝑟 is the risk-free interest rate. Solution: Suppose that over the life of the futures contract the stock pays dividends 𝛿𝑖 at time 𝑡𝑖 , 𝑖 = 1, 2, … , 𝑛 where 𝑡 < 𝑡1 < 𝑡2 < ⋯ < 𝑡𝑛 < 𝑇 . When dividends are paid, the stock price 𝑆𝑡 is reduced by the present values of all the dividends paid, that is 𝑆𝑡 −

𝑛 ∑ 𝑖=1

𝛿𝑖 𝑒−𝑟(𝑡𝑖 −𝑡) .

1.2.1 Forward and Futures Contracts

13

Hence, using the same steps as discussed in Problem 1.2.1.9 (page 12), the futures price is ( 𝐹 (𝑡, 𝑇 ) =

𝑆𝑡 −

𝑛 ∑ 𝑖=1

) 𝛿𝑖 𝑒

= 𝑆𝑡 𝑒𝑟(𝑇 −𝑡) −

−𝑟(𝑡𝑖 −𝑡)

𝑛 ∑ 𝑖=1

𝑒𝑟(𝑇 −𝑡)

𝛿𝑖 𝑒𝑟(𝑇 −𝑡𝑖 ) .

11. Consider the number of stocks owned by an investor at time 𝑡 as 𝐴𝑡 where each of the stocks pays a continuous dividend yield 𝐷. Assume that all the dividend payments are reinvested in the stock. Show that the number of stocks owned by time 𝑇 (𝑡 < 𝑇 ) is 𝐴𝑇 = 𝐴𝑡 𝑒𝐷(𝑇 −𝑡) . Next consider the price of a futures contract 𝐹 (𝑡, 𝑇 ) with delivery time 𝑇 on a stock with price 𝑆𝑡 at time 𝑡 (𝑡 < 𝑇 ). Suppose the stock pays a continuous dividend yield 𝐷. Using the above result, show that under the no-arbitrage condition the futures contract price is 𝐹 (𝑡, 𝑇 ) = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) where 𝑟 is the risk-free interest rate. Solution: We first divide the time interval [𝑡, 𝑇 ] into 𝑛 sub-intervals such that 𝑡𝑖 = 𝑡 + 𝑖(𝑇 − 𝑡) , 𝑖 = 1, 2, … , 𝑛 with 𝑡0 = 𝑡 and 𝑡𝑛 = 𝑇 . By letting the dividend payment at time 𝑛 𝑡𝑖 be 𝛿𝑖 =

𝐷(𝑇 − 𝑡) 𝑆𝑡 𝑛

for 𝑖 = 1, 2, … , 𝑛, and because all the dividends are reinvested in the stock, the number of stocks held becomes ] [ 𝐷(𝑇 − 𝑡) 𝐴 𝑡1 = 𝐴 𝑡0 1 + 𝑛 ] ]2 [ [ 𝐷(𝑇 − 𝑡) 𝐷(𝑇 − 𝑡) 𝐴𝑡2 = 𝐴𝑡1 1 + = 𝐴𝑡 0 1 + 𝑛 𝑛 ] ]3 [ [ 𝐷(𝑇 − 𝑡) 𝐷(𝑇 − 𝑡) 𝐴 𝑡3 = 𝐴 𝑡2 1 + = 𝐴𝑡 0 1 + 𝑛 𝑛 ⋮ ] ]𝑛 [ [ 𝐷(𝑇 − 𝑡) 𝐷(𝑇 − 𝑡) . = 𝐴𝑡 0 1 + 𝐴𝑡𝑛 = 𝐴𝑡𝑛−1 1 + 𝑛 𝑛

14

1.2.1 Forward and Futures Contracts

Because 𝐴𝑡0 = 𝐴𝑡 and 𝐴𝑡𝑛 = 𝐴𝑇 , therefore ]𝑛 [ 𝐷(𝑇 − 𝑡) 𝐴𝑇 = 𝐴𝑡 1 + 𝑛 and taking limits 𝑛 → ∞ we have ]𝑛 [ 𝐷(𝑇 − 𝑡) = 𝐴𝑡 𝑒𝐷(𝑇 −𝑡) . lim 𝐴𝑇 = 𝐴𝑡 lim 1 + 𝑛→∞ 𝑛→∞ 𝑛 From the above result we can deduce that investing one stock at time 𝑡 will lead to a total growth of 𝑒𝐷(𝑇 −𝑡) by time 𝑇 . Hence, if we start by buying 𝑒−𝐷(𝑇 −𝑡) number of stocks 𝑆𝑡 at time 𝑡 it will grow to one stock at time 𝑇 . The total value of the stock at time 𝑡 is therefore 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) and following the arguments in Problem 1.2.1.9 (page 12) the futures price is 𝐹 (𝑡, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑒𝑟(𝑇 −𝑡) = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) .

12. Suppose an asset is currently worth $20 and the 6-month futures price of this asset is $22.50. By assuming the stock does not pay any dividends and the risk-free interest rate is the same for all maturities, calculate the 1-year futures price of this asset. Solution: By definition the futures price is 𝐹 (𝑡, 𝑇 ) = 𝑆𝑡 𝑒𝑟(𝑇 −𝑡) where 𝑡 is the time of the start of the contract, 𝑇 is the delivery time, 𝑆𝑡 is the spot price at time 𝑡 and 𝑟 is the risk-free interest rate. By setting 𝑡 = 0, 𝑆0 = $20 and 𝑇1 = 0.5 years we have 𝐹 (0, 𝑇1 ) = 𝑆0 𝑒𝑟𝑇1 = $22.50. Hence, 𝑟 = 2 log

(

22.50 20

) = 2 log 1.125.

Therefore, for a 1-year futures price, 𝑇2 = 1 year 𝐹 (0, 𝑇2 ) = 𝑆0 𝑒𝑟𝑇2 = $20𝑒2 log 1.125×1 = $25.31.

13. Assume an investor buys 100,000 stocks of XYZ company and holds them for 3 years. Each of the stocks held pays a continuous dividend yield of 4% per annum and the investor

1.2.2 Options Theory

15

reinvests all the dividends when they are paid. Calculate the additional number of shares the investor would have at the end of 3 years. Solution: Let 𝐴0 = 100,000, 𝐷 = 0.04 and 𝑇 = 3 years. Therefore, by the end of 3 years, the number of shares owned by the investor is 𝐴𝑇 = 𝐴0 𝑒𝐷𝑇 = 100,000𝑒0.04×3 = 112,749.69. Therefore, the additional number of shares the investor has by the end of year 3 is 𝐴𝑇 − 𝐴0 = 112,749.69 − 100,000 = 12,749.69 ≃ 12,749.

14. Let the current stock price be $30 with two dividend payments in 6 months and 9 months from today of $1.50 and $1.80, respectively. The continuously compounded risk-free interest rate is 5% per annum. Find the price of a 1-year futures contract. 6 = 0.5 years, 𝑡2 = Solution: Let 𝑆0 = $30, 𝑡1 = 12 $1.80, 𝑟 = 0.05 and 𝑇 = 1 year. Therefore, the price of a 1-year futures contract is

9 12

= 0.75 years, 𝛿1 = $1.50, 𝛿2 =

𝐹 (0, 𝑇 ) = 𝑆0 𝑒𝑟𝑇 − 𝛿1 𝑒𝑟(𝑇 −𝑡1 ) − 𝛿2 𝑒𝑟(𝑇 −𝑡2 ) = 30𝑒0.05×1 − 1.50𝑒0.05×(1−0.5) − 1.80𝑒0.05×(1−0.75) = $28.18.

15. Let the current price of a stock be $12.75 that pays a continuous dividend yield 𝐷. Suppose the risk-free interest rate is 6% per annum and the price of a 6-month forward contract is $13.25. Find 𝐷. Solution: Let 𝑆0 = $12.50, 𝑟 = 0.06, 𝑇 = 0.5 years and 𝐹 (0, 𝑇 ) = $13.25. Since 𝐹 (0, 𝑇 ) = 𝑆0 𝑒(𝑟−𝐷)𝑇 , 12.75𝑒(0.06−𝐷)×0.5 = 13.25 𝐷 = 0.06 − log

(

) 1 13.25 × 12.75 0.5

= 0.020395. Hence, the dividend yield is 𝐷 = 2.0395% per annum.

1.2.2

Options Theory

1. Consider a long call option with strike price 𝐾 = $100. The current stock price is 𝑆𝑡 = $105 and the call premium is $10. What is the intrinsic value of the call option at time 𝑡? Find the payoff and profit if the spot price at the option expiration date 𝑇 is 𝑆𝑇 = $120. Draw the payoff and profit diagrams.

16

1.2.2 Options Theory

Solution: By defining 𝑆𝑡 = $105, 𝑆𝑇 = $120, 𝐾 = $100 and the call premium as 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = $10, the intrinsic value of the call option at time 𝑡 is Ψ(𝑆𝑡 ) = max{𝑆𝑡 − 𝐾, 0} = max{105 − 100, 0} = $5. At expiry time 𝑇 , the payoff is Ψ(𝑆𝑇 ) = max{𝑆𝑇 − 𝐾, 0} = max{120 − 100, 0} = $20 and the profit is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = $20 − $10 = $10. Figure 1.6 shows the payoff and profit diagrams for a long call option at the expiry time 𝑇 . Here the profit diagram is a vertical shift of the call payoff based on the premium paid. Payoff/Profit

Payoff

Profit

$100

$110

−$10

Figure 1.6

Long call option payoff and profit diagrams.

2. Consider a long put option with strike price 𝐾 = $100. The current stock price is 𝑆𝑡 = $80 and the put premium is $5. What is the intrinsic value of the put option at time 𝑡? Find the payoff and profit if the spot price at the option expiration date 𝑇 is 𝑆𝑇 = $75. Draw the payoff and profit diagrams. Solution: By defining 𝑆𝑡 = $80, 𝑆𝑇 = $75, 𝐾 = $100 and the put premium as 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = $5, the intrinsic value of the call option at time 𝑡 is Ψ(𝑆𝑡 ) = max{𝐾 − 𝑆𝑡 , 0} = max{100 − 80, 0} = $20.

1.2.2 Options Theory

17

At expiry time 𝑇 , the payoff is Ψ(𝑆𝑇 ) = max{𝐾 − 𝑆𝑇 , 0} = max{100 − 75, 0} = $25 and the profit is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = $25 − $5 = $20. Figure 1.7 shows the payoff and profit diagrams for a long put option at the expiry time 𝑇 . Here the profit diagram is a vertical shift of the put payoff based on the premium paid. Payoff/Profit

$100 Payoff $95

$95

$100

−$5 Profit

Figure 1.7

Long put option payoff and profit diagrams.

3. At time 𝑡 we consider a long call option with a strike price 𝐾 and a long forward contract with price 𝐾 on the same underlying asset 𝑆𝑡 . The premium for the call option is 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). Draw the profit diagram for these two financial instruments at the option expiry date 𝑇 . Under what conditions is the call option more profitable than the forward contract, and vice versa? Solution: Figure 1.8 shows the profit diagram for a long call and a long forward contract at expiry date 𝑇 . At time 𝑇 the break even at the profit level is at 𝑆𝑇 = 𝐾 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). Therefore, if the stock 𝑆𝑇 ≤ 𝐾 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) then the call option is more profitable as the loss is fixed with the amount of premium paid. However, if 𝑆𝑇 > 𝐾 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) then the forward contract is more profitable since there is no cost in entering a forward contract. 4. At time 𝑡 we consider a long put option with strike price 𝐾 and a short forward contract with price 𝐾 on the same underlying asset 𝑆𝑡 . The premium for the put option is 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). Draw the profit diagram for these two financial instruments at the option expiry date 𝑇 .

18

1.2.2 Options Theory Profit

Long Forward Long Call Option

− (

;

+ (

)

;

)

−

Figure 1.8

Long call option and long forward profit diagrams.

Under what conditions is the put option more profitable than the forward contract, and vice versa? Solution: Figure 1.9 shows the profit diagram for a long put and a short forward contract at expiry date 𝑇 . Profit

Short Forward − (

;

)

− (

;

)

− (

;

) Long Put Option

Figure 1.9

Long put option and short forward profit diagrams.

At time 𝑇 the break-even point is at 𝑆𝑇 = 𝐾 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). Therefore, if the stock 𝑆𝑇 ≥ 𝐾 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) then the put option is more profitable as the loss is fixed with the amount of premium paid. However, if 𝑆𝑇 < 𝐾 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) then the forward contract is more profitable since there is no cost in entering a forward contract.

1.2.2 Options Theory

19

5. Consider a short call option with strike price 𝐾 = $100. The current stock price is 𝑆𝑡 = $105 and the call premium is $10. What is the intrinsic value of the call option at time 𝑡? Find the payoff and profit if the spot price at the option expiration date 𝑇 is 𝑆𝑇 = $120. Draw the payoff and profit diagrams. Solution: By defining 𝑆𝑡 = $105, 𝑆𝑇 = $120, 𝐾 = $100 and the call premium as 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = $10, the intrinsic value of the short call option at time 𝑡 is Ψ(𝑆𝑡 ) = − max{𝑆𝑡 − 𝐾, 0} = min{𝐾 − 𝑆𝑡 , 0} = min{100 − 105, 0} = −$5. At expiry time 𝑇 , the payoff is Ψ(𝑆𝑇 ) = − max{𝑆𝑇 − 𝐾, 0} = min{𝐾 − 𝑆𝑇 , 0} = − min{100 − 120, 0} = −$20 and the profit is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = −$20 + $10 = −$10. Figure 1.10 shows the payoff and profit diagram for a short call option at the expiry time 𝑇 . Here the profit diagram is a vertical shift of the short call payoff based on the premium received. Payoff/Payoff

Profit $10 $100

$110

Payoff

Figure 1.10

Short call option payoff and profit diagrams.

6. Consider a short put option with strike price 𝐾 = $100. The current stock price is 𝑆𝑡 = $80 and the put premium is $5. What is the intrinsic value of the put option at time 𝑡? Find the payoff and profit if the spot price at the option expiration date 𝑇 is 𝑆𝑇 = $75. Draw the payoff and profit diagrams.

20

1.2.2 Options Theory

Solution: By defining 𝑆𝑡 = $80, 𝑆𝑇 = $75, 𝐾 = $100 and the put premium as 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = $5, the intrinsic value of the short put option at time 𝑡 is Ψ(𝑆𝑡 ) = − max{𝐾 − 𝑆𝑡 , 0} = min{𝑆𝑡 − 𝐾, 0} = min{80 − 100, 0} = −$20. At expiry time 𝑇 , the payoff is Ψ(𝑆𝑇 ) = − max{𝐾 − 𝑆𝑇 , 0} = min{𝑆𝑇 − 𝐾, 0} = min{75 − 100, 0} = −$25 and the profit is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = −$25 + $5 = −$20. Figure 1.11 shows the payoff and profit diagrams for a short put option at the expiry time 𝑇 . Here the profit diagram is a vertical shift of the short put payoff based on the premium received. Payoff/Profit

Profit $5 $95 $100

Payoff

−$95 −$100

Figure 1.11

Short put option payoff and profit diagrams.

7. At time 𝑡 we consider a short call option with strike price 𝐾 and a short forward contract with price 𝐾 on the same underlying asset 𝑆𝑡 and expiry time 𝑇 > 𝑡. The premium for the call option is 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) at time 𝑡. Draw the profit diagram for these two financial instruments at the option expiry time 𝑇 . Under what conditions is the call option more profitable than the forward contract, and vice versa? Solution: Figure 1.12 shows the profit diagram for a short call and a short forward contract at expiry time 𝑇 .

1.2.2 Options Theory

21 Profit

Short Forward

(

;

)

+ (

;

)

Short Call Option

Figure 1.12

Short call option and short forward profit diagrams.

At time 𝑇 the break-even point is at 𝑆𝑇 = 𝐾 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). Therefore, if the stock 𝑆𝑇 ≤ 𝐾 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) then the forward contract is more profitable as the short call profit is fixed with the amount of premium received. However, if 𝑆𝑇 > 𝐾 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) then the forward contract is less profitable since the short call is augmented by the amount of premium paid to it.

8. At time 𝑡 we consider a short put option with strike price 𝐾 and a long forward contract with price 𝐾 on the same underlying asset 𝑆𝑡 and expiry time 𝑇 > 𝑡. The premium for the put option is 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). Draw the profit diagram for these two financial instruments at the option expiry time 𝑇 . Under what conditions is the put option more profitable than the forward contract, and vice versa? Solution: Figure 1.13 shows the profit diagram for a short put and a long forward contract at expiry time 𝑇 . At time 𝑇 the break-even point is at 𝑆𝑇 = 𝐾 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). Therefore, if the stock 𝑆𝑇 ≥ 𝐾 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) then the forward contract is more profitable as the short put profit is fixed by the amount of premium received. However, if 𝑆𝑇 < 𝐾 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) then the forward contract is less profitable since the short put is augmented by the amount of premium paid to it.

9. Put–Call Parity I. At time 𝑡 we consider a non-dividend-paying stock with spot price 𝑆𝑡 and a risk-free interest rate 𝑟. Show that by taking a long European call option price at time 𝑡, 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and a short European put option price at time 𝑡, 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) on the same

22

1.2.2 Options Theory Profit

Long Forward

Short Put Option

−

(

;

)

+ (

;

− (

;

)

) −

Figure 1.13

Short put option and long forward profit diagrams.

underlying stock 𝑆𝑡 , strike price 𝐾 and expiry time 𝑇 (𝑡 < 𝑇 ) we have 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 − 𝐾𝑒−𝑟(𝑇 −𝑡) . Solution: At time 𝑡 we define the call option price as 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and the put option price as 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ), and we set the portfolio Π𝑡 as Π𝑡 = 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). At expiry time 𝑇 Π𝑇 = 𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0} − max{𝐾 − 𝑆𝑇 , 0} ⎧𝑆 − 𝐾 ⎪ 𝑇 =⎨ ⎪ 𝑆𝑇 − 𝐾 ⎩ = 𝑆𝑇 − 𝐾.

if 𝑆𝑇 ≥ 𝐾 if 𝑆𝑇 < 𝐾

In order for the portfolio to generate a guaranteed 𝐾 at expiry time 𝑇 , at time 𝑡 we can discount the final value of the portfolio to 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 − 𝐾𝑒−𝑟(𝑇 −𝑡) since a share valued at 𝑆𝑡 will be worth 𝑆𝑇 at expiry time 𝑇 . 10. Put–Call Parity II. At time 𝑡 we consider a discrete dividend-paying stock with spot price 𝑆𝑡 where the stock pays dividend 𝛿𝑖 ≥ 0 at time 𝑡𝑖 , 𝑖 = 1, 2, … , 𝑛 for 𝑡 < 𝑡1 < 𝑡2 < ⋯
𝐾, the intrinsic value of the call option is ITM. As for 𝑆𝑡 = $100 Ψ(𝑆𝑡 ) = max{100 − 100, 0} = 0 and because Ψ(𝑆𝑡 ) = 0 and 𝑆𝑡 = 𝐾, the intrinsic value of the call option is ATM. Finally, for 𝑆𝑡 = $95 Ψ(𝑆𝑡 ) = max{95 − 100, 0} = 0 and since 𝑆𝑡 < 𝐾, the intrinsic value of the call option is OTM. 13. At time 𝑡 we consider a put option with strike price 𝐾 = $100. Compute the intrinsic value of this option if the current spot price is 𝑆𝑡 = $105, 𝑆𝑡 = $100 or 𝑆𝑡 = $95 and state whether it is ITM, OTM or ATM. Solution: At time 𝑡 the intrinsic put option value is Ψ(𝑆𝑡 ) = max{𝐾 − 𝑆𝑡 , 0}. Hence, if 𝑆𝑡 = $105 Ψ(𝑆𝑡 ) = max{100 − 105, 0} = 0 and since 𝑆𝑡 > 𝐾, the intrinsic value of the put option is OTM. As for 𝑆𝑡 = $100 Ψ(𝑆𝑡 ) = max{100 − 100, 0} = 0 and because Ψ(𝑆𝑡 ) = 0 and 𝑆𝑡 = 𝐾, the intrinsic value of the put option is ATM.

1.2.2 Options Theory

25

Finally, for 𝑆𝑡 = $95 Ψ(𝑆𝑡 ) = max{100 − 95, 0} = $5 and since 𝑆𝑡 < 𝐾, the intrinsic value of the put option is OTM. 14. Suppose we have a quote for a 3-month European put option, with a strike price 𝐾 = $60 of $1.25. The current stock price 𝑆0 = $62 and the risk-free interest rate 𝑟 = 5% per annum. Owing to small trading in call options, there is no listing for the 3-month $60 call (a call option price with strike $60 expiring in 3 months). Suppose the stock does not pay any dividends then find the price of the 3-month European call option. Solution: We first denote 𝐶(𝑆0 , 0; 𝐾, 𝑇 ) and 𝑃 (𝑆0 , 0; 𝐾, 𝑇 ) as the call and put option prices, respectively at time 𝑡 = 0 with strike price 𝐾 and option expiry time 𝑇 = 3 months. 3 Given 𝑃 (𝑆0 , 0; 𝐾, 𝑇 ) = $1.25 and 𝑇 = 12 = 0.25 years, by rearranging the put–call parity we can write 𝐶(𝑆0 , 0; 𝐾, 𝑇 ) = 𝑃 (𝑆0 , 0; 𝐾, 𝑇 ) + 𝑆0 − 𝐾𝑒−𝑟𝑇 = $1.25 + $62 − 60𝑒−0.05×0.25 = $4.00. Hence, if the 3-month $60 call is available, it should be priced at $4.00. 15. At time 𝑡 we consider a European call option 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and a European put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) on the same underlying asset 𝑆𝑡 , strike price 𝐾 and expiry time 𝑇 . Suppose the underlying asset pays a continuous dividend yield 𝐷 and there is a risk-free interest rate 𝑟, then under what condition is a European call option more expensive than a European put option? Solution: From the put–call parity 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) then 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) > 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) if 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) > 0 or 𝑆𝑡 > 𝐾𝑒−(𝑟−𝐷)(𝑇 −𝑡) .

16. Suppose that a 6-month European call option, with a strike price of 𝐾 = $85, has a premium of $2.75. The futures price for a 6-month contract is worth $75 and the risk-free rate 𝑟 = 5% per annum. Find the price of a 6-month European put option with the same strike price. Solution: At initial time 𝑡 = 0 we denote 𝐶(𝑆0 , 0; 𝐾, 𝑇 ) and 𝑃 (𝑆0 , 0; 𝐾, 𝑇 ) as the call and put option prices, respectively on the underlying asset 𝑆0 , strike 𝐾 and expiry time 𝑇 = 6

26

1.2.2 Options Theory 6 months. Let the expiry time 𝑇 = 12 = 0.5 years, 𝐶(𝑆0 , 0; 𝐾, 𝑇 ) = $2.75 and set the futures price 𝐹 (0, 𝑇 ) = $75. From the put–call parity we have

𝐶(𝑆0 , 0; 𝐾, 𝑇 ) − 𝑃 (𝑆0 , 0; 𝐾, 𝑇 ) = 𝑆0 𝑒−𝐷𝑇 − 𝐾𝑒−𝑟𝑇 where 𝐷 is the continuous dividend yield. Since 𝐹 (0, 𝑇 ) = 𝑆0 𝑒(𝑟−𝐷)𝑇 we can write 𝑃 (𝑆0 , 0; 𝐾, 𝑇 ) = 𝐶(𝑆0 , 0; 𝐾, 𝑇 ) − (𝐹 (0, 𝑇 ) − 𝐾)𝑒−𝑟𝑇 and by substituting 𝐶(𝑆0 , 0; 𝐾, 𝑇 ) = $2.75, 𝐹 (0, 𝑇 ) = $75, 𝐾 = $85, 𝑟 = 0.05 and 𝑇 = 0.5 years, the put option price is 𝑃 (𝑆0 , 0; 𝐾, 𝑇 ) = $2.75 − ($75 − $85)𝑒−0.05×0.5 = $12.50.

17. Suppose a 12-month European call option, with a strike price of 𝐾 = $35, has a premium of $2.15. The stock pays a dividend valued at $1.50 four months from now and another dividend valued at $1.75 eight months from now. Given that the current stock price is 𝑆0 = $32 and the risk-free rate 𝑟 = 5% per annum, find the price of a 12-month European put option with the same strike price. Solution: Using the put–call parity for a stock with discrete dividends at time 𝑡 = 0 we have 𝐶(𝑆0 , 0; 𝐾, 𝑇 ) − 𝑃 (𝑆0 , 0; 𝐾, 𝑇 ) = 𝑆0 − 𝛿1 𝑒−𝑟𝑡1 − 𝛿2 𝑒−𝑟𝑡2 − 𝐾𝑒−𝑟𝑇 4 8 where 𝑆0 = $32, 𝛿1 = $1.50, 𝑡1 = 12 = 13 years, 𝛿2 = $1.75, 𝑡2 = 12 = 23 years, 𝐾 = $35, 𝑟 = 0.05 and 𝑇 = 1 year with European call option price 𝐶(𝑆0 , 0; 𝐾, 𝑇 ) = $2.15 and unknown European put option price 𝑃 (𝑆0 , 0; 𝐾, 𝑇 ). Thus,

𝑃 (𝑆0 , 0; 𝐾, 𝑇 ) = 𝐶(𝑆0 , 0; 𝐾, 𝑇 ) − 𝑆0 + 𝛿1 𝑒−𝑟𝑡1 + 𝛿2 𝑒−𝑟𝑡2 + 𝐾𝑒−𝑟𝑇 1

2

= 2.15 − 32 + 1.50𝑒−0.05× 3 + 1.75𝑒−0.05× 3 + 35𝑒−0.05 = $6.61.

18. Consider a European put option priced at $2.50 with strike price $22 and a European call option priced at $4.75 with strike price $30. What are the maximum losses to the writer of the put and the buyer of the call?

1.2.3 Hedging Strategies

27

Solution: At time 𝑡 the maximum loss of a short put is −𝐾 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) where 𝐾 is the strike price, 𝑇 > 𝑡 is the expiry date and 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is the put option price. Therefore, the maximum loss of a short put is −$22 + $2.50 = −$19.50. In contrast, the maximum loss to the buyer of the call option is the premium paid, that is −$4.75. 19. Let the current stock price be $35 and the European put option is ITM by $3.50. Find the corresponding strike price. Solution: At current time 𝑡, the intrinsic value of the put option is defined as Ψ(𝑆𝑡 ) = max{𝐾 − 𝑆𝑡 , 0} where 𝑆𝑡 and 𝐾 are the current stock price and strike price, respectively. Given 𝑆𝑡 = $35, Ψ(𝑆𝑡 ) = $3.50 and since 𝐾 > 𝑆𝑡 (intrinsic value is ITM), then 𝐾 − $35 = $3.50

or

𝐾 = $38.50.

20. Given the current spot price 𝑆0 = $55 we consider a European call option and a European put option with premiums $1.98 and $0.79, respectively on a common strike price 𝐾 = $58 and having the same expiry time 𝑇 . By setting the risk-free interest rate 𝑟 = 3% per annum, find 𝑇 . Solution: From the put–call parity at time 𝑡 = 0, 𝐶(𝑆0 , 0; 𝐾, 𝑇 ) + 𝑃 (𝑆0 , 0; 𝐾, 𝑇 ) = 𝑆0 − 𝐾𝑒−𝑟𝑇 where the call option 𝐶(𝑆0 , 0; 𝐾, 𝑇 ) = $1.98 and the put option 𝑃 (𝑆0 , 0; 𝐾, 𝑇 ) = $0.79. Substituting 𝑆0 = $55, 𝐾 = $58 and 𝑟 = 0.03, we have 1.98 − 0.79 = 55 − 58𝑒−0.03×𝑇 1.19 = 55 − 58𝑒−0.03×𝑇 and solving the equation, we have 𝑒−0.03×𝑇 = 0.9278 or 𝑇 ≃ 2.5 years.

1.2.3

Hedging Strategies

1. Covered Call. A covered call is an investment strategy constructed by buying a stock and selling an OTM call option on the same stock.

28

1.2.3 Hedging Strategies

Explain why a call writer would set up this portfolio trading strategy and show that this strategy is undertaken for 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝑆𝑡 − 𝐾 where 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is the call option price written on stock 𝑆𝑡 with strike price 𝐾 at time 𝑡 with option expiry time 𝑇 > 𝑡. Draw the profit diagram of this strategy at expiry time 𝑇 (𝑇 > 𝑡). Solution: In writing a covered call where the writer owns the stock, the writer can cover the obligation of delivering the stock if the holder of the call exercises the option at expiry date. In addition, by writing a covered call, the writer assumes the stock price will not be higher than the strike price and thus enhance his income by receiving the call option’s premium. In contrast, if the stock price declines in value then the writer will lose money. At time 𝑇 the payoff of this portfolio is Ψ(𝑆𝑇 ) = 𝑆𝑇 − 𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 𝑆𝑇 − max{𝑆𝑇 − 𝐾, 0} ⎧𝐾 ⎪ =⎨ ⎪ 𝑆𝑇 ⎩

if 𝑆𝑇 ≥ 𝐾 if 𝑆𝑇 < 𝐾

whilst the profit is { Υ(𝑆𝑇 ) =

𝐾 − 𝑆𝑡 + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 )

if 𝑆𝑇 ≥ 𝐾

𝑆𝑇 − 𝑆𝑡 + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) if 𝑆𝑇 < 𝐾

where we need to deduct the cost of acquiring 𝑆𝑡 and also to add the call option premium at the start of the contract. Since the break-even point occurs when 𝑆𝑇 = 𝑆𝑡 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) where Υ(𝑆𝑇 ) = 0, in order for the strategy to take place we require 𝑆𝑡 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝐾 or 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑆𝑡 − 𝐾. Figure 1.14 shows the profit diagram of a covered call. Since 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑆𝑡 − 𝐾, the maximum gain from this strategy is 𝐾 − 𝑆𝑡 + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0 whilst the maximum loss is −𝑆𝑡 + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 0. 2. Covered Put. A covered put is a hedging strategy constructed by selling a stock and selling an OTM put option on the same stock. Explain why a put writer would set up this portfolio trading strategy and show that this strategy is undertaken for 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝐾 − 𝑆𝑡 where 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is the put option price written on stock 𝑆𝑡 with strike price 𝐾 at time 𝑡 and expiry time 𝑇 > 𝑡. Draw the profit diagram of this strategy at expiry time 𝑇 (𝑇 > 𝑡). Solution: In writing a covered put the writer expects the stock price will decline in value relative to the strike and thus enhance his income by receiving the put option’s premium. By selling the stock short, the writer does not need to worry if the stock price drops further. However, if the stock price is much greater than the strike price at expiry then the writer will lose money.

1.2.3 Hedging Strategies

29

Profit

Profit Long Stock Short Call Option (

;

)

+

−

Profit

=

Covered Call −

−

+ (

;

)

+ (

;

)

Figure 1.14

Construction of a covered call.

At time 𝑇 the payoff of this portfolio is Ψ(𝑆𝑇 ) = −𝑆𝑇 − 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = −𝑆𝑇 − max{𝐾 − 𝑆𝑇 , 0} ⎧ −𝑆 ⎪ 𝑇 =⎨ ⎪ −𝐾 ⎩

if 𝑆𝑇 ≥ 𝐾 if 𝑆𝑇 < 𝐾

whilst the profit is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) + 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎧ 𝑆 − 𝑆 + 𝑃 (𝑆 , 𝑡; 𝐾, 𝑇 ) 𝑇 𝑡 ⎪ 𝑡 =⎨ ⎪ 𝑆𝑡 − 𝐾 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎩

if 𝑆𝑇 ≥ 𝐾 if 𝑆𝑇 < 𝐾

where we need to add the sale of 𝑆𝑡 and also the put option premium received at the start of the contract. Since the break-even point occurs when 𝑆𝑇 = 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) where Υ(𝑆𝑇 ) = 0, this strategy is undertaken when 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝐾 or 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝑆𝑡 − 𝐾. For a detailed construction of a covered put, see Figure 1.15.

30

1.2.3 Hedging Strategies Profit

Profit Short Stock Short Put Option (

;

)

;

)

+

−

+ (

=

Profit

Covered Put −

+ (

;

)

Figure 1.15

Construction of a covered put.

Since 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝐾 − 𝑆𝑡 , the maximum gain from this strategy is capped at 𝑆𝑡 − 𝐾 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0 whilst the loss is unlimited. 3. Protective Call. A protective call is an investment strategy constructed by selling a stock and buying an OTM call option on the same stock. Explain why a call holder would set up this portfolio trading strategy and show that this strategy is undertaken for 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝑆𝑡 − 𝐾 where 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is the call option price written on stock 𝑆𝑡 with strike price 𝐾 at time 𝑡 and expiry time 𝑇 > 𝑡. Draw the profit diagram of this strategy at expiry time 𝑇 (𝑇 > 𝑡). Solution: In buying a protective call the investor strategy is to protect profits from the rising stock price with respect to the strike price. By selling the stock the call holder assumes that the stock price will decline further. If the stock price is less than the strike at expiry time then the option will not be exercised and the call buyer will only lose the premium paid. At time 𝑇 the payoff of this portfolio is Ψ(𝑆𝑇 ) = −𝑆𝑇 + 𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = −𝑆𝑇 + max{𝑆𝑇 − 𝐾, 0} { −𝐾 if 𝑆𝑇 ≥ 𝐾 = −𝑆𝑇 if 𝑆𝑇 < 𝐾

1.2.3 Hedging Strategies

31

Profit

Profit Long Call Option

Short Stock

+ − (

−

;

)

− (

;

)

)

=

Profit

− (

;

Protective Call

Figure 1.16

Construction of a protective call.

whilst the profit is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) + 𝑆𝑡 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎧ 𝑆 − 𝐾 − 𝐶(𝑆 , 𝑡; 𝐾, 𝑇 ) if 𝑆 ≥ 𝐾 𝑡 𝑇 ⎪ 𝑡 =⎨ ⎪ 𝑆𝑡 − 𝑆𝑇 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) if 𝑆𝑇 < 𝐾 ⎩ where we need to add the sale of the stock at time 𝑡 and deduct the call option premium paid at the beginning of the contract. Since the break-even point occurs when 𝑆𝑇 = 𝑆𝑡 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) so that Υ(𝑆𝑇 ) = 0, this strategy should be undertaken when 𝑆𝑡 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐾 or 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝑆𝑡 − 𝐾. Figure 1.16 shows the profit diagram of a protective call. From the profit formula we can see that the protective call has a maximum upside gain of 𝑆𝑡 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ), which is the difference between the sale of the stock and the call option premium, and a limited loss of 𝑆𝑡 − 𝐾 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 0.

32

1.2.3 Hedging Strategies

4. Protective Put. A protective put is a hedging strategy constructed by buying a stock and buying an OTM put option on the same stock. Explain why a put buyer would set up this portfolio trading strategy and show that this strategy is undertaken for 𝑃 (𝑆𝑡 , 𝑡) ≥ 𝐾 − 𝑆𝑡 where 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is the put option price written on stock 𝑆𝑡 with strike price 𝐾 at time 𝑡 and expiry time 𝑇 > 𝑡. Draw the profit diagram of this strategy at expiry time 𝑇 (𝑇 > 𝑡). Solution: In buying a protective put the investor strategy is to protect profits from the stock declining in value with respect to the strike price. By owning the stock the put holder can cover the obligation of delivering the stock to the put writer if the option is exercised at expiry time. If the stock price is above the strike at expiry time then the option will not be exercised and the put buyer will only lose the premium paid. At time 𝑇 the payoff of this portfolio is Ψ(𝑆𝑇 ) = 𝑆𝑇 + 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 𝑆𝑇 + max{𝐾 − 𝑆𝑇 , 0} ⎧𝑆 ⎪ 𝑇 =⎨ ⎪𝐾 ⎩

if 𝑆𝑇 ≥ 𝐾 if 𝑆𝑇 < 𝐾

whilst the profit is ⎧ 𝑆 − 𝑆 − 𝑃 (𝑆 , 𝑡; 𝐾, 𝑇 ) 𝑡 𝑡 ⎪ 𝑇 Υ(𝑆𝑇 ) = ⎨ ⎪ 𝐾 − 𝑆𝑡 − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎩

if 𝑆𝑇 ≥ 𝐾 if 𝑆𝑇 < 𝐾

where we need to deduct both the cost of acquiring 𝑆𝑡 and the put option premium paid at the beginning of the contract. Since the break-even point occurs when 𝑆𝑇 = 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) so that Υ(𝑆𝑇 ) = 0, this strategy is undertaken if 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝐾 or 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝐾 − 𝑆𝑡 . Figure 1.17 shows the profit diagram of a protective put. From the profit formula we can see that the protective put has an unlimited upside gain and, because 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝐾 − 𝑆𝑡 , it has a limited loss of 𝐾 − 𝑆𝑡 − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 0. 5. At time 𝑡 consider a writer of a covered call on a $35 stock with a strike price of $40. The premium of the call option is $1.75. Calculate the writer’s maximum gain and loss at expiry time 𝑇 > 𝑡. Solution: At time 𝑡 at the start of the contract let 𝑆𝑡 = $35, 𝐾 = $40 and the call option price 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = $1.75. From Problem 1.2.3.1 (page 28) the maximum gain of a covered call at expiry time 𝑇 (𝑇 > 𝑡) is Υ𝐺 (𝑆𝑇 ) = 𝐾 − 𝑆𝑡 + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 40 − 35 + 1.75 = $6.75

1.2.3 Hedging Strategies

33

Profit

Profit Long Stock

− (

;

)

+ − (

;

)

Long Put Option

−

Profit

−

− (

;

=

) Protective Put

Figure 1.17

Construction of a protective put.

and the maximum loss is Υ𝐿 (𝑆𝑇 ) = −𝑆𝑡 + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = −35 + 1.75 = −$33.25. 6. At time 𝑡 a writer short sells a stock for $33 and sells a put option with strike price $25 for $1.75. What is the maximum gain and loss for the writer of this protective put at expiry time 𝑇 > 𝑡? Solution: At current time 𝑡 we let the spot price of the stock be 𝑆𝑡 = $33, strike 𝐾 = $25 and put option price 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = $1.75. Therefore, from Problem 1.2.3.4 (page 33) Maximum Gain of Protective Put = +∞ and Maximum Loss of Protective Put = 𝐾 − 𝑆𝑡 − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = $25 − $33 − $1.75 = −$9.75.

7. Bull Call Spread. A bull call spread is a hedging position designed to buy a call option 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) with strike 𝐾1 and simultaneously sell a call option 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) with

34

1.2.3 Hedging Strategies

strike 𝐾2 , 𝐾1 ≤ 𝐾2 on the same underlying asset 𝑆𝑡 and having the same expiry time 𝑇 (𝑇 > 𝑡). Show that 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) ≥ 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ),

𝐾1 ≤ 𝐾2

and draw the payoff and profit diagrams of a bull call spread. Discuss under what conditions an investor should invest in such a hedging strategy. Solution: We first assume 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) < 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) and we set up a portfolio Π𝑡 = 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) < 0. At expiry time 𝑇 Π𝑇 = 𝐶(𝑆𝑇 , 𝑇 ; 𝐾1 ) − 𝐶(𝑆𝑇 , 𝑇 ; 𝐾2 ) = max{𝑆𝑇 − 𝐾1 , 0} − max{𝑆𝑇 − 𝐾2 , 0} ⎧ ⎪0 ⎪ = ⎨ 𝑆𝑇 − 𝐾1 ⎪ ⎪ 𝐾2 − 𝐾1 ⎩ ≥0

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2

which constitutes an arbitrage opportunity. Therefore, 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) ≥ 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ), 𝐾1 ≤ 𝐾2 . At time 𝑇 the payoff of this hedging strategy is Ψ(𝑆𝑇 ) = 𝐶(𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) = max{𝑆𝑇 − 𝐾1 , 0} − max{𝑆𝑇 − 𝐾2 , 0} ⎧0 ⎪ ⎪ = ⎨ 𝑆𝑇 − 𝐾1 ⎪ ⎪ 𝐾2 − 𝐾1 ⎩

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2

and the corresponding profit is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ⎧ 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) if 𝑆𝑇 ≤ 𝐾1 ⎪ ⎪ = ⎨ 𝑆𝑇 − 𝐾1 + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 ⎪ ⎪ 𝐾 − 𝐾 + 𝐶(𝑆 , 𝑡; 𝐾 , 𝑇 ) − 𝐶(𝑆 , 𝑡; 𝐾 , 𝑇 ) if 𝑆 > 𝐾 1 𝑡 2 𝑡 1 𝑇 2 ⎩ 2 where the break-even point is 𝑆𝑇 = 𝐾1 + 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) so that Υ(𝑆𝑇 ) = 0. Figure 1.18 shows the payoff and profit diagrams of a bull call spread. Based on the payoff and profit diagrams of a bull call spread, this hedging strategy would appeal to investors who have a bullish sentiment that the stock price

1.2.3 Hedging Strategies

35 Payoff/Profit

Payoff 2 2

−

1

−

1

+ (

;

2

)− (

;

1

)

(

;

2

)− (

;

1

)

1

2

Profit

Figure 1.18

Construction of a bull call spread.

will increase in value relative to the strike prices 𝐾1 and 𝐾2 . Hence, if 𝑆𝑇 > 𝐾1 + 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) then the investor would make a profit but capped at a maximum gain of 𝐾2 − 𝐾1 + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) > 0. However, if 𝑆𝑇 < 𝐾1 + 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) then the investor would make a loss but limited to the difference between premiums received and paid. 8. Bull Put Spread. A bull put spread is an investment strategy constructed by buying a put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) with strike 𝐾1 and simultaneously selling a put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) with strike 𝐾2 , 𝐾1 ≤ 𝐾2 on the same underlying asset 𝑆𝑡 and having the same expiry time 𝑇 (𝑇 > 𝑡). Show that 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≥ 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ), 𝐾1 ≤ 𝐾2 and draw the payoff and profit diagrams of a bull put spread. Discuss under what conditions an investor should invest in such an investment strategy. Solution: We first assume 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) < 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) and we set up a portfolio Π𝑡 = 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) < 0. At expiry time 𝑇 Π𝑇 = 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) = max{𝐾2 − 𝑆𝑇 , 0} − max{𝐾1 − 𝑆𝑇 , 0} ⎧ 𝐾2 − 𝐾1 ⎪ ⎪ = ⎨ 𝐾 2 − 𝑆𝑇 ⎪ ⎪0 ⎩

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2

≥0 which constitutes an arbitrage opportunity. Therefore, 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≥ 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ), 𝐾1 ≤ 𝐾2 .

36

1.2.3 Hedging Strategies

At time 𝑇 the payoff of this hedging strategy is Ψ(𝑆𝑇 ) = 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) = max{𝐾1 − 𝑆𝑇 , 0} − max{𝐾2 − 𝑆𝑇 , 0} ⎧ 𝐾1 − 𝐾 2 ⎪ ⎪ = ⎨ 𝑆𝑇 − 𝐾2 ⎪ ⎪0 ⎩

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2

and the corresponding profit is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ⎧ 𝐾1 − 𝐾2 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) ⎪ ⎪ = ⎨ 𝑆𝑇 − 𝐾2 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) ⎪ ⎪ 𝑃 (𝑆 , 𝑡; 𝐾 , 𝑇 ) − 𝑃 (𝑆 , 𝑡; 𝐾 , 𝑇 ) 𝑡 2 𝑡 1 ⎩

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2

where the break-even point is 𝑆𝑇 = 𝐾2 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) so that Υ(𝑆𝑇 ) = 0. Figure 1.19 shows the payoff and profit diagrams of a bull put spread. Payoff/Profit

(

;

2

)− (

;

1

)

+ (

;

2

)− (

;

1

)

1 1

−

2

2

Profit Payoff

1

−

2

Figure 1.19

Construction of a bull put spread.

Based on the payoff and profit diagrams of a bull put spread, this hedging strategy would appeal to investors who have a bullish sentiment that the stock price will increase in value relative to the strike prices 𝐾1 and 𝐾2 . Hence, if 𝑆𝑇 > 𝐾2 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) then the investor would make a profit but capped at a maximum gain based on the difference between the put premiums received and paid. However, if 𝑆𝑇 < 𝐾2 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) then the investor would make a loss but limited to a maximum loss of 𝐾1 − 𝐾2 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) < 0.

1.2.3 Hedging Strategies

37

9. Bear Call Spread. A bear call spread is a hedging position designed to sell a call option 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) with strike 𝐾1 and simultaneously buy a call option 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) with strike 𝐾2 , 𝐾1 ≤ 𝐾2 on the same underlying asset 𝑆𝑡 and having the same expiry time 𝑇 (𝑇 > 𝑡). Show that 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) ≥ 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ), 𝐾1 ≤ 𝐾2 and draw the payoff and profit diagrams of a bear call spread. Discuss under what conditions an investor should invest in such a hedging strategy. Solution: To show that 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) ≥ 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) for 𝐾1 ≤ 𝐾2 see Problem 1.2.3.7 (page 34). At time 𝑇 the payoff of this hedging strategy is Ψ(𝑆𝑇 ) = −𝐶(𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) = − max{𝑆𝑇 − 𝐾1 , 0} + max{𝑆𝑇 − 𝐾2 , 0} ⎧0 ⎪ ⎪ = ⎨ 𝐾1 − 𝑆𝑇 ⎪ ⎪𝐾 − 𝐾 2 ⎩ 1

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2

and the corresponding profit is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ⎧ 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ⎪ ⎪ = ⎨ 𝐾1 − 𝑆𝑇 + 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ⎪ ⎪ 𝐾 − 𝐾 + 𝐶(𝑆 , 𝑡; 𝐾 , 𝑇 ) − 𝐶(𝑆 , 𝑡; 𝐾 , 𝑇 ) 2 𝑡 1 𝑡 2 ⎩ 1

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2

where the break-even point is 𝑆𝑇 = 𝐾1 + 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) so that Υ(𝑆𝑇 ) = 0. Figure 1.20 is the payoff and profit diagrams of a bear call spread. Based on the payoff and profit diagrams of a bear call spread, this hedging strategy would appeal to investors who have a bearish attitude that the stock price will decrease in value relative to the strike prices 𝐾1 and 𝐾2 . Hence, if 𝑆𝑇 < 𝐾1 + 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) then the investor would make a profit but capped at a maximum gain based on the difference between the call premiums received and paid. On the other hand, if 𝑆𝑇 > 𝐾1 + 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) then the investor would make a loss but limited to a maximum loss of 𝐾1 − 𝐾2 + 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) < 0. 10. Bear Put Spread. A bear put spread is an investment strategy constructed by selling a put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) with strike 𝐾1 and simultaneously buying a put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) with strike 𝐾2 , 𝐾1 ≤ 𝐾2 on the same underlying asset 𝑆𝑡 and having the same expiry time 𝑇 (𝑇 > 𝑡).

38

1.2.3 Hedging Strategies Payoff/Profit

(

;

2+ (

;

1

)− (

;

1

)− (

;

2

)

2

)

Profit

1 1−

2

Payoff 1

−

Figure 1.20

2

Construction of a bear call spread.

Show that 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≥ 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ),

𝐾1 ≤ 𝐾2

and draw the payoff and profit diagrams of a bear put spread. Discuss under what conditions an investor should invest in such an investment strategy. Solution: To show that 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≥ 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) for 𝐾1 ≤ 𝐾2 see Problem 1.2.3.8 (page 36). At time 𝑇 the payoff of this hedging strategy is Ψ(𝑆𝑇 ) = −𝑃 (𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) + 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) = − max{𝐾1 − 𝑆𝑇 , 0} + max{𝐾2 − 𝑆𝑇 , 0} ⎧ 𝐾2 − 𝐾1 ⎪ ⎪ = ⎨ 𝐾2 − 𝑆𝑇 ⎪ ⎪0 ⎩

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2

and the corresponding profit is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ⎧ 𝐾2 − 𝐾1 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ⎪ ⎪ = ⎨ 𝐾2 − 𝑆𝑇 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ⎪ ⎪ 𝑃 (𝑆 , 𝑡; 𝐾 , 𝑇 ) − 𝑃 (𝑆 , 𝑡; 𝐾 , 𝑇 ) 𝑡 1 𝑡 2 ⎩

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2

where the break-even point is 𝑆𝑇 = 𝐾2 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) so that Υ(𝑆𝑇 ) = 0. Figure 1.21 shows the payoff and profit diagrams of a bear put spread.

1.2.3 Hedging Strategies

39 Payoff/Profit

2

2

−

1

+ (

;

(

;

1

)− (

;

1

)− (

;

Payoff

−

1

2

)

2

)

1

2

Profit

Figure 1.21

Construction of a bear put spread.

From the payoff and profit diagrams of a bear put spread we can see that this hedging strategy would appeal to investors who have a bearish attitude that the stock price will decrease in value relative to the strike prices 𝐾1 and 𝐾2 . Hence, if 𝑆𝑇 < 𝐾2 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) then the investor would make a profit but capped at a maximum gain of 𝐾2 − 𝐾1 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ). However, if 𝑆𝑇 > 𝐾2 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) then the investor would make a loss but limited to a maximum loss of 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) < 0. 11. Box Spread. A long box spread is an investment strategy constructed by buying a bull call spread 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) and buying a bear put spread 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) with strikes 𝐾1 and 𝐾2 , 𝐾1 ≤ 𝐾2 on the same underlying asset 𝑆𝑡 and having the same expiry time 𝑇 (𝑇 > 𝑡). Let the risk-free interest rate be 𝑟 and the stock pays a continuous dividend 𝐷. Show that the premium paid to enter into this portfolio strategy is the same as the present value of the payoff. Show also that at time 𝑇 the profit based on this strategy is independent of the terminal stock price 𝑆𝑇 . Give a financial interpretation of this investment strategy. Solution: At initial time 𝑡 the portfolio is worth Π𝑡 = 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) and from the put–call parity 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾1 𝑒−𝑟(𝑇 −𝑡) and 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾2 𝑒−𝑟(𝑇 −𝑡) .

40

1.2.3 Hedging Strategies

Therefore, Π𝑡 = (𝐾2 − 𝐾1 )𝑒−𝑟(𝑇 −𝑡) . At expiry time 𝑇 the portfolio is Π𝑇 = 𝐶(𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) +𝑃 (𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) = max{𝑆𝑇 − 𝐾1 , 0} − max{𝐾1 − 𝑆𝑇 , 0} − max{𝑆𝑇 − 𝐾2 , 0} + max{𝐾2 − 𝑆𝑇 , 0} ⎧ 𝐾2 − 𝐾1 ⎪ ⎪ = ⎨ 𝐾2 − 𝐾1 ⎪ ⎪𝐾 − 𝐾 1 ⎩ 2

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2

= 𝐾2 − 𝐾 1 . Thus, at time 𝑡 we can discount back the final value of the portfolio to become Π𝑡 = (𝐾2 − 𝐾1 )𝑒−𝑟(𝑇 −𝑡) which is also the price of the premium paid. The profit of entering such a hedging strategy is therefore Υ(𝑆𝑇 ) = Π𝑇 − Π𝑡 = (𝐾2 − 𝐾1 )(1 − 𝑒−𝑟(𝑇 −𝑡) ) >0 which guarantees a positive cash flow irrespective of the terminal stock price value 𝑆𝑇 . Thus, the box spread is clearly an arbitrage opportunity provided the transaction cost is low. 12. At current time 𝑡 = 0 a stock is trading at $20 and for a risk-free interest rate of 4% per annum the prices of 6-month European options are given in Table 1.4. Table 1.4 Strike Price $20 $27

European option prices for different strikes. European Call

European Put

$1.98 $1.21

$1.58 $7.68

Determine the payoff, premium paid and profit at the expiry time for a box spread constructed by buying a 20-strike European call, selling a 27-strike European call, selling a 20-strike European put and buying a 27-strike European put.

1.2.3 Hedging Strategies

41

Solution: By setting 𝑆0 = $20, 𝐾1 = $20, 𝐾2 = $27, 𝑟 = 0.04 and 𝑇 = from Problem 1.2.3.11 (page 40) the payoff of a box spread is

6 12

= 0.5 years,

Ψ(𝑆𝑇 ) = 𝐾2 − 𝐾1 = $27 − $20 = $7. The premium paid is therefore Ψ(𝑆0 ) = (𝐾2 − 𝐾1 )𝑒−𝑟𝑇 = $7𝑒−0.04×0.5 = $6.86 and the profit is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) − Ψ(𝑆0 ) = $7 − $6.86 = $0.14.

13. Purchased Collar. A purchased collar is a hedging strategy whereby at time 𝑡 an investor buys an asset 𝑆𝑡 , buys a put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) with strike price 𝐾1 and sells a call option 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) with strike price 𝐾2 , 𝐾1 < 𝐾2 on the same underlying 𝑆𝑡 and having the same expiry time 𝑇 (𝑇 > 𝑡). Show that in order to prevent any arbitrage opportunity 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≥ 0. Draw the profit diagram and give a financial interpretation of this hedging strategy. Solution: We assume at time 𝑡 that 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) < 0 and we set up a portfolio Ψ(𝑆𝑡 ) = 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) < 0. At expiry time 𝑇 Ψ(𝑆𝑇 ) = 𝑆𝑇 + 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) = 𝑆𝑇 + max{𝐾1 − 𝑆𝑇 , 0} − max{𝑆𝑇 − 𝐾2 , 0} ⎧ 𝐾1 ⎪ ⎪ = ⎨ 𝑆𝑇 ⎪ ⎪𝐾 ⎩ 2

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2

≥0 which is an arbitrage opportunity. Hence, 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≥ 0. Given that the investor has paid for both the stock 𝑆𝑡 and the put premium 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) and simultaneously received 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ), the profit at expiry time 𝑇 is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑆𝑡

42

1.2.3 Hedging Strategies

where Ψ(𝑆𝑇 ) = 𝑆𝑇 + 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) is the payoff of the purchased collar contract. Thus, ⎧ 𝐾1 + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑆𝑡 ⎪ ⎪ Υ(𝑆𝑇 ) = ⎨ 𝑆𝑇 + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑆𝑡 ⎪ ⎪ 𝐾 + 𝐶(𝑆 , 𝑡; 𝐾 , 𝑇 ) − 𝑃 (𝑆 , 𝑡; 𝐾 , 𝑇 ) − 𝑆 𝑡 2 𝑡 1 𝑡 ⎩ 2

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2

with break even at 𝑆𝑇 = 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≥ 0. Figure 1.22 shows the payoff and profit diagrams of a purchased collar. Payoff/Payoff Payoff 2

Profit 2+ (

;

2

)− (

;

1

)− 1 1

1

+ (

;

2

)− (

;

1

2

)−

Figure 1.22

Construction of a purchased collar.

From the combination of options and asset we can see that the purchased collar is a hedging strategy consisting of buying a protective put and selling a call option. By buying a protective put the investor is able to insure the asset, whilst selling a call reduces the cost of insurance. Therefore, this position would be beneficial when the asset price 𝑆𝑇 > 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) (up to a maximum gain of 𝐾2 + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑆𝑡 ) but if 𝑆𝑇 < 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) then the investor would lose money (up to a maximum loss of 𝐾1 + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑆𝑡 ). 14. Written Collar. A written collar is an investment strategy whereby at time 𝑡 an investor would sell an asset 𝑆𝑡 , sell a put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) with strike price 𝐾1 and buy a call option 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) with strike price 𝐾2 , 𝐾1 < 𝐾2 on the same underlying 𝑆𝑡 and having the same expiry time 𝑇 > 𝑡. Show that in order to prevent any arbitrage opportunity 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≥ 0. Draw the payoff and profit diagrams and give a financial interpretation of this hedging strategy.

1.2.3 Hedging Strategies

43

Solution: To show that at time 𝑡, 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≥ 0 see Problem 1.2.3.13 (page 42). At expiry time 𝑇 the payoff is Ψ(𝑆𝑇 ) = −𝑆𝑇 − 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) = −𝑆𝑇 − max{𝐾1 − 𝑆𝑇 , 0} + max{𝑆𝑇 − 𝐾2 , 0} ⎧ −𝐾1 ⎪ ⎪ = ⎨ −𝑆𝑇 ⎪ ⎪ −𝐾 2 ⎩

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2

while the profit at expiry time 𝑇 is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) + 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ⎧ −𝐾1 + 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) if 𝑆𝑇 ≤ 𝐾1 ⎪ ⎪ = ⎨ −𝑆𝑇 + 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 ⎪ ⎪ −𝐾 + 𝑆 + 𝑃 (𝑆 , 𝑡; 𝐾 , 𝑇 ) − 𝐶(𝑆 , 𝑡; 𝐾 , 𝑇 ) if 𝑆 > 𝐾 2 𝑡 𝑡 1 𝑡 2 𝑇 2 ⎩ with break even at 𝑆𝑇 = 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≥ 0. Figure 1.23 shows the payoff and profit diagrams of a written collar. Payoff/Payoff

−

2

+

+ (

;

1

)− (

;

2

)

−

2

2

)

−

1

Profit 1

−

1

+

+ (

;

1

)− (

;

Figure 1.23

2

Payoff

Construction of a written collar.

Given the combination of options and asset we can see that the written collar is a hedging strategy consisting of buying a protective call and selling a put option. By buying a protective call the investor is able to insure the short sale of the asset, whilst selling a put reduces the cost of insurance. Hence, this position would be beneficial when the asset price declines 𝑆𝑇 < 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) (up to a maximum

44

1.2.3 Hedging Strategies

gain of −𝐾2 + 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 )) but if 𝑆𝑇 > 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) then the investor would lose money (up to a maximum loss of −𝐾1 + 𝑆𝑡 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 )). 15. Long Straddle. A long straddle is an investment strategy whereby at time 𝑡 an investor would buy a call option 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and buy a put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) on the same strike 𝐾 using the same stock 𝑆𝑡 and having the same expiry time 𝑇 (𝑡 < 𝑇 ). Draw the payoff and profit diagrams of this investment strategy and give a financial interpretation based on this combination of options portfolio. Solution: We consider at time 𝑡 that there is a stock worth 𝑆𝑡 and for a strike price 𝐾 the investor buys a call option 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and a put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) on the same stock 𝑆𝑡 with expiry time 𝑇 . The payoff at time 𝑇 is Ψ(𝑆𝑇 ) = 𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) + 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0} + max{𝐾 − 𝑆𝑇 , 0} ⎧𝐾 − 𝑆 𝑇 ⎪ =⎨ ⎪ 𝑆𝑇 − 𝐾 ⎩

if 𝑆𝑇 ≤ 𝐾 if 𝑆𝑇 > 𝐾

and the profit is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎧ 𝐾 − 𝑆 − 𝐶(𝑆 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆 , 𝑡; 𝐾, 𝑇 ) 𝑇 𝑡 𝑡 ⎪ =⎨ ⎪ 𝑆𝑇 − 𝐾 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎩

if 𝑆𝑇 ≤ 𝐾 if 𝑆𝑇 > 𝐾

with break even at the profit level occurring at 𝑆𝑇 = 𝐾 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) (provided 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐾) and 𝑆𝑇 = 𝐾 + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). Figure 1.24 shows the payoff and profit diagrams of a long straddle. From the profit diagram we can see that the investor would make an unlimited profit if 𝑆𝑇 > 𝐾 + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) or a limited profit if 𝑆𝑇 < 𝐾 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). In contrast, the maximum loss for the investor is the cost of purchasing the option. Hence, this strategy is dependent on how high the volatility of the stock is rather than the direction of the stock price. In short, the profit at expiry time relies on how much the stock price moves instead of whether it is increasing or decreasing in value. 16. Short Straddle. A short straddle is an investment strategy where at time 𝑡 an investor would sell a call option 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and sell a put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) on the same strike 𝐾 using the same stock 𝑆𝑡 and having the same expiry time 𝑇 (𝑡 < 𝑇 ).

1.2.3 Hedging Strategies

45

Payoff/Payoff

Payoff − (

;

)− (

;

)

− (

;

)− (

;

)

Figure 1.24

Profit

Construction of a long straddle.

Draw the payoff and profit diagrams of this investment strategy and give a financial interpretation based on this combination of options portfolio. Solution: We consider at time 𝑡 that there is a stock worth 𝑆𝑡 and for a strike price 𝐾 the writer sells a call option 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and a put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) on the same stock 𝑆𝑡 with expiry time 𝑇 . The payoff at time 𝑇 is Ψ(𝑆𝑇 ) = −𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = − max{𝑆𝑇 − 𝐾, 0} − max{𝐾 − 𝑆𝑇 , 0} ⎧𝑆 − 𝐾 ⎪ 𝑇 =⎨ ⎪ 𝐾 − 𝑆𝑇 ⎩

if 𝑆𝑇 ≤ 𝐾 if 𝑆𝑇 > 𝐾

and the profit is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎧ 𝑆 − 𝐾 + 𝐶(𝑆 , 𝑡; 𝐾, 𝑇 ) + 𝑃 (𝑆 , 𝑡; 𝐾, 𝑇 ) 𝑡 𝑡 ⎪ 𝑇 =⎨ ⎪ 𝐾 − 𝑆𝑇 + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎩

if 𝑆𝑇 ≤ 𝐾 if 𝑆𝑇 > 𝐾

with break even at the profit level occurring either at 𝑆𝑇 = 𝐾 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) (provided 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐾) or 𝑆𝑇 = 𝐾 + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). Figure 1.25 illustrates the payoff and profit diagrams of a short straddle. From the profit diagram we can see that the writer would make an unlimited loss if 𝑆𝑇 > 𝐾 + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) or a limited loss if 𝑆𝑇 < 𝐾 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). In contrast, the maximum gain for the writer is only the options premium received. Thus, unlike the long straddle, the short straddle depends very much on low

46

1.2.3 Hedging Strategies Payoff/Payoff

−

(

;

)+ (

;

)

+ (

;

)+ (

;

)

Profit

−

Payoff

Figure 1.25

Construction of a short straddle.

volatility and is most profitable if 𝑆𝑇 = 𝐾. Hence, this strategy is dependent on the volatility of the stock rather than the direction of the stock price, and the writer who sells this portfolio of options would bet on the low volatility of the stock price. 17. Consider an investor buying a 35-strike call option and a 25-strike put option on a stock for prices of $0.69 and $0.52, respectively. Both of the options have the same expiry time of 6 months from now. Given that the current price of a stock is $28 and the risk-free interest rate is 𝑟 = 3% per annum, what is the position of this investment strategy? Find the profit and determine the break-even price of this hedging position at expiry time. Solution: Let 𝑆0 = $28, 𝐾1 = $25, 𝐾2 = $35, 𝑇 = portfolio at time 𝑡 = 0 as

6 12

=

1 2

years and we can write the

Π0 = 𝑃 (𝑆0 , 0; 𝐾1 , 𝑇 ) + 𝐶(𝑆0 , 0; 𝐾2 , 𝑇 ) where 𝑃 (𝑆0 , 0; 𝐾1 , 𝑇 ) = $0.52 is the put option price with strike 𝐾1 = $25 and 𝐶(𝑆0 , 0; 𝐾2 , 𝑇 ) = $0.69 is the call option price with strike price 𝐾2 = $35. Since 𝐾1 < 𝐾2 and the options have the same expiry time on the same underlying asset price, the position is a long straddle. At expiry time 𝑇 the payoff is Ψ(𝑆𝑇 ) = 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) = max{𝐾1 − 𝑆𝑇 , 0} + max{𝑆𝑇 − 𝐾2 , 0} ⎧ 𝐾1 − 𝑆 𝑇 ⎪ ⎪ = ⎨ 𝐾1 − 𝐾2 ⎪ ⎪𝑆 − 𝐾 2 ⎩ 𝑇

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2 .

1.2.3 Hedging Strategies

47

Given the total cost of the premium paid is 𝑃 (𝑆0 , 0; 𝐾1 , 𝑇 ) + 𝐶(𝑆0 , 0; 𝐾2 , 𝑇 ) = $0.52 + $0.69 = $1.21, the profit at expiry time 𝑇 is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) − $1.21 ⎧ $25 − 𝑆𝑇 − $1.21 ⎪ ⎪ = ⎨ $25 − $35 − $1.21 ⎪ ⎪ 𝑆 − $35 − $1.21 ⎩ 𝑇 ⎧ $23.79 − 𝑆𝑇 ⎪ ⎪ = ⎨ −$11.21 ⎪ ⎪ 𝑆 − $36.21 ⎩ 𝑇

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2

with break-even points occurring at 𝑆𝑇 = $23.79 and 𝑆𝑇 = $36.21. 18. Long Strangle. A long strangle is a hedging technique where at time 𝑡 an investor would buy a put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) with strike 𝐾1 and buy a call option 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) with strike 𝐾2 , 𝐾1 < 𝐾2 on the same stock 𝑆𝑡 and having the same expiry time 𝑇 (𝑡 < 𝑇 ). Draw the payoff and profit diagrams of this investment strategy and give a financial interpretation based on this combination of options portfolio. Solution: At expiry time 𝑇 the payoff of a long strangle portfolio is Ψ(𝑆𝑇 ) = 𝑃 (𝑆𝑇 ; 𝑇 ; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑇 ; 𝑇 ; 𝐾2 , 𝑇 ) = max{𝐾1 − 𝑆𝑇 , 0} + max{𝑆𝑇 − 𝐾2 , 0} ⎧ 𝐾1 − 𝑆𝑇 ⎪ ⎪ = ⎨0 ⎪ ⎪𝑆 − 𝐾 2 ⎩ 𝑇

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2 .

Given that the investor paid for the call and put premiums at time 𝑡, the profit at expiry time 𝑇 is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ⎧ 𝐾1 − 𝑆𝑇 − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) if 𝑆𝑇 ≤ 𝐾1 ⎪ ⎪ if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 = ⎨ −𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ⎪ ⎪ 𝑆 − 𝐾 − 𝑃 (𝑆 , 𝑡; 𝐾 ) − 𝐶(𝑆 , 𝑡; 𝐾 ) if 𝑆𝑇 > 𝐾2 2 𝑡 1 𝑡 2 ⎩ 𝑇

48

1.2.3 Hedging Strategies

with break even occurring at 𝑆𝑇 = 𝐾1 − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) (provided 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≤ 𝐾1 ) and 𝑆𝑇 = 𝐾2 + 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ). Figure 1.26 shows the payoff and profit diagrams of a long strangle. Payoff/Payoff

1

1

− (

;

− (

;

1

)− (

;

1

)− (

;

2

)

2

)

Payoff

1

Figure 1.26

2

Profit

Construction of a long strangle.

Like the long straddle, the long strangle also exploits the volatility of the stock price where the profit is based on how much the price of the stock moves instead of its direction. Here the investor makes a positive gain at the expiry time 𝑇 if 𝑆𝑇 > 𝐾2 + 𝑃 (𝑆𝑡 ; 𝑡; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) (unlimited gain) and if 𝑆𝑇 < 𝐾1 − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) (gain limited to 𝐾1 − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 )). As for the downward risk, this strategy has a limited risk which is only the cost of the premiums paid and there is a range between the strikes in which the loss is unaffected by the change in stock price. Hence, this strategy is more suitable for high-volatility stocks. 19. Short Strangle. A short strangle is a hedging technique where at time 𝑡 a writer would sell a put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) with strike 𝐾1 and sell a call option 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) with strike 𝐾2 , 𝐾1 < 𝐾2 on the same stock 𝑆𝑡 and having the same expiry time 𝑇 (𝑡 < 𝑇 ). Draw the payoff and profit diagrams of this investment strategy and give a financial interpretation based on this combination of options portfolio. Solution: At expiry time 𝑇 the payoff of a short strangle portfolio is Ψ(𝑆𝑇 ) = −𝑃 (𝑆𝑇 ; 𝑇 ; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑇 ; 𝑇 ; 𝐾2 , 𝑇 ) = − max{𝐾1 − 𝑆𝑇 , 0} − max{𝑆𝑇 − 𝐾2 , 0} ⎧ 𝑆𝑇 − 𝐾1 ⎪ ⎪ = ⎨0 ⎪ ⎪𝐾 − 𝑆 𝑇 ⎩ 2

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2 .

1.2.3 Hedging Strategies

49

Given that the investor received the call and put premiums at time 𝑡, the profit at expiry time 𝑇 is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ⎧ 𝑆𝑇 − 𝐾1 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) if 𝑆𝑇 ≤ 𝐾1 ⎪ ⎪ if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 = ⎨ 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ⎪ ⎪ 𝐾 − 𝑆 + 𝑃 (𝑆 , 𝑡; 𝐾 , 𝑇 ) + 𝐶(𝑆 , 𝑡; 𝐾 , 𝑇 ) if 𝑆 > 𝐾 𝑇 𝑡 1 𝑡 2 𝑇 2 ⎩ 2 with break even occurring at 𝑆𝑇 = 𝐾1 − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) (provided 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≤ 𝐾1 ) and 𝑆𝑇 = 𝐾2 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ). Figure 1.27 shows the payoff and profit diagrams of a short strangle. Payoff/Payoff

(

;

1

)+ (

;

2

Profit

) 1

−

1

+ (

;

1

)+ (

;

2

2

) Payoff

−

1

Figure 1.27

Construction of a short strangle.

Like the short straddle, the short strangle also exploits the low volatility of the stock price where the maximum profit is attained from the premiums received. But unlike the short straddle, this contract has a range between the strikes in which the gain is a constant value and unaffected by the change in strike price 𝑆𝑇 . In contrast, by taking a short position of a strangle, the writer is exposed to a limited loss if 𝑆𝑇 < 𝐾1 − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) and an unlimited loss if 𝑆𝑇 > 𝐾2 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ). Hence, this strategy is more suitable for low-volatility stocks. 20. Long Strip. A long strip is an investment strategy where at time 𝑡 an investor would buy a call option 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and buy two put options 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) with strike 𝐾 on the same stock 𝑆𝑡 and having the same expiry time 𝑇 (𝑡 < 𝑇 ). Draw the payoff and profit diagrams of this investment strategy and give a financial interpretation based on this combination of options portfolio.

50

1.2.3 Hedging Strategies

Solution: At expiry time 𝑇 the payoff of this portfolio of options is Ψ(𝑆𝑇 ) = 𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) + 2𝑃 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0} + 2 max{𝐾 − 𝑆𝑇 , 0} ⎧ 2(𝐾 − 𝑆 ) 𝑇 ⎪ =⎨ ⎪ 𝑆𝑇 − 𝐾 ⎩

if 𝑆𝑇 ≤ 𝐾 if 𝑆𝑇 > 𝐾.

Given that the investor paid for the options premiums, the profit at expiry time 𝑇 is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 2𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎧ 2(𝐾 − 𝑆 ) − 𝐶(𝑆 , 𝑡; 𝐾, 𝑇 ) − 2𝑃 (𝑆 , 𝑡; 𝐾, 𝑇 ) 𝑇 𝑡 𝑡 ⎪ =⎨ ⎪ 𝑆𝑇 − 𝐾 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 2𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎩

if 𝑆𝑇 ≤ 𝐾 if 𝑆𝑇 > 𝐾

( ) (provided with break even at 𝑆𝑇 = 𝐾 − 12 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 2𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 2𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 2𝐾) and 𝑆𝑇 = 𝐾 + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 2𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). Figure 1.28 shows the payoff and profit diagrams of a long strip. Payoff/Payoff 2

2

− (

;

)−2 (

;

Payoff

)

Profit

− (

;

)−2 (

;

)

Figure 1.28

Construction of a long strip.

From the profit diagram we can see that the investor would make an unlimited profit if 𝑆𝑇 > 𝐾 + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 2𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) or a limited profit if 1 𝑆𝑇 < 𝐾 − (𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 2𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 )). 2

1.2.3 Hedging Strategies

51

In contrast, the maximum loss for the investor is the cost of purchasing the options. Hence, like the long straddle, this strategy is also dependent on how high the volatility of the stock is rather than the direction of the stock price. However, the only difference between a long straddle and a long strip is that in the latter the strategy exploits more the fall in the stock price as the profit is much higher. 21. Short Strip. A short strip is an investment strategy where at time 𝑡 an investor would sell a call option 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and sell two put options 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) with strike 𝐾 on the same stock 𝑆𝑡 and having the same expiry time 𝑇 (𝑡 < 𝑇 ). Draw the payoff and profit diagrams of this investment strategy and give a financial interpretation based on this combination of options portfolio. Solution: At expiry time 𝑇 the payoff of this portfolio of options is Ψ(𝑆𝑇 ) = −𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) − 2𝑃 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = − max{𝑆𝑇 − 𝐾, 0} − 2 max{𝐾 − 𝑆𝑇 , 0} ⎧ 2(𝑆 − 𝐾) if 𝑆 ≤ 𝐾 𝑇 𝑇 ⎪ =⎨ ⎪ 𝐾 − 𝑆𝑇 if 𝑆𝑇 > 𝐾. ⎩ Given that the writer received the options premium, the profit at expiry time 𝑇 is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 2𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎧ 2(𝑆 − 𝐾) + 𝐶(𝑆 , 𝑡; 𝐾, 𝑇 ) + 2𝑃 (𝑆 , 𝑡; 𝐾, 𝑇 ) 𝑇 𝑡 𝑡 ⎪ =⎨ ⎪ 𝐾 − 𝑆𝑇 + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 2𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎩

if 𝑆𝑇 ≤ 𝐾 if 𝑆𝑇 > 𝐾

( ) with break even at 𝑆𝑇 = 𝐾 − 12 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 2𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) (provided 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 2𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 2𝐾) and 𝑆𝑇 = 𝐾 + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 2𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). Figure 1.29 illustrates the payoff and profit diagrams of a short strip. From the profit diagram we can see that the writer would make an unlimited loss if 𝑆𝑇 > 𝐾 + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 2𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) or a limited loss if 1 𝑆𝑇 < 𝐾 − (𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 2𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 )). 2 In contrast, the maximum gain is the premium received from the options. Hence, like the short straddle, this strategy is also dependent on how high the volatility of the stock is rather than the direction of the stock price. Here the writer bets that the stock price will remain stagnant but incurs a smaller loss if the stock price rises in value.

52

1.2.3 Hedging Strategies

Payoff/Payoff

−2

(

;

)+2 (

;

)

+ (

;

)+2 (

;

)

Profit Payoff

−2

Figure 1.29

Construction of a short strip.

22. Long Strap. A long strap is a hedging technique where at time 𝑡 an investor buys two call options 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and one put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) with strike 𝐾 on the same stock 𝑆𝑡 and having the same expiry time 𝑇 (𝑡 < 𝑇 ). Draw the payoff and profit diagrams of this investment strategy and give a financial interpretation based on this combination of options portfolio. Solution: At expiry time 𝑇 the payoff of this portfolio of options is Ψ(𝑆𝑇 ) = 2𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) + 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 2 max{𝑆𝑇 − 𝐾, 0} + max{𝐾 − 𝑆𝑇 , 0} ⎧𝐾 − 𝑆 if 𝑆𝑇 ≤ 𝐾 𝑇 ⎪ =⎨ ⎪ 2(𝑆𝑇 − 𝐾) if 𝑆𝑇 > 𝐾. ⎩ Given that the investor paid for an options premium, the profit at expiry time 𝑇 is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) − 2𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎧ 𝐾 − 𝑆 − 2𝐶(𝑆 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆 , 𝑡; 𝐾, 𝑇 ) 𝑇 𝑡 𝑡 ⎪ =⎨ ⎪ 2(𝑆𝑇 − 𝐾) − 2𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎩

if 𝑆𝑇 ≤ 𝐾 if 𝑆𝑇 > 𝐾

with break even at 𝑆𝑇 = 𝐾 − 2𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) (provided 2𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + ( ) 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐾) and 𝑆𝑇 = 𝐾 + 12 2𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) . Figure 1.30 shows the payoff and profit diagrams of a long strap.

1.2.3 Hedging Strategies

53 Payoff/Payoff Payoff

Profit −2 (

;

)− (

;

)

−2 (

;

)− (

;

)

Figure 1.30

Construction of a long strap.

From the profit diagram we can see that the investor would make an unlimited profit if 1 𝑆𝑇 > 𝐾 + (2𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 )) 2 or a limited profit if 𝑆𝑇 < 𝐾 − 2𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) whilst the maximum loss for the investor is the cost of purchasing the options. Thus, like the long straddle, this strategy is also dependent on how high the volatility of the stock is rather than the direction of the stock price. However, the only difference between a long straddle and a long strap is that in the latter the strategy exploits more the rise in the stock price as the profit is much higher. 23. Short Strap. A short strap is a hedging technique where at time 𝑡 a writer sells two call options 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and one put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) with strike 𝐾 on the same stock 𝑆𝑡 and having the same expiry time 𝑇 (𝑡 < 𝑇 ). Draw the payoff and profit diagrams of this investment strategy and give a financial interpretation based on this combination of options portfolio. Solution: At expiry time 𝑇 the payoff of this portfolio of options is Ψ(𝑆𝑇 ) = −2𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = −2 max{𝑆𝑇 − 𝐾, 0} − max{𝐾 − 𝑆𝑇 , 0} ⎧𝑆 − 𝐾 if 𝑆𝑇 ≤ 𝐾 ⎪ 𝑇 =⎨ ⎪ 2(𝐾 − 𝑆𝑇 ) if 𝑆𝑇 > 𝐾. ⎩

54

1.2.3 Hedging Strategies

Given that the writer received the options premium, the profit at expiry time 𝑇 is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) + 2𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎧ 𝑆 − 𝐾 + 2𝐶(𝑆 , 𝑡; 𝐾, 𝑇 ) + 𝑃 (𝑆 , 𝑡; 𝐾, 𝑇 ) 𝑡 𝑡 ⎪ 𝑇 =⎨ ⎪ 2(𝐾 − 𝑆𝑇 ) + 2𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎩

if 𝑆𝑇 ≤ 𝐾 if 𝑆𝑇 > 𝐾

with break even at 𝑆𝑇 = 𝐾 − 2𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) (provided 2𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐾) and 𝑆𝑇 = 𝐾 + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 12 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). Figure 1.31 shows the payoff and profit diagrams of a short strap. Payoff/Payoff

−

2 (

;

)+ (

;

)

+2 (

;

)+ (

;

) Profit

−

Payoff

Figure 1.31

Construction of a short strap.

Based on the profit diagram we can see that the writer would make an unlimited loss if 1 𝑆𝑇 > 𝐾 + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) 2 or a limited loss if 𝑆𝑇 < 𝐾 − 2𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). In contrast, the maximum gain is the premium received from the options. Hence, like the short straddle, this strategy is also dependent on how high the volatility of the stock is rather than the direction of the stock price. Here, the writer bets that the stock is price will remain stagnant but incurs a smaller loss if the stock price falls in value. 24. Butterfly Spread (Using Call Options). A butterfly spread is a hedging technique that, at time 𝑡 and for strikes 𝐾1 < 𝐾2 < 𝐾3 , is constructed by buying one call option 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ), buying one call option 𝐶(𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) and selling two call options 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) on the same stock 𝑆𝑡 and having the same expiry time 𝑇 (𝑡 < 𝑇 ). Draw the payoff and profit diagrams of this hedging strategy with 𝐾2 = 12 (𝐾1 + 𝐾3 ).

1.2.3 Hedging Strategies

55

Solution: At expiry time 𝑇 the payoff of a butterfly spread is Ψ(𝑆𝑇 ) = 𝐶(𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑇 , 𝑇 ; 𝐾3 , 𝑇 ) − 2𝐶(𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) = max{𝑆𝑇 − 𝐾1 , 0} + max{𝑆𝑇 − 𝐾3 , 0} − 2 max{𝑆𝑇 − 𝐾2 , 0} if 𝑆𝑇 ≤ 𝐾1 ⎧0 ⎪ ⎪𝑆 − 𝐾 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 1 ⎪ 𝑇 =⎨ ⎪ 2𝐾2 − 𝐾1 − 𝑆𝑇 if 𝐾2 < 𝑆𝑇 ≤ 𝐾3 ⎪ ⎪ ⎩ 2𝐾2 − 𝐾1 − 𝐾3 if 𝑆𝑇 > 𝐾3 and by setting Π𝑡 = 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) − 2𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) the corresponding profit is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) − Π𝑡 . Hence, provided −𝐾1 ≤ 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) − 2𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≤ 2𝐾2 − 1 then break even occurs at 𝑆𝑇 = 𝐾1 + 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) − 2𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) and 𝑆𝑇 = 2𝐾2 − 𝐾1 − 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) + 2𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ). Figure 1.32 shows the payoff and profit diagrams of a butterfly spread using call options with 𝐾2 = 12 (𝐾1 + 𝐾3 ). Payoff/Payoff

2

−

Payoff

1

1 2−

Figure 1.32

1−Π

2

3

Profit

Construction of a butterfly spread with 𝐾2 = 12 (𝐾1 + 𝐾3 ) and Π𝑡 the net premium paid.

56

1.2.3 Hedging Strategies

25. Butterfly Spread (Using Put Options). A butterfly spread is a hedging technique that, at time 𝑡 and for strikes 𝐾1 < 𝐾2 < 𝐾3 , is constructed by buying one put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ), buying one put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) and selling two put options 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) on the same stock 𝑆𝑡 and having the same expiry time 𝑇 (𝑡 < 𝑇 ). Draw the payoff and profit diagrams of this hedging strategy with 𝐾2 = 12 (𝐾1 + 𝐾3 ). Solution: At expiry time 𝑇 the payoff of a butterfly spread is Ψ(𝑆𝑇 ) = 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) + 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾3 , 𝑇 ) − 2𝑃 (𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) = max{𝐾1 − 𝑆𝑇 , 0} + max{𝐾3 − 𝑆𝑇 , 0} − 2 max{𝐾2 − 𝑆𝑇 , 0} ⎧ 𝐾1 + 𝐾3 − 2𝐾2 if 𝑆𝑇 ≤ 𝐾1 ⎪ ⎪ 𝑆 + 𝐾 − 2𝐾 if 𝐾 < 𝑆 ≤ 𝐾 3 2 1 𝑇 2 ⎪ 𝑇 =⎨ ⎪ 𝐾 3 − 𝑆𝑇 if 𝐾2 < 𝑆𝑇 ≤ 𝐾3 ⎪ ⎪ if 𝑆𝑇 > 𝐾3 ⎩0 and by setting Π𝑡 = 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) − 2𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) the corresponding profit is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) − Π𝑡 . Hence, provided 𝐾3 − 2𝐾2 ≤ 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) − 2𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≤ 𝐾3 then break even occurs at 𝑆𝑇 = 2𝐾2 − 𝐾3 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) − 2𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) and 𝑆𝑇 = 𝐾3 − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) + 2𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ). Figure 1.33 shows the payoff and profit diagrams of a butterfly spread using put options with 𝐾2 = 12 (𝐾1 + 𝐾3 ). 26. Butterfly Spread (Using Straddle and Strangle). A butterfly spread is a hedging technique that, at time 𝑡 and for strikes 𝐾1 < 𝐾2 < 𝐾3 , is constructed by selling a straddle: s sell a call option 𝐶(𝑆 , 𝑡; 𝐾 , 𝑇 ) with strike 𝐾 𝑡 2 2 s sell a put option 𝑃 (𝑆 , 𝑡; 𝐾 , 𝑇 ) with strike 𝐾 𝑡 2 2 and buying a strangle: s buy a call option 𝐶(𝑆 , 𝑡; 𝐾 , 𝑇 ) with strike 𝐾 𝑡 1 1 s buy a put option 𝑃 (𝑆 , 𝑡; 𝐾 , 𝑇 ) with strike 𝐾 𝑡 3 3 on the same stock 𝑆𝑡 and having the same option expiry time 𝑇 (𝑡 < 𝑇 ). Draw the payoff and profit diagrams of this investment strategy with 𝐾2 = 12 (𝐾1 + 𝐾3 ).

1.2.3 Hedging Strategies

57

Payoff/Payoff

3

−

2

−Π

Payoff

2

1 3

Figure 1.33

−

2

3

Profit

Construction of a butterfly spread with 𝐾2 = 12 (𝐾1 + 𝐾3 ) and Π𝑡 the net premium paid.

Solution: At expiry time 𝑇 the payoff of a butterfly spread is Ψ(𝑆𝑇 ) = 𝐶(𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) + 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾3 , 𝑇 ) −𝐶(𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) = max{𝑆𝑇 − 𝐾1 , 0} + max{𝐾3 − 𝑆𝑇 , 0} − max{𝑆𝑇 − 𝐾2 , 0} − max{𝐾2 − 𝑆𝑇 , 0} if 𝑆𝑇 ≤ 𝐾1 ⎧ 𝐾3 − 𝐾1 ⎪ ⎪𝑆 + 𝐾 − 𝐾 − 𝐾 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 3 1 2 ⎪ 𝑇 =⎨ ⎪ −𝑆𝑇 + 𝐾2 + 𝐾3 − 𝐾1 if 𝐾2 < 𝑆𝑇 ≤ 𝐾3 ⎪ ⎪ if 𝑆𝑇 > 𝐾3 ⎩ 𝐾2 − 𝐾1 and by setting Π𝑡 = 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) the corresponding profit is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) − Π𝑡 . Hence, provided −𝐾1 − 𝐾2 + 𝐾3 ≤

𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) ≤ −𝐾1 + 𝐾2 + 𝐾3 −𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 )

break even occurs at 𝑆𝑇 = 𝐾2 + 𝐾3 − 𝐾1 − 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 )

58

1.2.3 Hedging Strategies

and 𝑆𝑇 = 𝐾1 + 𝐾2 − 𝐾3 + 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ). Figure 1.34 shows the payoff and profit diagrams of a butterfly spread with 𝐾2 = 1 (𝐾1 + 𝐾3 ). 2 Payoff/Payoff

3

−

1

2

−

1

Payoff 1

2

Figure 1.34

−

1

−Π

2

3

Profit

Construction of a butterfly spread with 𝐾2 = 12 (𝐾1 + 𝐾3 ) and Π𝑡 the net premium paid.

27. What is the financial motivation for an investor to enter into a butterfly spread contract? Solution: Given three strike prices 𝐾1 < 𝐾2 < 𝐾3 , a butterfly spread can be constructed with a combination of call options, put options or by combining a straddle and a strangle. At expiry time 𝑇 for 0 < 𝑆𝑇min < 𝑆𝑇max such that the profit of a butterfly spread Υ(𝑆𝑇min ) = 0 and Υ(𝑆𝑇max ) = 0 then the investor would make a limited loss if the stock price 𝑆𝑇 > 𝑆𝑇max or 𝑆𝑇 < 𝑆𝑇min . In contrast, the maximum gain for the investor occurs when 𝑆𝑇 = 𝐾2 . Thus, the investor buying a butterfly spread would speculate that the stock price at expiry time 𝑇 will be between 𝐾1 and 𝐾3 in which the strategy will be most profitable. In essence, by exploiting simultaneously both the low and high volatilities of the stock price based on the combination of options, the investor purchasing a butterfly spread would bet that the stock price will stay close to 𝐾2 . 28. Condor Spread (Using Call Options). A condor spread is a hedging technique that, at time 𝑡 and for strikes 𝐾1 < 𝐾2 < 𝐾3 < 𝐾4 , is constructed by buying one call option 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ), buying one call option 𝐶(𝑆𝑡 , 𝑡; 𝐾4 , 𝑇 ), selling one call option 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) and selling one call option 𝐶(𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) on the same stock 𝑆𝑡 and having the same option expiry time 𝑇 (𝑡 < 𝑇 ). Draw the payoff and profit diagrams of this hedging strategy with 𝐾2 − 𝐾1 = 𝐾4 − 𝐾3 .

1.2.3 Hedging Strategies

59

Solution: Based on the construction of the options, the payoff at expiry time 𝑇 is Ψ(𝑆𝑇 ) = 𝐶(𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) −𝐶(𝑆𝑇 , 𝑇 ; 𝐾3 , 𝑇 ) + 𝐶(𝑆𝑇 , 𝑇 ; 𝐾4 , 𝑇 ) = max{𝑆𝑇 − 𝐾1 , 0} − max{𝑆𝑇 − 𝐾2 , 0} − max{𝑆𝑇 − 𝐾3 , 0} + max{𝑆𝑇 − 𝐾4 , 0} ⎧0 ⎪ ⎪ ⎪ 𝑆𝑇 − 𝐾1 ⎪ ⎪ = ⎨ 𝐾2 − 𝐾1 ⎪ ⎪𝐾 − 𝐾 + 𝐾 − 𝑆 1 3 𝑇 ⎪ 2 ⎪ ⎪ 𝐾2 − 𝐾1 + 𝐾3 − 𝐾4 ⎩

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝐾2 < 𝑆𝑇 ≤ 𝐾3 if 𝐾3 < 𝑆𝑇 ≤ 𝐾4 if 𝑆𝑇 > 𝐾4

and by setting Π𝑡 = 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾4 , 𝑇 ) the corresponding profit is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) − Π𝑡 . Hence, provided −𝐾1 ≤

𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≤ −𝐾1 + 𝐾2 + 𝐾3 −𝐶(𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾4 , 𝑇 )

then break even occurs at 𝑆𝑇 = 𝐾1 + 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾4 , 𝑇 ) and 𝑆𝑇 = 𝐾2 − 𝐾1 + 𝐾3 − 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) +𝐶(𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾4 , 𝑇 ). Figure 1.35 shows the payoff and profit diagrams of a condor spread using call options with 𝐾2 − 𝐾1 = 𝐾4 − 𝐾3 . 29. Condor Spread (Using Put Options). A condor spread is a hedging technique that, at time 𝑡 and for strikes 𝐾1 < 𝐾2 < 𝐾3 < 𝐾4 , is constructed by buying one put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ), buying one put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾4 , 𝑇 ), selling one put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) and selling one put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) on the same stock 𝑆𝑡 and having the same expiry time 𝑇 (𝑡 < 𝑇 ). Draw the payoff and profit diagrams of this hedging strategy with 𝐾2 − 𝐾1 = 𝐾4 − 𝐾3 .

60

1.2.3 Hedging Strategies

Payoff/Payoff

Payoff

2

Figure 1.35

−

2

−

1

−Π

1

1

2

3

4

Profit

Construction of a condor spread with 𝐾2 − 𝐾1 = 𝐾4 − 𝐾3 and Π𝑡 the net premium paid.

Solution: Based on the construction of the put options, the payoff at expiry time 𝑇 is Ψ(𝑆𝑇 ) = 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) −𝑃 (𝑆𝑇 , 𝑇 ; 𝐾3 , 𝑇 ) + 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾4 , 𝑇 ) = max{𝐾1 − 𝑆𝑇 , 0} − max{𝐾2 − 𝑆𝑇 , 0} − max{𝐾3 − 𝑆𝑇 , 0} + max{𝐾4 − 𝑆𝑇 , 0} ⎧𝐾 − 𝐾 − 𝐾 + 𝐾 2 3 4 ⎪ 1 ⎪ ⎪ 𝑆𝑇 − 𝐾2 − 𝐾3 + 𝐾4 ⎪ ⎪ = ⎨ 𝐾4 − 𝐾3 ⎪ ⎪𝐾 − 𝑆 𝑇 ⎪ 4 ⎪ ⎪0 ⎩

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝐾2 < 𝑆𝑇 ≤ 𝐾3 if 𝐾3 < 𝑆𝑇 ≤ 𝐾4 if 𝑆𝑇 > 𝐾4

and by setting Π𝑡 = 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾4 , 𝑇 ) the corresponding profit is Υ(𝑆𝑇 ) = Ψ(𝑆𝑇 ) − Π𝑡 . Hence, provided

−𝐾2 − 𝐾3 + 𝐾4 ≤

𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≤ 𝐾4 −𝑃 (𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾4 , 𝑇 )

1.2.3 Hedging Strategies

61

then break even occurs at 𝑆𝑇 = 𝐾2 + 𝐾3 − 𝐾4 + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) −𝑃 (𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾4 , 𝑇 ) and 𝑆𝑇 = 𝐾4 − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾3 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾4 , 𝑇 ). Figure 1.36 shows the payoff and profit diagrams of a condor spread using put options with 𝐾2 − 𝐾1 = 𝐾4 − 𝐾3 . Payoff/Payoff

Payoff

4

Figure 1.36

−

4

−

3

−Π

3

1

2

3

4

Profit

Construction of a condor spread with 𝐾2 − 𝐾1 = 𝐾4 − 𝐾3 and Π𝑡 the net premium paid.

30. Explain the financial motivation for an investor to enter into a condor spread contract. Solution: Using four different strike prices 𝐾1 < 𝐾2 < 𝐾3 < 𝐾4 , a condor spread can be constructed with a combination of call options or put options. At expiry time 𝑇 for 0 < 𝑆𝑇min < 𝑆𝑇max such that the profit of a condor spread Υ(𝑆𝑇min ) = 0 and Υ(𝑆𝑇max ) = 0 then the investor would make a limited loss if the stock price 𝑆𝑇 > 𝑆𝑇max or 𝑆𝑇 < 𝑆𝑇min . Thus, the investor who invests in a condor spread speculates that the stock price at expiry time 𝑇 will be between 𝐾2 and 𝐾3 in which the strategy will be most profitable. But unlike a butterfly spread, a condor spread has a much wider profit range at the expense of a higher premium paid. Thus, this contract is profitable for the investor who has a neutral outlook on the market.

2 European Options Options are one of the basic building blocks in finance and European options are the most common type of equity derivatives. They give the holder of the contract the right but not the obligation to enter into a future transaction only at the expiry date of the contract. In short, for buyers of European options, their right of exercise is only at expiration of the contract. As discussed in Chapter 1, a combination of such options with other products provides a multitude of trading strategies for hedgers, investors, traders and speculators.

2.1 INTRODUCTION Central to the pricing of European options are the Black–Scholes (or Black–Scholes–Merton) formula and the martingale pricing theory. In the Black–Scholes–Merton formulation, there are two major approaches which arrive at the same partial differentiation equation formula: s delta hedging strategy s self-financing trading strategy Delta Hedging Strategy In 1973, Fischer Black and Myron Scholes published their paper “The pricing of options and corporate liabilities” in The Journal of Political Economy with the aim of deriving a theoretical valuation formula to price European options. The principal idea behind their theory is to hedge the option by buying/selling the underlying asset in such a way as to eliminate risk. This type of hedging is called delta hedging. Conceptually, at time 𝑡 the hedging portfolio Π𝑡 can be expressed by Π𝑡 = 𝑉 − Δ𝑆𝑡 consisting of a long position in the option 𝑉 and a Δ number of short positions in the asset 𝑆𝑡 . Before the formula was derived, they made the following assumptions: s s s s s s s

The risk-free interest rate is known and is constant over time. The asset price follows a geometric Brownian motion. The asset pays no dividends during the life of the option. The option is European-style, which can only be exercised at expiry date. No transaction costs are associated with buying or selling the asset/option. Trading of the asset can take place continuously. Short selling is permitted.

Under these assumptions the option value depends only on the price of the asset and time, and on parameters which are taken to be known constants (i.e, 𝑉 = 𝑉 (𝑆𝑡 , 𝑡)). In addition,

64

2.1 Introduction

over “short” time intervals, the stochastic part of the change in the option price is perfectly correlated with changes in the stock price, i.e., 𝑑Π𝑡 = 𝑑𝑉 − Δ𝑑𝑆𝑡 . By writing the stochastic process of the asset price 𝑆𝑡 as a geometric Brownian motion 𝑑𝑆𝑡 = 𝜇𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift or expected rate of return, 𝜎 is a constant volatility and 𝑊𝑡 is the standard Wiener process on the probability space (Ω, ℱ, ℙ), and expanding 𝑉 (𝑆𝑡 , 𝑡) in a Taylor series about (𝑆𝑡 , 𝑡) and subsequently apply It¯o’s lemma we eventually have 1 𝜕2𝑉 𝜕𝑉 𝜕𝑉 𝑑𝑆𝑡 + (𝑑𝑆𝑡 )2 + … 𝑑𝑡 + 𝜕𝑡 𝜕𝑆𝑡 2 𝜕𝑆𝑡2 ( ) 𝜕𝑉 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 = + 𝜇𝑆𝑡 𝑑𝑊𝑡 . 𝑑𝑡 + 𝜎𝑆𝑡 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡2

𝑑𝑉 =

As a result ( 𝑑Π𝑡 =

𝜕𝑉 𝜕2 𝑉 1 + 𝜎 2 𝑆𝑡2 2 + 𝜇𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡

(

)) ) ( 𝜕𝑉 𝜕𝑉 −Δ 𝑑𝑡 + 𝜎𝑆𝑡 − Δ 𝑑𝑊𝑡 . 𝜕𝑆𝑡 𝜕𝑆𝑡

𝜕𝑉 to eliminate all market risk, and from the assump𝜕𝑆𝑡 tion that the risk-free portfolio must have an expected return equal to the risk-free rate 𝑟, the governing partial-differential (or Black–Scholes) equation for an option price at time 𝑡, 𝑉 (𝑆𝑡 , 𝑡) is By choosing the portfolio weight Δ =

𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0. + 𝜎 2 𝑆𝑡2 2 + 𝑟 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 Note that in the Black–Scholes equation the drift parameter is not present, and this shows that the value of an option is independent of the rate of change of an underlying asset. The only parameters that affect the option price are the risk-free interest rate and the volatility of the asset underlying. As a result of this important argument, we may have a situation whereby two parties may differ in their estimate of the asset price growth, yet still agree on the price of an option. Self-Financing Trading Strategy In the same year, Robert C. Merton published “Theory of rational option pricing” in The Bell Journal of Economics and Management Science where an alternative derivation of the Black– Scholes model is presented via the construction of a self-financing trading strategy.

2.1 Introduction

65

By definition, at time 𝑡 ∈ [0, 𝑇 ], the trading strategy (𝜙𝑡 , 𝜓𝑡 ) of holding 𝜙𝑡 shares of risky asset 𝑆𝑡 and 𝜓𝑡 units of risk-free asset 𝐵𝑡 having a portfolio value Π𝑡 = 𝜙𝑡 𝑆𝑡 + 𝜓𝑡 𝐵𝑡 is called self-financing (or a self-financing portfolio) if and only if 𝑑Π𝑡 = 𝜙𝑡 𝑑𝑆𝑡 + 𝜓𝑡 𝑑𝐵𝑡 implying that the change in portfolio value is due to changes in market conditions and not to either infusion or extraction of funds. By definition, the contingent claim 𝑉 (𝑆𝑇 , 𝑇 ) is said to be attainable if there exists an admissible strategy worth Π𝑇 = 𝑉 (𝑆𝑇 , 𝑇 ) at the option expiry time 𝑇 . Here, the trading strategy (𝜙𝑡 , 𝜓𝑡 ) is admissible if the portfolio Π𝑡 = 𝜙𝑡 𝑆𝑡 + 𝜓𝑡 𝐵𝑡 is self-financing and if Π𝑡 ≥ −𝛼 almost surely for some 𝛼 > 0 (i.e., a finite credit line). In the absence of arbitrage, at every time 𝑡 ∈ [0, 𝑇 ], 𝑉 (𝑆𝑡 , 𝑡) must be equal to the portfolio value Π𝑡 such that Π𝑡 = 𝑉 (𝑆𝑡 , 𝑡). Otherwise, an arbitrage opportunity occurs. At time 𝑡 ∈ [0, 𝑇 ], to replicate the option 𝑉 (𝑆𝑡 , 𝑡), we can now form the following self-financing portfolio 𝑉 (𝑆𝑡 , 𝑡) = 𝜙𝑡 𝑆𝑡 + 𝜓𝑡 𝐵𝑡 . Let the risk-free asset price 𝐵𝑡 and the risky asset price 𝑆𝑡 have the following diffusion process 𝑑𝐵𝑡 = 𝑟𝐵𝑡 𝑑𝑡

and

𝑑𝑆𝑡 = 𝜇 𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡

where 𝑟 is the risk-free interest rate, 𝜇 is the asset drift rate, 𝜎 is the asset price volatility which are all constants and {𝑊𝑡 : 0 ≤ 𝑡 ≤ 𝑇 } is a standard Wiener process on the probability space (Ω, ℱ, ℙ). Applying Taylor series on 𝑑𝑉 and using It¯o’s lemma, we eventually have 𝜕𝑉 𝑑𝑆 + 𝑑𝑉 = 𝜕𝑆𝑡 𝑡

(

𝜕𝑉 𝜕2 𝑉 1 + 𝜎 2 𝑆𝑡2 2 𝜕𝑡 2 𝜕𝑆𝑡

) 𝑑𝑡.

Since the trading strategy (𝜙𝑡 , 𝜓𝑡 ) is self-financing if and only if 𝑑 𝑉 = 𝜙𝑡 𝑑𝑆𝑡 + 𝜓𝑡 𝑑𝐵𝑡 = 𝜙𝑡 𝑑𝑆𝑡 + 𝑟 𝐵𝑡 𝜓𝑡 𝑑𝑡 equating both of the equations we have 𝑟𝐵𝑡 𝜓𝑡 =

𝜕𝑉 𝜕2𝑉 1 + 𝜎 2 𝑆𝑡2 2 𝜕𝑡 2 𝜕𝑆𝑡

and

𝜙𝑡 =

𝜕𝑉 𝜕𝑆𝑡

66

2.1 Introduction

and substituting the results into 𝑉 (𝑆𝑡 , 𝑡) = 𝜙𝑡 𝑆𝑡 + 𝜓𝑡 𝐵𝑡 we have the Black–Scholes equation 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0. + 𝜎 2 𝑆𝑡2 2 + 𝑟 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 For a constant strike 𝐾 > 0, if the terminal payoff at time 𝑇 > 𝑡 is ⎧max{𝑆 − 𝐾, 0} for European call option 𝑇 ⎪ 𝑉 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = ⎨ ⎪max{𝐾 − 𝑆𝑇 , 0} for European put option ⎩ with boundary conditions ⎧0 ⎪ 𝑉 (0, 𝑡; 𝐾, 𝑇 ) = ⎨ ⎪𝐾𝑒−𝑟(𝑇 −𝑡) ⎩

for European call option for European put option

and ⎧𝑆 ⎪ 𝑡 lim 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ 𝑆𝑡 →∞ ⎪0 ⎩

for European call option for European put option

then by solving the Black–Scholes equation via the the heat equation, the closed-form solution for the European call or put option price at time 𝑡 is ⎧𝑆 Φ(𝑑 ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑 ) + − ⎪ 𝑡 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ ⎪𝐾𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑− ) − 𝑆𝑡 Φ(−𝑑+ ) ⎩

for European call option for European put option

where log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝑡) 𝑑± = √ 𝜎 𝑇 −𝑡 𝑥

1 2 1 and Φ(𝑥) = √ 𝑒− 2 𝑢 𝑑𝑢 is the cumulative distribution function (cdf) of a standard 2𝜋 ∫−∞ normal.

2.1 Introduction

67

Given that the values of the European call and put options contain the function for the cdf of a standard normal distribution, we can see that the option prices have a relationship with the probability density function of the asset 𝑆𝑡 which is log normally distributed. From the Feynman–Kac formula, we can infer that the option price can be interpreted as the discounted expected value of the payoff at expiry time 𝑇 . This leads us to the subject of risk-neutral valuation of contingent claims via the martingale pricing theory. Martingale Pricing Theory As the name suggests, martingale pricing is based on the notion of equivalent martingale measure and risk-neutral valuation. Without loss of generality, assume we are in the Black–Scholes world with an economy consisting of a risky asset or stock 𝑆𝑡 following a geometric Brownian motion and a risk-free asset 𝐵𝑡 growing at a continuously compounded interest rate 𝑟 of the form 𝑑𝐵𝑡 = 𝑟 𝑑𝑡 𝐵𝑡

𝑑𝑆𝑡 = 𝜇𝑑𝑡 + 𝜎𝑑𝑊𝑡 , 𝑆𝑡

where 𝜇 is the stock drift, 𝜎 is the stock volatility and 𝑊𝑡 is a standard Wiener process on the probability space (Ω, ℱ, ℙ). Definition 2.1 (Equivalent Martingale Measure) Let (Ω, ℱ, ℙ) be the probability space satisfying the usual conditions and let ℚ be another probability measure on (Ω, ℱ, ℚ). The probability measure ℚ is said to be an equivalent measure or risk-neutral measure if it satisfies s ℚ ∼ ℙ. s The discounted price process {𝐵 −1 𝑆 (𝑖) }, 𝑖 = 1, 2, … , 𝑚 are martingales under ℚ, that is 𝑡 𝑡 ( ) | 𝔼ℚ 𝐵𝑢−1 𝑆𝑢(𝑖) | ℱ𝑡 = 𝐵𝑡−1 𝑆𝑡(𝑖) | for all 0 ≤ 𝑡 ≤ 𝑢 ≤ 𝑇 . From the definition of an equivalent martingale measure, we can now state Girsanov’s theorem which tells us how to convert from the physical measure ℙ to the risk-neutral measure ℚ. Theorem 2.2 (Girsanov’s Theorem) Let {𝑊𝑡 : 0 ≤ 𝑡 ≤ 𝑇 } be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let ℱ𝑡 , 0 ≤ 𝑡 ≤ 𝑇 be the associated Wiener process filtration. Suppose 𝜃𝑡 is an adapted process, 0 ≤ 𝑡 ≤ 𝑇 and consider 𝑡

1

𝑡 2

𝑍𝑡 = 𝑒− ∫0 𝜃𝑠 𝑑𝑊𝑠 − 2 ∫0 𝜃𝑠 𝑑𝑠 . If ( 1 𝑇 2 ) 𝔼ℙ 𝑒 2 ∫0 𝜃𝑡 𝑑𝑡 < ∞

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2.1 Introduction

then 𝑍𝑡 is a positive ℙ-martingale for 0 ≤ 𝑡 ≤ 𝑇 . By changing the measure ℙ to a measure ℚ such that ( ) 𝑑ℚ || 𝑑ℚ || ℙ = 𝑍𝑡 ℱ = 𝔼 𝑑ℙ || 𝑡 𝑑ℙ ||ℱ𝑡 ̃𝑡 = 𝑊𝑡 + then 𝑊

𝑡

∫0

𝜃𝑢 𝑑𝑢 is a ℚ-standard Wiener process.

Finally, as far as pricing of contingent claims 𝑉 (𝑆𝑇 , 𝑇 ) we can obtain the risk-neutral valuation approach given in the following theorem. Theorem 2.3 Assume that an equivalent martingale measure ℚ exists, and let 𝑉 (𝑆𝑇 , 𝑇 ) be an attainable contingent claim generated by an admissible self-financing trading strategy. Under the filtration ℱ𝑡 , the European option price at time 𝑡 is given by [ ] | 𝑉 (𝑆𝑡 , 𝑡) = 𝐵𝑡 𝔼ℚ 𝐵𝑇−1 𝑉 (𝑆𝑇 , 𝑇 )| ℱ𝑡 = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ [𝑉 (𝑆𝑇 , 𝑇 )|ℱ𝑡 ]. | Remark 2.4 Consider the stochastic differential equation (SDE) 𝑑𝑋𝑡 = 𝜇(𝑋𝑡 , 𝑡)𝑑𝑡 + 𝜎(𝑋𝑡 , 𝑡)𝑑𝑊𝑡 [( with 𝔼

𝑡

∫0 𝜇(𝑋𝑡 , 𝑡) = 0). Since

)2 ] 𝜎(𝑋𝑠 , 𝑠) 𝑑𝑠 2

< ∞, then 𝑋𝑡 is a martingale if and only if 𝑋𝑡 is driftless (i.e.,

(

) 𝑆𝑡 ( ) 𝐵𝑡 𝑑𝑆𝑡 𝑑𝐵𝑡 𝑑𝑆𝑡 𝑑𝐵𝑡 𝑑𝐵𝑡 2 − − + ( ) = 𝑆𝑡 𝐵𝑡 𝑆𝑡 𝐵𝑡 𝐵𝑡 𝑆𝑡 𝐵𝑡 = (𝜇 − 𝑟)𝑑 𝑡 + 𝜎𝑑 𝑊𝑡

𝑑

̃𝑡 = 𝑑 𝑊𝑡 + 𝜃𝑑𝑡 where 𝜃 is a constant and 𝑊 ̃𝑡 a ℚ-standard Wiener process, and by writing 𝑑 𝑊 ( )/ ( ) 𝑆𝑡 𝑆𝑡 and applying Girsanov’s theorem to 𝑑 , we obtain 𝐵𝑡 𝐵𝑡 ( ) 𝑆𝑡 𝑑 𝐵𝑡 ̃𝑡 . ( ) = (𝜇 − 𝑟 − 𝜃𝜎)𝑑𝑡 + 𝜎𝑑 𝑊 𝑆𝑡 𝐵𝑡 To make 𝑆𝑡 ∕𝐵𝑡 a ℚ-martingale, we set 𝜃=

𝜇−𝑟 . 𝜎

2.1 Introduction

69

Thus, there is a unique 𝜃 which makes the discounted asset price process driftless, which is equivalent to saying that there is a unique change of measure which makes the discounted asset price a martingale under the risk-neutral measure. Hence, under the risk-neutral measure ℚ, the asset price follows the diffusion process 𝑑𝑆𝑡 ̃𝑡 = 𝑟 𝑑 𝑡 + 𝜎𝑑 𝑊 𝑆𝑡 and solving the SDE for 𝑇 > 𝑡 and under the filtration ℱ𝑡 we have ( )| ) (( ) 𝑆𝑇 | 1 log | ℱ𝑡 ∼  𝑟 − 𝜎 2 (𝑇 − 𝑡), 𝜎 2 (𝑇 − 𝑡) . 𝑆𝑡 || 2 Thus, the European call option at time 𝑡 with strike price 𝐾 and expiry time 𝑇 > 𝑡 is [ ] 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑆𝑇 − 𝐾, 0}|| ℱ𝑡 [ ] | = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ (𝑆𝑇 − 𝐾)1I{𝑆𝑇 ≥𝐾} | ℱ𝑡 | { [ ] [ ]} | | −𝑟(𝑇 −𝑡) ℚ 𝔼 𝑆𝑇 1I{𝑆𝑇 ≥𝐾} | ℱ𝑡 − 𝐾𝔼ℚ 1I{𝑆𝑇 ≥𝐾} | ℱ𝑡 =𝑒 | | { [ ] )} ( | = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 𝑆𝑇 1I{𝑆𝑇 ≥𝐾} | ℱ𝑡 − 𝐾ℚ 𝑆𝑇 ≥ 𝐾 || ℱ𝑡 | whilst the European put option at time 𝑡 with strike price 𝐾 and expiry time 𝑇 > 𝑡 is [ ] 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐾 − 𝑆𝑇 , 0}|| ℱ𝑡 [ ] | = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ (𝐾 − 𝑆𝑇 )1I{𝑆𝑇 ≤𝐾} | ℱ𝑡 | { [ ]} [ ] | | −𝑟(𝑇 −𝑡) ℚ 𝐾𝔼 1I{𝑆𝑇 ≤𝐾} | ℱ𝑡 − 𝔼ℚ 𝑆𝑇 1I{𝑆𝑇 ≤𝐾} | ℱ𝑡 =𝑒 | | { [ ]} ) ( | −𝑟(𝑇 −𝑡) ℚ | 𝐾ℚ 𝑆𝑇 ≤ 𝐾 | ℱ𝑡 − 𝔼 𝑆𝑇 1I{𝑆𝑇 ≤𝐾} | ℱ𝑡 . =𝑒 | Continuous-Time Limit of Binomial Model The binomial model or lattice approach is an alternative way to describe the evolution of the stochastic asset price process in discrete time. Following the seminal work of Cox, Ross and Rubinstein (1979), for each time step Δ𝑡, if the asset price begins at 𝑆𝑡 at time 𝑡, then at time 𝑡 + Δ𝑡 the asset can either move up to a value 𝑢𝑆𝑡 or move down to a value 𝑑𝑆𝑡 with riskneutral probabilities 𝜋 and 1 − 𝜋, respectively where 𝑢 and 𝑑 = 1∕𝑢 are fixed constants such that 0 < 𝑑 < 𝑒𝑟Δ𝑡 < 𝑢. The condition 𝑑 = 1∕𝑢 is used to ensure that the lattice is symmetrical and is recombinant. Here, an up movement in the asset price for one node followed by a down movement in the asset price for the next node generates the same asset price as a down movement in the first node followed by an up movement in the next node. Assuming that the asset price follows a geometric Brownian motion under the risk-neutral measure ℚ ( log

𝑆𝑡+Δ𝑡 𝑆𝑡

) ∼

[(

) ] 1 𝑟 − 𝜎 2 Δ𝑡, 𝜎 2 Δ𝑡 2

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2.1 Introduction

and by matching the first and second moments up to 𝑂(Δ𝑡) ( ) 𝔼ℚ 𝑆𝑡+Δ𝑡 || 𝑆𝑡 = 𝜋𝑢𝑆𝑡 + (1 − 𝜋)𝑑𝑆𝑡 ( ) | 2 𝔼ℚ 𝑆𝑡+Δ𝑡 | 𝑆𝑡 = 𝜋𝑢2 𝑆𝑡2 + (1 − 𝜋)𝑑 2 𝑆𝑡2 | we eventually have 𝑢 = 𝑒𝜎Δ𝑡 ,

𝑑 = 𝑒−𝜎Δ𝑡

and

𝜋=

𝑒𝑟Δ𝑡 − 𝑑 . 𝑢−𝑑

Once we have built up a lattice of possible asset prices up to the option expiry time 𝑇 , 𝑇 > 𝑡 we can work backwards to price the option. Under the risk-neutral probability measure ℚ, the European call option price at any node is given as the discounted expected value using the move up option and the move down option values multiplied by their respective risk-neutral probabilities. Thus, the option price corresponding to a node for time 𝑡 is calculated as [ ] 𝑉 (𝑆𝑡 , 𝑡) = 𝑒−𝑟Δ𝑡 𝜋𝑉 (𝑢 𝑆𝑡 , 𝑡 + Δ𝑡) + (1 − 𝜋)𝑉 (𝑑𝑆𝑡 , 𝑡 + Δ𝑡) . To show that the Black–Scholes equation can be obtained as a limiting case in continuous time of the binomial model, we first take Taylor expansions of 𝑉 (𝑢 𝑆𝑡 , 𝑡 + Δ𝑡) and 𝑉 (𝑑𝑆𝑡 , 𝑡 + Δ𝑡) so that 𝑉 (𝑢 𝑆𝑡 , 𝑡 + Δ𝑡) = 𝑉 (𝑆𝑡 + (𝑢 − 1)𝑆𝑡 , 𝑡 + Δ𝑡) 𝜕𝑉 1 𝜕2𝑉 𝜕𝑉 (𝑢 − 1)𝑆𝑡 + (𝑢 − 1)2 𝑆𝑡2 Δ𝑡 + 𝜕𝑡 𝜕𝑆𝑡 2 𝜕𝑆𝑡2 ) ( ) ( +𝑂 (Δ𝑡)2 + 𝑂 (𝑢 − 1)3 𝑆𝑡3

= 𝑉 (𝑆𝑡 , 𝑡) +

and 𝑉 (𝑑𝑆𝑡 , 𝑡 + Δ𝑡) = 𝑉 (𝑆𝑡 + (𝑑 − 1)𝑆𝑡 , 𝑡 + Δ𝑡) 𝜕𝑉 1 𝜕2 𝑉 𝜕𝑉 (𝑑 − 1)𝑆𝑡 + (𝑑 − 1)2 𝑆𝑡2 Δ𝑡 + 𝜕𝑡 𝜕𝑆𝑡 2 𝜕𝑆𝑡2 ) ( ) ( +𝑂 (Δ𝑡)2 + 𝑂 (𝑑 − 1)3 𝑆𝑡3 .

= 𝑉 (𝑆𝑡 , 𝑡) +

Hence, 𝜋𝑉 (𝑢 𝑆𝑡 , 𝑡 + Δ𝑡) + (1 − 𝜋)𝑉 (𝑑𝑆𝑡 , 𝑡 + Δ𝑡) − 𝑒𝑟Δ𝑡 𝑉 (𝑆𝑡 , 𝑡) = 0 becomes ] 𝜕2𝑉 𝜕𝑉 1[ 𝜕𝑉 𝜋(𝑢 − 1)2 + (1 − 𝜋)(𝑑 − 1)2 𝑆𝑡2 2 + [𝜋(𝑢 − 1) + (1 − 𝜋)(𝑑 − 1)] 𝑆𝑡 Δ𝑡 + 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 ) ) ( ) ( ) ( ( + 1 − 𝑒𝑟Δ𝑡 𝑉 (𝑆𝑡 , 𝑡) + 𝑂 (Δ𝑡)2 + 𝑂 (𝑢 − 1)3 𝑆𝑡3 + 𝑂 (𝑑 − 1)3 𝑆𝑡3 = 0.

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71

Taking first-order approximations 1 − 𝑒𝑟Δ𝑡 = −𝑟Δ𝑡 + 𝑂((Δ𝑡)2 ) 𝜋(𝑢 − 1) + (1 − 𝜋)(𝑑 − 1) = 𝑟Δ𝑡 + 𝑂((Δ𝑡)2 ) 𝜋(𝑢 − 1)2 + (1 − 𝜋)(𝑑 − 1)2 = 𝜎 2 Δ𝑡 + 𝑂((Δ𝑡)2 ) and because ) ) ( ( 𝑂 (𝑢 − 1)3 𝑆𝑡3 = 𝑂((Δ𝑡)3 ), 𝑂 (𝑑 − 1)3 𝑆𝑡3 = 𝑂((Δ𝑡)3 ) and by substituting the above information into the equation and dividing it by Δ𝑡, we have 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡) + 𝑂(Δ𝑡) = 0. + 𝜎 2 𝑆𝑡2 2 + 𝑟𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 Taking limits Δ𝑡 → 0, we finally obtain the Black–Scholes equation 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0. + 𝜎 2 𝑆𝑡2 2 + 𝑟𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 The Greeks In this section we present the “Greeks”, which are the sensitivities of the option prices with respect to the variables (i.e., asset price and time) and the parameters used as inputs to the Black–Scholes formula. The importance of Greeks lies mainly in the construction of portfolio and dynamic hedging, since the sign and magnitude of the Greeks provide an indication of how much the option price increases or decreases in value when the variables/parameters change. Note that the Greek for a written option is opposite in sign to its purchased option counterpart. Tables 2.1 and 2.2 list the most common Greeks used for risk management. Table 2.1 Name Delta

Delta and gamma for European options. Symbol Δ

Definition 𝜕𝑉 𝜕𝑆𝑡

Description

s Calculates the rate of change of the theoretical Black– Scholes formula with respect to asset price.

s Commonly used as a percentage of the total number of shares in an option contract.

s Delta for a call option is always positive. Thus, an increase in stock price increases the call option value.

s Delta for a put option is always negative. Thus, an increase in strike price decreases the put option value.

s Maximum delta value occurs when it is near the strike price. Gamma

Γ

𝜕2 𝑉 𝜕𝑆𝑡2

s Calculates the rate of change of the delta with respect to the asset price or second partial derivatives of the option with respect to the asset price. s Gamma is identical for call and put options, and is positive. Thus, both call and put options are convex functions. s Gamma is maximum when it is near the strike price and decreases its value when it is away from the strike price.

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2.1 Introduction

Table 2.2

Vega, rho and theta for European options.

Name

Symbol

Vega



Definition 𝜕𝑉 𝜕𝜎

Description

s Calculates the rate of change of the theoretical Black– Scholes formula with respect to the asset price volatility.

s Vega is identical for call and put options and is positive. Thus, higher volatility increases both call and put option prices. s Usually measures how much an option price will change as volatility increases or decreases by 1%.

𝜌

Rho

s Calculates the rate of change of the theoretical Black–

𝜕𝑉 𝜕𝑟

Scholes formula with respect to the risk-free interest rate.

s Higher interest rates increase (decrease) call (put) option price.

s Lower interest rates increase (decrease) put (call) option price. Theta

𝜃

s Calculates the rate of change of the theoretical Black–

𝜕𝑉 𝜕𝑡

Scholes formula with respect to the passage of time.

s Theta has its maximum absolute value when the call option is ATM.

Extension of Black–Scholes Model The assumptions of the Black–Scholes model can be relaxed further to obtain closed-form formulas for European options on the following. s Asset-paying dividends (fixed discrete dividends, discrete or continuous dividend yield). For asset 𝑆𝑡 which pays a sequence of fixed discrete dividends 𝐷𝑖 at time 𝑡𝐷𝑖 , 𝑖 = 1, 2, … , 𝑛 such that 𝑡 < 𝑡𝐷1 < 𝑡𝐷2 < ⋯ < 𝑡𝐷𝑛 , the Black–Scholes equation becomes 𝜕𝑉 𝜕2𝑉 1 + 𝜎 2 𝑆𝑡2 2 + 𝜕𝑡 2 𝜕𝑆𝑡

( 𝑟 𝑆𝑡 −

𝑛 ∑ 𝑖=1

) 𝐷𝑖 𝛿(𝑡 − 𝑡𝐷𝑖 )

𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0 𝜕𝑆𝑡

where 𝛿(𝑡 − 𝑡𝐷𝑖 ) is the Dirac delta function centred at 𝑡𝐷𝑖 . For asset 𝑆𝑡 paying continuous dividend yield 𝐷, the Black–Scholes equation becomes 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0. + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 s Time-dependent continuous dividend yield, risk-free interest rate and volatility. For asset 𝑆𝑡 paying time-dependent continuous dividend yield 𝐷𝑡 with time-dependent volatility 𝜎𝑡 and risk-free interest rate 𝑟𝑡 , the Black–Scholes equation becomes 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 − 𝑟𝑡 𝑉 (𝑆𝑡 , 𝑡) = 0. + 𝜎𝑡2 𝑆𝑡2 2 + (𝑟𝑡 − 𝐷𝑡 )𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡

2.1 Introduction

73

s Incorporating the effects of transaction costs in the hedging portfolio. By setting the following transaction costs in buying or selling an asset 𝑆𝑡 which pays continuous dividend yield 𝐷: (i) a fixed cost at each transaction, 𝜅1 (ii) a cost proportional to the number of assets traded, 𝜅2 |𝜈| (iii) a cost proportional to the value of the assets traded, 𝜅3 |𝜈|𝑆𝑡 where 𝜅1 , 𝜅2 , 𝜅3 > 0, 𝜈 > 0 (for buying assets) and 𝜈 < 0 (for selling assets), the Black– Scholes equation with transaction costs at time interval 𝛿𝑡 becomes 𝜅 1 2 2 𝜕2𝑉 𝜕𝑉 ̃ 𝑡 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡) − 1 = 0 + (𝑟 − 𝐷)𝑆 + 𝜎 ̃ 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝛿𝑡 𝜕𝑆𝑡 where √ ̃ = 𝐷 + 𝜅2 𝜎 𝐷

( )−1 2 || 𝜕 2 𝑉 || 𝜕𝑉 | | 𝜋𝛿𝑡 || 𝜕𝑆𝑡2 || 𝜕𝑆𝑡

√ and

𝜎 ̃ = 𝜎 − 𝜅3 𝜎 2

2

( 2 sgn 𝜋𝛿𝑡

𝜕2 𝑉 𝜕𝑆𝑡2

)

𝑥 . |𝑥| s Discontinuous jumps in asset price. For an asset 𝑆𝑡 paying continuous dividend yield 𝐷 and having a jump process such that 𝐽𝑡 is the jump size variable, the Black–Scholes equation becomes such that sgn(𝑥) =

] 𝜕𝑉 𝜕𝑉 𝜕2 𝑉 [ 1 − 𝑟𝑉 (𝑆𝑡 , 𝑡) + 𝜎 2 𝑆𝑡2 2 + 𝑟 − 𝐷 − 𝔼𝐽 (𝐽𝑡 − 1) 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 [ ] +𝜆𝔼𝐽 𝑉 (𝐽𝑡 𝑆𝑡 , 𝑡) − 𝑉 (𝑆𝑡 , 𝑡) = 0 where the expectation 𝔼𝐽 (⋅) is taken over the jump component. Criticisms of Black–Scholes Model Given that the Black–Scholes model with European payoffs has closed-form solutions, one of the main advantages is to be able to calculate option prices analytically. In addition, the Black–Scholes formula is also easy to understand intuitively without the need to evaluate it using expensive numerical methods. Unfortunately, some of the assumptions in the formula are questionable under the current economic and market conditions. One major criticism is the assumption of stock prices moving in a manner referred to as a random walk (stock price moves up and down with the same probability) and the stock returns following a normal distribution. Empirical tests have shown that there are significant fat tails (leptokurtic) and asymmetry (skewness) in the stock returns. Another major weakness of the Black–Scholes framework is that the volatility at the time of buying/selling the option remains unchanged until the option expiry date. This is simply not true. Nevertheless, even with the availability of option valuation models incorporating either stochastic volatility or fat tails which are more expensive to evaluate, the Black–Scholes model still continues to be used due to its robustness and tractability. After all, the model has only one unobservable parameter and that is the volatility.

74

2.2.1 Basic Properties

2.2 PROBLEMS AND SOLUTIONS 2.2.1

Basic Properties

1. Consider a European call option with price 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) where 𝑆𝑡 is the spot price at time 𝑡 < 𝑇 , 𝑇 is the expiry date of the option and 𝐾 is the strike price. Show that the call option is greater than or equal to its intrinsic value 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝑆𝑡 − 𝐾, 0}. Solution: We assume 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) < max{𝑆𝑡 − 𝐾, 0} and at time 𝑡 we set up the following portfolio Π𝑡 = 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − max{𝑆𝑡 − 𝐾, 0} < 0. At expiry time 𝑇 Π𝑇 = 𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) − max{𝑆𝑇 − 𝐾, 0} = max{𝑆𝑇 − 𝐾, 0} − max{𝑆𝑇 − 𝐾, 0} =0 which is an arbitrage opportunity. Therefore, 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝑆𝑡 − 𝐾, 0}. 2. Consider a European put option with price 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) where 𝑆𝑡 is the spot price at time 𝑡 < 𝑇 , 𝑇 is the expiry date of the option and 𝐾 is the strike price. Show that the put option is greater than or equal to its intrinsic value 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝐾 − 𝑆𝑡 , 0}. Solution: We assume 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) < max{𝐾 − 𝑆𝑡 , 0} and at time 𝑡 we set up the following portfolio Π𝑡 = 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − max{𝐾 − 𝑆𝑡 , 0} < 0. At expiry time 𝑇 Π𝑇 = 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) − max{𝐾 − 𝑆𝑇 , 0} = max{𝐾 − 𝑆𝑇 , 0} − max{𝐾 − 𝑆𝑇 , 0} =0 which is an arbitrage opportunity. Therefore, 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝐾 − 𝑆𝑡 , 0}. 3. Let 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of a European call option where 𝑆𝑡 is the spot price at time 𝑡 < 𝑇 , 𝑇 is the expiry date of the option and 𝐾 is the strike price. Show that 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑆𝑡 .

2.2.1 Basic Properties

75

Solution: We assume at time 𝑡, 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) > 𝑆𝑡 and we set up the portfolio Π𝑡 = 𝑆𝑡 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) < 0. At time 𝑇 Π𝑇 = 𝑆𝑇 − 𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 𝑆𝑇 − max{𝑆𝑇 − 𝐾, 0} ⎧𝑆 ⎪ 𝑇 =⎨ ⎪𝐾 ⎩ >0

if 𝑆𝑇 ≤ 𝐾 if 𝑆𝑇 > 𝐾

which is an arbitrage opportunity. Therefore, 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑆𝑡 . 4. Let 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of a European put option where 𝑆𝑡 is the spot price at time 𝑡 < 𝑇 , 𝑇 is the expiry date of the option and 𝐾 is the strike price. Show that 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐾. Solution: We assume at time 𝑡, 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) > 𝐾 and we set up the portfolio Π𝑡 = 𝐾 − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) < 0. At time 𝑇 Π𝑇 = 𝐾 − 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 𝐾 − max{𝐾 − 𝑆𝑇 , 0} ⎧𝑆 ⎪ 𝑇 =⎨ ⎪𝐾 ⎩ >0

if 𝑆𝑇 ≤ 𝐾 if 𝑆𝑇 > 𝐾

which is an arbitrage opportunity. Therefore, 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐾. 5. Consider a European call option 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) with strike price 𝐾 where 𝑆𝑡 is the spot price of a risky asset at time 𝑡 < 𝑇 , with 𝑇 the expiry time of the option. Assume that 𝑆𝑡 pays a continuous dividend yield 𝐷, then show that 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾 𝑒−𝑟(𝑇 −𝑡) , 0} where 𝑟 is the risk-free interest rate.

76

2.2.1 Basic Properties

Solution: We assume 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) < max{𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) , 0} and we set up a portfolio Π𝑡 = 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − max{𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) , 0} < 0. At time 𝑇 Π𝑇 = 𝐶(𝑆𝑇 , 𝑡; 𝐾, 𝑇 ) − max{𝑆𝑇 − 𝐾, 0} = max{𝑆𝑇 − 𝐾, 0} − max{𝑆𝑇 − 𝐾, 0} =0 which is an arbitrage opportunity and hence a contradiction. Therefore, 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) , 0}. N.B. We can also show this result using the put–call parity 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) , where 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is the European put option on the same strike price 𝐾 and expiry time 𝑇 . Thus, 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) ≥ 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) ≥ max{𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) , 0} since 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0 and 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0. 6. Consider a European put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) with strike price 𝐾 where 𝑆𝑡 is the spot price of a risky asset at time 𝑡 < 𝑇 , with 𝑇 the expiry time of the option. Assume that 𝑆𝑡 pays a continuous dividend yield 𝐷, then show that 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) , 0} where 𝑟 is the risk-free interest rate. Solution: We assume 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) < max{𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) , 0} and we set up a portfolio Π𝑡 = 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − max{𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) , 0} < 0. At time 𝑇 Π𝑇 = 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) − max{𝐾 − 𝑆𝑇 , 0} = max{𝐾 − 𝑆𝑇 , 0} − max{𝐾 − 𝑆𝑇 , 0} =0 which is an arbitrage opportunity and hence a contradiction. Therefore, 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) , 0}.

2.2.1 Basic Properties

77

N.B. Like its European call option counterpart, we can also show this result using the put– call parity 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) where 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is the European call option on the same strike price 𝐾 and expiry time 𝑇 . Thus, 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) ≥ 𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) ≥ max{𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) , 0} since 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0 and 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0. 7. Consider a European call option 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) with strike price 𝐾 where 𝑆𝑡 is the spot price of a risky asset at time 𝑡 < 𝑇 , with 𝑇 the expiry time of the option. If the stock pays a sequence of discrete dividends 𝐷𝑖 at time 𝑡𝑖 , 𝑖 = 1, 2, … , 𝑛, 𝑡 < 𝑡1 < 𝑡2 < ⋯ < 𝑡𝑛 < 𝑇 and by denoting 𝑟 as the risk-free interest rate show that { 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max

𝑆𝑡 −

𝑛 ∑ 𝑖=1

} 𝐷𝑖 𝑒−𝑟(𝑡𝑖 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) , 0

.

Solution: Using the put–call parity 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 − ∑𝑛 −𝑟(𝑡𝑖 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) where 𝑃 (𝑆 , 𝑡; 𝐾, 𝑇 ) is the European put option on the 𝑡 𝑖=1 𝐷𝑖 𝑒 same strike price 𝐾 and expiry time 𝑇 . Therefore, 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑆𝑡 − ≥ 𝑆𝑡 −

𝑛 ∑ 𝑖=1

𝑖=1

𝐷𝑖 𝑒−𝑟(𝑡𝑖 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡)

𝐷𝑖 𝑒−𝑟(𝑡𝑖 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡)

{ ≥ max

𝑛 ∑

𝑆𝑡 −

𝑛 ∑ 𝑖=1

} 𝐷𝑖 𝑒

−𝑟(𝑡𝑖 −𝑡)

− 𝐾𝑒

−𝑟(𝑇 −𝑡)

,0

since 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0 and 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0. 8. Consider a European put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) with strike price 𝐾 where 𝑆𝑡 is the spot price of a risky asset at time 𝑡 < 𝑇 , with 𝑇 the expiry time of the option. If the stock pays a sequence of discrete dividends 𝐷𝑖 at time 𝑡𝑖 , 𝑖 = 1, 2, … , 𝑛, 𝑡 < 𝑡1 < 𝑡2 < ⋯ < 𝑡𝑛 < 𝑇 and by denoting 𝑟 as the risk-free interest rate show that { 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max

𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 +

𝑛 ∑ 𝑖=1

} 𝐷𝑖 𝑒−𝑟(𝑡𝑖 −𝑡) , 0

.

Solution: Using the put–call parity 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 − ∑𝑛 −𝑟(𝑡𝑖 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) where 𝐶(𝑆 , 𝑡; 𝐾, 𝑇 ) is the European call option on the 𝑡 𝑖=1 𝐷𝑖 𝑒

78

2.2.1 Basic Properties

same strike price 𝐾 and expiry time 𝑇 . Thus, 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 + ≥ 𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 + { ≥ max

𝑛 ∑ 𝑖=1

𝑛 ∑ 𝑖=1

𝐷𝑖 𝑒−𝑟(𝑡𝑖 −𝑡)

𝐷𝑖 𝑒−𝑟(𝑡𝑖 −𝑡)

𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 +

𝑛 ∑ 𝑖=1

} 𝐷𝑖 𝑒−𝑟(𝑡𝑖 −𝑡) , 0

since 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0 and 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0. 9. A stock price is currently worth $25 and it pays a continuous dividend yield of 2.29%. Given that the risk-free interest rate is 4% find the lower bound for an 8-month European put option on the stock with strike price $30. Solution: At initial time 𝑡 = 0 we set the current stock price 𝑆0 = $25, strike 𝐾 = $30, div8 idend yield 𝐷 = 2.29%, risk-free interest rate 𝑟 = 4% and expiry time 𝑇 = 12 = 23 years. By setting 𝑃 (𝑆0 , 0; 𝐾, 𝑇 ) as the European put option at time 𝑡 = 0 its lower bound is 𝑃 (𝑆0 , 0; 𝐾, 𝑇 ) ≥ max{𝐾 𝑒−𝑟𝑇 − 𝑆0 𝑒−𝐷𝑇 , 0} 2

2

= max{30𝑒−0.04× 3 − 25𝑒−0.029× 3 , 0} = max{4.69, 0} = $4.69.

10. Consider a 6-month European call option on a discrete dividend-paying stock having a strike price of $33. The current stock price is worth $35 and dividends of $1.00 and $1.25 are expected to be paid in two months and four months, respectively. Given that the riskfree interest rate is 2.5% find the lower bound for the European call option price. Solution: At initial time 𝑡 = 0 we let the current stock price 𝑆0 = $35, strike 𝐾 = $33, 2 4 discrete dividends 𝐷1 = $1.00 at time 𝑡1 = 12 = 16 years, 𝐷2 = $1.25 at time 𝑡2 = 12 = 13

6 years, risk-free interest rate 𝑟 = 2.5% and expiry time 𝑇 = 12 = 12 years. By denoting 𝐶(𝑆0 , 0; 𝐾, 𝑇 ) as the European call option at time 𝑡 = 0, its lower bound is

𝐶(𝑆0 , 0; 𝐾, 𝑇 ) ≥ max{𝑆0 − 𝐷1 𝑒−𝑟𝑡1 − 𝐷2 𝑒−𝑟𝑡2 − 𝐾𝑒−𝑟𝑇 , 0} 1

1

1

= max{35 − 𝑒−0.025× 6 − 1.25𝑒−0.025× 3 − 33𝑒−0.025× 2 , 0} = max{0.1745, 0} = $0.17.

2.2.1 Basic Properties

79

11. Consider two European call options 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇1 ) and 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇2 ) having expiry times 𝑇1 and 𝑇2 with 𝑇1 < 𝑇2 , 𝑆𝑡 is the spot price at time 𝑡 and 𝐾 is the common strike price. Show that for a non-dividend-paying stock 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇1 ) ≤ 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇2 ) for 𝑇1 < 𝑇2 . Solution: We assume 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇1 ) > 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇2 ) where at time 𝑡 we can set up a portfolio Π𝑡 = 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇2 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇1 ) < 0. At time 𝑇1 , the portfolio is now worth Π𝑇1 = 𝐶(𝑆𝑇1 , 𝑇1 ; 𝐾, 𝑇2 ) − max{𝑆𝑇1 − 𝐾, 0} ⎧𝐶(𝑆 , 𝑇 ; 𝐾, 𝑇 ) 𝑇1 1 2 ⎪ =⎨ ⎪𝐶(𝑆𝑇 , 𝑇1 ; 𝐾, 𝑇2 ) − 𝑆𝑇 + 𝐾 1 1 ⎩

if 𝑆𝑇1 ≤ 𝐾 if 𝑆𝑇1 > 𝐾.

Since 𝐶(𝑆𝑇1 , 𝑇1 ; 𝐾, 𝑇2 ) ≥ 𝑆𝑇1 − 𝐾 we have Π𝑇1 ≥ 0, which is a contradiction. Therefore, 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇1 ) ≤ 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇2 ) for 𝑇1 < 𝑇2 . 12. Consider two European put options 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇1 ) and 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇2 ) having expiry times 𝑇1 and 𝑇2 with 𝑇1 < 𝑇2 , 𝑆𝑡 is the spot price at time 𝑡 and 𝐾 is the common strike price. Show that for a non-dividend-paying stock 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇1 ) ≤ 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇2 ) for 𝑇1 < 𝑇2 . Solution: We assume 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇1 ) > 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇2 ) where at time 𝑡 we can set up a portfolio Π𝑡 = 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇2 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇1 ) < 0. At time 𝑇1 , the portfolio is now worth Π𝑇1 = 𝑃 (𝑆𝑇1 , 𝑇1 ; 𝐾, 𝑇2 ) − max{𝐾 − 𝑆𝑇1 , 0} ⎧𝑃 (𝑆 , 𝑇 ; 𝐾, 𝑇 ) 𝑇1 1 2 ⎪ =⎨ ⎪𝑃 (𝑆𝑇 , 𝑇1 ; 𝐾, 𝑇2 ) − 𝐾 + 𝑆𝑇 1 1 ⎩

if 𝑆𝑇1 ≥ 𝐾 if 𝑆𝑇1 < 𝐾.

Since 𝑃 (𝑆𝑇1 , 𝑇1 ; 𝐾, 𝑇2 ) ≥ 𝐾 − 𝑆𝑇1 we have Π𝑇1 ≥ 0, which is a contradiction. Therefore, 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇1 ) ≤ 𝑃 (𝑆𝑡 , 𝑡; 𝑇2 ) for 𝑇1 < 𝑇2 .

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13. For a non-dividend-paying stock we consider a 3-month European call option with strike $51 and a 6-month European call with strike $52 with premiums $4.50 and $4.17, respectively. Let the risk-free interest rate be 10% and show that we can construct an arbitrage opportunity portfolio. Solution: At time 𝑡 = 0 we denote 𝑆0 as the current stock price, strikes 𝐾1 = $4.50, 3 6 𝐾2 = $4.17, expiry times 𝑇1 = 12 = 14 years, 𝑇2 = 12 = 12 years and risk-free interest rate 𝑟 = 10%. We denote the 3-month European call option with strike 𝐾1 = $51 and the 6-month European call option with strike 𝐾2 = $52 as 𝐶(𝑆0 , 0; 𝐾1 , 𝑇1 ) = $4.50 and 𝐶(𝑆0 , 0; 𝐾2 , 𝑇2 ) = $4.17, respectively. At time 𝑡 = 0 we set up the following portfolio Π0 = 𝐶(𝑆0 , 0; 𝐾2 , 𝑇2 ) − 𝐶(𝑆0 , 0; 𝐾1 , 𝑇1 ) = $4.17 − $4.50 = −$0.33. At time 𝑇1 Π𝑇1 = 𝐶(𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) − 𝐶(𝑆𝑇1 , 𝑇1 ; 𝐾1 , 𝑇1 ) = 𝐶(𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) − max{𝑆𝑇1 − 𝐾1 , 0} ⎧𝐶(𝑆 , 𝑇 ; 𝐾 , 𝑇 ) 𝑇1 1 2 2 ⎪ =⎨ ⎪𝐶(𝑆𝑇 , 𝑇1 ; 𝐾2 , 𝑇2 ) − 𝑆𝑇 + 𝐾1 1 1 ⎩

if 𝑆𝑇1 ≤ 𝐾1 if 𝑆𝑇1 > 𝐾1

⎧𝐶(𝑆 , 𝑇 ; 𝐾 , 𝑇 ) 𝑇1 1 2 2 ⎪ ≥⎨ ⎪𝑆𝑇 − 𝐾2 𝑒−𝑟(𝑇2 −𝑇1 ) − 𝑆𝑇 + 𝐾1 1 ⎩ 1 ⎧𝐶(𝑆 , 𝑇 ; 𝐾 , 𝑇 ) 𝑇1 1 2 2 ⎪ =⎨ ⎪𝐾1 − 𝐾2 𝑒−𝑟(𝑇2 −𝑇1 ) ⎩

if 𝑆𝑇1 ≤ 𝐾1 if 𝑆𝑇1 > 𝐾1

if 𝑆𝑇1 ≤ 𝐾1 if 𝑆𝑇1 > 𝐾1 . 1

Given 𝐶(𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) ≥ 0 and 𝐾1 − 𝐾2 𝑒−𝑟(𝑇2 −𝑇1 ) = 51 − 52𝑒−0.1× 4 = $0.28 > 0, the portfolio Π𝑇1 ≥ 0 which is an arbitrage opportunity. In contrast, at time 𝑇2 Π𝑇2 = 𝐶(𝑆𝑇2 , 𝑇2 ; 𝐾2 , 𝑇2 ) = max{𝑆𝑇2 − 𝐾2 , 0} ≥ 0 which is also an arbitrage opportunity. Thus, by setting up the portfolio at time 𝑡 = 0 Π0 = 𝐶(𝑆0 , 0; 𝐾2 , 𝑇2 ) − 𝐶(𝑆0 , 0; 𝐾1 , 𝑇1 ) we can guarantee an arbitrage opportunity.

2.2.1 Basic Properties

81

14. Consider two European call options 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) and 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) having strike prices 𝐾1 and 𝐾2 , respectively with 𝐾1 < 𝐾2 , 𝑆𝑡 is the spot price at time 𝑡 and 𝑇 > 𝑡 is the expiry time. Show that 0 ≤ 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≤ 𝐾2 − 𝐾1 for 𝐾1 < 𝐾2 . Solution: We first assume 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) < 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) where at time 𝑡 we can set up a portfolio Π𝑡 = 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) < 0. At expiry time 𝑇 Π𝑇 = 𝐶(𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) = max{𝑆𝑇 − 𝐾1 , 0} − max{𝑆𝑇 − 𝐾2 , 0} ⎧0 ⎪ ⎪ = ⎨𝑆𝑇 − 𝐾1 ⎪ ⎪𝐾 − 𝐾 1 ⎩ 2

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2 .

Hence, Π𝑇 ≥ 0 which is a contradiction. We next assume 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) > 𝐾2 − 𝐾1 where at time 𝑡 we set up the following portfolio Π𝑡 = 𝐾2 − 𝐾1 − 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) < 0. At expiry time 𝑇 Π𝑇 = 𝐾2 − 𝐾1 − 𝐶(𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) = 𝐾2 − 𝐾1 − max{𝑆𝑇 − 𝐾1 , 0} + max{𝑆𝑇 − 𝐾2 , 0} ⎧𝐾2 − 𝐾1 ⎪ ⎪ = ⎨𝐾2 − 𝑆𝑇 ⎪ ⎪0 ⎩

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2 .

Hence, Π𝑇 ≥ 0 which is also a contradiction. Therefore, 0 ≤ 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≤ 𝐾2 − 𝐾1 for 𝐾1 < 𝐾2 .

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15. Consider two European put options 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) and 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) having strike prices 𝐾1 and 𝐾2 , respectively with 𝐾1 < 𝐾2 , 𝑆𝑡 is the spot price at time 𝑡 and 𝑇 > 𝑡 is the expiry time. Show that 0 ≤ 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) ≤ 𝐾2 − 𝐾1 for 𝐾1 < 𝐾2 . Solution: We first assume 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) < 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) where at time 𝑡 we can set up a portfolio Π𝑡 = 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) < 0. At expiry time 𝑇 Π𝑇 = 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) = max{𝐾2 − 𝑆𝑇 , 0} − max{𝐾1 − 𝑆𝑇 , 0} ⎧𝐾2 − 𝐾1 ⎪ ⎪ = ⎨𝐾2 − 𝑆𝑇 ⎪ ⎪0 ⎩

if 𝑆𝑇 < 𝐾1 if 𝐾1 ≤ 𝑆𝑇 < 𝐾2 if 𝑆𝑇 ≥ 𝐾2 .

Hence, Π𝑇 ≥ 0 which is a contradiction. We next assume 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) > 𝐾2 − 𝐾1 where at time 𝑡 we set up the following portfolio Π𝑡 = 𝐾2 − 𝐾1 − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) < 0. At expiry time 𝑇 Π𝑇 = 𝐾2 − 𝐾1 − 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) + 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) = 𝐾2 − 𝐾1 − max{𝐾2 − 𝑆𝑇 , 0} + max{𝐾1 − 𝑆𝑇 , 0} ⎧0 ⎪ ⎪ = ⎨𝑆𝑇 − 𝐾1 ⎪ ⎪𝐾 − 𝐾 1 ⎩ 2

if 𝑆𝑇 < 𝐾1 if 𝐾1 ≤ 𝑆𝑇 < 𝐾2 if 𝑆𝑇 ≥ 𝐾2 .

Hence, Π𝑇 ≥ 0 which is also a contradiction. Therefore, 0 ≤ 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) ≤ 𝐾2 − 𝐾1 for 𝐾1 < 𝐾2 .

2.2.1 Basic Properties

83

16. Consider two European call options 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) and 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) having strike prices 𝐾1 and 𝐾2 , respectively with 𝐾1 < 𝐾2 , 𝑆𝑡 is the spot price at time 𝑡 and 𝑇 > 𝑡 is the expiry time. Given the risk-free interest rate 𝑟 show that 0 ≤ 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≤ (𝐾2 − 𝐾1 )𝑒−𝑟(𝑇 −𝑡) for 𝐾1 < 𝐾2 . Solution: From Problem 2.2.1.14 (page 81) we have shown that 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≥ 0. In order to show that 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≤ (𝐾2 − 𝐾1 )𝑒−𝑟(𝑇 −𝑡) we note that from the put–call parity 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) = 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝑆𝑡 − 𝐾1 𝑒−𝑟(𝑇 −𝑡) 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) = 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) + 𝑆𝑡 − 𝐾2 𝑒−𝑟(𝑇 −𝑡) . Taking differences, we have 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) = (𝐾2 − 𝐾1 )𝑒−𝑟(𝑇 −𝑡) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) −𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) and since 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≤ 0 (see Problem 2.2.1.15, page 82) 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≤ (𝐾2 − 𝐾1 )𝑒−𝑟(𝑇 −𝑡) for 𝐾1 < 𝐾2 . 17. Consider two European put options 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) and 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) having strike prices 𝐾1 and 𝐾2 , respectively with 𝐾1 < 𝐾2 , 𝑆𝑡 is the spot price at time 𝑡 and 𝑇 > 𝑡 is the expiry time. Given the risk-free interest rate 𝑟 show that 0 ≤ 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) ≤ (𝐾2 − 𝐾1 )𝑒−𝑟(𝑇 −𝑡) for 𝐾1 < 𝐾2 . Solution: From Problem 2.2.1.15 (page 82) we have 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) ≥ 0. In order to show that 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) ≤ (𝐾2 − 𝐾1 )𝑒−𝑟(𝑇 −𝑡)

84

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we note that from the put–call parity 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) = 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑆𝑡 + 𝐾2 𝑒−𝑟(𝑇 −𝑡) 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) = 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑆𝑡 + 𝐾1 𝑒−𝑟(𝑇 −𝑡) . Taking differences, we have 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) = (𝐾2 − 𝐾1 )𝑒−𝑟(𝑇 −𝑡) + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) −𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) and since 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) ≤ 0 (see Problem 2.2.1.14, page 81) 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) ≤ (𝐾2 − 𝐾1 )𝑒−𝑟(𝑇 −𝑡) for 𝐾1 < 𝐾2 .

18. Let the prices of two European put options with strikes $50 and $53 be $1.50 and $5.00, respectively where both options have the same time to expiry. (a) Is the no-arbitrage condition violated? (b) Suggest a spread position so that the portfolio will ensure an arbitrage opportunity. Solution: We assume the options are priced at time 𝑡 with spot price 𝑆𝑡 and we let 𝐾1 = $50 and 𝐾2 = $53 so that 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) = $1.50 and 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) = $5.00 where 𝑇 > 𝑡 are the options’ expiry time. (a) Since 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) = $5.00 − $1.50 > 𝐾2 − 𝐾1 = $53 − $50 for 𝐾1 < 𝐾2 , this violates the condition 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) ≤ 𝐾2 − 𝐾1 . (b) We let the portfolio at time 𝑡 be Π𝑡 = 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) = $1.50 − $5.00 = −$3.50 where at time 𝑇 , the portfolio will be worth Π𝑇 = max{50 − 𝑆𝑇 , 0} − max{53 − 𝑆𝑇 , 0} ⎧−3 ⎪ ⎪ = ⎨𝑆𝑇 − 53 ⎪ ⎪0 ⎩

if 𝑆𝑇 ≤ 50 if 50 < 𝑆𝑇 ≤ 53 if 𝑆𝑇 > 53.

2.2.1 Basic Properties

85

Hence, the profit at time 𝑇 is Υ𝑇 = Π𝑇 − Π𝑡 ⎧0.50 if 𝑆𝑇 ≤ 50 ⎪ ⎪ = ⎨𝑆𝑇 − 49.50 if 50 < 𝑆𝑇 ≤ 53 ⎪ ⎪3.50 if 𝑆𝑇 > 53 ⎩ > 0. Thus, Υ𝑇 > 0, which shows that the portfolio Π𝑡 = 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) provides an arbitrage opportunity. 19. The Black–Scholes formulae for the value of a European call option 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and a European put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) are 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑− ) 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑− ) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑+ ) with 𝑑± given by

𝑑± =

log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝑡) . √ 𝜎 𝑇 −𝑡

Φ(⋅) is the cdf of a standard normal, 𝑆𝑡 is the spot price at time 𝑡 < 𝑇 , 𝑇 is the expiry date of the option, 𝐾 is the strike price, 𝑟 is the risk-free interest rate, 𝐷 is the continuous dividend yield and 𝜎 is the spot price volatility. In the following limits show that the values of the call and put prices satisfy (a) lim 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0}, lim 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = max{𝐾 − 𝑆𝑇 , 0} 𝑡→𝑇

𝑡→𝑇

⎧𝑆 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) ⎪ 𝑡 (b) lim 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ 𝜎→0 ⎪0 ⎩ ⎧0 ⎪ lim 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ 𝜎→0 ⎪𝐾 −𝑟(𝑇 −𝑡) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) ⎩

if 𝑆𝑡 > 𝐾𝑒−(𝑟−𝐷)(𝑇 −𝑡) if 𝑆𝑡 ≤ 𝐾𝑒−(𝑟−𝐷)(𝑇 −𝑡) if 𝑆𝑡 ≥ 𝐾𝑒−(𝑟−𝐷)(𝑇 −𝑡) if 𝑆𝑡 < 𝐾𝑒−(𝑟−𝐷)(𝑇 −𝑡)

86

2.2.1 Basic Properties

(c) lim 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) , lim 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑒−𝑟(𝑇 −𝑡) 𝜎→∞

(d)

lim

𝐷→∞,𝜎>0

𝜎→∞

𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0,

lim

𝐷→∞,𝜎>0

⎧𝑆 ⎪ 𝑡 (e) lim 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ 𝑇 →∞,𝜎>0 ⎪0 ⎩ ⎧𝐾 ⎪ lim 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ 𝑇 →∞,𝜎>0 ⎪0 ⎩

𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑒−𝑟(𝑇 −𝑡)

if 𝐷 = 0 if 𝐷 > 0 if 𝑟 = 0 if 𝑟 > 0.

Solution: (a) For the case 𝑡 → 𝑇 { lim 𝑑± = lim

𝑡→𝑇

𝑡→𝑇

⎧+∞ ⎪ ⎪ =⎨ 0 ⎪ ⎪−∞ ⎩

log(𝑆𝑡 ∕𝐾) ( 𝑟 − 𝐷 1 ) √ 𝑇 −𝑡 + ± 𝜎 √ 𝜎 2 𝜎 𝑇 −𝑡 if 𝑆𝑇 > 𝐾 if 𝑆𝑇 = 𝐾 if 𝑆𝑇 < 𝐾.

Hence, ⎧𝑆 − 𝐾 ⎪ 𝑇 lim 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ 𝑡→𝑇 ⎪0 ⎩ ⎧𝐾 − 𝑆 𝑇 ⎪ lim 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ 𝑡→𝑇 ⎪0 ⎩

if 𝑆𝑇 > 𝐾 if 𝑆𝑇 ≤ 𝐾

if 𝑆𝑇 < 𝐾 if 𝑆𝑇 ≥ 𝐾

or lim 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0}

𝑡→𝑇

lim 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = max{𝐾 − 𝑆𝑇 , 0}.

𝑡→𝑇

}

2.2.1 Basic Properties

87

(b) For the case 𝜎 → 0 { lim 𝑑± = lim

𝜎→0

𝜎→0

log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷)(𝑇 − 𝑡) 1 √ ± 𝜎 𝑇 −𝑡 √ 2 𝜎 𝑇 −𝑡

}

⎧+∞ if 𝑆𝑡 > 𝐾𝑒−(𝑟−𝐷)(𝑇 −𝑡) ⎪ ⎪ if 𝑆𝑡 = 𝐾𝑒−(𝑟−𝐷)(𝑇 −𝑡) =⎨ 0 ⎪ ⎪−∞ if 𝑆 < 𝐾𝑒−(𝑟−𝐷)(𝑇 −𝑡) . 𝑡 ⎩ Therefore, ⎧𝑆 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) ⎪ 𝑡 lim 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ 𝜎→0 ⎪0 ⎩

if 𝑆𝑡 > 𝐾𝑒−(𝑟−𝐷)(𝑇 −𝑡) if 𝑆𝑡 ≤ 𝐾𝑒−(𝑟−𝐷)(𝑇 −𝑡)

and ⎧0 ⎪ lim 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ 𝜎→0 ⎪𝐾 −𝑟(𝑇 −𝑡) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) ⎩

if 𝑆𝑡 ≥ 𝐾𝑒−(𝑟−𝐷)(𝑇 −𝑡) if 𝑆𝑡 < 𝐾𝑒−(𝑟−𝐷)(𝑇 −𝑡) .

(c) For the case 𝜎 → ∞ { lim 𝑑 𝜎→∞ ±

= lim

𝜎→∞

} log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷)(𝑇 − 𝑡) 1 √ ± 𝜎 𝑇 − 𝑡 = ±∞. √ 2 𝜎 𝑇 −𝑡

Therefore, lim 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡)

𝜎→∞

and lim 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑒−𝑟(𝑇 −𝑡) .

𝜎→∞

(d) For the case 𝐷 → ∞ and 𝜎 > 0 { lim

𝐷→∞,𝜎>0

𝑑± =

lim

𝐷→∞,𝜎>0

= −∞.

} log(𝑆𝑡 ∕𝐾) + (𝑟 ± 12 𝜎 2 )(𝑇 − 𝑡) 𝐷 √ 𝑇 −𝑡 − √ 𝜎 𝜎 𝑇 −𝑡

88

2.2.1 Basic Properties

Hence, lim

𝐷→∞,𝜎>0

𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0

and lim

𝐷→∞,𝜎>0

𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑒−𝑟(𝑇 −𝑡) .

(e) For the case 𝑇 → ∞, 𝜎 > 0 and if 𝐷 > 0, 𝑟 > 0 and because 0 ≤ Φ(𝑑± ) ≤ 1, 0 ≤ Φ(−𝑑± ) ≤ 1 lim

𝑇 →∞,𝜎>0

𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0, lim 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0. 𝑇 →∞

For the case 𝑇 → ∞, 𝜎 > 0 and if 𝐷 = 0, 𝑟 > 0 we have { 𝑑+ =

lim

𝑇 →∞,𝜎>0

lim

𝑇 →∞,𝜎>0

log(𝑆𝑡 ∕𝐾) ( 𝑟 1 ) √ 𝑇 −𝑡 + + 𝜎 √ 𝜎 2 𝜎 𝑇 −𝑡

}

= +∞. Hence, lim

𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡

lim

𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0.

𝑇 →∞,𝜎>0

and 𝑇 →∞,𝜎>0

Finally, for the case 𝑇 → ∞, 𝜎 > 0 and if 𝐷 > 0, 𝑟 = 0 we have { lim

𝑇 →∞,𝜎>0

𝑑− =

lim

𝑇 →∞,𝜎>0

} log(𝑆𝑡 ∕𝐾) ( 𝐷 1 ) √ − 𝑇 −𝑡 + 𝜎 √ 𝜎 2 𝜎 𝑇 −𝑡

= −∞. Therefore, lim

𝑇 →∞,𝜎>0

𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0,

lim

𝑇 →∞,𝜎>0

𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾.

Collectively, we can deduce ⎧𝑆 ⎪ 𝑡 lim 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ 𝑇 →∞,𝜎>0 ⎪0 ⎩

if 𝐷 = 0 if 𝐷 > 0

2.2.2 Black–Scholes Model

89

and ⎧𝐾 ⎪ lim 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ 𝑇 →∞,𝜎>0 ⎪0 ⎩

2.2.2

if 𝑟 = 0 if 𝑟 > 0.

Black–Scholes Model

{ } 1. Black–Scholes Equation with Stock Paying Continuous Dividend Yield I. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the stock price 𝑆𝑡 follow a geometric Brownian motion (GBM) with the following SDE 𝑑𝑆𝑡 = 𝜇𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter and 𝜎 is the volatility parameter. In addition, let 𝐵𝑡 be the risk-free asset having the following differential equation 𝑑𝐵𝑡 = 𝑟 𝐵𝑡 𝑑𝑡 where 𝑟 denotes the risk-free interest rate. Assume that the stock 𝑆𝑡 pays a continuous dividend yield 𝐷. Intuitively, explain why the above SDE can be written as 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 . By considering a delta-hedging portfolio involving both an option 𝑉 (𝑆𝑡 , 𝑡) which can only be exercised at expiry time 𝑇 and a stock 𝑆𝑡 , derive the Black–Scholes equation for a stock that pays continuous dividend yield. Solution: Given that dividend yield is defined as the proportion of asset price paid out per unit time, when a dividend is paid out, the stock price 𝑆𝑡 is reduced by 𝐷𝑆𝑡 𝑑𝑡. Therefore, we can write 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 . At time 𝑡 we let the value of a portfolio Π𝑡 be Π𝑡 = 𝑉 (𝑆𝑡 , 𝑡) − Δ 𝑆𝑡 where it involves buying one unit of option 𝑉 (𝑆𝑡 , 𝑡) and selling Δ units of 𝑆𝑡 . Since the investor receives 𝐷𝑆𝑡 𝑑𝑡 for every unit of asset held, and because the investor holds −Δ 𝑆𝑡 ,

90

2.2.2 Black–Scholes Model

the portfolio changes by an amount −Δ𝐷 𝑆𝑡 𝑑𝑡 and therefore the change in portfolio Π𝑡 is 𝑑Π𝑡 = 𝑑𝑉 − Δ(𝑑𝑆𝑡 + 𝐷 𝑆𝑡 𝑑𝑡) = 𝑑𝑉 − Δ𝑑𝑆𝑡 − Δ𝐷 𝑆𝑡 𝑑𝑡. Expanding 𝑉 (𝑆𝑡 , 𝑡) using Taylor’s theorem 𝑑𝑉 =

)2 𝜕𝑉 1 𝜕2𝑉 ( 𝜕𝑉 𝑑𝑆𝑡 + … 𝑑𝑆𝑡 + 𝑑𝑡 + 𝜕𝑡 𝜕𝑆𝑡 2 𝜕𝑆𝑡2

and by substituting 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 and subsequently applying It¯o’s lemma we have [ 𝑑𝑉 =

] 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 𝜕𝑉 + (𝜇 − 𝐷) 𝑆𝑡 𝑑𝑊𝑡 . + 𝜎 𝑆𝑡 𝑑 𝑡 + 𝜎𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡2

Substituting back into 𝑑Π𝑡 and rearranging the terms we have (

) 𝜕𝑉 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 𝑑Π𝑡 = + (𝜇 − 𝐷) 𝑆𝑡 𝑑𝑊𝑡 𝑑𝑡 + 𝜎𝑆𝑡 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆 𝜕𝑆 𝜕𝑆𝑡 𝑡 𝑡 ] [ −Δ (𝜇 − 𝐷) 𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 − Δ𝐷 𝑆𝑡 𝑑𝑡 ( ) 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 + (𝜇 − 𝐷) 𝑆𝑡 − 𝜇Δ𝑆𝑡 𝑑𝑡 = + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 ) ( 𝜕𝑉 +𝜎𝑆𝑡 − Δ 𝑑𝑊𝑡 . 𝜕𝑆𝑡 To eliminate the random component we choose Δ=

𝜕𝑉 𝜕𝑆𝑡

which leads to ( 𝑑Π𝑡 =

𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 + 𝜎 2 𝑆𝑡2 2 − 𝐷𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡

) 𝑑𝑡.

Under the no-arbitrage condition the return on the amount Π𝑡 invested at a risk-free interest rate would see a growth of 𝑑Π𝑡 = 𝑟Π𝑡 𝑑𝑡

2.2.2 Black–Scholes Model

91

where 𝑟 is the risk-free rate and hence we have ( ) 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2 𝑉 − 𝐷𝑆𝑡 𝑟Π𝑡 𝑑𝑡 = + 𝜎 𝑆𝑡 𝑑𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 ( ) ) ( 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2 𝑉 𝑟 𝑉 (𝑆𝑡 , 𝑡) − Δ𝑆𝑡 𝑑𝑡 = − 𝐷𝑆𝑡 𝑑𝑡 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 ( ) 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 𝑟 𝑉 (𝑆𝑡 , 𝑡) − 𝑆𝑡 + 𝜎 2 𝑆𝑡2 2 − 𝐷𝑆𝑡 = 𝜕𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 and finally 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷) 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 which is the Black–Scholes equation for a stock which pays continuous dividend yield. { } 2. Black–Scholes Equation with Stock Paying Continuous Dividend Yield II. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the stock price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. In addition, let 𝐵𝑡 be the risk-free asset having the following differential equation 𝑑𝐵𝑡 = 𝑟 𝐵𝑡 𝑑𝑡 where 𝑟 denotes the risk-free interest rate. At time 𝑡 we consider a trader who has a portfolio valued at Π𝑡 , given as Π𝑡 = 𝜙𝑡 𝑆𝑡 + 𝜓𝑡 𝐵𝑡 holding 𝜙𝑡 shares of stock and 𝜓𝑡 units being invested in a risk-free asset 𝐵𝑡 . Let the option price at time 𝑡 be 𝑉 (𝑆𝑡 , 𝑡); the contingent claim 𝑉 (𝑆𝑇 , 𝑇 ) is said to be attainable if there exists an admissible strategy worth Π𝑇 = 𝑉 (𝑆𝑇 , 𝑇 ) at the option expiry time 𝑇 , 𝑇 > 𝑡. In the absence of arbitrage, show that Π𝑡 = 𝑉 (𝑆𝑡 , 𝑡), 𝑡 ∈ [0, 𝑇 ]. Finally, show that the portfolio (𝜙𝑡 , 𝜓𝑡 ) is self-financing if and only if 𝑉 (𝑆𝑡 , 𝑡) satisfies the Black–Scholes equation for a stock that pays continuous dividend yield. Solution: Assume there is a risk-free interest rate 𝑟 where an investor can invest in a money market account. At time 𝑡, if Π𝑡 > 𝑉 (𝑆𝑡 , 𝑡) then the investor can sell the portfolio Π𝑡 and use the proceeds to buy the option 𝑉 (𝑆𝑡 , 𝑡), making a profit Π𝑡 − 𝑉 (𝑆𝑡 , 𝑡) > 0 and saving it in a risk-free money-market account. At time 𝑇 both of the asset values are identical,

92

2.2.2 Black–Scholes Model

Π𝑇 = 𝑉 (𝑆𝑇 , 𝑇 ), with the value of the bought option covering the sold portfolio and making a risk-free profit (Π𝑡 − 𝑉 (𝑆𝑡 , 𝑡))𝑒𝑟(𝑇 −𝑡) . Alternatively, at time 𝑡 if Π𝑡 < 𝑉 (𝑆𝑡 , 𝑡) then an investor can sell the option 𝑉 (𝑆𝑡 , 𝑡) and buy the portfolio Π𝑡 , also making an instant profit 𝑉 (𝑆𝑡 , 𝑡) − Π𝑡 > 0, and the proceeds can go into the money-market account. At time 𝑇 the asset values are equal, Π𝑇 = 𝑉 (𝑆𝑇 , 𝑇 ), with the value of the bought portfolio covering the sold option and making a risk-free profit (𝑉 (𝑆𝑡 , 𝑡) − Π𝑡 )𝑒𝑟(𝑇 −𝑡) . Therefore, under the no-arbitrage condition we must have Π𝑡 = 𝑉 (𝑆𝑡 , 𝑡). At time 𝑡 ∈ [0, 𝑇 ], to replicate the option 𝑉 (𝑆𝑡 , 𝑡), let the self-financing portfolio be 𝑉 (𝑆𝑡 , 𝑡) = 𝜙𝑆𝑡 + 𝜓𝑡 𝐵𝑡 . By applying Taylor’s theorem on 𝑑𝑉 𝑑𝑉 =

𝜕𝑉 1 𝜕2𝑉 𝜕𝑉 𝑑𝑆𝑡 + (𝑑𝑆𝑡 )2 + … 𝑑𝑡 + 𝜕𝑡 𝜕𝑆𝑡 2 𝜕𝑆𝑡2

and since (𝑑𝑆𝑡 )2 = ((𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 )2 = 𝜎 2 𝑆𝑡2 𝑑𝑡 such that (𝑑𝑡)𝜈 = 0, 𝜈 ≥ 2 we have 1 𝜕2𝑉 𝜕𝑉 𝜕𝑉 𝑑𝑆𝑡 + 𝜎 2 𝑆𝑡2 2 𝑑𝑡 𝑑𝑡 + 𝜕𝑡 𝜕𝑆𝑡 2 𝜕𝑆𝑡 ( ) 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 = 𝑑𝑆 + 𝑑𝑡. + 𝜎 𝑆𝑡 𝜕𝑆𝑡 𝑡 𝜕𝑡 2 𝜕𝑆𝑡2

𝑑𝑉 =

Since the trader will receive 𝐷𝑆𝑡 𝑑𝑡 for every stock held, the portfolio (𝜙𝑡 , 𝜓𝑡 ) is selffinancing if and only if 𝑑𝑉 = 𝜙𝑡 𝑑𝑆𝑡 + 𝜓𝑡 𝑑𝐵𝑡 + 𝜙𝑡 𝐷𝑆𝑡 𝑑𝑡 = 𝜙𝑡 𝑑𝑆𝑡 + (𝑟𝐵𝑡 𝜓𝑡 + 𝜙𝑡 𝐷𝑆𝑡 )𝑑𝑡. By equating both of the equations we have 𝑟𝐵𝑡 𝜓𝑡 + 𝜙𝑡 𝐷𝑆𝑡 =

𝜕𝑉 𝜕2 𝑉 1 + 𝜎 2 𝑆𝑡2 2 𝜕𝑡 2 𝜕𝑆𝑡

and

𝜙𝑡 =

𝜕𝑉 𝜕𝑆𝑡

and substituting the above two equations into 𝑉 = 𝜙𝑡 𝑆𝑡 + 𝜓𝑡 𝐵𝑡 we have 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0. + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡

2.2.2 Black–Scholes Model

93

{ } 3. Black–Scholes Equation with Stock Paying Discrete Dividends. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙstandard Wiener process on the probability space (Ω, ℱ, ℙ) and let the stock price 𝑆𝑡 follow the SDE 𝑑𝑆𝑡 = 𝜇𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter and 𝜎 is the volatility parameter. In addition, let 𝐵𝑡 be the risk-free asset having the following differential equation 𝑑𝐵𝑡 = 𝑟𝐵𝑡 𝑑𝑡 where 𝑟 denotes the risk-free interest rate. Assume the asset 𝑆𝑡 pays a sequence of fixed dividends 𝐷𝑖 at time 𝑡𝐷𝑖 , 𝑖 = 1, 2, … , 𝑛 such that 𝑡 < 𝑡𝐷𝑖 < 𝑡𝐷2 < ⋯ < 𝑡𝐷𝑛 . Explain intuitively why the SDE for 𝑆𝑡 can be written as ( 𝜇𝑆𝑡 −

𝑑𝑆𝑡 =

𝑛 ∑ 𝑖=1

) 𝐷𝑖 𝛿(𝑡 − 𝑡𝐷𝑖 ) 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡

where 𝛿(𝑡 − 𝑡𝐷𝑖 ) is the Dirac delta function centred at 𝑡𝐷𝑖 . By considering a hedging portfolio involving both an option 𝑉 (𝑆𝑡 , 𝑡) which can only be exercised at expiry time 𝑇 > 𝑡𝐷𝑛 and a stock 𝑆𝑡 , show that 𝑉 (𝑆𝑡 , 𝑡) satisfies the following differential equation 𝜕2𝑉 𝜕𝑉 1 + 𝜎 2 𝑆𝑡2 2 + 𝜕𝑡 2 𝜕𝑆𝑡

( 𝑟𝑆𝑡 −

𝑛 ∑ 𝑖=1

) 𝐷𝑖 𝛿(𝑡 − 𝑡𝐷𝑖 )

𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0. 𝜕𝑆𝑡

Solution: When a discrete dividend 𝐷𝑖 is paid at time 𝑡𝐷𝑖 , the asset price 𝑆𝑡 is reduced by the same amount. However, for the time between dividend payment dates, 𝑆𝑡 still follows a GBM process. Hence, we can write ( 𝑑𝑆𝑡 =

𝜇𝑆𝑡 −

𝑛 ∑ 𝑖=1

) 𝐷𝑖 𝛿(𝑡 − 𝑡𝐷𝑖 ) 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 .

At time 𝑡 we let the value of a portfolio Π𝑡 be Π𝑡 = 𝑉 (𝑆𝑡 , 𝑡) − Δ𝑆𝑡 where it involves buying one unit of option 𝑉 (𝑆𝑡 , 𝑡) and selling Δ units of 𝑆𝑡 . At times 𝑡𝐷1 , 𝑡𝐷2 , … , 𝑡𝐷𝑛 , the owner of the portfolio receives 𝐷1 , 𝐷2 , … , 𝐷𝑛 dividend payments

94

2.2.2 Black–Scholes Model

for every asset held. Therefore, the change in portfolio Π𝑡 becomes ( 𝑑Π𝑡 = 𝑑𝑉 − Δ 𝑑𝑆𝑡 +

𝑛 ∑ 𝑖=1

) 𝐷𝑖 𝛿(𝑡 − 𝑡𝐷𝑖 )𝑑𝑡

= 𝑑𝑉 − Δ(𝜇𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 ). Expanding 𝑉 (𝑆𝑡 , 𝑡) using Taylor’s theorem 𝑑𝑉 =

)2 1 𝜕2𝑉 ( 𝜕𝑉 𝜕𝑉 𝑑𝑆𝑡 + … 𝑑𝑆𝑡 + 𝑑𝑡 + 𝜕𝑡 𝜕𝑆𝑡 2 𝜕𝑆𝑡2

( ) ∑ and by substituting 𝑑𝑆𝑡 = 𝜇𝑆𝑡 − 𝑛𝑖=1 𝐷𝑖 𝛿(𝑡 − 𝑡𝐷𝑖 ) 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 and applying It¯o’s lemma we have [ 𝑑𝑉 =

𝜕𝑉 𝜕2𝑉 1 + 𝜎 2 𝑆𝑡2 2 + 𝜕𝑡 2 𝜕𝑆𝑡

( 𝜇𝑆𝑡 −

𝑛 ∑ 𝑖=1

) 𝐷𝑖 𝛿(𝑡 − 𝑡𝐷𝑖 )

] 𝜕𝑉 𝜕𝑉 𝑑𝑊𝑡 . 𝑑𝑡 + 𝜎𝑆𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡

Hence, (

( ) ) 𝑛 ∑ 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2 𝑉 + 𝜇𝑆𝑡 − 𝐷𝑖 𝛿(𝑡 − 𝑡𝐷𝑖 ) − 𝜇Δ𝑆𝑡 𝑑𝑡 + 𝜎 𝑆𝑡 𝑑Π𝑡 = 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 𝑖=1 ) ( 𝜕𝑉 +𝜎𝑆𝑡 − Δ 𝑑𝑊𝑡 . 𝜕𝑆𝑡

Setting Δ =

𝜕𝑉 to eliminate the random component leads to 𝜕𝑆𝑡 ( 𝑑Π𝑡 =

𝑛

𝜕𝑉 𝜕𝑉 𝜕2 𝑉 ∑ 1 𝐷𝑖 𝛿(𝑡 − 𝑡𝐷𝑖 ) + 𝜎 2 𝑆𝑡2 2 − 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 𝑖=1

) 𝑑𝑡.

Under the no-arbitrage condition the return on the portfolio Π𝑡 invested in a risk-free interest rate would see a growth of 𝑑Π𝑡 = 𝑟Π𝑡 𝑑𝑡

2.2.2 Black–Scholes Model

95

where 𝑟 is the risk-free interest rate and hence we have ( 𝑟Π𝑡 𝑑𝑡 = ) ( 𝑟 𝑉 (𝑆𝑡 , 𝑡) − Δ𝑆𝑡 𝑑𝑡 = (

𝜕𝑉 𝑟 𝑉 (𝑆𝑡 , 𝑡) − 𝑆𝑡 𝜕𝑆𝑡

)

(

𝑛

𝜕𝑉 𝜕𝑉 𝜕2𝑉 ∑ 1 𝐷𝑖 𝛿(𝑡 − 𝑡𝐷𝑖 ) + 𝜎 2 𝑆𝑡2 2 − 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝑖=1 𝑛

𝜕𝑉 𝜕𝑉 𝜕2𝑉 ∑ 1 𝐷𝑖 𝛿(𝑡 − 𝑡𝐷𝑖 ) + 𝜎 2 𝑆𝑡2 2 − 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 𝑖=1

) 𝑑𝑡 ) 𝑑𝑡

𝑛

=

𝜕𝑉 𝜕2𝑉 ∑ 1 𝜕𝑉 𝐷𝑖 𝛿(𝑡 − 𝑡𝐷𝑖 ) + 𝜎 2 𝑆𝑡2 2 − 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 𝑖=1

which eventually leads to 𝜕𝑉 𝜕2𝑉 1 + 𝜎 2 𝑆𝑡2 2 + 𝜕𝑡 2 𝜕𝑆𝑡

( 𝑟𝑆𝑡 −

𝑛 ∑ 𝑖=1

) 𝐷𝑖 𝛿(𝑡 − 𝑡𝐷𝑖 )

𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0 𝜕𝑆𝑡

which is a Black–Scholes equation with stock paying discrete dividend payments. { } 4. European Option Valuation (PDE Approach). Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the stock price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the dividend yield, 𝜎 is the volatility parameter and let 𝑟 denote the risk-free interest rate. Let 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of a European call option with strike 𝐾 and expiry time 𝑇 > 𝑡, satisfying the Black–Scholes equation 𝜕𝐶 𝜕𝐶 1 2 2 𝜕 2 𝐶 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 with boundary conditions 𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0}, 𝐶(0, 𝑡; 𝐾, 𝑇 ) = 0 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ∼ 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) as 𝑆𝑡 → ∞. With the introduction of variables 𝑥, 𝜏 and 𝑣(𝑥, 𝜏) defined by 𝑆𝑡 = 𝐾𝑒𝑥 , 𝑡 = 𝑇 −

𝜏 1 2 𝜎 2

, 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑣(𝑥, 𝜏)

96

2.2.2 Black–Scholes Model

show that the Black–Scholes equation is reduced to 𝜕𝑣 𝜕2𝑣 𝜕𝑣 = 2 + 𝑘1 − 𝑘0 𝑣 𝜕𝜏 𝜕𝑥 𝜕𝑥 with boundary condition 𝑣(𝑥, 0) = max{𝑒𝑥 − 1} where 𝑘1 =

(𝑟 − 𝐷 − 12 𝜎 2 ) 1 2 𝜎 2

and 𝑘0 =

Using a change of variables 𝑣(𝑥, 𝜏) 1 𝛼 = − 𝑘1 2

𝑟

.

1 2 𝜎 2 = 𝑒𝛼𝑥+𝛽𝜏 𝑢(𝑥, 𝜏)

and

show that by setting

1 𝛽 = − 𝑘21 − 𝑘0 4

the problem is reduced to a diffusion equation of the form 𝜕2𝑢 𝜕𝑢 = 2 , 𝑢(𝑥, 0) = 𝑓 (𝑥) 𝜕𝜏 𝜕𝑥 1

1

where 𝑓 (𝑥) = max{𝑒 2 (𝑘1 +1)𝑥 − 𝑒 2 𝑘1 𝑥 , 0} for −∞ < 𝑥 < ∞ and 𝜏 > 0. Given the solution ∞

(𝑥−𝑧)2 1 𝑓 (𝑧)𝑒− 4𝜏 𝑑𝑧 𝑢(𝑥, 𝜏) = √ 2 𝜋𝜏 ∫−∞

deduce that ( ) ( ) 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ 𝑑+ − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ 𝑑− where 𝑑± =

log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

and Φ(𝑥) is the standard normal cdf 𝑥

Φ(𝑥) =

∫−∞

1 − 1 𝑢2 √ 𝑒 2 𝑑𝑢. 2𝜋

Solution: A European call option with value 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) satisfies the Black–Scholes equation 𝜕𝐶 𝜕𝐶 1 2 2 𝜕 2 𝐶 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

2.2.2 Black–Scholes Model

97

with conditions 𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0}, 𝐶(0, 𝑡; 𝐾, 𝑇 ) = 0 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ∼ 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) as 𝑆𝑡 → ∞. First we transform the Black–Scholes equation by making it dimensionless, setting 𝑆𝑡 = 𝐾𝑒𝑥 , 𝑡 = 𝑇 −

𝜏 1 2 𝜎 2

, 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑣(𝑥, 𝜏)

and write 𝜕𝐶𝑡 𝜕𝑥 𝜕𝐶 𝐾 𝜕𝑣 𝜕𝑣 𝜕𝐶 𝜕𝐶 𝜕𝜏 1 = = = = − 𝐾𝜎 2 , 𝜕𝑡 𝜕𝜏 𝜕𝑡 2 𝜕𝜏 𝜕𝑆𝑡 𝜕𝑥 𝜕𝑆𝑡 𝑆𝑡 𝜕𝑥 and 𝜕 𝜕2𝐶 = 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 = = =

= =

(

)

) 𝜕𝐶 𝜕𝑥 𝜕𝑥 𝜕𝑆𝑡 ( ) ) ( 𝜕𝐶 𝜕𝑥 𝜕 𝜕 𝜕𝑥 𝜕𝐶 + 𝜕𝑆𝑡 𝜕𝑥 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑥 ( ) ) ( 𝜕 𝜕𝑥 𝜕𝐶 𝜕𝑥 1 𝜕𝐶 − 𝜕𝑥 𝜕𝑆𝑡 𝜕𝑥 𝜕𝑆𝑡 𝑆𝑡2 𝜕𝑥 ( )2 ( ) 𝜕 2 𝐶 𝜕𝑥 1 𝜕𝐶 − 2 2 𝜕𝑆𝑡 𝜕𝑥 𝜕𝑥 𝑆𝑡 ) ( 2 𝐾 𝜕 𝑣 𝜕𝑣 − . 𝑆𝑡2 𝜕𝑥2 𝜕𝑥 𝜕 𝜕𝑆𝑡

(

𝜕𝐶 𝜕𝑆𝑡

By substituting the above expressions back into the Black–Scholes equation we have 𝜕𝑣 𝜕 2 𝑣 𝜕𝑣 𝑟 − 𝐷 𝜕𝑣 𝑟 = 2− + 1 − 1 𝑣(𝑥, 𝜏) 2 𝜕𝜏 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝜎 𝜎2 2 2 or 𝜕𝑣 𝜕𝑣 𝜕2 𝑣 = 2 + 𝑘1 − 𝑘0 𝑣(𝑥, 𝜏) 𝜕𝜏 𝜕𝑥 𝜕𝑥 where 𝑘1 =

(𝑟 − 𝐷 − 12 𝜎 2 ) 1 2 𝜎 2

and 𝑘0 =

𝑟 1 2 𝜎 2

.

98

2.2.2 Black–Scholes Model

As for the boundary condition { } 𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max 𝑆𝑇 − 𝐾, 0 it becomes

or

} { 𝐾𝑣 (𝑥, 0) = max 𝐾𝑒𝑥 − 𝐾, 0 } { 𝑣(𝑥, 0) = max 𝑒𝑥 − 1, 0 .

In order to simplify the partial differential equation (PDE) we let 𝑣(𝑥, 𝜏) = 𝑒𝛼𝑥+𝛽𝜏 𝑢(𝑥, 𝜏) with 𝜕𝑢 𝜕𝑣 = 𝛽𝑒𝛼𝑥+𝛽𝜏 𝑢(𝑥, 𝜏) + 𝑒𝛼𝑥+𝛽𝜏 𝜕𝜏 𝜕𝜏 𝜕𝑢 𝜕𝑣 = 𝛼𝑒𝛼𝑥+𝛽𝜏 𝑢(𝑥, 𝜏) + 𝑒𝛼𝑥+𝛽𝜏 𝜕𝑥 𝜕𝑥 2 𝜕2 𝑣 2 𝛼𝑥+𝛽𝜏 𝛼𝑥+𝛽𝜏 𝜕𝑢 𝛼𝑥+𝛽𝜏 𝜕𝑢 𝛼𝑥+𝛽𝜏 𝜕 𝑢 = 𝛼 𝑒 𝑢(𝑥, 𝜏) + 𝛼𝑒 + 𝛼𝑒 + 𝑒 𝜕𝑥 𝜕𝑥 𝜕𝑥2 𝜕𝑥2 ( ) 2𝑢 𝜕𝑢 𝜕 = 𝑒𝛼𝑥+𝛽𝜏 𝛼 2 𝑢(𝑥, 𝜏) + 2𝛼 + . 𝜕𝑥 𝜕𝑥2 Substituting the above expressions back into the PDE yields ) ( 𝜕𝑢 𝜕𝑢 𝜕 2 𝑢 𝜕𝑢 − 𝑘0 𝑢(𝑥, 𝜏) 𝛽𝑢(𝑥, 𝜏) + = 𝛼 2 𝑢 + 2𝛼 + 2 + 𝑘1 𝛼𝑢 + 𝜕𝜏 𝜕𝑥 𝜕𝑥 𝜕𝑥 or ) 𝜕𝑢 ( 2 ) 𝜕𝑢 𝜕2𝑢 ( + 2𝛼 + 𝑘1 + 𝛼 + 𝑘1 𝛼 − 𝑘0 − 𝛽 𝑢(𝑥, 𝜏) = . 2 𝜕𝑥 𝜕𝜏 𝜕𝑥 In order to reduce the above problem to the form 𝜕2𝑢 𝜕𝑢 = 𝜕𝜏 𝜕𝑥2 we let 𝛽 = 𝛼 2 + 𝑘1 𝛼 − 𝑘0

and

2𝛼 + 𝑘1 = 0

which gives 1 𝛼 = − 𝑘1 2

and

1 𝛽 = − 𝑘21 − 𝑘0 . 4

2.2.2 Black–Scholes Model

99

Therefore, 𝑣(𝑥, 𝜏) = 𝑒𝛼𝑥+𝛽𝜏 𝑢(𝑥, 𝜏) 1

1 2

= 𝑒− 2 𝑘1 𝑥−( 4 𝑘1 +𝑘0 )𝜏 𝑢(𝑥, 𝜏) where 𝜕2𝑢 𝜕𝑢 = , 𝑥 ∈ (−∞, ∞), 𝜏 > 0 2 𝜕𝜏 𝜕𝑥 with 𝑢(𝑥, 0) = 𝑒𝛼𝑥 𝑣(𝑥, 0)

1 { } = 𝑒 2 𝑘1 𝑥 max 𝑒𝑥 − 1, 0 1

1

= max{𝑒( 2 𝑘1 +1)𝑥 − 𝑒 2 𝑘1 𝑥 , 0} or 1

1

𝑓 (𝑥) = max{𝑒( 2 𝑘1 +1)𝑥 − 𝑒 2 𝑘1 𝑥 , 0}. The solution to the diffusion equation 𝜕𝑢 𝜕2𝑢 = , 𝑥 ∈ (−∞, ∞), 𝜏 > 0 2 𝜕𝜏 𝜕𝑥 is given by ∞

(𝑥−𝑧)2 1 𝑓 (𝑧)𝑒− 4𝜏 𝑑𝑧. 𝑢(𝑥, 𝜏) = √ 2 𝜋𝜏 ∫−∞

𝑧−𝑥 Using the changing of variables we let 𝑦 = √ 2𝜏 √ 1 2√ 1 𝑢(𝑥, 𝜏) = √ 𝑓 (𝑥 + 2𝜏𝑦)𝑒− 2 𝑦 2𝜏𝑑𝑦. 2 𝜋𝜏 ∫−∞ ∞

√ √ √ 1 𝑥 2𝜏 𝑦) = min{𝑒 2 𝑘1 (𝑥+ 2𝜏𝑦) (𝑒𝑥+ 2𝜏𝑦 − 1), 0} > 0 if 𝑦 > − √ 2𝜏 ] √ √ ∞ [ 1 1 1 2 1 𝑒( 2 𝑘1 +1)(𝑥+ 2𝜏𝑦) − 𝑒 2 𝑘1 (𝑥+ 2𝜏𝑦) 𝑒− 2 𝑦 𝑑𝑦 𝑢(𝑥, 𝜏) = √ 𝑥 ∫ 2𝜋 − √2𝜏

Since 𝑓 (𝑥 +

= 𝐼1 − 𝐼2 where ∞

1 1 𝑒( 2 𝑘1 +1)(𝑥+ 𝐼1 = √ 𝑥 2𝜋 ∫− √2𝜏

√

2𝜏𝑦)− 21 𝑦2

𝑑𝑦

100

2.2.2 Black–Scholes Model

and ∞

1 1 𝑒 2 𝑘1 (𝑥+ 𝐼2 = √ 𝑥 ∫ 2𝜋 − √2𝜏

√ 2𝜏𝑦)− 21 𝑦2

𝑑𝑦.

To calculate 𝐼1 we have ∞

1 1 𝐼1 = √ 𝑒( 2 𝑘1 +1)(𝑥+ 𝑥 2𝜋 ∫− √2𝜏 1

=

1

𝑒( 2 𝑘1 +1)𝑥+( 2 𝑘1 +1) √ 2𝜋

2𝜏

√ 2𝜏𝑦)− 12 𝑦2

∞

𝑑𝑦

1

1

𝑒− 2 (𝑦−( 2 𝑘1 +1)

∫− √𝑥

√ 2𝜏)2

𝑑𝑦.

2𝜏

√ Let 𝑠 = 𝑦 − ( 12 𝑘1 + 1) 2𝜏 then 1

𝐼1 =

1

𝑒( 2 𝑘1 +1)𝑥+( 2 𝑘1 +1) √ 2𝜋

2𝜏

( 12 𝑘1 +1)𝑥+( 12 𝑘1 +1)2 𝜏

=𝑒

∞

1 2

∫− √𝑥 −( 1 𝑘1 +1)√2𝜏 2 (

𝑒− 2 𝑠 𝑑𝑠

2𝜏

Φ 𝑑+

)

( )√ 𝑑+ 1 2 ( ) 1 1 𝑥 where Φ 𝑑+ = √ 𝑒− 2 𝑠 𝑑𝑠 and 𝑑+ = √ + 2𝜏. 𝑘1 + 1 2 2𝜋 ∫−∞ 2𝜏 Using the same techniques we can write 𝐼2 as 1 1 2 ( ) 𝐼2 = 𝑒 2 𝑘1 𝑥+ 4 𝑘1 𝜏 Φ 𝑑−

1 √ 𝑥 where 𝑑− = √ + 𝑘1 2𝜏. 2𝜏 2 Thus, 1

1

1

1 2

𝑢(𝑥, 𝜏) = 𝑒( 2 𝑘1 +1)𝑥+( 2 𝑘1 +1) 𝜏 Φ(𝑑+ ) + 𝑒 2 𝑘1 𝑥+ 4 𝑘1 𝜏 Φ(𝑑− ) 2

and from 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑣(𝑥, 𝜏) = 𝐾𝑒𝛼𝑥+𝛽𝜏 𝑢(𝑥, 𝜏) we eventually have 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑒𝑥+(𝑘1 −𝑘0 +1)𝜏 Φ(𝑑+ ) − 𝐾𝑒−𝑘0 𝜏 Φ(𝑑− ) where 𝛼 = − 12 𝑘1 and 𝛽 = − 14 𝑘21 − 𝑘0 . Finally, by substituting 𝑘1 =

(𝑟 − 𝐷 − 12 𝜎 2 ) 1 2 𝜎 2

and 𝑘0 =

𝑟 1 2 𝜎 2

we have

𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑− )

2.2.2 Black–Scholes Model

101

where 𝑑± =

log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝑡) . √ 𝜎 𝑇 −𝑡

N.B. Using the put–call parity we can show that the price of a European put option is 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑− ) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑+ ). { } 5. European Option Valuation (Probabilistic Approach). Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. Using It¯o’s lemma show that under the risk-neutral measure ℚ the conditional distribution of 𝑆𝑇 given 𝑆𝑡 is [ ] 1 𝑆𝑇 |𝑆𝑡 ∼ log - log 𝑆𝑡 + (𝑟 − 𝐷 − 𝜎 2 )(𝑇 − 𝑡), 𝜎 2 (𝑇 − 𝑡) 2 where 𝑟 is the risk-free interest rate and 𝑇 > 𝑡. From the Feynman–Kac formula the European call option with strike 𝐾 and expiry time 𝑇 is given by [ ] 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑆𝑇 − 𝐾, 0}|| ℱ𝑡 where 𝔼ℚ (⋅) is the expectation under the risk-neutral measure ℚ. Using the risk-neutral valuation approach show that the European call option price is 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑− ) where ) ( log 𝑆𝑡 ∕𝐾 + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝑡) 𝑑± = √ 𝜎 𝑇 −𝑡 and Φ(𝑥) is the standard normal cdf 𝑥

Φ(𝑥) =

∫−∞

1 − 1 𝑢2 √ 𝑒 2 𝑑𝑢. 2𝜋

102

2.2.2 Black–Scholes Model

Finally, deduce that the European put option price with strike 𝐾 and expiry time 𝑇 is 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑− ) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑+ ). Solution: From Girsanov’s theorem, under the risk-neutral measure ℚ we can write 𝑑𝑆𝑡 = (𝑟 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡 (𝜇 − 𝑟) 𝑡 is a ℚ-standard Wiener process. Using It̄o’s lemma we such that 𝑊𝑡ℚ = 𝑊𝑡 + 𝜎 can show that for 𝑡 < 𝑇 𝑑𝑆𝑡 1 𝑑(log 𝑆𝑡 ) = − 𝑆𝑡 2

(

𝑑𝑆𝑡 𝑆𝑡

)2 +…

1 = (𝑟 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ − 𝜎 2 𝑑𝑡 2 ) ( 1 2 = 𝑟 − 𝐷 − 𝜎 𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ . 2 Integrating over time we have ) 𝑇 ( 𝑇 1 𝑟 − 𝐷 − 𝜎 2 𝑑𝑢 + 𝑑(log 𝑆𝑢 ) = 𝜎 𝑑𝑊𝑢ℚ ∫𝑡 ∫𝑡 ∫𝑡 2 ( ) ( ) 𝑆𝑇 1 log = 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡) + 𝜎𝑊𝑇ℚ−𝑡 𝑆𝑡 2 𝑇

where 𝑊𝑇ℚ−𝑡 ∼  (0, 𝑇 − 𝑡). Hence, conditional on 𝑆𝑡 we can write ) [ ( ] 1 𝑆𝑇 |𝑆𝑡 ∼ log - log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡), 𝜎 2 (𝑇 − 𝑡) . 2 Under the risk-neutral measure ℚ we can write the European call option price as [ ] ( ) 𝐶 𝑆𝑡 , 𝑡; 𝐾, 𝑇 = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑆𝑇 − 𝐾, 0}|ℱ𝑡 = 𝑒−𝑟(𝑇 −𝑡)

∞

∫0

max{𝑆𝑇 − 𝐾, 0}𝑓 (𝑆𝑇 |𝑆𝑡 )𝑑𝑆𝑇 .

) ( Here, for a log normally distributed random variable log 𝑋 ∼  𝜇, 𝜎 2 the probability density function (pdf) is 𝑓𝑋 (𝑥; 𝜇, 𝜎) =

1 √

𝑥𝜎 2𝜋

𝑒

− 21

(

) log𝑥−𝜇 2 𝜎

, 𝑥>0

2.2.2 Black–Scholes Model

103

and we can thus write (

) ( 𝑓 𝑆𝑇 |𝑆𝑡 =

√

− 21

1

𝑆𝑇 𝜎 2𝜋(𝑇 − 𝑡)

𝑒

log 𝑆𝑇 −log 𝑆𝑡 −(𝑟−𝐷− 1 𝜎 2 )(𝑇 −𝑡) 2 √ 𝜎 𝑇 −𝑡

)2

, 𝑆𝑇 > 0

or ) ( 𝑓 𝑆𝑇 |𝑆𝑡 =

( − 21

1

𝑒 √ 𝑆𝑇 𝜎 2𝜋(𝑇 − 𝑡)

log 𝑆𝑇 −𝑚 √ 𝜎 𝑇 −𝑡

)2

, 𝑆𝑇 > 0

) ( where 𝑚 = log 𝑆𝑡 + 𝑟 − 𝐷 − 12 𝜎 2 (𝑇 − 𝑡). Therefore, ( ) 𝐶 𝑆𝑡 , 𝑡; 𝐾, 𝑇 = 𝑒−𝑟(𝑇 −𝑡)

𝐾

max{𝑆𝑇 − 𝐾, 0}𝑓 (𝑆𝑇 |𝑆𝑡 ) 𝑑𝑆𝑇

∫0

∞

+𝑒−𝑟(𝑇 −𝑡) = 𝑒−𝑟(𝑇 −𝑡) = 𝐼1 − 𝐼2

∫𝐾 ∞

∫𝐾

max{𝑆𝑇 − 𝐾, 0}𝑓 (𝑆𝑇 |𝑆𝑡 ) 𝑑𝑆𝑇

(𝑆𝑇 − 𝐾)𝑓 (𝑆𝑇 |𝑆𝑡 ) 𝑑𝑆𝑇

where 𝐼1 = 𝑒−𝑟(𝑇 −𝑡)

∞

∫𝐾

𝑆𝑇 𝑓 (𝑆𝑇 |𝑆𝑡 ) 𝑑𝑆𝑇

and

𝐼2 = 𝑒−𝑟(𝑇 −𝑡)

∞

∫𝐾

𝐾𝑓 (𝑆𝑇 |𝑆𝑡 ) 𝑑𝑆𝑇 .

Solving 𝐼1 we have 𝐼1 = 𝑒−𝑟(𝑇 −𝑡) = 𝑒−𝑟(𝑇 −𝑡)

and by letting 𝑢 =

∞

∫𝐾

𝑆𝑇 𝑓 (𝑆𝑇 |𝑆𝑡 ) 𝑑𝑆𝑇 (

∞

∫𝐾

√

− 21

1

𝜎 2𝜋(𝑇 − 𝑡)

𝑒

log 𝑆𝑇 −𝑚 √ 𝜎 𝑇 −𝑡

)2

𝑑𝑆𝑇

log𝑆𝑇 − 𝑚 we then have √ 𝜎 𝑇 −𝑡 𝐼1 =

1 2 𝑒𝑚−𝑟(𝑇 −𝑡) 𝑒− 2 𝑢 +𝜎𝑢 √ log 𝐾−𝑚 2𝜋 ∫ 𝜎 √𝑇 −𝑡

∞

√ 𝑇 −𝑡

𝑑𝑢.

Using the sum of squares √ 1 1 − 𝑢2 + 𝜎𝑢 𝑇 − 𝑡 = − 2 2

] )2 2 𝑢 − 𝜎 𝑇 − 𝑡 − 𝜎 (𝑇 − 𝑡)

[(

√

104

2.2.2 Black–Scholes Model

we can simplify 𝐼1 to become − 𝑒𝑚−𝑟(𝑇 −𝑡) 𝑒 2 𝐼1 = √ log 𝐾−𝑚 2𝜋 ∫ 𝜎 √𝑇 −𝑡 ∞

𝑚−𝑟(𝑇 −𝑡)+ 21 𝜎 2 (𝑇 −𝑡)

=𝑒

1

[(

] )2 √ 𝑢−𝜎 𝑇 −𝑡 −𝜎 2 (𝑇 −𝑡)

∞

∫ log𝐾−𝑚 √ 𝜎 𝑇 −𝑡

1 −1 √ 𝑒 2 2𝜋

(

𝑑𝑢

)2 √ 𝑢−𝜎 𝑇 −𝑡

𝑑 𝑢.

) ( √ By setting 𝑣 = 𝑢 − 𝜎 𝑇 − 𝑡 and substituting 𝑚 = log 𝑆𝑡 + 𝑟 − 𝐷 − 12 𝜎 2 (𝑇 − 𝑡) we eventually have ⎛ log(𝑆 ∕𝐾) + (𝑟 − 𝐷 + 1 𝜎 2 )(𝑇 − 𝑡) ⎞ 𝑡 ⎜ ⎟ 2 𝐼1 = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ ⎜ √ ⎟. 𝜎 𝑇 −𝑡 ⎜ ⎟ ⎝ ⎠ Similarly for 𝐼2 we have 𝐼2 = 𝑒−𝑟(𝑇 −𝑡) = 𝑒−𝑟(𝑇 −𝑡)

and by setting 𝑢 =

∞

∫𝐾

𝐾𝑓 (𝑆𝑇 ) 𝑑𝑆𝑇 (

∞

∫𝐾

√

1

𝑆𝑇 𝜎 2𝜋(𝑇 − 𝑡)

− 21

𝑒

log𝑆𝑇 −𝑚 √ 𝜎 𝑇 −𝑡

)2

𝑑𝑆𝑇

) ( log 𝑆𝑇 − 𝑚 and substituting 𝑚 = log 𝑆𝑡 + 𝑟 − 𝐷 − 12 𝜎 2 (𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡 ∞

1 − 21 𝑢2 𝑒 𝑑𝑢 √ ∫ log√𝐾−𝑚 2𝜋 𝜎 𝑇 −𝑡 )] [ ( log𝐾 − 𝑚 = 𝐾𝑒−𝑟(𝑇 −𝑡) 1 − Φ √ 𝜎 𝑇 −𝑡

𝐼2 = 𝑒−𝑟(𝑇 −𝑡) 𝐾

= 𝐾𝑒

−𝑟(𝑇 −𝑡)

⎛ log(𝑆 ∕𝐾) + (𝑟 − 𝐷 − 1 𝜎 2 )(𝑇 − 𝑡) ⎞ 𝑡 ⎟ ⎜ 2 Φ⎜ √ ⎟. 𝜎 𝑇 −𝑡 ⎟ ⎜ ⎠ ⎝

Therefore, ( ) ( ) 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ 𝑑+ − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ 𝑑−

2.2.2 Black–Scholes Model

105

where 1 log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 𝜎 2 ) (𝑇 − 𝑡) 2 𝑑± = . √ 𝜎 𝑇 −𝑡 Using the put–call parity 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) the European put option is 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑆𝑡 + 𝐾𝑒−𝑟(𝑇 −𝑡) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑− ) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) + 𝐾𝑒−𝑟(𝑇 −𝑡) [ [ ] ] = −𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) 1 − Φ(𝑑+ ) + 𝐾𝑒−𝑟(𝑇 −𝑡) 1 − Φ(𝑑− ) = 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑− ) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑+ ).

6. Martingale Property. Let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. In addition, let 𝑟 be the risk-free interest rate. We define the price of a European call option priced at time 𝑡 with strike price 𝐾 and expiry time 𝑇 , 𝑡 < 𝑇 as 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑− ) where 𝑑± =

log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

and Φ(⋅) is the cdf of a standard normal. Show that 𝑒−𝑟𝑡 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is a martingale under the risk-neutral measure ℚ. Solution: Under the risk-neutral measure ℚ, 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡 where 𝑊𝑡ℚ = 𝑊𝑡 +

(𝜇 − 𝑟) 𝜎

𝑡 is a ℚ-standard Wiener process.

106

2.2.2 Black–Scholes Model

Using It¯o’s lemma we can easily show for 𝑇 > 𝑡 ( log

𝑆𝑇 𝑆𝑡

) ∼

[(

) ] 1 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡), 𝜎 2 (𝑇 − 𝑡) 2

with density function for 𝑆𝑇 (

𝑓 (𝑆𝑇 |𝑆𝑡 ) =

√

− 21

1

𝑆𝑇 𝜎 2𝜋(𝑇 − 𝑡)

𝑒

log(𝑆𝑇 ∕𝑆𝑡 )−(𝑟−𝐷− 1 𝜎 2 )(𝑇 −𝑡) 2 √ 𝜎 𝑇 −𝑡

)2

.

To show that 𝑒−𝑟𝑡 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is a ℚ-martingale we note the following. log(𝑆𝜏 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝜏) (a) For 𝑡 < 𝜏 < 𝑇 and by setting 𝑑±𝜏 = √ 𝜎 𝑇 −𝜏 [ ] [ { 𝔼ℚ 𝑒−𝑟𝜏 𝐶(𝑆𝜏 , 𝜏; 𝐾, 𝑇 )|| ℱ𝑡 = 𝔼ℚ 𝑒−𝑟𝜏 𝑆𝜏 𝑒−𝐷(𝑇 −𝜏) Φ(𝑑+𝜏 ) }| ] − 𝐾𝑒−𝑟(𝑇 −𝜏) Φ(𝑑−𝜏 ) | ℱ𝑡 | [ ] | −𝑟𝜏−𝐷(𝑇 −𝜏) ℚ 𝔼 𝑆𝜏 Φ(𝑑+𝜏 )| ℱ𝑡 =𝑒 | [ ] −𝑟𝑇 ℚ 𝜏 | −𝐾𝑒 𝔼 Φ(𝑑− )| ℱ𝑡 = 𝐼1 − 𝐼2 [ ] [ ] | where 𝐼1 = 𝑒−𝑟𝜏−𝐷(𝑇 −𝜏) 𝔼ℚ 𝑆𝜏 Φ(𝑑+𝜏 )| ℱ𝑡 and 𝐼2 = 𝐾𝑒−𝑟𝑇 𝔼ℚ Φ(𝑑−𝜏 )|| ℱ𝑡 . | For the case [ ] | 𝐼1 = 𝑒−𝑟𝜏−𝐷(𝑇 −𝜏) 𝔼ℚ 𝑆𝜏 Φ(𝑑+𝜏 )| ℱ𝑡 | = 𝑒−𝑟𝜏−𝐷(𝑇 −𝜏) ) ( ∞ log(𝑆𝜏 ∕𝐾) + (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝜏) 𝑓 (𝑆𝜏 |𝑆𝑡 ) 𝑑𝑆𝜏 × 𝑆𝜏 Φ √ ∫0 𝜎 𝑇 −𝜏 log(𝑆𝜏 ∕𝑆𝑡 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝜏 − 𝑡) we have and by setting 𝑥 = √ 𝜎 𝜏 −𝑡 1 2

𝐼1 = 𝑆𝑡 𝑒−𝑟𝜏−𝐷(𝑇 −𝜏)+(𝑟−𝐷− 2 𝜎 )(𝜏−𝑡) ) ( √ √ ∞ 𝑚 + 𝑥𝜎 𝜏 − 𝑡 1 − 1 (𝑥2 −2𝑥𝜎 𝜏−𝑡) × Φ 𝑑𝑥 √ 𝑒 2 √ ∫−∞ 𝜎 𝑇 −𝜏 2𝜋

2.2.2 Black–Scholes Model

107

or 𝐼1 = 𝑆𝑡 𝑒−𝑟𝜏−𝐷(𝑇 −𝜏)+(𝑟−𝐷)(𝜏−𝑡) ) ( √ √ ∞ 𝑚 + 𝑥𝜎 𝜏 − 𝑡 1 − 1 (𝑥−𝜎 𝜏−𝑡)2 × Φ 𝑑𝑥 √ 𝑒 2 √ ∫−∞ 𝜎 𝑇 −𝜏 2𝜋

where 𝑚 = log(𝑆𝑡 ∕𝐾) +√(𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝜏) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝜏 − 𝑡). By setting 𝑦 = 𝑥 − 𝜎 𝜏 − 𝑡 we have 𝐼1 = 𝑆𝑡 𝑒−𝑟𝜏−𝐷(𝑇 −𝜏)+(𝑟−𝐷)(𝜏−𝑡) ( ) √ ∞ 𝑚 + 𝜎 2 (𝜏 − 𝑡) + 𝑦𝜎 𝜏 − 𝑡 1 − 1 𝑦2 × Φ √ 𝑒 2 𝑑𝑦 √ ∫−∞ 𝜎 𝑇 −𝜏 2𝜋 or

𝐼1 = 𝑆𝑡 𝑒

(

∞ −𝑟𝜏−𝐷(𝑇 −𝑡)

∫−∞

Φ

𝑑+ + 𝜌𝑦 √ 1 − 𝜌2

) 1 − 1 𝑦2 √ 𝑒 2 𝑑𝑦 2𝜋

√

𝜏 −𝑡 . From Problem 1.2.2.14 of Problems and Solutions in Mathemat𝑇 −𝑡 ical Finance, Volume 1: Stochastic Calculus we can deduce where 𝜌 =

𝐼1 = 𝑆𝑡 𝑒−𝑟𝑡−𝐷(𝑇 −𝑡) 𝚽(∞, 𝑑+ , −𝜌) = 𝑆𝑡 𝑒−𝑟𝑡−𝐷(𝑇 −𝑡) Φ(𝑑+ )

log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) where 𝑑+ = . √ 𝜎 𝑇 −𝑡 In contrast, for [ ] 𝐼2 = 𝐾𝑒−𝑟𝑇 𝔼ℚ Φ(𝑑−𝜏 )|| ℱ𝑡 ) ( ∞ log(𝑆𝜏 ∕𝐾) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝜏) −𝑟𝑇 = 𝐾𝑒 𝑓 (𝑆𝜏 |𝑆𝑡 ) 𝑑𝑆𝜏 Φ √ ∫0 𝜎 𝑇 −𝜏

108

2.2.2 Black–Scholes Model

log(𝑆𝜏 ∕𝑆𝑡 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝜏 − 𝑡) we have and by setting 𝑥 = √ 𝜎 𝜏 −𝑡 𝐼2 = 𝐾𝑒−𝑟𝑇

√ ⎛ log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 − 1 𝜎 2 )(𝑇 − 𝑡) + 𝑥𝜎 𝜏 − 𝑡 ⎞ 1 2 ⎟ √1 𝑒− 2 𝑥2 𝑑𝑥 × Φ⎜ √ ∫−∞ ⎜ ⎟ 2𝜋 𝜎 𝑇 −𝜏 ⎝ ⎠ ( ) ∞ 𝑑 + 𝜌𝑥 1 − 1 𝑥2 Φ √− = 𝐾𝑒−𝑟𝑇 √ 𝑒 2 𝑑𝑥 ∫−∞ 1 − 𝜌2 2𝜋 ∞

= 𝐾𝑒−𝑟𝑇 𝚽(∞, 𝑑− , −𝜌) = 𝐾𝑒−𝑟𝑇 Φ(𝑑− ) √ where 𝜌 = Therefore,

log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) 𝜏 −𝑡 . and 𝑑− = √ 𝑇 −𝑡 𝜎 𝑇 −𝑡

[ ] [ ] 𝔼ℚ 𝑒−𝑟𝜏 𝐶(𝑆𝜏 , 𝜏; 𝐾, 𝑇 )|| ℱ𝑡 = 𝑒−𝑟𝑡 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑− ) = 𝑒−𝑟𝑡 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). (b) Since ||𝑒−𝑟𝑡 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 )|| = 𝑒−𝑟𝑡 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and because 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) < 𝑆𝑡 < ∞ therefore [ [ ] ] 𝔼ℚ ||𝑒−𝑟𝑡 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 )|| = 𝔼ℚ 𝑒−𝑟𝑡 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) < 𝑒−𝑟𝑡 𝑆𝑡 < ∞. (c) Given that 𝑒−𝑟𝑡 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is a function of 𝑆𝑡 , hence it is ℱ𝑡 -adapted. From the results of (a) – (c) we have shown that 𝑒−𝑟𝑡 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is a ℚ-martingale. N.B. Using the same steps we can also show that 𝑒−𝑟𝑡 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is a ℚ-martingale where 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is the European put option price with strike 𝐾 and expiry time 𝑇 where the stock pays a continuous dividend yield. 7. Invariance Property I. Let the price of a European option 𝑉 (𝑆𝑡 , 𝑡) satisfy the Black– Scholes equation 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 where 𝑆𝑡 is the spot price of a stock at time 𝑡, 𝜎 is the stock volatility, 𝑟 is the constant risk-free interest rate, 𝐷 is the continuous dividend yield and 𝑇 is the option expiry time. Show that 𝑉 (𝜆𝑆𝑡 , 𝑡) is also a solution of the Black–Scholes equation for 𝜆 > 0. Give a financial interpretation of this property.

2.2.2 Black–Scholes Model

109

Solution: Let 𝑊 (𝑆𝑡 , 𝑡) = 𝑉 (𝜆𝑆𝑡 , 𝑡) and by setting 𝑆̂𝑡 = 𝜆𝑆𝑡 we have 𝜕𝑉 𝜕 𝑆̂𝑡 𝜕𝑉 𝜕𝑉 𝜕𝑊 𝜕𝑊 = =𝜆 = , 𝜕𝑡 𝜕𝑡 𝜕𝑆𝑡 𝜕 𝑆̂𝑡 𝜕𝑆𝑡 𝜕 𝑆̂𝑡 𝜕 𝜕2𝑊 = 2 𝜕𝑆 𝜕𝑆𝑡 𝑡

(

𝜕𝑊 𝜕𝑆𝑡

)

𝜕 = 𝜕𝑆𝑡

(

𝜕𝑉 𝜆 𝜕 𝑆̂𝑡

) =𝜆

𝜕2𝑉 𝜕 2 𝑉 𝜕 𝑆̂𝑡 = 𝜆2 . 𝜕 𝑆̂𝑡2 𝜕𝑆𝑡 𝜕 𝑆̂𝑡2

By substituting the above expressions into 𝜕2𝑊 𝜕𝑊 1 𝜕𝑊 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑊 (𝑆𝑡 , 𝑡) + 𝜎 2 𝑆𝑡2 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 we have 𝜕2 𝑊 𝜕𝑊 1 𝜕𝑊 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑊 (𝑆𝑡 , 𝑡) + 𝜎 2 𝑆𝑡2 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 =

𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 + (𝑟 − 𝐷)𝜆𝑆𝑡 − 𝑟𝑉 (𝑆̂𝑡 , 𝑡) + 𝜎 2 𝜆2 𝑆𝑡2 𝜕𝑡 2 𝜕 𝑆̂ 2 𝜕 𝑆̂𝑡 𝑡

𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 + (𝑟 − 𝐷)𝑆̂𝑡 − 𝑟𝑉 (𝑆̂𝑡 , 𝑡) + 𝜎 2 𝑆̂𝑡2 = 2 ̂ 𝜕𝑡 2 𝜕𝑆 𝜕 𝑆̂𝑡 𝑡

= 0. Thus, the Black–Scholes equation is invariant under the change of variable 𝑆̂𝑡 = 𝜆𝑆𝑡 for 𝜆 > 0. Given that both 𝑉 (𝑆𝑡 , 𝑡) and 𝑉 (𝜆𝑆𝑡 , 𝑡), 𝜆 > 0 satisfy the Black–Scholes equation, the price of a European option scales with the underlying stock price. 8. Invariance Property II. Let 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) denote the Black–Scholes formula at time 𝑡 for a European option with expiry 𝑇 , 𝑇 > 𝑡, strike 𝐾, given 𝑆𝑡 is the spot price of a stock at time 𝑡, 𝜎 is the stock volatility, 𝐷 is the continuous dividend yield, and let 𝑟 be the constant risk-free interest rate. Prove that for any constant 𝛼 > 0, 𝑉𝑏𝑠 (𝛼𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝛼𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾∕𝛼, 𝑇 ). Solution: By definition 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝛿𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝛿𝑑+ ) − 𝛿𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝛿𝑑− ) log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝑡) 𝑑± = √ 𝜎 𝑇 −𝑡 { +1 for European call option 𝛿= −1 for European put option.

110

2.2.2 Black–Scholes Model

For any constant 𝛼 > 0, 𝑉𝑏𝑠 (𝛼𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝛼𝛿𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝛿𝑑+ ) − 𝛿𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝛿𝑑− ) 𝑑± =

=

log(𝛼𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡 log(𝑆𝑡 ∕(𝐾∕𝛼)) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝑡) . √ 𝜎 𝑇 −𝑡

Thus, [ ] 𝑉𝑏𝑠 (𝛼𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝛼 𝛿𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝛿𝑑+ ) − 𝛿(𝐾∕𝛼)𝑒−𝑟(𝑇 −𝑡) Φ(𝛿𝑑− ) = 𝛼𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾∕𝛼, 𝑇 ). N.B. For example in a one-for-two stock split (holder of one share before the split will hold two shares after the split), with the introduction of new shares the value of the stock price and the strike price will be scaled by half. Thus, the value of the option prices will also be halved. 9. Higher Derivatives Property. We consider the value of a European option 𝑉 (𝑆𝑡 , 𝑡) satisfying the following Black–Scholes equation 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 where 𝑆𝑡 is the spot price of a stock at time 𝑡, 𝜎 is the stock volatility, 𝑟 is the constant risk-free interest rate, 𝐷 is the continuous dividend yield and 𝑇 > 𝑡 is the option expiry time. Using mathematical induction show that 𝑊 (𝑛) (𝑆𝑡 , 𝑡) = 𝑆𝑡𝑛

𝜕𝑛𝑉 𝜕𝑆𝑡𝑛

also satisfy the Black–Scholes equation for any 𝑛 = 1, 2, … 𝜕𝑛𝑉 , 𝑛 = 1, 2, … 𝜕𝑆𝑡𝑛 𝜕𝑉 and by differentiation For 𝑛 = 1 we have 𝑊 (1) (𝑆𝑡 , 𝑡) = 𝑆𝑡 𝜕𝑆𝑡 Solution: Let 𝑊 (𝑛) (𝑆𝑡 , 𝑡) = 𝑆𝑡𝑛

𝜕 𝜕𝑊 (1) = 𝑆𝑡 𝜕𝑡 𝜕𝑡

(

𝜕𝑉 𝜕𝑆𝑡

) ,

𝜕𝑊 (1) 𝜕 = 𝑆𝑡 𝑆𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡

(

𝜕𝑉 𝑆𝑡 𝜕𝑆𝑡

(

) = 𝑆𝑡

𝜕2𝑉 𝜕𝑉 + 𝑆𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡

)

2.2.2 Black–Scholes Model

111

and 𝜕 𝑆𝑡2

2 𝑊 (1)

𝜕𝑆𝑡2

[ =

𝑆𝑡2

𝜕 𝜕𝑆𝑡

(

𝜕2𝑉 𝜕𝑉 + 𝑆𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡

)]

𝜕2𝑉 𝜕3𝑉 + 𝑆𝑡3 3 2 𝜕𝑆𝑡 𝜕𝑆𝑡 ( ) 2𝑉 𝜕 𝜕 = 𝑆𝑡 𝑆𝑡2 2 . 𝜕𝑆𝑡 𝜕𝑆𝑡

= 2𝑆𝑡2

Substituting

𝜕𝑊 (1) 𝜕𝑊 (1) 𝜕 2 𝑊 (1) and 𝑆𝑡2 into , 𝑆𝑡 𝜕𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡2 𝜕𝑊 (1) 1 2 2 𝜕 2 𝑊 (1) 𝜕𝑊 (1) + (𝑟 − 𝐷)𝑆 − 𝑟𝑊 (1) (𝑆𝑡 , 𝑡) + 𝜎 𝑆𝑡 𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

we have 𝜕𝑊 (1) 𝜕𝑊 (1) 1 2 2 𝜕 2 𝑊 (1) + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑊 (1) (𝑆𝑡 , 𝑡) + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 ( ) ( ) ( ) 2 𝜕 𝜕𝑉 𝜕 𝜕 𝜕𝑉 𝜕𝑉 1 2 2𝜕 𝑉 = 𝑆𝑡 𝑆𝑡 + (𝑟 − 𝐷)𝑆 + 𝜎 𝑆𝑡 𝑆 − 𝑟𝑆𝑡 𝑡 𝑡 2 𝜕𝑡 𝜕𝑆𝑡 2 𝜕𝑆𝑡 𝜕𝑆 𝜕𝑆 𝜕𝑆 𝜕𝑆𝑡 𝑡 𝑡 𝑡 ) ( 𝜕 𝜕2 𝑉 𝜕𝑉 1 𝜕𝑉 = 𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑡) + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 = 0. 𝜕𝑉 is a solution of the Black–Scholes equation. 𝜕𝑆𝑡 𝜕𝑛𝑉 Assume that 𝑊 (𝑛) (𝑆𝑡 , 𝑡) = 𝑆𝑡𝑛 𝑛 is also a solution of the Black–Scholes equation such 𝜕𝑆𝑡 that

Hence, 𝑊 (1) (𝑆𝑡 , 𝑡) = 𝑆𝑡

𝜕 𝑆𝑡𝑛 𝑛 𝜕𝑆𝑡

(

) 𝜕𝑉 1 2 2 𝜕2𝑉 𝜕𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0. + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

For the case of 𝑊 (𝑛+1) (𝑆𝑡 , 𝑡) = 𝑆𝑡𝑛+1

𝜕 𝑛+1 𝑉 we first differentiate 𝜕𝑆𝑡𝑛+1

)] [ ( 𝜕 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 𝑛 𝜕 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑡) =0 𝑆𝑡 + 𝜎 𝑆𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡𝑛 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

112

2.2.2 Black–Scholes Model

and we then have 𝜕𝑛 𝑛𝑆𝑡𝑛−1 𝑛 𝜕𝑆𝑡 +𝑆 𝑛

𝜕 𝑛+1 𝜕𝑆𝑡𝑛+1

) 𝜕𝑉 1 2 2 𝜕2𝑉 𝜕𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑡) + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 ) ( 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑡) + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

(

= 0. By multiplying the above equation with 𝑆𝑡 we obtain ) 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑡) + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 ) ( 𝑛+1 𝜕𝑉 1 2 2 𝜕2𝑉 𝜕𝑉 𝑛+1 𝜕 +𝑆 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑡) + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 𝜕𝑆 𝑛+1 𝜕𝑛 𝑛𝑆𝑡𝑛 𝑛 𝜕𝑆𝑡

(

𝑡

=0 and since 𝜕 𝑆𝑡𝑛 𝑛 𝜕𝑆𝑡

(

) 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

we therefore have 𝑆

𝑛+1

𝜕 𝑛+1 𝜕𝑆𝑡𝑛+1

(

) 𝜕𝑉 1 2 2 𝜕2𝑉 𝜕𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0. + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

𝜕 𝑛+1 𝑉 is also a solution of the Black–Scholes equation. 𝜕𝑆𝑡𝑛+1 𝜕𝑛𝑉 Thus, from mathematical induction we have shown that 𝑆𝑡𝑛 𝑛 satisfies the Black– 𝜕𝑆𝑡 Scholes equation for 𝑛 = 1, 2, … Hence, 𝑊 (𝑛+1) (𝑆𝑡 , 𝑡) = 𝑆𝑡𝑛+1

10. Let the Black–Scholes equation for the price of a European option 𝑉 (𝑆𝑡 , 𝑡) on an underlying asset priced 𝑆𝑡 at time 𝑡 be 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 where 𝜎 is the constant volatility, 𝑟 is the constant risk-free interest rate and 𝐷 is the constant dividend yield.

2.2.2 Black–Scholes Model

113

Show that if 𝑉 (𝑆𝑡 , 𝑡) = 𝑓 (𝑆𝑡 )𝑔(𝑡) then 𝑔 ′ (𝑡) =− 𝑔(𝑡)

[1

𝜎 2 𝑆𝑡2 𝑓 ′′ (𝑆𝑡 ) + (𝑟 − 𝐷)𝑆𝑡 𝑓 ′ (𝑆𝑡 ) − 𝑟𝑓 (𝑆𝑡 ) 2

] .

𝑓 (𝑆𝑡 )

Explain why 𝑔 and 𝑓 satisfy the following ordinary differential equations (ODEs) 𝑔 ′ (𝑡) − 𝜆𝑔(𝑡) = 0 and 1 2 2 ′′ 𝜎 𝑆𝑡 𝑓 (𝑆𝑡 ) + (𝑟 − 𝐷)𝑆𝑡 𝑓 ′ (𝑆𝑡 ) − (𝑟 − 𝜆)𝑓 (𝑆𝑡 ) = 0 2 for a constant 𝜆. Hence, conditional on 𝜆 find all solutions of the Black–Scholes equation. Solution: If 𝑉 (𝑆𝑡 , 𝑡) can be separated as a product 𝑉 (𝑆𝑡 , 𝑡) = 𝑓 (𝑆𝑡 )𝑔(𝑡) then 𝜕𝑉 = 𝑓 (𝑆𝑡 )𝑔 ′ (𝑡), 𝜕𝑡

𝜕𝑉 = 𝑓 ′ (𝑆𝑡 )𝑔(𝑡) 𝜕𝑆𝑡

and

𝜕2𝑉 = 𝑓 ′′ (𝑆𝑡 )𝑔(𝑡) 𝜕𝑆𝑡2

and by substituting them into the Black–Scholes equation we have 𝜕2 𝑉 𝜕𝑉 𝜕𝑉 1 − 𝑟𝑉 (𝑆𝑡 , 𝑡) + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 1 = 𝑓 (𝑆𝑡 )𝑔 ′ (𝑡) + 𝜎 2 𝑆𝑡2 𝑓 ′′ (𝑆𝑡 )𝑔(𝑡) + (𝑟 − 𝐷)𝑆𝑡 𝑓 ′ (𝑆𝑡 )𝑔(𝑡) − 𝑟𝑓 (𝑆𝑡 )𝑔(𝑡) 2 [ ] 1 = 𝑓 (𝑆𝑡 )𝑔 ′ (𝑡) + 𝑔(𝑡) 𝜎 2 𝑆𝑡2 𝑓 ′′ (𝑆𝑡 ) + (𝑟 − 𝐷)𝑆𝑡 𝑓 ′ (𝑆𝑡 ) − 𝑟𝑓 (𝑆𝑡 ) 2 = 0. Hence, 𝑔 ′ (𝑡) =− 𝑔(𝑡)

[1 2

𝜎 2 𝑆𝑡2 𝑓 ′′ (𝑆𝑡 ) + (𝑟 − 𝐷)𝑆𝑡 𝑓 ′ (𝑆𝑡 ) − 𝑟𝑓 (𝑆𝑡 ) 𝑓 (𝑆𝑡 )

] .

Given that the left-hand side of the equation is a function of 𝑡 whilst the right-hand side is a function of 𝑆𝑡 , both equations must be equal to a constant 𝜆. Thus, 𝑔 ′ (𝑡) − 𝜆𝑔(𝑡) = 0

114

2.2.2 Black–Scholes Model

and 1 2 2 ′′ 𝜎 𝑆𝑡 𝑓 (𝑆𝑡 ) + (𝑟 − 𝐷)𝑆𝑡 𝑓 ′ (𝑆𝑡 ) − (𝑟 − 𝜆) 𝑓 (𝑆𝑡 ) = 0. 2 For the first-order ODE 𝑔 ′ (𝑡) − 𝜆𝑔(𝑡) = 0, by setting the integrating factor 𝐼 = 𝑒− ∫

𝜆𝑑𝑡

= 𝑒−𝜆𝑡

we have ) 𝑑 ( −𝜆𝑡 𝑒 𝑔(𝑡) = 0 𝑑𝑡 or 𝑔(𝑡) = 𝐶𝑔 𝑒𝜆𝑡 where 𝐶𝑔 is a constant value. As for solving 12 𝜎 2 𝑆𝑡2 𝑓 ′′ (𝑆𝑡 ) + (𝑟 − 𝐷)𝑆𝑡 𝑓 ′ (𝑆𝑡 ) − (𝑟 − 𝜆)𝑓 (𝑆𝑡 ) = 0, which is a secondorder ODE, we let 𝑓 (𝑆𝑡 ) = 𝐶𝑆𝑡𝑚 where 𝐶 is a constant. By substituting 𝑓 (𝑆𝑡 ) = 𝐶𝑆𝑡𝑚 , 𝑓 ′ (𝑆𝑡 ) = 𝑚𝐶𝑆𝑡𝑚−1

and

𝑓 ′′ (𝑆𝑡 ) = 𝑚(𝑚 − 1)𝐶𝑆𝑡𝑚−2

we eventually have 1 1 2 2 𝜎 𝑚 + (𝑟 − 𝐷 − 𝜎 2 )𝑚 − (𝑟 − 𝜆) = 0. 2 2 Hence, by solving the above quadratic equation

𝑚=

−(𝑟 − 𝐷 − 12 𝜎 2 ) ±

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 (𝑟 − 𝜆) 𝜎2

where we need to consider three cases of roots of 𝑚. 1 1 First, if 𝜆 < 𝑟 + 2 (𝑟 − 𝐷 − 𝜎 2 )2 then the solution of the ODE is of the form 2 2𝜎 𝑚+

𝑓 (𝑆𝑡 ) = 𝐴𝑓 𝑆𝑡

𝑚−

+ 𝐵𝑓 𝑆𝑡

where

𝑚+ =

𝑚− =

−(𝑟 − 𝐷 − 12 𝜎 2 ) +

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 (𝑟 − 𝜆) 𝜎2

−(𝑟 − 𝐷 − 12 𝜎 2 ) −

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 (𝑟 − 𝜆) 𝜎2

>0

𝑟 +

1 1 (𝑟 − 𝐷 − 𝜎 2 )2 then we have complex roots for 𝑚 so that 2 2𝜎 2 𝑓 (𝑆𝑡 ) = 𝐾1 𝑆𝑡𝛼+𝑖𝛽 + 𝐾2 𝑆𝑡𝛼−𝑖𝛽

√ 1 1 2 1 1 where 𝛼 = − 2 (𝑟 − 𝐷 − 𝜎 ), 𝛽 = 2 −(𝑟 − 𝐷 − 𝜎 2 )2 − 2𝜎 2 (𝑟 − 𝜆) and 𝐾1 , 𝐾2 are 2 2 𝜎 𝜎 constants. From the identity 𝑒𝑖𝜃 = cos 𝜃 + 𝑖 sin 𝜃 we can write the solution of the second-order ODE as ) ) ( ( 𝑓 (𝑆𝑡 ) = 𝐶𝑓(1) 𝑆𝑡𝛼 cos 𝛽 log 𝑆𝑡 + 𝐶𝑓(2) 𝑆𝑡𝛼 sin 𝛽 log 𝑆𝑡 where 𝐶𝑓(1) and 𝐶𝑓(2) are constants. Therefore, ) ) ( ( 𝑉 (𝑆𝑡 , 𝑡) = 𝑓 (𝑆𝑡 )𝑔(𝑡) = 𝐶1 𝑒𝜆𝑡 𝑆𝑡𝛼 cos 𝛽 log 𝑆𝑡 + 𝐶2 𝑒𝜆𝑡 𝑆𝑡𝛼 sin 𝛽 log 𝑆𝑡 where 𝐶1 = 𝐶𝑓(1) 𝐶𝑔 and 𝐶2 = 𝐶𝑓(2) 𝐶𝑔 . 11. Discrete Dividends I – Escrowed Dividend Model. We consider at time 𝑡 a European option 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) with strike 𝐾 written on a stock priced at 𝑆𝑡 paying a dividend 𝛿 at time 𝑡𝛿 , 𝑡𝛿 < 𝑇 where 𝑇 > 𝑡 is the option expiry time. Using no-arbitrage arguments explain why 𝑆𝑡+ = 𝑆𝑡− − 𝛿, 𝛿

𝛿

0 ≤ 𝛿 ≤ 𝑆 𝑡− 𝛿

+ − 𝑉 (𝑆𝑡− , 𝑡− 𝛿 ; 𝐾, 𝑇 ) = 𝑉 (𝑆𝑡 − 𝛿, 𝑡𝛿 ; 𝐾, 𝑇 ) 𝛿

𝛿

where 𝑡− and 𝑡+ are the time immediately before and after the dividend is paid, respectively. 𝛿 𝛿

116

2.2.2 Black–Scholes Model

Hence, deduce that ⎧𝑉 (𝑆 − 𝛿𝑒−𝑟(𝑡𝛿 −𝑡) , 𝑡; 𝐾, 𝑇 ) ⎪ 𝑏𝑠 𝑡 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ ⎪𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎩

if 0 < 𝑡 < 𝑡𝛿 if 𝑡𝛿 < 𝑡 ≤ 𝑇

where 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is a European option with strike 𝐾 and expiry time 𝑇 written on a non-dividend-paying stock 𝑆𝑡 satisfying the Black–Scholes equation 𝜕𝑉 𝜕𝑉𝑏𝑠 1 2 2 𝜕 2 𝑉𝑏𝑠 + 𝑟𝑆𝑡 𝑏𝑠 − 𝑟𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 ⎧max{𝑆 − 𝐾, 0} if option is a call 𝑇 ⎪ 𝑉𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = ⎨ ⎪max{𝐾 − 𝑆𝑇 , 0} if option is a put ⎩ where 𝜎 is the stock volatility and 𝑟 is the risk-free interest rate. Finally, deduce that if there are 𝑛 > 1 dividends 𝛿1 , 𝛿2 , … , 𝛿𝑛 to be paid by the stock at time 𝑡𝛿1 < 𝑡𝛿2 < ⋯ < 𝑡𝛿𝑛 , respectively where 𝑡 < 𝑡𝛿1 and 𝑡𝛿𝑛 < 𝑇 then ) ⎧ ( 𝑛 ∑ −𝑟(𝑡𝛿𝑖 −𝑡) ⎪𝑉 𝛿𝑖 𝑒 , 𝑡; 𝐾, 𝑇 if 0 ≤ 𝑡 < 𝑡𝛿1 < 𝑡𝛿2 < ⋯ < 𝑡𝛿𝑛 𝑆𝑡 − ⎪ 𝑏𝑠 𝑖=1 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ ⎪ ⎪𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) if 𝑡𝛿𝑛 < 𝑡 ≤ 𝑇 . ⎩ Solution: At time 𝑡𝛿 the stock pays out a discrete dividend 𝛿 and to ensure there is no arbitrage opportunity we need to show 𝑆𝑡+ = 𝑆𝑡− − 𝛿. 𝛿

𝛿

Assume that 𝑆𝑡+ < 𝑆𝑡− − 𝛿 then we can sell the stock at 𝑡 = 𝑡− , forfeiting the dividend at 𝛿 𝛿

𝛿

. Hence, the profit is time 𝑡 = 𝑡𝛿 , and buy back the stock at time 𝑡+ 𝛿 Profit = 𝑆𝑡− − 𝛿 − 𝑆𝑡+ > 0 𝛿

𝛿

which is an arbitrage opportunity. , collecting the In contrast, if 𝑆𝑡+ > 𝑆𝑡− − 𝛿 then the strategy is to buy the stock at 𝑡 = 𝑡− 𝛿 𝛿 𝛿 dividend at time 𝑡 = 𝑡𝛿 and selling it afterwards. Thus, the profit is Profit = −𝑆𝑡− + 𝛿 + 𝑆𝑡+ > 0 𝛿

𝛿

which is also an arbitrage opportunity. Hence, 𝑆𝑡+ = 𝑆𝑡− − 𝛿 and since 𝑆𝑡+ ≥ 0 therefore 𝛿 𝛿 𝛿 0 ≤ 𝛿 ≤ 𝑆 𝑡− . 𝛿

2.2.2 Black–Scholes Model

117

Given that the option holder does not receive the dividend, the option value cannot jump across time 𝑡𝛿 . Therefore, + + 𝑉 (𝑆𝑡− , 𝑡− 𝛿 ; 𝐾, 𝑇 ) = 𝑉 (𝑆𝑡 , 𝑡𝛿 ; 𝐾, 𝑇 ). 𝛿

𝛿

That is, the value of the option is the same immediately before the dividend date as well as immediately after the dividend date. Thus, the option price remains continuous in 𝑆𝑡𝛿 even though 𝑆𝑡𝛿 is not continuous. Since 𝑆𝑡+ = 𝑆𝑡− − 𝛿 therefore 𝛿

𝛿

+ − 𝑉 (𝑆𝑡− , 𝑡− 𝛿 ; 𝐾, 𝑇 ) = 𝑉 (𝑆𝑡 − 𝛿, 𝑡𝛿 ; 𝐾, 𝑇 ). 𝛿

𝛿

If the option is priced at time 0 ≤ 𝑡 < 𝑡𝛿 then at expiry 𝑇 𝑉 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑇 − 𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) , 𝑇 ; 𝐾, 𝑇 ) since 𝛿 being paid at time 𝑡𝛿 will grow at a risk-free interest rate 𝑟. In contrast, if the option is priced at 𝑡 > 𝑡𝛿 and because no dividends are paid thereafter, the option payoff at time 𝑇 is 𝑉 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ). Therefore, by discounting the option payoff back to time 𝑡 we can write ⎧𝑉 (𝑆 − 𝛿𝑒−𝑟(𝑡𝛿 −𝑡) , 𝑡; 𝐾, 𝑇 ) ⎪ 𝑏𝑠 𝑡 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ ⎪𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎩

if 0 ≤ 𝑡 < 𝑡𝛿 if 𝑡𝛿 < 𝑡 ≤ 𝑇 .

Finally, for the case when we have 𝑛 > 1 dividends, we use an iterative method to prove the desired results. For the case when 𝑡𝛿𝑛 < 𝑡 < 𝑇 and because there are no more dividends being paid, we have 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). For 𝑡 < 𝑡𝛿𝑛 < 𝑇 and given that a dividend 𝛿𝑛 is paid at time 𝑡𝛿𝑛 , then 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 − 𝛿𝑛 𝑒−𝑟(𝑡𝛿𝑛 −𝑡) , 𝑡; 𝐾, 𝑇 ) with spot price reset to 𝑆̃𝑡 = 𝑆𝑡 − 𝛿𝑛 𝑒−𝑟(𝑡𝛿𝑛 −𝑡) . When 𝑡 < 𝑡𝛿𝑛−1 < 𝑡𝛿𝑛 < 𝑇 then, following the same arguments as before 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑉𝑏𝑠 (𝑆̃𝑡 − 𝛿𝑛−1 𝑒−𝑟(𝑡𝛿𝑛−1 −𝑡) , 𝑡; 𝐾, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 − 𝛿𝑛−1 𝑒−𝑟(𝑡𝛿𝑛−1 −𝑡) − 𝛿𝑛 𝑒−𝑟(𝑡𝛿𝑛 −𝑡) , 𝑡; 𝐾, 𝑇 ) with spot price now reset to 𝑆̃𝑡 = 𝑆𝑡 − 𝛿𝑛−1 𝑒−𝑟(𝑡𝛿𝑛−1 −𝑡) − 𝛿𝑛 𝑒−𝑟(𝑡𝛿𝑛 −𝑡) .

118

2.2.2 Black–Scholes Model

Hence, for the case when 𝑡 < 𝑡𝛿1 < 𝑡𝛿2 < ⋯ < 𝑡𝛿𝑛 < 𝑇 we can deduce that 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑉𝑏𝑠 (𝑆̃𝑡 − 𝛿1 𝑒−𝑟(𝑡𝛿𝑛−1 −𝑡) , 𝑡; 𝐾, 𝑇 ) where 𝑆̃𝑡 = 𝑆𝑡 −

𝑛 ∑ 𝑖=2

𝛿𝑖 𝑒−𝑟(𝑡𝛿𝑖 −𝑡) . Thus,

) ⎧ ( 𝑛 ∑ −𝑟(𝑡𝛿𝑖 −𝑡) ⎪𝑉 𝛿𝑖 𝑒 , 𝑡; 𝐾, 𝑇 if 0 ≤ 𝑡 < 𝑡𝛿1 < 𝑡𝛿2 < ⋯ < 𝑡𝛿𝑛 𝑆𝑡 − ⎪ 𝑏𝑠 𝑖=1 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ ⎪ ⎪𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) if 𝑡𝛿𝑛 < 𝑡 ≤ 𝑇 . ⎩

12. Discrete Dividends II – Forward Dividend Model. We consider at time 𝑡 a European option 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) with strike 𝐾 written on a stock 𝑆𝑡 paying a dividend 𝛿 at time 𝑡𝛿 , 𝑡𝛿 < 𝑇 where 𝑇 > 𝑡 is the option expiry time. Using no-arbitrage arguments explain why 𝑆𝑡+ = 𝑆𝑡− − 𝛿, 𝛿

𝛿

0 ≤ 𝛿 ≤ 𝑆 𝑡− 𝛿

+ − 𝑉 (𝑆𝑡− , 𝑡− 𝛿 ; 𝐾, 𝑇 ) = 𝑉 (𝑆𝑡 − 𝛿, 𝑡𝛿 ; 𝐾, 𝑇 ) 𝛿

𝛿

and 𝑡+ are the time immediately before and after the dividend is paid, respectively. where 𝑡− 𝛿 𝛿 Hence, deduce that ⎧𝑉 (𝑆 , 𝑡; 𝐾 + 𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) , 𝑇 ) if 0 < 𝑡 < 𝑡 𝛿 ⎪ 𝑏𝑠 𝑡 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ ⎪𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) if 𝑡𝛿 < 𝑡 ≤ 𝑇 ⎩ where 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is a European option with strike 𝐾 and expiry time 𝑇 written on a non-dividend-paying stock 𝑆𝑡 satisfying the Black–Scholes equation 𝜕𝑉𝑏𝑠 1 2 2 𝜕 2 𝑉𝑏𝑠 𝜕𝑉𝑏𝑠 + 𝑟𝑆 − 𝑟𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 ⎧max{𝑆 − 𝐾, 0} if option is a call 𝑇 ⎪ 𝑉𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = ⎨ ⎪max{𝐾 − 𝑆𝑇 , 0} if option is a put ⎩ where 𝜎 is the stock volatility and 𝑟 is the risk-free interest rate.

2.2.2 Black–Scholes Model

119

Finally, deduce that if there are 𝑛 > 1 dividends 𝛿1 , 𝛿2 , … , 𝛿𝑛 to be paid by the stock at time 𝑡𝛿1 < 𝑡𝛿2 < ⋯ < 𝑡𝛿𝑛 , respectively where 𝑡 < 𝑡𝛿1 and 𝑡𝛿𝑛 < 𝑇 then ) ⎧ ( 𝑛 ∑ 𝑟(𝑇 −𝑡𝛿𝑖 ) ⎪𝑉 𝛿𝑖 𝑒 ,𝑇 if 0 ≤ 𝑡 < 𝑡𝛿1 < 𝑡𝛿2 < ⋯ < 𝑡𝛿𝑛 𝑆𝑡 , 𝑡; 𝐾 + ⎪ 𝑏𝑠 𝑖=1 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ ⎪ ⎪𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) if 𝑡𝛿𝑛 < 𝑡 ≤ 𝑇 . ⎩ Solution: See Problem 2.2.2.11 (page 115) for the first part of the solutions. Using the same arguments as in Problem 2.2.2.11 (page 115) we note that if the option is priced at time 0 ≤ 𝑡 < 𝑡𝛿 then at expiry time 𝑇 𝑉 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑇 − 𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) , 𝑇 ; 𝐾, 𝑇 ) ⎧max {𝑆 − 𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) − 𝐾, 0} for a European call option 𝑇 ⎪ =⎨ } { ⎪max 𝐾 − 𝑆𝑇 + 𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) , 0 for a European put option ⎩ ⎧max {𝑆 − (𝐾 + 𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) ) , 0} for a European call option 𝑇 ⎪ =⎨ ) {( } ⎪max 𝐾 + 𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) − 𝑆𝑇 , 0 for a European put option ⎩ = 𝑉𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾 + 𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) , 𝑇 ). By discounting the option payoff back to time 𝑡, we eventually have ⎧𝑉 (𝑆 , 𝑡; 𝐾 + 𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) , 𝑇 ) if 0 < 𝑡 < 𝑡 𝛿 ⎪ 𝑏𝑠 𝑡 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ ⎪𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) if 𝑡𝛿 < 𝑡 ≤ 𝑇 . ⎩ Following the same iterative analysis as presented in Problem 2.2.2.11 (page 115) we can easily deduce that for 𝑛 > 1 number of dividends the option price is ) ⎧ ( 𝑛 ∑ 𝑟(𝑇 −𝑡 ) ⎪𝑉 𝛿𝑖 , 𝑇 𝛿𝑖 𝑒 if 0 ≤ 𝑡 < 𝑡𝛿1 < 𝑡𝛿2 < ⋯ < 𝑡𝛿𝑛 𝑆𝑡 , 𝑡; 𝐾 + ⎪ 𝑏𝑠 𝑖=1 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ ⎪ ⎪𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) if 𝑡𝛿𝑛 < 𝑡 ≤ 𝑇 . ⎩

120

2.2.2 Black–Scholes Model

13. Discrete Dividends III – Bos–Vandermark Model. We consider at time 𝑡 a European option 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) with strike 𝐾 written on a stock 𝑆𝑡 paying a dividend 𝛿 at time 𝑡𝛿 , 𝑡𝛿 < 𝑇 where 𝑇 > 𝑡 is the option expiry time. Using no-arbitrage arguments explain why 𝑆𝑡+ = 𝑆𝑡− − 𝛿, 𝛿

𝛿

0 ≤ 𝛿 ≤ 𝑆 𝑡− 𝛿

+ − 𝑉 (𝑆𝑡− , 𝑡− 𝛿 ; 𝐾, 𝑇 ) = 𝑉 (𝑆𝑡 − 𝛿, 𝑡𝛿 ; 𝐾, 𝑇 ) 𝛿

𝛿

and 𝑡+ are the time immediately before and after the dividend is paid, respectively. where 𝑡− 𝛿 𝛿 By setting 𝜆𝑡 ∈ [0, 1], deduce that ⎧𝑉 (𝑆 − 𝜆 𝛿𝑒−𝑟(𝑡𝛿 −𝑡) , 𝑡; 𝐾 + (1 − 𝜆 )𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) , 𝑇 ) if 0 < 𝑡 < 𝑡 𝑡 𝑡 𝛿 ⎪ 𝑏𝑠 𝑡 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ ⎪𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) if 𝑡𝛿 < 𝑡 ≤ 𝑇 ⎩ where 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is a European option with strike 𝐾 and expiry time 𝑇 written on a non-dividend-paying stock 𝑆𝑡 satisfying the Black–Scholes equation 𝜕𝑉𝑏𝑠 1 2 2 𝜕 2 𝑉𝑏𝑠 𝜕𝑉 + 𝑟𝑆𝑡 𝑏𝑠 − 𝑟𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 ⎧max{𝑆 − 𝐾, 0} if option is a call 𝑇 ⎪ 𝑉𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = ⎨ ⎪max{𝐾 − 𝑆𝑇 , 0} if option is a put ⎩ where 𝜎 is the stock volatility and 𝑟 is the risk-free interest rate. Finally, find the option price if there are 𝑛 > 1 dividends 𝛿1 , 𝛿2 , … , 𝛿𝑛 to be paid by the stock at time 𝑡𝛿1 < 𝑡𝛿2 < ⋯ < 𝑡𝛿𝑛 , respectively where 𝑡 < 𝑡𝛿1 and 𝑡𝛿𝑛 < 𝑇 , and 𝜆(𝑖) 𝑡 ∈ [0, 1], 𝑖 = 1, 2, … , 𝑛. Solution: See Problem 2.2.2.11 (page 115) for the first part of the solutions. Using the same arguments as in Problems 2.2.2.11 (page 115) and 2.2.2.12 (page 118) we note that if the option is priced at time 0 ≤ 𝑡 < 𝑡𝛿 then at expiry time 𝑇 𝑉 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑇 − 𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) , 𝑇 ; 𝐾, 𝑇 ) ⎧max {𝑆 − 𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) − 𝐾, 0} for a call option 𝑇 ⎪ =⎨ } { ⎪max 𝐾 − 𝑆𝑇 + 𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) , 0 for a put option ⎩

2.2.2 Black–Scholes Model

121

⎧max {𝑆 − 𝜆 𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) − (1 − 𝜆 )𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) − 𝐾, 0} for a call option 𝑇 𝑡 𝑡 ⎪ =⎨ } { ⎪max 𝐾 − 𝑆𝑇 + 𝜆𝑡 𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) + (1 − 𝜆𝑡 )𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) , 0 for a put option ⎩ ⎧max {𝑆 − 𝜆 𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) − (𝐾 + (1 − 𝜆 )𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) ) , 0} 𝑇 𝑡 𝑡 ⎪ =⎨ ) ( ) } {( ⎪max 𝐾 + (1 − 𝜆𝑡 )𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) − 𝑆𝑇 − 𝜆𝑡 𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) , 0 ⎩

for a call option for a put option

= 𝑉𝑏𝑠 (𝑆𝑇 − 𝜆𝑡 𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) , 𝑇 ; 𝐾 + (1 − 𝜆𝑡 )𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) , 𝑇 ). By discounting the option payoff back to time 𝑡, we eventually have ⎧𝑉 (𝑆 − 𝜆 𝛿𝑒−𝑟(𝑡𝛿 −𝑡) , 𝑡; 𝐾 + (1 − 𝜆 )𝛿𝑒𝑟(𝑇 −𝑡𝛿 ) , 𝑇 ) if 0 < 𝑡 < 𝑡 𝑡 𝑡 𝛿 ⎪ 𝑏𝑠 𝑡 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ ⎪𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) if 𝑡𝛿 < 𝑡 ≤ 𝑇 . ⎩ Following the same iterative analysis as presented in Problem 2.2.2.11 (page 115) we can easily deduce that for 𝑛 > 1 number of dividends and 𝜆(𝑖) 𝑡 ∈ [0, 1], 𝑖 = 1, 2, … , 𝑛, the option price is ( 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑉𝑏𝑠

𝑆𝑡 −

𝑛 ∑ 𝑖=1

−𝑟(𝑡𝛿𝑖 −𝑡) 𝜆(𝑖) , 𝑡; 𝐾 𝑡 𝛿𝑖 𝑒

𝑛 ∑ 𝑟(𝑇 −𝑡𝛿𝑖 ) + (1 − 𝜆(𝑖) ,𝑇 𝑡 )𝛿𝑖 𝑒

)

𝑖=1

if 0 ≤ 𝑡 < 𝑡𝛿1 < ⋯ < 𝑡𝛿𝑛 and 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) if 𝑡𝛿𝑛 < 𝑡 ≤ 𝑇 . 14. Discrete Dividend Yields. We consider at time 𝑡 a European option 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) with strike 𝐾 written on a stock 𝑆𝑡 paying a discrete dividend yield 𝐷 at time 𝑡𝐷 , 𝑡𝐷 < 𝑇 where 𝑇 is the option expiry time. Using no-arbitrage arguments explain why 𝑆𝑡+ = (1 − 𝐷)𝑆𝑡− , 𝐷

𝐷

0≤𝐷≤1

+ − 𝑉 (𝑆𝑡− , 𝑡− 𝐷 ; 𝐾, 𝑇 ) = 𝑉 ((1 − 𝐷)𝑆𝑡 , 𝑡𝐷 ; 𝐾, 𝑇 ) 𝐷

𝐷

and 𝑡+ are the times immediately before and after the dividend yield is paid, where 𝑡− 𝐷 𝐷 respectively.

122

2.2.2 Black–Scholes Model

Hence, deduce that ⎧𝑉 ((1 − 𝐷)𝑆 , 𝑡; 𝐾, 𝑇 ) 𝑡 ⎪ 𝑏𝑠 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ ⎪𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎩

if 0 ≤ 𝑡 < 𝑡𝐷 if 𝑡𝐷 < 𝑡 ≤ 𝑇

where 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is a European option with strike 𝐾 and expiry time 𝑇 written on a non-dividend-paying stock 𝑆𝑡 satisfying the Black–Scholes equation 𝜕𝑉𝑏𝑠 1 2 2 𝜕 2 𝑉𝑏𝑠 𝜕𝑉 + 𝑟𝑆𝑡 𝑏𝑠 − 𝑟𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 ⎧max{𝑆 − 𝐾, 0} if option is a call 𝑇 ⎪ 𝑉𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = ⎨ ⎪max{𝐾 − 𝑆𝑇 , 0} if option is a put ⎩ where 𝜎 is the stock volatility and 𝑟 is the risk-free interest rate. Finally, deduce if there are 𝑛 > 1 dividend yields 𝐷1 , 𝐷2 , … , 𝐷𝑛 to be paid by the stock at time 𝑡𝐷1 < 𝑡𝐷2 < ⋯ < 𝑡𝐷𝑛 , respectively where 𝑡 < 𝑡𝐷1 and 𝑡𝐷𝑛 < 𝑇 then ( ) 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑉𝑏𝑠 𝛽𝑛 𝑆𝑡 , 𝑡; 𝐾, 𝑇 where 𝛽𝑛 =

𝑛 ∏ 𝑖=1

(1 − 𝐷𝑖 ).

Solution: At time 𝑡𝐷 the stock pays out a discrete dividend 𝐷𝑆𝑡𝐷 and to ensure there is no arbitrage opportunity we need to show 𝑆𝑡+ = (1 − 𝐷)𝑆𝑡− . 𝐷

𝐷

, forfeiting the Assume that 𝑆𝑡+ < (1 − 𝐷)𝑆𝑡− then an investor can sell the stock at 𝑡 = 𝑡− 𝐷 𝐷

𝐷

dividend at time 𝑡 = 𝑡𝐷 , and buy back the stock at time 𝑡+ . Hence, the profit is 𝛿 Profit = 𝑆𝑡− − 𝐷𝑆𝑡− − 𝑆𝑡+ > 0 𝐷

𝐷

𝐷

which is an arbitrage opportunity. , collecting In contrast, if 𝑆𝑡+ > (1 − 𝐷)𝑆𝑡− then the investor can buy the stock at 𝑡 = 𝑡− 𝐷 𝐷 𝐷 the dividend at time 𝑡 = 𝑡𝐷 and selling it afterwards. Thus, the profit is Profit = −𝑆𝑡− + 𝐷𝑆𝑡− + 𝑆𝑡+ > 0 𝐷

𝐷

𝐷

which is also an arbitrage opportunity. Hence, 𝑆𝑡+ = (1 − 𝐷)𝑆𝑡− and since 𝑆𝑡+ ≥ 0 there𝐷 𝐷 𝐷 fore 0 ≤ 𝐷𝑆𝑡− ≤ 𝑆𝑡− or 0 ≤ 𝐷 ≤ 1. 𝐷

𝐷

2.2.2 Black–Scholes Model

123

Given that the option holder does not receive the dividend, the option value cannot jump across time 𝑡𝐷 . Therefore, + + 𝑉 (𝑆𝑡− , 𝑡− 𝐷 ; 𝐾, 𝑇 ) = 𝑉 (𝑆𝑡 , 𝑡𝐷 ; 𝐾, 𝑇 ). 𝐷

𝐷

That is, the value of the option is the same immediately before the dividend date as well as immediately after the dividend date. Hence, the option price remains continuous in 𝑆𝑡𝐷 even though 𝑆𝑡𝐷 is not continuous. Since 𝑆𝑡+ = (1 − 𝐷)𝑆𝑡− therefore 𝐷

𝐷

+ − 𝑉 (𝑆𝑡− , 𝑡− 𝐷 ; 𝐾, 𝑇 ) = 𝑉 ((1 − 𝐷)𝑆𝑡 , 𝑡𝐷 ; 𝐾, 𝑇 ). 𝐷

𝐷

If the option is priced at time 0 < 𝑡 < 𝑡𝐷 then at expiry 𝑇 we can set 𝑉 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 𝑉𝑏𝑠 ((1 − 𝐷)𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ). In contrast, if the option is priced at 𝑡 > 𝑡𝐷 and because no dividends are paid thereafter, the payoff at time 𝑇 is 𝑉 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ). Therefore, by discounting the option payoff back to time 𝑡 we can write ⎧𝑉 ((1 − 𝐷)𝑆 , 𝑡; 𝐾, 𝑇 ) 𝑡 ⎪ 𝑏𝑠 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ ⎪𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎩

if 0 ≤ 𝑡 < 𝑡𝛿 if 𝑡𝛿 < 𝑡 ≤ 𝑇 .

Finally, for the case when we have 𝑛 > 1 dividends, we use an iterative method to prove the desired results. For the case when 𝑡𝐷𝑛 < 𝑡 < 𝑇 and because there are no more dividends being paid, we have 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). For 𝑡 < 𝑡𝐷𝑛 < 𝑇 and given that a dividend 𝐷𝑛 is paid at time 𝑡𝐷𝑛 , then 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑉𝑏𝑠 ((1 − 𝐷𝑛 )𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) with spot price reset to 𝑆̃𝑡 = (1 − 𝐷𝑛 )𝑆𝑡 . When 𝑡 < 𝑡𝐷𝑛−1 < 𝑡𝐷𝑛 < 𝑇 then, following the same arguments as before 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑉𝑏𝑠 ((1 − 𝐷𝑛−1 )𝑆̃𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑉𝑏𝑠 ((1 − 𝐷𝑛−1 )(1 − 𝐷𝑛 )𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) with spot price now reset to 𝑆̃𝑡 = (1 − 𝐷𝑛−1 )(1 − 𝐷𝑛 )𝑆𝑡 . Hence, for the case when 𝑡 < 𝑡𝐷1 < 𝑡𝐷2 < ⋯ < 𝑡𝐷𝑛 < 𝑇 we can deduce that 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑉𝑏𝑠 ((1 − 𝐷1 )𝑆̃𝑡 , 𝑡; 𝐾, 𝑇 )

124

2.2.2 Black–Scholes Model

where 𝑆̃𝑡 = 𝑆𝑡

𝑛 ∏ 𝑖=2

(1 − 𝐷𝑖 ). Thus, ( 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑉𝑏𝑠

𝑆𝑡

𝑛 ∏ 𝑖=1

) (1 − 𝐷𝑖 ), 𝑡; 𝐾, 𝑇

for 𝑡 < 𝑡𝐷1 < 𝑡𝐷2 < ⋯ < 𝑡𝐷𝑛 < 𝑇 . { } 15. Market Price of Risk. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ). Suppose the stock price 𝑆𝑡 follows the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter, and let 𝑟 denote the risk-free interest rate. By considering a portfolio Π𝑡 holding only one option 𝑉 (𝑆𝑡 , 𝑡) which is only exercised at expiry 𝑇 > 𝑡 and using the Black–Scholes equation show that 𝑑Π𝑡 − 𝑟Π𝑡 𝑑𝑡 = 𝜎𝑆𝑡

) 𝜕𝑉 ( 𝑑𝑊𝑡 + 𝜆𝑑𝑡 𝜕𝑆𝑡

𝜇−𝑟 is known as the market price of risk. where 𝜆 = 𝜎 Is the portfolio riskless? Interpret the function 𝜆. Solution: At time 𝑡 we let the value of a portfolio Π𝑡 be Π𝑡 = 𝑉 (𝑆𝑡 , 𝑡) and the change in portfolio Π𝑡 becomes 𝑑Π𝑡 = 𝑑𝑉 . Expanding 𝑉 (𝑆𝑡 , 𝑡) using Taylor’s theorem 𝑑𝑉 =

)2 1 𝜕2𝑉 ( 𝜕𝑉 𝜕𝑉 𝑑𝑆𝑡 + … 𝑑𝑆𝑡 + 𝑑𝑡 + 2 𝜕𝑡 𝜕𝑆𝑡 2 𝜕𝑆𝑡

and by substituting 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 and applying It¯o’s lemma we have [ 𝑑Π𝑡 =

] 𝜕𝑉 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2 𝑉 + (𝜇 − 𝐷) 𝑆𝑡 𝑑𝑊𝑡 . 𝑑𝑡 + 𝜎𝑆𝑡 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡2

2.2.2 Black–Scholes Model

125

Hence, [ 𝑑Π𝑡 − 𝑟Π𝑡 𝑑𝑡 =

] 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 + (𝜇 − 𝐷) 𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑡) 𝑑𝑡 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

+𝜎𝑆𝑡

𝜕𝑉 𝑑𝑊𝑡 𝜕𝑆𝑡

and from the Black–Scholes equation 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷) 𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 we have 𝜕𝑉 𝜕𝑉 𝑑Π𝑡 − 𝑟Π𝑡 𝑑𝑡 = (𝜇 − 𝑟) 𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 ( 𝜇−𝑟 ) 𝜕𝑉 𝑑𝑊𝑡 + = 𝜎𝑆𝑡 𝑑𝑡 𝜕𝑆𝑡 𝜎 ) 𝜕𝑉 ( 𝑑𝑊𝑡 + 𝜆𝑑𝑡 = 𝜎𝑆𝑡 𝜕𝑆𝑡 𝜇−𝑟 where 𝜆 = . 𝜎 Given the presence of 𝑑𝑊𝑡 , the portfolio is not riskless. 𝜇−𝑟 By accepting a certain level of risk, 𝜆 = can be interpreted as the excess return 𝜎 above the risk-free interest rate. { } 16. Black–Scholes Model with Transaction Costs. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter, and let 𝑟 be the risk-free interest rate from a money-market account. We consider incorporating a transaction cost in buying or selling the asset which has three components: (i) a fixed cost at each transaction, 𝜅1 (ii) a cost proportional to the number of assets traded, 𝜅2 |𝜈| (iii) a cost proportional to the value of the assets traded, 𝜅3 |𝜈|𝑆𝑡 where 𝜅1 , 𝜅2 , 𝜅3 > 0, 𝜈 >√ 0 (for buying assets) and 𝜈 < 0 (for selling assets). ) ( 2𝑡 . Show that 𝔼 |𝑊𝑡 | = 𝜋

126

2.2.2 Black–Scholes Model

By considering a hedging portfolio involving buying an option 𝑉 (𝑆𝑡 , 𝑡) which can only be exercised at expiry time 𝑇 > 𝑡, selling Δ numbers of asset 𝑆𝑡 plus transaction costs, using It¯o’s lemma show that after a time interval 𝛿𝑡 > 0, 𝑉 (𝑆𝑡 , 𝑡) satisfies 𝜅 1 2 2 𝜕2𝑉 𝜕𝑉 ̃ 𝑡 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡) − 1 = 0 + (𝑟 − 𝐷)𝑆 + 𝜎 ̃ 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆 𝛿𝑡 𝜕𝑆𝑡 𝑡 where √ ̃ = 𝐷 + 𝜅2 𝜎 𝐷

( )−1 2 || 𝜕 2 𝑉 || 𝜕𝑉 | 2| 𝜋𝛿𝑡 || 𝜕𝑆𝑡 || 𝜕𝑆𝑡

√ and

𝜎 ̃ = 𝜎 − 𝜅3 𝜎 2

2

( 2 sgn 𝜋𝛿𝑡

𝜕2 𝑉 𝜕𝑆𝑡2

)

𝑥 . |𝑥| Can the option price following a transaction cost model be negative?

such that sgn(𝑥) =

Solution: For the first part of the problem, given 𝑊𝑡 ∼  (0, 𝑡) then from Problem 1.2.2.11 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus, |𝑊𝑡 | √ 2𝑡 follows a folded normal distribution, |𝑊𝑡 | ∼ 𝑓 (0, 𝑡). Therefore, 𝔼(|𝑊𝑡 |) = . 𝜋 At time 𝑡 we let the value of a portfolio Π𝑡 be Π𝑡 = 𝑉 (𝑆𝑡 , 𝑡) − Δ𝑆𝑡 where it involves buying one unit of option 𝑉 (𝑆𝑡 , 𝑡) and selling Δ units of 𝑆𝑡 . The continuous dividend yield is defined as the proportion of the asset price paid out per unit time. At time interval 𝛿𝑡 > 0, since we receive 𝐷𝑆𝑡 𝛿𝑡 for every asset held and the change in transaction costs is 𝜅1 + 𝜅2 |𝛿Δ| + 𝜅3 |𝛿Δ|𝑆𝑡 , the change in portfolio Π𝑡 is 𝛿Π𝑡 = 𝛿𝑉 − Δ(𝛿𝑆𝑡 + 𝐷𝑆𝑡 𝛿𝑡) − (𝜅1 + 𝜅2 |𝛿Δ| + 𝜅3 |𝛿Δ|𝑆𝑡 ). Expanding 𝑉 (𝑆𝑡 , 𝑡) using Taylor’s theorem 𝛿𝑉 =

1 𝜕 2 𝑉 ( )2 𝜕𝑉 𝜕𝑉 𝛿𝑆𝑡 + … 𝛿𝑆𝑡 + 𝛿𝑡 + 𝜕𝑡 𝜕𝑆𝑡 2 𝜕𝑆𝑡2

and by substituting 𝛿𝑆𝑡 = (𝜇 − 𝐷) 𝑆𝑡 𝛿𝑡 + 𝜎𝑆𝑡 𝛿𝑊𝑡 and subsequently applying It¯o’s lemma we have [ 𝛿𝑉 =

] 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 𝜕𝑉 + (𝜇 − 𝐷) 𝑆𝑡 𝛿𝑊𝑡 . 𝛿𝑡 + 𝜎𝑆𝑡 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡2

2.2.2 Black–Scholes Model

127

Substituting back into 𝛿Π𝑡 and rearranging the terms we have (

) 𝜕𝑉 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2 𝑉 + (𝜇 − 𝐷) 𝑆𝑡 𝛿𝑊𝑡 𝛿Π𝑡 = 𝛿𝑡 + 𝜎𝑆𝑡 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡2 ] [ −Δ (𝜇 − 𝐷) 𝑆𝑡 𝛿𝑡 + 𝜎𝑆𝑡 𝛿𝑊𝑡 − Δ𝐷𝑆𝑡 𝛿𝑡 −(𝜅1 + 𝜅2 |𝛿Δ| + 𝜅3 |𝛿Δ|𝑆𝑡 ) ( ) 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2 𝑉 = + (𝜇 − 𝐷) 𝑆𝑡 − 𝜇Δ𝑆𝑡 𝛿𝑡 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 ) ( 𝜕𝑉 +𝜎𝑆𝑡 − Δ 𝛿𝑊𝑡 − (𝜅1 + 𝜅2 |𝛿Δ| + 𝜅3 |𝛿Δ|𝑆𝑡 ). 𝜕𝑆𝑡 To eliminate the random component we choose Δ=

𝜕𝑉 𝜕𝑆𝑡

which leads to ( 𝛿Π𝑡 =

𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 + 𝜎 2 𝑆𝑡2 2 − 𝐷𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡

) 𝛿𝑡 − (𝜅1 + 𝜅2 |𝛿Δ| + 𝜅3 |𝛿Δ|𝑆𝑡 ).

To find |𝛿Δ| we note that 𝛿Δ = =

𝜕𝑉 𝜕𝑉 (𝑆 + 𝛿𝑆𝑡 , 𝑡 + 𝛿𝑡) − (𝑆 , 𝑡) 𝜕𝑆𝑡 𝑡 𝜕𝑆𝑡 𝑡 𝜕𝑉 𝜕2𝑉 𝜕𝑉 𝜕2𝑉 (𝑆𝑡 , 𝑡) + 𝛿𝑆𝑡 2 (𝑆𝑡 , 𝑡) + 𝛿𝑡 (𝑆 , 𝑡) + … − (𝑆 , 𝑡) 𝜕𝑆𝑡 𝜕𝑡𝜕𝑆𝑡 𝑡 𝜕𝑆𝑡 𝑡 𝜕𝑆𝑡

= 𝛿𝑆𝑡

𝜕2𝑉 𝜕2𝑉 (𝑆 , 𝑡) + 𝛿𝑡 (𝑆 , 𝑡) + … 𝑡 𝜕𝑡𝜕𝑆𝑡 𝑡 𝜕𝑆𝑡2

√ and because 𝛿𝑆𝑡 = 𝜎𝑆𝑡 𝜙 𝛿𝑡 + (𝛿𝑡) such that 𝜙 ∼  (0, 1) we have 𝛿Δ = 𝜎𝑆𝑡

𝜕2𝑉 √ 𝜕2𝑉 √ 𝜙 𝛿𝑡 + (𝛿𝑡) ≈ 𝜎𝑆𝑡 2 𝜙 𝛿𝑡. 2 𝜕𝑆𝑡 𝜕𝑆𝑡

Thus, the expected number of assets bought or sold becomes √ 𝔼(|𝛿Δ|) = 𝜎𝑆𝑡

2𝛿𝑡 || 𝜕 2 𝑉 || |. | 𝜋 || 𝜕𝑆𝑡2 ||

128

2.2.2 Black–Scholes Model

Under the no-arbitrage condition the expected return on the amount Π𝑡 invested in a riskfree interest rate would see a growth of 𝔼(𝛿Π𝑡 ) = 𝑟𝔼(Π𝑡 )𝛿𝑡 and hence we have ( 𝑟𝔼(Π𝑡 )𝛿𝑡 =

𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 + 𝜎 2 𝑆𝑡2 2 − 𝐷𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡

) 𝛿𝑡

−(𝜅1 + 𝜅2 𝔼(|𝛿Δ|) + 𝜅3 𝔼(|𝛿Δ|)𝑆𝑡 ) ( ) ) ( 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 𝑟 𝑉 (𝑆𝑡 , 𝑡) − Δ𝑆𝑡 𝛿𝑡 = − 𝐷𝑆𝑡 + 𝜎 𝑆𝑡 𝛿𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 ( ) √ √ 2𝛿𝑡 || 𝜕 2 𝑉 || 2𝛿𝑡 || 𝜕 2 𝑉 || 2 − 𝜅1 + 𝜅2 𝜎𝑆𝑡 | + 𝜅3 𝜎𝑆𝑡 | | | 𝜋 || 𝜕𝑆𝑡2 || 𝜋 || 𝜕𝑆𝑡2 || ( ) 𝜕𝑉 𝜕2 𝑉 𝜕𝑉 1 𝜕𝑉 𝑟 𝑉 (𝑆𝑡 , 𝑡) − 𝑆𝑡 + 𝜎 2 𝑆𝑡2 2 − 𝐷𝑆𝑡 = 𝜕𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 √ √ | 𝜅1 2 | 𝜕 2 𝑉 || 2 || 𝜕 2 𝑉 || 2 − − 𝜅2 𝜎𝑆𝑡 | | 2 | − 𝜅3 𝜎𝑆𝑡 | 𝛿𝑡 𝜋𝛿𝑡 || 𝜕𝑆𝑡 || 𝜋𝛿𝑡 || 𝜕𝑆𝑡2 || and after rearranging terms we finally have the Black–Scholes equation with transaction costs 𝜅 𝜕𝑉 1 2 2 𝜕2𝑉 ̃ 𝑡 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡) − 1 = 0 + (𝑟 − 𝐷)𝑆 + 𝜎 ̃ 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝛿𝑡 𝜕𝑆𝑡2 where √ ̃ = 𝐷 + 𝜅2 𝜎 𝐷

( )−1 2 || 𝜕 2 𝑉 || 𝜕𝑉 | | 𝜋𝛿𝑡 || 𝜕𝑆𝑡2 || 𝜕𝑆𝑡

√ and

𝜎 ̃2 = 𝜎 2 − 𝜅3 𝜎

( 2 sgn 𝜋𝛿𝑡

𝜕2 𝑉 𝜕𝑆𝑡2

) .

By considering the case when 𝜅2 = 𝜅3 = 0, the Black–Scholes equation with transaction costs becomes 𝜅 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡) − 1 = 0. + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝛿𝑡 𝜕𝑆𝑡 The solution to the above equation can be negative by setting 𝑉 (𝑆𝑡 , 𝑡) = −

𝜅1 < 0. 𝑟𝛿𝑡

{ } 17. Merton Model. Let (Ω, ℱ, ℙ) be a probability space and let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard } { Wiener process and 𝑁𝑡 : 𝑡 ≥ 0 be a Poisson process with intensity 𝜆 > 0 relative to the same filtration ℱ𝑡 , 𝑡 ≥ 0. Suppose the stock price 𝑆𝑡 follows a jump diffusion process with

2.2.2 Black–Scholes Model

129

the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 + (𝐽𝑡 − 1)𝑑𝑁𝑡 𝑆 𝑡− where 𝑑𝑁𝑡 =

{ 1 with probability 𝜆𝑑𝑡 0 with probability 1 − 𝜆𝑑𝑡

with 𝜇 being the drift parameter, 𝐷 the continuous dividend yield, 𝜎 the volatility parameter and 𝐽𝑡 the jump size variable. Let 𝑟 denote the risk-free interest rate and assume that 𝐽𝑡 , 𝑊𝑡 and 𝑁𝑡 are mutually independent. By considering a hedging portfolio involving both an option 𝑉 (𝑆𝑡 , 𝑡) which can only be exercised at expiry time 𝑇 ≥ 𝑡 and a stock 𝑆𝑡 , and assuming that the jump component is uncorrelated with the market, show that 𝑉 (𝑆𝑡 , 𝑡) satisfies ] 𝜕𝑉 𝜕2𝑉 [ 1 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡) + 𝜎 2 𝑆𝑡2 2 + 𝑟 − 𝐷 − 𝔼𝐽 (𝐽𝑡 − 1) 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 [ ] +𝜆𝔼𝐽 𝑉 (𝐽𝑡 𝑆𝑡 , 𝑡) − 𝑉 (𝑆𝑡 , 𝑡) = 0 where the expectation 𝔼𝐽 (⋅) is taken over the jump component. Solution: At time 𝑡 we let the value of a portfolio Π𝑡 be Π𝑡 = 𝑉 (𝑆𝑡 , 𝑡) − Δ𝑆𝑡 where it involves buying one unit of option 𝑉 (𝑆𝑡 , 𝑡) and selling Δ units of 𝑆𝑡 . Since we receive 𝐷𝑆𝑡 𝑑𝑡 for every asset held and because we hold −Δ𝑆𝑡 , our portfolio changes by an amount Δ𝐷𝑆𝑡 𝑑𝑡 and therefore the change in portfolio Π𝑡 is 𝑑Π𝑡 = 𝑑𝑉 − Δ(𝑑𝑆𝑡 + 𝐷𝑆𝑡 𝑑𝑡) = 𝑑𝑉 − Δ𝑑𝑆𝑡 − Δ𝐷𝑆𝑡 𝑑𝑡. Expanding 𝑉 (𝑆𝑡 , 𝑡) using Taylor’s theorem 𝑑𝑉 =

)2 1 𝜕 3 𝑉 ( )3 1 𝜕2𝑉 ( 𝜕𝑉 𝜕𝑉 𝑑𝑆 𝑑𝑆𝑡 + … 𝑑𝑆𝑡 + + 𝑑𝑡 + 𝑡 𝜕𝑡 𝜕𝑆𝑡 2! 𝜕𝑆𝑡2 3! 𝜕𝑆𝑡3

𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 + (𝐽𝑡 − 1)𝑑𝑁𝑡 where 𝑆𝑡− = 𝑆𝑡 and subse𝑆𝑡 quently applying It¯o’s lemma we have and substituting

130

2.2.2 Black–Scholes Model

] 𝜕𝑉 𝜕𝑉 [ (𝜇 − 𝐷) 𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 + (𝐽𝑡 − 1)𝑆𝑡 𝑑𝑁𝑡 𝑑𝑡 + 𝜕𝑡 𝜕𝑆𝑡 2 ] 1 𝜕3𝑉 [ ] 1 𝜕 𝑉 [ 2 2 𝜎 𝑆𝑡 𝑑𝑡 + (𝐽𝑡 − 1)2 𝑆𝑡2 𝑑𝑁𝑡 + (𝐽𝑡 − 1)3 𝑆𝑡3 𝑑𝑁𝑡 + … + 2 3 2! 𝜕𝑆𝑡 3! 𝜕𝑆𝑡 [ ] 𝜕𝑉 𝜕2𝑉 𝜕𝑉 𝜕𝑉 1 = 𝑑𝑊𝑡 + 𝜎 2 𝑆𝑡2 2 + (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝑆 𝜕𝑆𝑡 𝑡 𝑡 [ ] 2 3 1 1 𝜕𝑉 2 2𝜕 𝑉 3 3𝜕 𝑉 + (𝐽𝑡 − 1)𝑆𝑡 + (𝐽 − 1) 𝑆𝑡 + (𝐽𝑡 − 1) 𝑆𝑡 + … 𝑑𝑁𝑡 𝜕𝑆𝑡 2! 𝑡 𝜕𝑆𝑡2 3! 𝜕𝑆𝑡3 [ ] 𝜕𝑉 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 = + (𝜇 − 𝐷)𝑆𝑡 𝑑𝑊𝑡 𝑑𝑡 + 𝜎𝑆𝑡 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆 𝜕𝑆 𝜕𝑆𝑡 𝑡 𝑡 [ ] + 𝑉 (𝐽𝑡 𝑆𝑡 , 𝑡) − 𝑉 (𝑆𝑡 , 𝑡) 𝑑𝑁𝑡 .

𝑑𝑉 =

Substituting back into 𝑑Π𝑡 and rearranging terms we have [

] 𝜕𝑉 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 𝑑Π𝑡 = + (𝜇 − 𝐷) 𝑆𝑡 𝑑𝑊𝑡 + 𝜎 𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆 𝜕𝑆 𝜕𝑆𝑡 𝑡 𝑡 ] [ ] [ + 𝑉 (𝐽𝑡 𝑆𝑡 , 𝑡) − 𝑉 (𝑆𝑡 , 𝑡) 𝑑𝑁𝑡 − Δ (𝜇 − 𝐷) 𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 + (𝐽𝑡 − 1)𝑆𝑡 𝑑𝑁𝑡 −Δ𝐷𝑆𝑡 𝑑𝑡 ] [ ] [ 𝜕𝑉 1 2 2 𝜕2𝑉 𝜕𝑉 𝜕𝑉 + (𝜇 − 𝐷) 𝑆𝑡 − 𝜇Δ𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 − Δ 𝑑𝑊𝑡 + 𝜎 𝑆𝑡 = 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡2 ] [ + 𝑉 (𝐽𝑡 𝑆𝑡 , 𝑡) − 𝑉 (𝑆𝑡 , 𝑡) − Δ(𝐽𝑡 − 1)𝑆𝑡 𝑑𝑁𝑡 . To eliminate the random component so as to ensure the profit or loss is riskless, we choose Δ=

𝜕𝑉 𝜕𝑆𝑡

and hence [

] 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 − 𝐷𝑆𝑡 + 𝜎 𝑆𝑡 𝑑𝑡 𝑑Π𝑡 = 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 [ ] 𝜕𝑉 + 𝑉 (𝐽𝑡 𝑆𝑡 , 𝑡) − 𝑉 (𝑆𝑡 , 𝑡) − (𝐽𝑡 − 1)𝑆𝑡 𝑑𝑁𝑡 . 𝜕𝑆𝑡 Given that the jump component is uncorrelated with the market and under the no-arbitrage condition, the return on the amount Π𝑡 invested in a risk-free interest rate would see a growth of 𝔼𝐽 (𝑑Π𝑡 ) = 𝑟Π𝑡 𝑑𝑡

2.2.2 Black–Scholes Model

131

and because 𝐽𝑡 ⟂ ⟂ 𝑁𝑡 we have [

] 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2 𝑉 − 𝐷𝑆𝑡 𝑟Π𝑡 𝑑𝑡 = 𝑑𝑡 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 [ ] 𝜕𝑉 +𝔼𝐽 𝑉 (𝐽𝑡 𝑆𝑡 , 𝑡) − 𝑉 (𝑆𝑡 , 𝑡) − (𝐽𝑡 − 1)𝑆𝑡 𝔼 (𝑑𝑁𝑡 ) 𝜕𝑆𝑡 𝐽 { ] [ 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 𝑟 𝑉 (𝑆𝑡 , 𝑡) − Δ𝑆𝑡 𝑑𝑡 = + 𝜎 2 𝑆𝑡2 2 − 𝐷𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 } [ ] 𝜕𝑉 −𝜆𝔼𝐽 (𝐽𝑡 − 1)𝑆𝑡 + 𝜆𝔼𝐽 𝑉 (𝐽𝑡 𝑆𝑡 , 𝑡) − 𝑉 (𝑆𝑡 , 𝑡) 𝑑𝑡 𝜕𝑆𝑡 ] [ 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 + 𝜎 2 𝑆𝑡2 2 − 𝐷𝑆𝑡 = 𝑟 𝑉 (𝑆𝑡 , 𝑡) − 𝑆𝑡 𝜕𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 [ ] 𝜕𝑉 −𝜆𝔼𝐽 (𝐽𝑡 − 1) 𝑆𝑡 + 𝜆𝔼𝐽 𝑉 (𝐽𝑡 𝑆𝑡 , 𝑡) − 𝑉 (𝑆𝑡 , 𝑡) 𝜕𝑆𝑡 and finally ] 𝜕𝑉 𝜕2𝑉 [ 1 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡) + 𝜎 2 𝑆𝑡2 2 + 𝑟 − 𝐷 − 𝔼𝐽 (𝐽𝑡 − 1) 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 [ ] +𝜆𝔼𝐽 𝑉 (𝐽𝑡 𝑆𝑡 , 𝑡) − 𝑉 (𝑆𝑡 , 𝑡) = 0 which is the partial differentiation equation of 𝑉 (𝑆𝑡 , 𝑡) under the jump diffusion price process. 18. { European Option Valuation on Merton Model. Let {(Ω, ℱ, ℙ) be} a probability space and let } 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process and 𝑁𝑡 : 𝑡 ≥ 0 be a Poisson process with intensity 𝜆 > 0 relative to the same filtration ℱ𝑡 , 𝑡 ≥ 0. Suppose the asset price 𝑆𝑡 follows a jump diffusion process with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 + (𝐽𝑡 − 1)𝑑𝑁𝑡 𝑆 𝑡− where 𝑑𝑁𝑡 =

{ 1 with probability 𝜆𝑑𝑡 0 with probability 1 − 𝜆𝑑𝑡

with 𝜇 being the drift parameter, 𝐷 the continuous dividend yield, 𝜎 the volatility parameter and 𝐽𝑡 the jump size variable which follows a log-normal distribution log 𝐽𝑡 ∼  (𝜇𝐽 , 𝜎𝐽2 ). Let 𝑟 denote the risk-free interest rate and assume that 𝐽𝑡 , 𝑊𝑡 and 𝑁𝑡 are mutually independent.

132

2.2.2 Black–Scholes Model

By assuming that the jump component is uncorrelated with the market, using It¯o’s formula show that under the equivalent martingale measure ℚ𝑀 the conditional distribution of 𝑆𝑇 given 𝑆𝑡 and 𝑁𝑇 −𝑡 = 𝑛, 𝑛 = 0, 1, 2, … and 𝑡 < 𝑇 can be expressed as ) [ ( ] { } ̃ − 1𝜎 ̃2 (𝑇 − 𝑡) ̃2 (𝑇 − 𝑡), 𝜎 log 𝑆𝑇 || 𝑆𝑡 , 𝑁𝑇 −𝑡 = 𝑛 ∼  log 𝑆𝑡 + 𝑟 − 𝐷 2 ( ) ) ( 𝑛𝜎𝐽2 1 2 1 ̃ = 𝐷 + 𝜆 𝑒𝜇𝐽 + 2 𝜎𝐽 − 1 − 𝑛 𝜇𝐽 + 𝜎𝐽2 and 𝜎 ̃2 = 𝜎 2 + . where 𝐷 𝑇 −𝑡 2 𝑇 −𝑡 Hence, show that the European call option price at time 𝑡 ≤ 𝑇 with strike 𝐾 > 0 and expiry time 𝑇 is 𝐶𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡)

∞ −𝜆(𝑇 −𝑡) ∑ 𝑒 [𝜆(𝑇 − 𝑡)]𝑛 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) 𝑛! 𝑛=0

where ̃ 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑̃+ ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑̃− )

such that 𝑑̃± =

̃ ± 1𝜎 𝑥 ̃2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 1 − 1 𝑢2 2 and Φ(𝑥) = √ 𝑒 2 𝑑 𝑢. √ ∫ −∞ 𝜎 ̃ 𝑇 −𝑡 2𝜋

Solution: From Problem 5.2.4.4 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus, under the equivalent martingale measure ℚ𝑀 we can express the jump diffusion process as ) 𝑑𝑆𝑡 ( ̃𝑡 + (𝐽𝑡 − 1)𝑑𝑁𝑡 = 𝑟 − 𝐷 − 𝜆(𝐽̄ − 1) 𝑑𝑡 + 𝜎𝑑 𝑊 𝑆 𝑡− 1 2

̃𝑡 = 𝑊𝑡 − where 𝐽̄ = 𝔼ℙ (𝐽𝑡 ) = 𝑒𝜇𝐽 + 2 𝜎𝐽 , 𝑊

(

𝜇 − 𝑟 + 𝜆(𝐽̄ − 1) 𝜎

) 𝑡 is a ℚ𝑀 -standard

Wiener process and 𝑁𝑡 ∼ Poisson(𝜆𝑡). Using the results of Problem 5.2.2.3 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus we can write ̄

1 2 ̃𝑇 −𝑡 )(𝑇 −𝑡)+𝜎 𝑊

𝑆𝑇 = 𝑆𝑡 𝑒(𝑟−𝐷−𝜆(𝐽 −1)− 2 𝜎

𝑁𝑇 −𝑡

∏ 𝑖=1

𝐽𝑖

where 𝐽𝑖 ∼ log- (𝜇𝐽 , 𝜎𝐽2 ), 𝑖 = 1, 2, … , 𝑁𝑇 −𝑡 is a sequence of independent and ideñ𝑇 −𝑡 ∼  (0, 𝑇 − 𝑡). By independence of 𝐽𝑖 , tically distributed random variables and 𝑊 ̃ 𝑖 = 1, 2, … , 𝑁𝑇 −𝑡 , 𝑊𝑇 −𝑡 and 𝑁𝑇 −𝑡 we have ) [ ( ( ) 1 2 { } 1 log 𝑆𝑇 || 𝑆𝑡 , 𝑁𝑇 −𝑡 = 𝑛 ∼  log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜆 𝑒𝜇𝐽 + 2 𝜎𝐽 − 1 − 𝜎 2 (𝑇 − 𝑡) 2 ] +𝑛𝜇𝐽 , 𝜎 2 (𝑇 − 𝑡) + 𝑛𝜎𝐽2 .

2.2.2 Black–Scholes Model

133

By setting ) [ ( ] { } ̃ − 1𝜎 ̃2 (𝑇 − 𝑡) ̃2 (𝑇 − 𝑡), 𝜎 log 𝑆𝑇 || 𝑆𝑡 , 𝑁𝑇 −𝑡 = 𝑛 ∼  log 𝑆𝑡 + 𝑟 − 𝐷 2 ̃ and 𝜎 where 𝐷 ̃ are the “continuous dividend” and “volatility” of the asset price 𝑆𝑡 following a GBM process we can therefore set ) ( ( ) ( ) 1 2 1 ̃ − 1𝜎 𝑟−𝐷 ̃2 (𝑇 − 𝑡) = 𝑟 − 𝐷 − 𝜆 𝑒𝜇𝐽 + 2 𝜎𝐽 − 1 − 𝜎 2 (𝑇 − 𝑡) + 𝑛𝜇𝐽 2 2 2 2 2 𝜎 ̃ (𝑇 − 𝑡) = 𝜎 (𝑇 − 𝑡) + 𝑛𝜎𝐽 . Solving the equations simultaneously we have ) ( 1 2 ̃ = 𝐷 + 𝜆 𝑒 𝜇𝐽 + 2 𝜎𝐽 − 1 − 𝐷

( ) 𝑛 1 𝜇𝐽 + 𝜎𝐽2 𝑇 −𝑡 2

and

𝜎 ̃2 = 𝜎 2 +

𝑛𝜎𝐽2

𝑇 −𝑡

.

From the definition of a European call option price with strike 𝐾 > 0 and expiry time 𝑇 ≥𝑡 [ ] 𝐶𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ𝑀 max{𝑆𝑇 − 𝐾, 0}|| ℱ𝑡 = 𝑒−𝑟(𝑇 −𝑡)

∞ ∑ 𝑛=0

=𝑒

−𝑟(𝑇 −𝑡)

×𝔼

ℚ𝑀

[

∞ ∑ 𝑛=0

[ ] ℚ𝑀 (𝑁𝑇 −𝑡 = 𝑛)𝔼ℚ𝑀 max{𝑆𝑇 − 𝐾, 0}|| ℱ𝑡 , 𝑁𝑇 −𝑡 = 𝑛 {

𝑒−𝜆(𝑇 −𝑡) [𝜆(𝑇 − 𝑡)]𝑛 𝑛!

]} max{𝑆𝑇 − 𝐾, 0}|| ℱ𝑡 , 𝑁𝑇 −𝑡 = 𝑛

𝑒−𝜆(𝑇 −𝑡) [𝜆(𝑇 − 𝑡)]𝑛 is the probability of 𝑛 jumps in time period (𝑡, 𝑇 ]. Since where the term 𝑛! the conditional distribution of 𝑆𝑇 given 𝑆𝑡 and 𝑁𝑇 −𝑡 = 𝑛 can be written as ) [ ( ] { } ̃ − 1𝜎 ̃2 (𝑇 − 𝑡) log 𝑆𝑇 || 𝑆𝑡 , 𝑁𝑇 −𝑡 = 𝑛 ∼  log 𝑆𝑡 + 𝑟 − 𝐷 ̃2 (𝑇 − 𝑡), 𝜎 2 therefore from Problem 2.2.2.5 (page 101) the conditional call option [ ] 𝔼ℚ𝑀 max{𝑆𝑇 − 𝐾, 0}|| ℱ𝑡 , 𝑁𝑇 −𝑡 = 𝑛 ̃ and volatility 𝜎 is simply the Black–Scholes call option price with dividend 𝐷 ̃ [ ] 𝔼ℚ𝑀 max{𝑆𝑇 − 𝐾, 0}|| ℱ𝑡 , 𝑁𝑇 −𝑡 = 𝑛 = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 )

134

2.2.2 Black–Scholes Model

where ̃ 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑̃+ ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑̃− )

̃ ± 1𝜎 ̃2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 2 . √ 𝜎 ̃ 𝑇 −𝑡 Hence, the explicit solution is

with 𝑑̃± =

𝐶𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡)

∞ −𝜆(𝑇 −𝑡) ∑ 𝑒 [𝜆(𝑇 − 𝑡)]𝑛 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). 𝑛! 𝑛=0

{ } 19. Digital Option (PDE Approach). Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the stock price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility, parameter, and let 𝑟 denote the risk-free interest rate. A digital (or cash-or-nothing) call option is a contract that pays $1 at expiry time 𝑇 if the spot price 𝑆𝑇 > 𝐾 and nothing if 𝑆𝑇 ≤ 𝐾. By denoting 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) as the price of a European digital call option satisfying the following PDE with boundary conditions 𝜕𝐶𝑑 1 2 2 𝜕 2 𝐶𝑑 𝜕𝐶 + (𝑟 − 𝐷)𝑆𝑡 𝑑 − 𝑟𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝐶𝑑 (0, 𝑡; 𝐾, 𝑇 ) = 0,

𝐶𝑑 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 1I{𝑆𝑇 >𝐾}

and letting the solution of the SDE be in the form 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒𝛼𝑥+𝛽𝜏 𝑢(𝑥, 𝜏) where ( ) 𝑥 = log 𝑆𝑡 ∕𝐾 and 𝜏 = 12 𝜎 2 (𝑇 − 𝑡), show that by setting 1 𝛼 = − 𝑘1 2 where 𝑘1 =

𝑟−𝐷 1 2 𝜎 2

− 1 and 𝑘0 =

𝜕𝑢 𝜕2𝑢 = 2, 𝜕𝜏 𝜕𝑥 1

where 𝑓 (𝑥) = 𝑒 2 𝑘1 𝑥 1I{𝑥>0} .

𝑟 1 2 𝜎 2

and

1 𝛽 = − 𝑘21 − 𝑘0 4

, the Black–Scholes equation for 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is

𝑢(𝑥, 0) = 𝑓 (𝑥),

𝑥 ∈ (−∞, ∞),

𝜏>0

2.2.2 Black–Scholes Model

135

Given that the solution of 𝑢(𝑥, 𝜏) is 𝑢(𝑥, 𝜏) = √

1

∞

2𝜋𝜏 ∫−∞

𝑓 (𝑧)𝑒−

(𝑥−𝜏)2 4𝜏

𝑑𝑧

deduce that

𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒

−𝑟(𝑇 −𝑡)

⎛ log(𝑆 ∕𝐾) + (𝑟 − 𝐷 − 1 𝜎 2 )(𝑇 − 𝑡) ⎞ 𝑡 ⎜ ⎟ 2 Φ⎜ √ ⎟. 𝜎 𝑇 −𝑡 ⎜ ⎟ ⎝ ⎠

( ) Solution: Let 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒𝛼𝑥+𝛽𝜏 𝑢(𝑥, 𝜏) where 𝑥 = log 𝑆𝑡 ∕𝐾 and 𝜏 = 12 𝜎 2 (𝑇 − 𝑡). We have [ ] 𝜕𝐶𝑑 𝜕𝜏 𝜕𝐶𝑑 𝜕𝑢 1 = ⋅ = − 𝜎 2 𝑒𝛼𝑥+𝛽𝜏 + 𝛽𝑢(𝑥, 𝜏) 𝜕𝑡 𝜕𝜏 𝜕𝑡 2 𝜕𝜏 [ ] 𝜕𝐶𝑑 𝜕𝑥 𝜕𝐶𝑑 𝑒𝛼𝑥+𝛽𝜏 𝜕𝑢 = = ⋅ + 𝛼𝑢(𝑥, 𝜏) 𝜕𝑆𝑡 𝜕𝑥 𝜕𝑆𝑡 𝑆𝑡 𝜕𝑥 and 𝜕 2 𝐶𝑑 𝜕𝑆𝑡2

]} [ 𝜕𝑢 𝑒𝛼𝑥+𝛽𝜏 𝛼𝑢(𝑥, 𝜏) + 𝑆𝑡 𝜕𝑥 ] [ 𝑒𝛼𝑥+𝛽𝜏 𝜕 2 𝑢 𝜕𝑢 = + (2𝛼 − 1) + 𝛼(𝛼 − 1)𝑢(𝑥, 𝜏) . 𝜕𝑥 𝜕𝑥2 𝑆𝑡2

=

𝜕 𝜕𝑆𝑡

{

By substituting the above expressions into the Black–Scholes equation for 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ), we eventually have ) [ ( ] 𝜕𝑢 1 2 𝜕𝑢 1 2 𝜕𝑢 + − (2𝛼 − 1) + 𝑟 − 𝐷 𝜎 𝜎 2 𝜕𝑥 2 𝜕𝑥2 𝜕𝜏 ] [ 1 + 𝑢 𝜎 2 [𝛼(𝛼 − 1) − 𝛽] + 𝛼(𝑟 − 𝐷) − 𝑟 = 0. 2 To eliminate the terms

𝜕𝑢 and 𝑢 we set 𝜕𝑥 1 2 𝜎 (2𝛼 − 1) + 𝑟 − 𝐷 = 0 2 1 2 𝜎 [𝛼(𝛼 − 1) − 𝛽] + 𝛼(𝑟 − 𝐷) − 𝑟 = 0. 2

136

2.2.2 Black–Scholes Model

By solving the equations simultaneously we have 1 𝛼 = − 𝑘1 2 where 𝑘1 = Thus,

𝑟−𝐷 1 2 𝜎 2

− 1 and 𝑘0 =

𝑟 1 2 𝜎 2

1 𝛽 = − 𝑘21 − 𝑘0 4

and

.

1

1 2

𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒− 2 𝑘1 𝑥−( 4 𝑘1 +𝑘0 )𝜏 𝑢(𝑥, 𝜏) such that 1

1

𝑢(𝑥, 0) = 𝑒 2 𝑘1 𝑥 𝐶𝑑 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 𝑒 2 𝑘1 𝑥 1I{𝑥>0} . Writing the solution ∞

(𝑥−𝑧)2 1 𝑓 (𝑧)𝑒− 4𝜏 𝑑𝑧 𝑢(𝑥, 𝜏) = √ 2 𝜋𝜏 ∫−∞

𝜏

1 1 2 1 𝑒 2 𝑘1 𝑧− 4𝜏 (𝑥−𝑧) 𝑑𝑧 = √ ∫ 2 𝜋𝜏 0

(

=𝑒

(𝑥+𝑘1 𝜏)2 −𝑥2 4𝜏

(𝑥+𝑘1 𝜏)2 −𝑥2 4𝜏

∞

1 √ 𝑒 2 𝜋𝜏

∫0

− 21

𝑧−(𝑥+𝑘1 𝜏) √ 2𝜏

)2

𝑑𝑧

∞

1 − 1 𝑦2 √ 𝑒 2 𝑑𝑦 ∫ 2𝜋 ) ( (𝑥+𝑘1 𝜏)2 −𝑥2 𝑥 + 𝑘1 𝜏 4𝜏 =𝑒 Φ √ 2𝜏

=𝑒

(𝑥+𝑘 𝜏) − √1 2𝜏

hence 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒𝛼𝑥+𝛽𝜏 𝑢(𝑥, 𝜏) = 𝑒𝛼𝑥+𝛽𝜏 𝑒

(𝑥+𝑘1 𝜏)2 −𝑥2 4𝜏

( −𝑘0 𝜏

=𝑒

where 𝛼 = − 12 𝑘1 and 𝛽 = − 14 𝑘21 − 𝑘0 .

Φ

( Φ

𝑥 + 𝑘1 𝜏 √ 2𝜏

)

𝑥 + 𝑘1 𝜏 √ 2𝜏

)

2.2.2 Black–Scholes Model

137

( ) Finally, by setting 𝑥 = log 𝑆𝑡 ∕𝐾 and 𝜏 = 12 𝜎 2 (𝑇 − 𝑡) we have

𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒

−𝑟(𝑇 −𝑡)

⎛ log(𝑆 ∕𝐾) + (𝑟 − 𝐷 − 1 𝜎 2 )(𝑇 − 𝑡) ⎞ 𝑡 ⎜ ⎟ 2 Φ⎜ √ ⎟. 𝜎 𝑇 −𝑡 ⎜ ⎟ ⎝ ⎠

N.B. Using the same techniques, for a digital (or cash-or-nothing) put which pays $1 at expiry time 𝑇 if the spot price 𝑆𝑇 < 𝐾 and nothing if 𝑆𝑇 ≥ 𝐾 we can also show that the cash-or-nothing put price at time 𝑡 < 𝑇 is

𝑃𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒

−𝑟(𝑇 −𝑡)

⎛ − log(𝑆 ∕𝐾) − (𝑟 − 𝐷 − 1 𝜎 2 )(𝑇 − 𝑡) ⎞ 𝑡 ⎜ ⎟ 2 Φ⎜ √ ⎟. 𝜎 𝑇 −𝑡 ⎜ ⎟ ⎝ ⎠

{ } 20. Digital Option (Probabilistic Approach). Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the stock price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter, and let 𝑟 denote the risk-free interest rate. A digital (or cash-or-nothing) call option is a contract that pays $1 at expiry time 𝑇 if the spot price 𝑆𝑇 > 𝐾 and nothing if 𝑆𝑇 ≤ 𝐾. In contrast, a digital (or cash-or-nothing) put pays $1 at expiry time 𝑇 if the spot price 𝑆𝑇 < 𝐾 and nothing if 𝑆𝑇 ≥ 𝐾. By denoting 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑃𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) as the digital call and put option prices, respectively at time 𝑡, 𝑡 < 𝑇 show using the risk-neutral valuation approach that 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) Φ(𝑑− ) and

𝑃𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑− )

where 1 log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 − 𝜎 2 )(𝑇 − 𝑡) 2 𝑑− = √ 𝜎 𝑇 −𝑡

𝑥

and

1 2 1 𝑒− 2 𝑢 𝑑𝑢. Φ(𝑥) = √ 2𝜋 ∫−∞

Verify that the put–call parity for a digital option is 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑃𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) .

138

2.2.2 Black–Scholes Model

Solution: From Girsanov’s theorem, the spot price SDE under the risk-neutral measure ℚ is 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡 where 𝑊𝑡ℚ = 𝑊𝑡 + By definition

(𝜇 − 𝑟) 𝜎

𝑡 is a ℚ-standard Wiener process.

( ) | 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 1I{𝑆𝑇 >𝐾} | ℱ𝑡 | [ ( ) ( )] −𝑟(𝑇 −𝑡) 1 ⋅ ℚ 𝑆𝑇 > 𝐾 || ℱ𝑡 + 0 ⋅ ℚ 𝑆𝑇 ≤ 𝐾 || ℱ𝑡 =𝑒 ) ( = 𝑒−𝑟(𝑇 −𝑡) ℚ 𝑆𝑇 > 𝐾 || ℱ𝑡 . From It¯o’s lemma we can write 1 2 )(𝑇 −𝑡)+𝜎𝑊𝑇ℚ−𝑡

𝑆𝑇 = 𝑆𝑡 𝑒(𝑟−𝐷− 2 𝜎

,𝑡 < 𝑇.

Hence, ) ℚ 𝑆𝑇 > 𝐾 || ℱ𝑡 = ℚ (

( 𝑆𝑡 𝑒

(𝑟−𝐷− 21 𝜎 2 )(𝑇 −𝑡)+𝜎𝑊𝑇ℚ−𝑡

( =ℚ

𝑊𝑇ℚ−𝑡

| > 𝐾 || ℱ𝑡 |

)

) log(𝐾∕𝑆𝑡 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) || |ℱ . > | 𝑡 𝜎 | |

𝑊ℚ Given that 𝑊𝑇ℚ−𝑡 ∼  (0, 𝑇 − 𝑡), 𝑍 = √ 𝑇 −𝑡 ∼  (0, 1) and hence 𝑇 −𝑡 (

) log(𝐾∕𝑆𝑡 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) || 𝑊𝑇ℚ−𝑡 |ℱ > ℚ 𝑆𝑇 > 𝐾 || ℱ𝑡 = ℚ √ √ | 𝑡 | 𝑇 −𝑡 𝜎 𝑇 −𝑡 | ( ) 1 2 | log(𝐾∕𝑆𝑡 ) − (𝑟 − 𝐷 − 2 𝜎 )(𝑇 − 𝑡) | |ℱ =ℚ 𝑍> √ | 𝑡 | 𝜎 𝑇 −𝑡 | = Φ(𝑑− ) (

)

1 log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 − 𝜎 2 )(𝑇 − 𝑡) 2 . Therefore, where 𝑑− = √ 𝜎 𝑇 −𝑡 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) Φ(𝑑− ).

2.2.2 Black–Scholes Model

139

In contrast, for a digital put option price, by definition ( ) | 𝑃𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 1I{𝑆𝑇 0 at expiry time 𝑇 if the spot price 𝑆𝑇 > 𝐾 and nothing if 𝑆𝑇 ≤ 𝐾. In contrast, an asset-or-nothing put pays 𝑆𝑇 > 0 at expiry time 𝑇 if the spot price 𝑆𝑇 < 𝐾 and nothing if 𝑆𝑇 ≥ 𝐾. By denoting 𝐶𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) as the price of an asset-or-nothing call option at time 𝑡 < 𝑇 satisfying the following PDE with boundary conditions 𝜕𝐶𝑎 𝜕𝐶𝑎 1 2 2 𝜕 2 𝐶𝑎 + (𝑟 − 𝐷)𝑆 − 𝑟𝐶𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 𝐶𝑎 (0, 𝑡; 𝐾, 𝑇 ) = 0,

𝐶𝑎 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 𝑆𝑇 1I{𝑆𝑇 >𝐾}

and letting the solution of the PDE be in the form 𝐶𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑒𝛼𝑥+𝛽𝜏 𝑢(𝑥, 𝜏) where ( ) 𝑥 = log 𝑆𝑡 ∕𝐾 and 𝜏 = 12 𝜎 2 (𝑇 − 𝑡), show that by setting 1 𝛼 = − 𝑘1 2

and

1 𝛽 = − 𝑘21 − 𝑘0 4

140

2.2.2 Black–Scholes Model

where 𝑘1 =

𝑟−𝐷 1 2 𝜎 2

− 1 and 𝑘0 = 𝜕2𝑢 𝜕𝑢 = 2, 𝜕𝜏 𝜕𝑥

𝑟 1 2 𝜎 2

, the Black–Scholes equation for 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is

𝑢(𝑥, 0) = 𝑓 (𝑥),

𝑥 ∈ (−∞, ∞), 𝜏 > 0

1

where 𝑓 (𝑥) = 𝑒( 2 𝑘1 +1)𝑥 1I{𝑥>0} . Given that the solution ∞

(𝑥−𝜏)2 1 𝑢(𝑥, 𝜏) = √ 𝑓 (𝑧)𝑒− 4𝜏 𝑑𝑧 2𝜋𝜏 ∫−∞

deduce that

𝐶𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒

−𝐷(𝑇 −𝑡)

⎛ log(𝑆 ∕𝐾) + (𝑟 − 𝐷 + 1 𝜎 2 )(𝑇 − 𝑡) ⎞ 𝑡 ⎜ ⎟ 2 Φ⎜ √ ⎟. 𝜎 𝑇 −𝑡 ⎜ ⎟ ⎝ ⎠

( ) Solution: By setting 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑒𝛼𝑥+𝛽𝜏 𝑢(𝑥, 𝜏) such that 𝑥 = log 𝑆𝑡 ∕𝐾 , 𝜏 = 1 2 𝜎 (𝑇 − 𝑡) and using the same steps as described in Problem 2.2.2.19 (page 134), we 2 can show that 𝜕𝑢 𝜕2𝑢 = 2, 𝜕𝜏 𝜕𝑥

1

𝑢(𝑥, 0) = 𝑒( 2 𝑘1 +1)𝑥 1I{𝑥>0} ,

𝑥 ∈ (−∞, ∞), 𝜏 > 0

1 1 𝑟−𝐷 where 𝛼 = − 𝑘1 , 𝛽 = − 𝑘21 − 𝑘0 , 𝑘1 = 1 − 1 and 𝑘0 = 2 4 𝜎2 2 By solving the heat equation we have 1 𝑢(𝑥, 𝜏) = √ 2 𝜋𝜏 ∫0

∞

1 = √ 2 𝜋𝜏 ∫0

∞

=𝑒

𝑓 (𝑧)𝑒−

(𝑥−𝑧)2 4𝜏

1

𝑒( 2 𝑘1 +1)𝑧−

(𝑥+( 1 𝑘1 +1)𝜏)2 −𝑥2 2 4𝜏

(𝑥+( 1 𝑘1 +1)𝜏)2 −𝑥2 2 4𝜏

∫0

(𝑥−𝑧)2 4𝜏

𝑑𝑧

1 √ 𝑒 2 𝜋𝜏

∞

− 21

𝑧−(𝑥+( 1 𝑘1 +1)𝜏) √2 2𝜏

1 − 1 𝑦2 √ 𝑒 2 𝑑𝑦 ∫ 2𝜋 ) ( 1 (𝑥+( 1 𝑘1 +1)𝜏)2 −𝑥2 𝑥 + ( 2 𝑘1 + 1)𝜏 2 4𝜏 . =𝑒 Φ √ 2𝜏 =𝑒

.

𝑑𝑧

( ∞

𝑟 1 2 𝜎 2

𝑥+( 1 𝑘1 +1)𝜏 2√ − 2𝜏

)2

𝑑𝑧

2.2.2 Black–Scholes Model

141

Therefore, 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑒𝛼𝑥+𝛽𝜏 𝑢(𝑥, 𝜏) = 𝐾𝑒

𝑥+(𝑘1 −𝑘0 +1)𝜏

1 1 where 𝛼 = − 𝑘1 and 𝛽 = − 𝑘21 − 𝑘0 . 2 4 𝑟−𝐷 − 1 and 𝑘0 = By substituting 𝑘1 = 1 2 𝜎 2

𝑟 1 2 𝜎 2

Φ

(

𝑥 + ( 12 𝑘1 + 1)𝜏 √ 2𝜏

)

, the asset-or-nothing call option price is

⎛ log(𝑆 ∕𝐾) + (𝑟 − 𝐷 + 1 𝜎 2 )(𝑇 − 𝑡) ⎞ 𝑡 ⎜ ⎟ 2 𝐶𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ ⎜ √ ⎟. 𝜎 𝑇 −𝑡 ⎜ ⎟ ⎝ ⎠ N.B. Using the same techniques we can also show that the asset-or-nothing put option price is ) ( ) ( ⎞ ⎛ 1 2 − log 𝑆 (𝑇 − 𝑡) ⎟ ∕𝐾 − 𝑟 − 𝐷 + 𝜎 𝑡 2 −𝐷(𝑇 −𝑡) ⎜ Φ⎜ 𝑃𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒 √ ⎟. 𝜎 𝑇 −𝑡 ⎟ ⎜ ⎠ ⎝

{ } 22. Asset-or-Nothing Option (Probabilistic Approach). Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the stock price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter, and let 𝑟 denote the risk-free interest rate. An asset-or-nothing call option is a contract that pays 𝑆𝑇 at expiry time 𝑇 if the spot price 𝑆𝑇 > 𝐾 and nothing if 𝑆𝑇 ≤ 𝐾. In contrast, an asset-or-nothing put pays 𝑆𝑇 > 0 at expiry time 𝑇 if the spot price 𝑆𝑇 < 𝐾 and nothing if 𝑆𝑇 ≥ 𝐾. By denoting 𝐶𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑃𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) as the asset-or-nothing call and put option prices, respectively at time 𝑡, 𝑡 < 𝑇 show using the risk-neutral valuation approach that 𝐶𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑+ ) and

𝑃𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑+ )

142

2.2.2 Black–Scholes Model

where 1 log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 + 𝜎 2 )(𝑇 − 𝑡) 2 𝑑+ = √ 𝜎 𝑇 −𝑡

𝑥

and

1 2 1 𝑒− 2 𝑢 𝑑𝑢. Φ(𝑥) = √ 2𝜋 ∫−∞

Verify that the put–call parity for an asset-or-nothing option is 𝐶𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑃𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) . Solution: Using Girsanov’s theorem, the spot price SDE under the risk-neutral measure ℚ is 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡 where 𝑊𝑡ℚ = 𝑊𝑡 + write

(𝜇 − 𝑟) 𝜎

𝑡 is a ℚ-standard Wiener process. From It¯o’s lemma we can

1 2 )(𝑇 −𝑡)+𝜎𝑊𝑇ℚ−𝑡

𝑆𝑇 = 𝑆𝑡 𝑒(𝑟−𝐷− 2 𝜎

,

𝑡𝐾} | ℱ𝑡 | = 𝑒−𝑟(𝑇 −𝑡)

∞

∫𝐾

𝑆𝑇 𝑓 (𝑆𝑇 |𝑆𝑡 ) 𝑑𝑆𝑇

such that ) ( 𝑓 𝑆𝑇 |𝑆𝑡 =

(

√

− 21

1

𝑆𝑇 𝜎 2𝜋(𝑇 − 𝑡)

𝑒

log 𝑆𝑇 −𝑚 √ 𝜎 𝑇 −𝑡

)2

,

𝑆𝑇 > 0

) ( where 𝑚 = log 𝑆𝑡 + 𝑟 − 𝐷 − 12 𝜎 2 (𝑇 − 𝑡). Hence, we can write (

𝐶𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡)

∞

∫𝐾

− 21

1 𝑒 √ 𝜎 2𝜋(𝑇 − 𝑡)

log 𝑆𝑇 −𝑚 √ 𝜎 𝑇 −𝑡

)2

𝑑𝑆𝑇

2.2.2 Black–Scholes Model

and by letting 𝑢 =

143

log𝑆𝑇 − 𝑚 we have √ 𝜎 𝑇 −𝑡 𝐶𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) =

1 2 𝑒𝑚−𝑟(𝑇 −𝑡) 𝑒− 2 𝑢 +𝜎𝑢 √ log 𝐾−𝑚 2𝜋 ∫ 𝜎 √𝑇 −𝑡

∞

√ 𝑇 −𝑡

𝑑𝑢.

Using the sum of squares √ 1 1 − 𝑢2 + 𝜎𝑢 𝑇 − 𝑡 = − 2 2

] )2 √ 𝑢 − 𝜎 𝑇 − 𝑡 − 𝜎 2 (𝑇 − 𝑡)

[(

we can simplify the integral to become − 𝑒𝑚−𝑟(𝑇 −𝑡) 𝑒 2 𝐶𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = √ log 𝐾−𝑚 ∫ √ 2𝜋 𝜎 𝑇 −𝑡 ∞

𝑚−𝑟(𝑇 −𝑡)+ 21 𝜎 2 (𝑇 −𝑡)

=𝑒

1

] [( )2 √ 𝑢−𝜎 𝑇 −𝑡 −𝜎 2 (𝑇 −𝑡)

∞

∫ log𝐾−𝑚 √ 𝜎 𝑇 −𝑡

1 −1 √ 𝑒 2 2𝜋

𝑑𝑢

( )2 √ 𝑢−𝜎 𝑇 −𝑡

𝑑𝑢.

√ Finally, by changing the variable once again with 𝑣 = 𝑢 − 𝜎 𝑇 − 𝑡 and substituting 𝑚 = log 𝑆𝑡 + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) we eventually have ⎛ log(𝑆 ∕𝐾) + (𝑟 − 𝐷 + 1 𝜎 2 )(𝑇 − 𝑡) ⎞ 𝑡 ⎜ ⎟ 2 𝐶𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ ⎜ √ ⎟. 𝜎 𝑇 −𝑡 ⎜ ⎟ ⎝ ⎠ As for the asset-or-nothing put option price, using similar arguments we can easily show that ( ) | 𝑃𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 𝑆𝑇 1I{𝑆𝑇 𝑡 on an asset with price 𝑆𝑡 at time 𝑡 is 𝐹 (𝑡, 𝑇 ) = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) where 𝐷 is the continuous dividend yield and 𝑟 is the constant risk-free interest rate. Here, under the risk-neutral measure ℚ the asset price is assumed to follow a GBM process 𝑑𝑆𝑡 = (𝑟 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡 { } where 𝑊𝑡ℚ : 𝑡 ≥ 0 is a ℚ-standard Wiener process on the probability space (Ω, ℱ, ℚ) and 𝜎 is the asset volatility parameter. By considering a hedging portfolio involving both a European option on futures 𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡) and a futures contract 𝐹 (𝑡, 𝑇 ), show that the Black equation is 𝜕2𝑉 1 𝜕𝑉 + 𝜎 2 𝐹 (𝑡, 𝑇 )2 2 − 𝑟𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡) = 0. 𝜕𝑡 2 𝜕𝐹 Solution: Let the value of the portfolio at time 𝑡 be Π𝑡 = 𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡) − Δ𝐹 (𝑡, 𝑇 ) where it involves buying one unit of an option on futures 𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡) and selling Δ units of 𝐹 (𝑡, 𝑇 ). The jump in the value of the portfolio in a single time step is 𝑑Π𝑡 = 𝑑𝑉 − Δ𝑑𝐹 where, by expanding 𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡) using Taylor’s theorem 𝑑𝑉 =

𝜕𝑉 1 𝜕2𝑉 𝜕𝑉 (𝑑𝐹 )2 + … + 𝑑𝐹 + 𝜕𝑡 𝜕𝐹 2 𝜕𝐹 2

2.2.2 Black–Scholes Model

145

Given 𝐹 (𝑡, 𝑇 ) = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) and

𝑑𝑆𝑡 = (𝑟 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ we can write 𝑆𝑡

) ( 𝑑𝐹 = 𝑑 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) = 𝑒(𝑟−𝐷)(𝑇 −𝑡) 𝑑𝑆𝑡 − (𝑟 − 𝐷)𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) 𝑑𝑡 [ ] = 𝑒(𝑟−𝐷)(𝑇 −𝑡) (𝑟 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡ℚ − (𝑟 − 𝐷)𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) 𝑑𝑡 = 𝜎𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) 𝑑𝑊𝑡ℚ

= 𝜎𝐹 (𝑡, 𝑇 )𝑑𝑊𝑡ℚ . From It¯o’s lemma

(𝑑𝑊𝑡ℚ )2 = 𝑑𝑡,

(𝑑𝑡)2 = 𝑑𝑊𝑡ℚ 𝑑𝑡 = 𝑑𝑡 𝑑𝑊𝑡ℚ = 0

we have (𝑑𝐹 )2 = 𝜎 2 𝐹 (𝑡, 𝑇 )2 𝑑𝑡 and

(𝑑𝐹 )𝜈 = 0,

𝜈 > 2.

Therefore, 𝜕𝑉 𝜕𝑉 1 𝜕2𝑉 (𝑑𝐹 )2 + … 𝑑𝑡 + 𝑑𝐹 + 𝜕𝑡 𝜕𝐹 2 𝜕𝐹 2 ( ) 𝜕𝑉 𝜕2𝑉 1 𝜕𝑉 = + 𝜎 2 𝐹 (𝑡, 𝑇 )2 2 𝑑𝑡 + 𝜎𝐹 (𝑡, 𝑇 ) 𝑑𝑊𝑡ℚ . 𝜕𝑡 2 𝜕𝐹 𝜕𝐹

𝑑𝑉 =

By substituting 𝑑𝑉 into 𝑑Π𝑡 and rearranging terms, we have ( 𝑑Π𝑡 =

𝜕𝑉 𝜕2 𝑉 1 + 𝜎 2 𝐹 (𝑡, 𝑇 )2 2 𝜕𝑡 2 𝜕𝐹

) 𝑑𝑡 + 𝜎𝐹 (𝑡, 𝑇 )

(

) 𝜕𝑉 − Δ 𝑑𝑊𝑡ℚ . 𝜕𝐹

To ensure a risk-free portfolio we set Δ=

𝜕𝑉 𝜕𝐹

which leads to ( 𝑑Π𝑡 =

𝜕𝑉 𝜕2𝑉 1 + 𝜎 2 𝐹 (𝑡, 𝑇 )2 2 𝜕𝑡 2 𝜕𝐹

) 𝑑𝑡.

Since it costs nothing to enter into a futures contract the cost of setting up the portfolio at time 𝑡 is only 𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡), and therefore 𝑑Π𝑡 = 𝑟𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡)𝑑𝑡.

146

2.2.2 Black–Scholes Model

Thus, ( 𝑟𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡)𝑑𝑡 =

𝜕2𝑉 1 𝜕𝑉 + 𝜎 2 𝐹 (𝑡, 𝑇 )2 2 𝜕𝑡 2 𝜕𝐹

) 𝑑𝑡

or 𝜕𝑉 𝜕2𝑉 1 + 𝜎 2 𝐹 (𝑡, 𝑇 )2 2 − 𝑟𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡) = 0. 𝜕𝑡 2 𝜕𝐹

24. Consider the value of a European option 𝑉 (𝑆𝑡 , 𝑡) satisfying the following Black–Scholes equation 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 where 𝑆𝑡 is the spot price of a stock at time 𝑡, 𝜎 is the stock volatility, 𝑟 is the constant risk-free interest rate, 𝐷 is the continuous dividend yield and 𝑇 > 𝑡 is the option expiry time. Given that the price of a futures contract maturing at time 𝑇 on an asset with price 𝑆𝑡 at time 𝑡 is 𝐹 (𝑡, 𝑇 ) = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) derive the Black equation for a European option written on a futures contract 𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡). Solution: Let 𝑉̂ (𝑆𝑡 , 𝑡) = 𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡) so that 𝜕 𝑉̂ 𝜕𝑉 𝜕𝑉 𝜕𝐹 = + 𝜕𝑡 𝜕𝑡 𝜕𝐹 𝜕𝑡 𝜕𝑉 𝜕𝑉 = − (𝑟 − 𝐷)𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) 𝜕𝑡 𝜕𝐹 𝜕𝑉 𝜕𝑉 = − (𝑟 − 𝐷)𝐹 (𝑡, 𝑇 ) 𝜕𝑡 𝜕𝐹 𝜕 𝑉̂ 𝜕𝑉 𝜕𝐹 = 𝜕𝑆𝑡 𝜕𝐹 𝜕𝑆𝑡

𝜕𝑉 = 𝑒(𝑟−𝐷)(𝑇 −𝑡) 𝜕𝐹 ( ) 𝐹 (𝑡, 𝑇 ) 𝜕𝑉 = 𝑆𝑡 𝜕𝐹

2.2.2 Black–Scholes Model

147

and ) ( 𝜕 𝜕 2 𝑉̂ (𝑟−𝐷)(𝑇 −𝑡) 𝜕𝑉 𝑒 = 𝜕𝑆𝑡 𝜕𝐹 𝜕𝑆𝑡2 𝜕 2 𝑉 𝜕𝐹 𝜕𝐹 2 𝜕𝑆𝑡 𝜕2𝑉 = 𝑒2(𝑟−𝐷)(𝑇 −𝑡) 2 𝜕𝐹 ( )2 2 𝐹 (𝑡, 𝑇 ) 𝜕 𝑉 = . 𝑆𝑡 𝜕𝐹 2

= 𝑒(𝑟−𝐷)(𝑇 −𝑡)

𝜕 𝑉̂ 𝜕 𝑉̂ 𝜕 2 𝑉̂ By substituting 𝑉̂ (𝑆𝑡 , 𝑡), and into , 𝜕𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡2 𝜕 𝑉̂ 𝜕 2 𝑉̂ 𝜕 𝑉̂ 1 − 𝑟𝑉̂ (𝑆𝑡 , 𝑡) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 we have 𝜕𝑉 𝜕𝑉 − (𝑟 − 𝐷)𝐹 (𝑡, 𝑇 ) + 𝜕𝑡 𝜕𝐹 ( ) 𝐹 (𝑡, 𝑇 ) 𝜕𝑉 +(𝑟 − 𝐷)𝑆𝑡 𝑆𝑡 𝜕𝐹

1 2 2 𝜎 𝑆𝑡 2

(

𝐹 (𝑡, 𝑇 ) 𝑆𝑡

)2

𝜕2𝑉 𝜕𝐹 2

− 𝑟𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡) = 0

or 𝜕2𝑉 1 𝜕𝑉 + 𝜎 2 𝐹 (𝑡, 𝑇 )2 2 − 𝑟𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡) = 0. 𝜕𝑡 2 𝜕𝐹 25. Let the price of a futures contract maturing at time 𝑇 > 𝑡 on an asset with price 𝑆𝑡 at time 𝑡 be 𝐹 (𝑡, 𝑇 ) = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) where 𝑟 is the constant risk-free interest rate and 𝐷 is the continuous dividend yield. For a European option written on a futures contract 𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡) it satisfies the Black–Scholes equation 𝜕2𝑉 1 𝜕𝑉 + 𝜎 2 𝐹 (𝑡, 𝑇 )2 2 − 𝑟𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡) = 0. 𝜕𝑡 2 𝜕𝐹 ( ) 𝐹 (𝑡, 𝑇 ) 𝐵2 𝑉 , 𝑡 is also a solution of the Black–Scholes equation for any Show that 𝐵 𝐹 (𝑡, 𝑇 ) constant 𝐵.

148

2.2.2 Black–Scholes Model

Show also that the prices of a European call option 𝐶(𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) and put option 𝑃 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) on a futures contract with the same strike 𝐾 and expiry time 𝜏 < 𝑇 (i.e., the options expire before the futures contract maturity) have the following property 𝐶(𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) =

Solution: Let 𝑊 (𝐹 (𝑡, 𝑇 ), 𝑡) = entiation we have

𝐹 (𝑡, 𝑇 ) 𝑃 𝐾

(

) 𝐾2 , 𝑡; 𝐾, 𝜏 . 𝐹 (𝑡, 𝑇 )

) 𝐹 (𝑡, 𝑇 ) ( ̂ 𝐵2 𝑉 𝐹 (𝑡, 𝑇 ), 𝑡 where 𝐹̂ (𝑡, 𝑇 ) = . By differ𝐵 𝐹 (𝑡, 𝑇 ) 𝐹 (𝑡, 𝑇 ) 𝜕𝑉 𝜕𝑊 = 𝜕𝑡 𝐵 𝜕𝑡

𝐹 (𝑡, 𝑇 ) 𝜕𝑉 𝜕𝐹 𝜕𝑊 1 = 𝑉 (𝐹̂ (𝑡, 𝑇 ), 𝑡) + 𝜕𝐹 𝐵 𝐵 𝜕 𝐹̂ 𝜕𝐹 ) ( 𝐹 (𝑡, 𝑇 ) 𝜕𝑉 1 𝐵2 ̂ = 𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡) + − 𝐵 𝐵 𝜕 𝐹̂ 𝐹 (𝑡, 𝑇 ) 𝐵 𝜕𝑉 1 = 𝑉 (𝐹̂ (𝑡, 𝑇 ), 𝑡) − 𝐵 𝐹 (𝑡, 𝑇 ) 𝜕 𝐹̂ and 1 𝜕𝑉 𝜕𝐹 𝜕𝑉 𝐵 𝜕2 𝑊 𝐵 𝜕 2 𝑉 𝜕 𝐹̂ = + − 𝐵 𝜕 𝐹̂ 𝜕𝐹 𝐹 (𝑡, 𝑇 ) 𝜕 𝐹̂ 2 𝜕𝐹 𝜕𝐹 2 𝐹 (𝑡, 𝑇 )2 𝜕 𝐹̂ ( )3 2 ) ( 1 𝜕 𝑉 𝐵2 𝜕𝑉 𝐵 2 𝜕𝑉 𝐵 = − + + 2 2 𝐵 𝐹 (𝑡, 𝑇 ) 𝐹 (𝑡, 𝑇 ) 𝐹 (𝑡, 𝑇 ) 𝜕 𝐹̂ 𝜕 𝐹̂ 𝜕 𝐹̂ 2 ( )3 2 𝐵 𝜕 𝑉 = . 𝐹 (𝑡, 𝑇 ) 𝜕 𝐹̂ 2 Since 𝜕𝑊 𝜕2𝑊 1 − 𝑟𝑊 (𝐹 (𝑡, 𝑇 ), 𝑡) = 0 + 𝜎 2 𝐹 (𝑡, 𝑇 )2 𝜕𝑡 2 𝜕𝐹 2 therefore ( )3 2 ) ( ( ) 𝐹 (𝑡, 𝑇 ) 𝐹 (𝑡, 𝑇 ) 𝜕𝑉 𝜕 𝑉 1 𝐵 −𝑟 + 𝜎 2 𝐹 (𝑡, 𝑇 )2 𝑉 𝐹̂ (𝑡, 𝑇 ), 𝑡 𝐵 𝜕𝑡 2 𝐹 (𝑡, 𝑇 ) 𝐵 𝜕 𝐹̂ 2 } { 2 𝐹 (𝑡, 𝑇 ) 𝜕𝑉 𝜕 𝑉 1 − 𝑟𝑉 (𝐹̂ (𝑡, 𝑇 ), 𝑡) + 𝜎 2 𝐹̂ (𝑡, 𝑇 )2 = 𝐵 𝜕𝑡 2 𝜕 𝐹̂ 2 = 0.

2.2.2 Black–Scholes Model

149

Hence, 𝜕2𝑉 1 𝜕𝑉 − 𝑟𝑉 (𝐹̂ (𝑡, 𝑇 ), 𝑡) = 0 + 𝜎 2 𝐹̂ (𝑡, 𝑇 )2 𝜕𝑡 2 𝜕 𝐹̂ 2 ( ) 𝐹 (𝑡, 𝑇 ) 𝐵2 which shows that 𝑉 , 𝑡 is also a solution of the Black–Scholes equation 𝐵 𝐹 (𝑡, 𝑇 ) for any constant 𝐵. ( ) 𝐹 (𝑡, 𝑇 ) 𝐾2 𝑃 , 𝑡; 𝐾, 𝜏 at the option Finally, to show that 𝐶(𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) = 𝐾 𝐹 (𝑡, 𝑇 ) expiry 𝜏 ( ) { } 𝐹 (𝜏, 𝑇 ) 𝐹 (𝜏, 𝑇 ) 𝐾2 𝐾2 𝑃 , 𝜏; 𝐾, 𝜏 = max 𝐾 − ,0 𝐾 𝐹 (𝜏, 𝑇 ) 𝐾 𝐹 (𝜏, 𝑇 ) = max{𝐹 (𝜏, 𝑇 ) − 𝐾} = 𝐶(𝐹 (𝜏, 𝑇 ), 𝜏; 𝐾, 𝜏). By discounting the payoffs back to time 𝑡 we have 𝐶(𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) =

𝐹 (𝑡, 𝑇 ) 𝑃 𝐾

(

) 𝐾2 , 𝑡; 𝐾, 𝜏 . 𝐹 (𝑡, 𝑇 )

26. Consider the price of a futures contract 𝐹 (𝑡, 𝑇 ) with delivery time 𝑇 on a stock with price 𝑆𝑡 at time 𝑡 (𝑡 < 𝑇 ). Throughout the life of the futures contract the stock pays discrete dividends 𝛿𝑖 , 𝑖 = 1, 2, … , 𝑛 where 𝑡 < 𝑡1 < 𝑡2 < ⋯ < 𝑡𝑛 < 𝑇 . Under the no-arbitrage condition the futures contract price is 𝐹 (𝑡, 𝑇 ) = 𝑆𝑡 𝑒𝑟(𝑇 −𝑡) −

𝑛 ∑ 𝑖=1

𝛿𝑖 𝑒𝑟(𝑇 −𝑡𝑖 )

where 𝑟 is the risk-free interest rate. By considering a European option on the futures 𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡), show that the Black– Scholes equation is 1 𝜕𝑉 + 𝜎2 𝜕𝑡 2

( 𝐹 (𝑡, 𝑇 ) +

𝑛 ∑ 𝑖=1

)2 𝛿𝑖 𝑒

𝑟(𝑇 −𝑡𝑖 )

𝜕2𝑉 − 𝑟𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡) = 0 𝜕𝐹 2

where 𝜎 is the stock volatility. Solution: Let 𝑉̂ (𝑆𝑡 , 𝑡) = 𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡); differentiating with respect to 𝑡 and 𝑆𝑡 we have 𝜕 𝑉̂ 𝜕𝑉 𝜕𝑉 𝜕𝑉 𝜕𝑉 𝜕𝐹 = = + − 𝑟𝑆𝑡 𝑒𝑟(𝑇 −𝑡) 𝜕𝑡 𝜕𝑡 𝜕𝐹 𝜕𝑆𝑡 𝜕𝑡 𝜕𝐹 𝜕𝑉 𝜕𝐹 𝜕𝑉 𝜕 𝑉̂ = = 𝑒𝑟(𝑇 −𝑡) 𝜕𝑆𝑡 𝜕𝐹 𝜕𝑆𝑡 𝜕𝐹

150

2.2.2 Black–Scholes Model

and ) ( 𝜕 𝜕 2 𝑉 𝜕𝐹 𝜕2𝑉 𝜕 2 𝑉̂ 𝑟(𝑇 −𝑡) 𝜕𝑉 = 𝑒𝑟(𝑇 −𝑡) 2 𝑒 = = 𝑒2𝑟(𝑇 −𝑡) 2 . 2 𝜕𝑆𝑡 𝜕𝐹 𝜕𝐹 𝜕𝑆𝑡 𝜕𝐹 𝜕𝑆𝑡 Given that 𝜕 2 𝑉̂ 𝜕 𝑉̂ 1 𝜕 𝑉̂ − 𝑟𝑉̂ (𝑆𝑡 , 𝑡) = 0 + 𝜎 2 𝑆𝑡2 2 + 𝑟𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 we have 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 − 𝑟𝑆𝑡 𝑒𝑟(𝑇 −𝑡) + 𝜎 2 𝑆𝑡2 𝑒2𝑟(𝑇 −𝑡) 2 + 𝑟𝑆𝑡 𝑒𝑟(𝑇 −𝑡) − 𝑟𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡) = 0 𝜕𝑡 𝜕𝐹 2 𝜕𝐹 𝜕𝐹 or 𝜕𝑉 1 + 𝜎2 𝜕𝑡 2

( 𝐹 (𝑡, 𝑇 ) +

𝑛 ∑ 𝑖=1

)2 𝛿𝑖 𝑒𝑟(𝑇 −𝑡𝑖 )

𝜕2𝑉 − 𝑟𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡) = 0. 𝜕𝐹 2

27. Let the price of a futures contract maturing at time 𝑇 on an asset with price 𝑆𝑡 at time 𝑡, 𝑡 < 𝑇 be 𝐹 (𝑡, 𝑇 ) = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) where 𝑟 is the constant risk-free interest rate, 𝐷 is the continuous dividend yield and 𝜎 is the asset volatility. For European call 𝐶(𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) and put 𝑃 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) options written on a futures contract with expiry time 𝑇 and strike price 𝐾, show that [ ] 𝐶(𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝐹 (𝑡, 𝑇 )Φ(𝑑1 ) − 𝐾Φ(𝑑2 ) and [ ] 𝑃 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝐾Φ(−𝑑2 ) − 𝐹 (𝑡, 𝑇 )Φ(−𝑑1 ) where 𝑑1,2 =

log (𝐹 (𝑡, 𝑇 )∕𝐾) ± 12 𝜎 2 (𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

𝑥

and

Φ(𝑥) =

∫−∞

1 − 1 𝑢2 √ 𝑒 2 𝑑𝑢. 2𝜋

Show that the put–call parity relationship for options on futures is 𝐶(𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) [𝐹 (𝑡, 𝑇 ) − 𝐾] .

2.2.2 Black–Scholes Model

151

Solution: For European call 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and put 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) options written on 𝑆𝑡 with strike price 𝐾 and expiry time 𝑇 , we have 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑− ) and 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑− ) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑+ ) where 1 log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 𝜎 2 )(𝑇 − 𝑡) 2 𝑑± = . √ 𝜎 𝑇 −𝑡 Since 𝐹 (𝑡, 𝑇 ) = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) , by substituting 𝑆𝑡 = 𝐹 (𝑡, 𝑇 )𝑒−(𝑟−𝐷)(𝑇 −𝑡) we have [ ] 𝐶(𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝐹 (𝑡, 𝑇 )Φ(𝑑1 ) − 𝐾Φ(𝑑2 ) and [ ] 𝑃 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝐾Φ(−𝑑2 ) − 𝐹 (𝑡, 𝑇 )Φ(−𝑑1 ) where 𝑑1,2

log (𝐹 (𝑡, 𝑇 )∕𝐾) ± 12 𝜎 2 (𝑇 − 𝑡) = . √ 𝜎 𝑇 −𝑡

Given the identity Φ(𝑑1 ) + Φ(−𝑑1 ) = 1 and Φ(𝑑2 ) + Φ(−𝑑2 ) = 1 therefore 𝐶(𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) [𝐹 (𝑡, 𝑇 ) − 𝐾] . N.B. From the expressions of 𝐶(𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) and 𝑃 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) we can deduce that options on futures contracts are independent on an asset that pays a continuous dividend yield. { } 28. Arithmetic Brownian Motion. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow an arithmetic Brownian motion (ABM) with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. Using It¯o’s lemma show that under the risk-neutral measure ℚ the conditional

152

2.2.2 Black–Scholes Model

distribution of 𝑆𝑇 given 𝑆𝑡 is ( 𝑆𝑇 |𝑆𝑡 ∼ 

)

𝑆𝑡 𝑒

(𝑟−𝐷)(𝑇 −𝑡)

[ 2(𝑟−𝐷)(𝑇 −𝑡) ] 𝜎2 𝑒 , −1 2(𝑟 − 𝐷)

where 𝑟 is the risk-free interest rate and 𝑇 > 𝑡. Using the risk-neutral valuation approach show that the European call option price at time 𝑡 with strike 𝐾 and expiry time 𝑇 is ) ( ̂ ′ (𝑑) 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ (𝑑) + 𝜎Φ and from the put–call parity deduce that the European put option price at time 𝑡 with strike 𝐾 and expiry time 𝑇 is ) ( ̂ ′ (−𝑑) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑) 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝜎Φ where 𝑆 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) 𝑑= 𝑡 , 𝜎̂ = 𝜎 𝜎̂

√

𝑒−2𝐷(𝑇 −𝑡) − 𝑒−2𝑟(𝑇 −𝑡) 2(𝑟 − 𝐷)

and Φ(𝑥) is the standard normal cdf 𝑥

Φ(𝑥) =

∫−∞

1 − 1 𝑢2 √ 𝑒 2 𝑑𝑢. 2𝜋

Solution: From Girsanov’s theorem, under the risk-neutral measure ℚ the asset price follows the process 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ (

) 𝜇 − 𝑟𝑆𝑡 𝑡 is a ℚ-standard. Further details on this change of mea𝜎 sure can be found in Problem 4.2.3.7 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus. From Taylor’s theorem and from the application of It¯o’s lemma

where 𝑊𝑡ℚ = 𝑊𝑡 +

𝑑(𝑒−(𝑟−𝐷)𝑡 𝑆𝑡 ) = −(𝑟 − 𝐷)𝑒−(𝑟−𝐷)𝑡 𝑆𝑡 𝑑𝑡 + 𝑒−(𝑟−𝐷)𝑡 𝑑𝑆𝑡 1 + (𝑟 − 𝐷)2 𝑒−(𝑟−𝐷)𝑡 𝑆𝑡 (𝑑𝑡)2 + … 2 = −(𝑟 − 𝐷)𝑒−(𝑟−𝐷)𝑡 𝑆𝑡 𝑑𝑡 + 𝑒−(𝑟−𝐷)𝑡 ((𝑟 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ ) = 𝜎𝑒−(𝑟−𝐷)𝑡 𝑑𝑊𝑡ℚ .

2.2.2 Black–Scholes Model

153

Taking integrals 𝑇

∫𝑡

𝑑(𝑒−(𝑟−𝐷)𝑢 𝑆𝑢 ) =

𝑇

𝜎𝑒−(𝑟−𝐷)𝑢 𝑑𝑊𝑢ℚ

∫𝑡

𝑒−(𝑟−𝐷)𝑇 𝑆𝑇 − 𝑒−(𝑟−𝐷)𝑡 𝑆𝑡 = 𝜎

𝑇

∫𝑡

𝑒−(𝑟−𝐷)𝑢 𝑑𝑊𝑢ℚ

or

𝑆𝑇 = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) + 𝜎 𝑇

Since

∫𝑡

𝑒(𝑟−𝐷)(𝑇 −𝑢) 𝑑𝑊𝑢ℚ ∼ 

(

𝑇

0,

∫𝑡

( 𝑆𝑇 |𝑆𝑡 ∼ 

𝑆𝑡 𝑒

(𝑟−𝐷)(𝑇 −𝑡)

𝑆𝑇 given 𝑆𝑡 is

∫𝑡

𝑒(𝑟−𝐷)(𝑇 −𝑢) 𝑑𝑊𝑢ℚ . )

𝑒2(𝑟−𝐷)(𝑇 −𝑢) 𝑑𝑢

therefore

) [ 2(𝑟−𝐷)(𝑇 −𝑡) ] 𝜎2 𝑒 , −1 . 2(𝑟 − 𝐷)

√ By writing 𝑚 = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) and 𝑠 = 𝜎

𝑇

𝑒2(𝑟−𝐷)(𝑇 −𝑡) − 1 so that the conditional pdf of 2(𝑟 − 𝐷)

−1 1 𝑓 (𝑆𝑇 |𝑆𝑡 ) = √ 𝑒 2 𝑠 2𝜋

(

) 𝑆𝑇 −𝑚 2 𝑠

then from the definition of a European call option [ ] 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑆𝑇 − 𝐾, 0}|| ℱ𝑡 = 𝑒−𝑟(𝑇 −𝑡)

∞

∫𝐾

(𝑆𝑇 − 𝐾)𝑓 (𝑆𝑇 |𝑆𝑡 ) 𝑑𝑆𝑇

= 𝐼1 − 𝐼2 where 𝐼1 = 𝑒−𝑟(𝑇 −𝑡)

∞

∫𝐾

𝑆𝑇 𝑓 (𝑆𝑇 |𝑆𝑡 ) 𝑑𝑆𝑇 and 𝐼2 = 𝐾𝑒−𝑟(𝑇 −𝑡)

∞

∫𝐾

𝑓 (𝑆𝑇 |𝑆𝑡 ) 𝑑𝑆𝑇 .

154

2.2.2 Black–Scholes Model

For case 𝐼1 and by setting 𝑧 =

𝑆𝑇 − 𝑚 we can write 𝑠 (

)2

𝑆𝑇 − 1 𝑆𝑇 −𝑚 𝑑𝑆𝑇 √ 𝑒 2 𝑠 ∫𝐾 𝑠 2𝜋 ∞ 𝑠𝑧 + 𝑚 − 21 𝑧2 𝑒 = 𝑒−𝑟(𝑇 −𝑡) 𝑑𝑧 √ 𝐾−𝑚 ∫ 2𝜋 𝑠 ∞ ∞ 𝑧 − 21 𝑧2 1 − 1 𝑧2 𝑒 𝑑𝑧 + 𝑚𝑒−𝑟(𝑇 −𝑡) = 𝑠𝑒−𝑟(𝑇 −𝑡) √ √ 𝑒 2 𝑑𝑧 ∫ 𝐾−𝑚 2𝜋 ∫ 𝐾−𝑚 2𝜋 𝑠 𝑠 ) )] [ ( ( 𝑚−𝐾 −𝑟(𝑇 −𝑡) ′ 𝑚−𝐾 + 𝑠Φ . 𝑚Φ =𝑒 𝑠 𝑠

𝐼1 = 𝑒−𝑟(𝑇 −𝑡)

∞

Using similar techniques for case 𝐼2 we have (

)2

𝑆 −𝑚 −1 𝑇 1 𝐼2 = 𝐾𝑒 𝑑𝑆𝑇 √ 𝑒 2 𝑠 ∫𝐾 𝑠 2𝜋 ∞ 1 − 12 𝑧2 𝑒 = 𝐾𝑒−𝑟(𝑇 −𝑡) 𝑑𝑧 √ ∫ 𝐾−𝑚 2𝜋 𝑠 ) ( 𝑚−𝐾 −𝑟(𝑇 −𝑡) . Φ = 𝐾𝑒 𝑠 √ 𝑒2(𝑟−𝐷)(𝑇 −𝑡) − 1 By substituting 𝑚 = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) and 𝑠 = 𝜎 we have 2(𝑟 − 𝐷)

∞

−𝑟(𝑇 −𝑡)

𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) − 𝐾 𝑚−𝐾 = √ 𝑠 𝑒2(𝑟−𝐷)(𝑇 −𝑡) − 1 𝜎 2(𝑟 − 𝐷) ( −𝐷(𝑇 −𝑡) ) − 𝐾𝑒−𝑟(𝑇 −𝑡) 𝑒𝑟(𝑇 −𝑡) 𝑆𝑡 𝑒 = √ 𝑒2(𝑟−𝐷)(𝑇 −𝑡) − 1 𝜎 2(𝑟 − 𝐷) = √ where 𝜎̂ = 𝜎

𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) 𝜎̂

𝑒−2𝐷(𝑇 −𝑡) − 𝑒−2𝑟(𝑇 −𝑡) and hence the European call option price at time 𝑡 is 2(𝑟 − 𝐷)

𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐼1 − 𝐼2

) )] [ ( ( 𝑚−𝐾 𝑚−𝐾 + 𝑠Φ′ = 𝑒−𝑟(𝑇 −𝑡) (𝑚 − 𝐾)Φ 𝑠 𝑠 ) ( ̂ ′ (𝑑) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ (𝑑) + 𝜎Φ

where 𝑑 =

𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) . 𝜎̂

2.2.2 Black–Scholes Model

155

Finally, from the put–call parity relationship of European options 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) the equivalent European put option price at time 𝑡 is 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) + 𝐾𝑒−𝑟(𝑇 −𝑡) ) ( = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) (Φ (𝑑) − 1) + 𝜎Φ ̂ ′ (𝑑) ) ( = 𝜎Φ ̂ ′ (−𝑑) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑) 1 2 1 since Φ′ (𝑥) = √ 𝑒− 2 𝑥 is an even function. 2𝜋

29. Let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the stock price 𝑆𝑡 satisfy the following GBM process 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter and let 𝑟 denote the risk-free interest rate. We consider a European-style option contract with strike 𝐾 written on the stock 𝑆𝑡 that pays amount ( Ψ(𝑆𝑇 ) = log

𝑆𝑇 𝑆0

) −𝐾

at the expiry time 𝑇 > 𝑡 where 𝑆0 is the initial stock price. Using It¯o’s lemma find the SDE for log 𝑆𝑡 under the risk-neutral measure ℚ. Hence, show that the price of the option struck at time 𝑡 is ] [ ( ) ( ) 𝑆𝑡 1 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) log + 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡) − 𝐾 . 𝑆0 2 Find the hedging ratio, the number of stocks to hold and the amount of cash needed to invest in a risk-free money market in order to hedge this option contract at time 𝑡. Finally, by considering a European contract that pays [ Ψ(𝑆𝑇 ) = log

(

𝑆𝑇 𝑆0

)

]2 −𝐾

at expiry time 𝑇 what is the option price of this contract at time 𝑡?

156

2.2.2 Black–Scholes Model

Solution: From Girsanov’s theorem the SDE of 𝑆𝑡 under the risk-neutral measure ℚ is 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡 (𝜇 − 𝑟) 𝑡 is a ℚ-standard Wiener process. where 𝑊𝑡ℚ = 𝑊𝑡 + 𝜎 From Taylor’s expansion 𝑑𝑆𝑡 1 𝑑(log 𝑆𝑡 ) = − 𝑆𝑡 2

(

𝑑𝑆𝑡 𝑆𝑡

)2 +…

and using It¯o’s lemma we eventually have ) ( 1 𝑑(log 𝑆𝑡 ) = 𝑟 − 𝐷 − 𝜎 2 𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ . 2 Taking integrals ) 𝑇 ( 𝑇 1 𝑟 − 𝐷 − 𝜎 2 𝑑𝑢 + 𝑑 log 𝑆𝑢 = 𝜎 𝑑𝑊𝑢ℚ ∫𝑡 ∫𝑡 ∫𝑡 2 ( ) ( ) 𝑆𝑇 1 log = 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡) + 𝜎𝑊𝑇ℚ−𝑡 𝑆𝑡 2 𝑇

such that ( log

𝑆𝑇 𝑆𝑡

) ∼

[(

) ] 1 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡), 𝜎 2 (𝑇 − 𝑡) . 2

Hence, the option price at time 𝑡 is [ ] 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ Ψ(𝑆𝑇 )|| ℱ𝑡 [ ] ( ) | 𝑆 | 𝑇 = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ log − 𝐾 | ℱ𝑡 | 𝑆0 | [ ] ( ) ( ) | 𝑆𝑡 𝑆𝑇 | −𝑟(𝑇 −𝑡) ℚ =𝑒 𝔼 log + log − 𝐾 | ℱ𝑡 | 𝑆𝑡 𝑆0 | ] [ ( ) ( ) 𝑆 1 𝑡 = 𝑒−𝑟(𝑇 −𝑡) log + 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡) − 𝐾 . 𝑆0 2 At time 𝑡, 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝜙𝑡 𝑆𝑡 + 𝜓𝑡

2.2.2 Black–Scholes Model

157

where 𝜙𝑡 is the number of units of 𝑆𝑡 and 𝜓𝑡 is the amount of cash invested in the risk-free money market. By partial differentiation with respect to 𝑆𝑡 we therefore have 𝜙𝑡 =

𝜕𝑉 𝑒−𝑟(𝑇 −𝑡) = 𝜕𝑆𝑡 𝑆𝑡

and hence 𝜓𝑡 = 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝜙𝑡 𝑆𝑡 ] [ ( ) ( ) 𝑆𝑡 1 2 −𝑟(𝑇 −𝑡) =𝑒 log + 𝑟 − 𝐷 − 𝜎 (𝑇 − 𝑡) − 𝐾 − 1 . 𝑆0 2 [

(

For the case when the payoff is Ψ(𝑆𝑇 ) = log [(

𝑆𝑇 𝑆0

)

]2 −𝐾

, the option price at time 𝑡 is

)2 | ] | log − 𝐾 | ℱ𝑡 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ | | [( ( ) ( ) )2 | ] 𝑆𝑡 𝑆𝑇 | −𝑟(𝑇 −𝑡) ℚ =𝑒 𝔼 log + log − 𝐾 | ℱ𝑡 | 𝑆𝑡 𝑆0 | [( ( ) ) ( 𝑆𝑇 1 = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ log − 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡) 𝑆𝑡 2 ( ) )2 | ] ( ) 𝑆𝑡 1 2 | + 𝑟 − 𝐷 − 𝜎 (𝑇 − 𝑡) + log − 𝐾 | ℱ𝑡 | 2 𝑆0 | [( ( ) )2 | ] ) ( 𝑆𝑇 1 2 | −𝑟(𝑇 −𝑡) ℚ =𝑒 𝔼 log − 𝑟 − 𝐷 − 𝜎 (𝑇 − 𝑡) | ℱ𝑡 | 𝑆𝑡 2 | [ ] ( ) ( ) | 𝑆𝑇 1 2 | −𝑟(𝑇 −𝑡) ℚ + 2𝑒 𝔼 log − 𝑟 − 𝐷 − 𝜎 (𝑇 − 𝑡)| ℱ𝑡 | 𝑆𝑡 2 | [ ] ( ) ( ) | 𝑆𝑡 1 | × 𝔼ℚ log + 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡) − 𝐾 | ℱ𝑡 | 𝑆0 2 | [( ( ) )2 | ] ) ( 𝑆𝑡 1 2 | −𝑟(𝑇 −𝑡) ℚ +𝑒 𝔼 log + 𝑟 − 𝐷 − 𝜎 (𝑇 − 𝑡) − 𝐾 | ℱ𝑡 . | 𝑆0 2 | (

𝑆𝑇 𝑆0

)

Since ( log

𝑆𝑇 𝑆𝑡

)

) ( − 𝑟 − 𝐷 − 12 𝜎 2 (𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

∼  (0, 1)

158

2.2.2 Black–Scholes Model

therefore ⎡ ⎢ log ⎢ ⎢ ⎢ ⎣

(

)

𝑆𝑇 𝑆𝑡

2 ) ( ⎤ − 𝑟 − 𝐷 − 12 𝜎 2 (𝑇 − 𝑡) ⎥ ⎥ ∼ 𝜒 2 (1) √ ⎥ 𝜎 𝑇 −𝑡 ⎥ ⎦

with ⎡⎛ ⎢ ⎜ log 𝔼ℚ ⎢ ⎜ ⎢⎜ ⎢⎜ ⎣⎝

(

𝑆𝑇 𝑆𝑡

)

(

− 𝑟−𝐷− √ 𝜎 𝑇 −𝑡

1 2 𝜎 2

)

⎞ (𝑇 − 𝑡) ⎟ ⎟ ⎟ ⎟ ⎠

2|

| ⎤ | ⎥ | |ℱ⎥ = 1 | 𝑡⎥ | | ⎥ | ⎦ |

or 𝔼ℚ

[(

( log

𝑆𝑇 𝑆𝑡

)

)2 | ] ) 1 2 | − 𝑟 − 𝐷 − 𝜎 (𝑇 − 𝑡) | ℱ𝑡 = 𝜎 2 (𝑇 − 𝑡). | 2 | (

Hence, { 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒

−𝑟(𝑇 −𝑡)

[ + log

(

𝜎 2 (𝑇 − 𝑡) 𝑆𝑡 𝑆0

)

]2 ) 1 2 + 𝑟 − 𝐷 − 𝜎 (𝑇 − 𝑡) − 𝐾 2 (

} .

30. Merton Model for Default of a Company I. At time 𝑡, we assume the asset 𝐴𝑡 of a company satisfies the SDE 𝑑𝐴𝑡 = 𝜇𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝐴𝑡 where 𝜇 is the drift parameter, 𝜎 is the volatility and {𝑊𝑡 : 𝑡 ≥ 0} is a standard Wiener process on the probability space (Ω, ℱ, ℙ). Let 𝑟 be the risk-free interest rate. In financial accounting the asset 𝐴𝑡 is a combination of equity 𝐸𝑡 and debt 𝐷𝑡 so that 𝐴𝑡 = 𝐸𝑡 + 𝐷𝑡 where at time 𝑇 , 𝑡 < 𝑇 the debt holders will receive an amount 𝐹 > 0 which is the face value of the debt if 𝐴𝑇 > 𝐹 and the equity holders will receive the rest of the value of the company. Otherwise, the company will be in default and the debt holders will receive 𝐴𝑇 and the equity holders will receive nothing. By constructing the payoff diagrams for 𝐸𝑇 and 𝐷𝑇 , find the values of 𝐸𝑡 and 𝐷𝑡 for all 𝑡 < 𝑇 under the Black–Scholes framework.

2.2.2 Black–Scholes Model

159

Solution: At time 𝑇 , for the debt holders ⎧𝐹 ⎪ 𝐷𝑇 = ⎨ ⎪𝐴𝑇 ⎩

if 𝐴𝑇 > 𝐹 if 𝐴𝑇 ≤ 𝐹

and hence, for the equity holders 𝐸𝑇 = 𝐴𝑇 − 𝐷𝑇 ⎧𝐹 ⎪ = 𝐴𝑇 − ⎨ ⎪𝐴𝑇 ⎩

if 𝐴𝑇 > 𝐹 if 𝐴𝑇 ≤ 𝐹

⎧𝐴 − 𝐹 if 𝐴 > 𝐹 𝑇 ⎪ 𝑇 =⎨ ⎪0 if 𝐴𝑇 ≤ 𝐹 ⎩ = max{𝐴𝑇 − 𝐹 , 0}. At terminal time 𝑇 , the payoff diagram for equity shareholders is given in Figure 2.1. Mathematically 𝐸𝑇 = max{𝐴𝑇 − 𝐹 , 0} is the payoff of a European call option on the assets with strike equal to the face value.

Payoff

Figure 2.1

Payment to equity holders at time 𝑇 .

160

2.2.2 Black–Scholes Model

Given that 𝐴𝑡 follows a GBM process, the amount of equity 𝐸𝑡 at time 𝑡 is 𝐸𝑡 = 𝐶(𝐴𝑡 , 𝑡; 𝐹 , 𝑇 ) where 𝐶(𝐴𝑡 , 𝑡; 𝐹 , 𝑇 ) = 𝐴𝑡 Φ(𝑑+ ) − 𝐹 𝑒−𝑟(𝑇 −𝑡) Φ(𝑑− ) such that 𝑑± =

log(𝐴𝑡 ∕𝐹 ) + (𝑟 ± 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

𝑥

and

Φ(𝑥) =

∫−∞

1 − 1 𝑢2 √ 𝑒 2 𝑑𝑢. 2𝜋

In contrast, the payment to debt holders at time 𝑇 is ⎧𝐹 if 𝐴𝑇 > 𝐹 ⎪ 𝐷𝑇 = ⎨ ⎪𝐴𝑇 if 𝐴𝑇 ≤ 𝐹 ⎩ = min{𝐹 , 𝐴𝑇 } or ⎧𝐹 ⎪ 𝐷𝑇 = ⎨ ⎪𝐴𝑇 ⎩

if 𝐴𝑇 > 𝐹 if 𝐴𝑇 ≤ 𝐹

⎧𝐹 if 𝐴 > 𝐹 ⎧0 𝑇 ⎪ ⎪ −⎨ =⎨ ⎪𝐹 if 𝐴𝑇 ≤ 𝐹 ⎪𝐹 − 𝐴𝑇 ⎩ ⎩ = 𝐹 − max{𝐹 − 𝑉𝑇 , 0}.

if 𝐴𝑇 > 𝐹 if 𝐴𝑇 ≤ 𝐹

Therefore, the payoff diagram for the debt holders is given in Figure 2.2. Thus, 𝐷𝑇 = min{𝐹 , 𝐴𝑇 } = 𝐹 − max{𝐹 − 𝐴𝑇 , 0} which is the difference between the face value and the payoff of a European put option with strike price equal to 𝐹 . Thus, at time 𝑡 the debt is equal to 𝐷𝑡 = 𝐹 𝑒−𝑟(𝑇 −𝑡) − 𝑃 (𝐴𝑡 , 𝑡; 𝐹 , 𝑇 ) = 𝐴𝑡 − 𝐶(𝐴𝑡 , 𝑡; 𝐹 , 𝑇 )

2.2.2 Black–Scholes Model

161

Payoff

Figure 2.2

Payment to debt holders at time 𝑇 .

where, from the put–call parity 𝑃 (𝐴𝑡 , 𝑡; 𝐹 , 𝑇 ) = 𝐶(𝐴𝑡 , 𝑡; 𝐹 , 𝑇 ) − 𝐴𝑡 + 𝐹 𝑒−𝑟(𝑇 −𝑡) .

31. Generalised Black–Scholes Formula. Let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 have the following diffusion process 𝑑𝑆𝑡 = (𝜇𝑡 − 𝐷𝑡 )𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡 𝑆𝑡 where 𝜇𝑡 , 𝐷𝑡 and 𝜎𝑡 are time-dependent drift, continuous dividend yield and volatility functions, and let 𝑟𝑡 be the time-dependent risk-free interest rate from a money-market account. Under the risk-neutral measure ℚ, deduce that the European option price at time 𝑡, written on 𝑆𝑡 with strike price 𝐾 expiring at 𝑇 , 𝑇 > 𝑡 is 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝛿𝑆𝑡 𝑒− ∫𝑡

𝑇

𝐷𝑢 𝑑𝑢

Φ(𝛿𝑑+ ) − 𝛿𝐾𝑒− ∫𝑡

𝑇

𝑟𝑢 𝑑𝑢

where 𝑇

log(𝑆𝑡 ∕𝐾) + ∫𝑡 (𝑟𝑢 − 𝐷𝑢 ± 12 𝜎𝑢2 ) 𝑑𝑢 √ 𝑇 ∫𝑡 𝜎𝑢2 𝑑𝑢 { +1 for call option 𝛿= −1 for put option.

𝑑± =

Φ(𝛿𝑑− )

162

2.2.2 Black–Scholes Model

Solution: Under the risk-neutral measure ℚ, 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟𝑡 − 𝐷𝑡 )𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡ℚ 𝑆𝑡 𝑡

𝜇𝑢 − 𝑟 𝑢 𝑑𝑢 is a ℚ-standard Wiener process. ∫0 𝜎𝑢 Hence, from It¯o’s lemma

where 𝑊𝑡ℚ = 𝑊𝑡 +

𝑑 log 𝑆𝑡 =

𝑑𝑆𝑡 1 − 𝑆𝑡 2

(

𝑑𝑆𝑡 𝑆𝑡

)2 +…

1 = (𝑟𝑡 − 𝐷𝑡 )𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡ℚ − 𝜎𝑡2 𝑑𝑡 2 ) ( 1 2 = 𝑟𝑡 − 𝐷𝑡 − 𝜎𝑡 𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡ℚ . 2 Taking integrals 𝑇

∫𝑡

𝑑 log 𝑆𝑢 =

𝑇

(

) 1 𝑟𝑢 − 𝐷𝑢 − 𝜎𝑢2 𝑑𝑢 + ∫𝑡 2

∫𝑡

𝑇

𝜎𝑢 𝑑𝑊𝑢ℚ

or ( log

𝑆𝑇 𝑆𝑡

)

(

) 1 𝑟𝑢 − 𝐷𝑢 − 𝜎𝑢2 𝑑𝑢 + ∫𝑡 2

𝑇

=

∫𝑡

𝑇

𝜎𝑢 𝑑𝑊𝑢ℚ .

Since we can also write the It¯o integral as 𝑇

∫𝑡

𝜎𝑢 𝑑𝑊𝑢ℚ = lim

𝑛→∞

𝑛−1 ∑ 𝑖=0

𝜎𝑡𝑖 (𝑊𝑡ℚ − 𝑊𝑡ℚ ) 𝑖+1

𝑖

where 𝑡𝑖 = 𝑡 + 𝑖(𝑇 − 𝑡)∕𝑛, 𝑡 = 𝑡0 < 𝑡1 < 𝑡2 < ⋯ < 𝑡𝑛−1 < 𝑡𝑛 = 𝑇 , 𝑛 ∈ ℕ and due to the stationary increment of a standard Wiener process, each term of 𝑊𝑡ℚ − 𝑊𝑡ℚ ∼ 𝑖+1 𝑖 ) (  0, 𝑇 𝑛−𝑡 is normally distributed multiplied by a deterministic term, therefore we can deduce that ( ) 𝑇 𝑇 ℚ 2 𝜎𝑢 𝑑𝑊𝑢 ∼  0, 𝜎𝑢 𝑑𝑢 ∫𝑡 ∫𝑡 and hence ( log

𝑆𝑇 𝑆𝑡

)

[ ∼

𝑇

∫𝑡

(

) 1 𝑟𝑢 − 𝐷𝑢 − 𝜎𝑢2 𝑑𝑢, ∫𝑡 2

𝑇

] 𝜎𝑢2 𝑑𝑢

or ( log

𝑆𝑇 𝑆𝑡

) ∼

) ] [( 1 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡), 𝜎 2 (𝑇 − 𝑡) 2

2.2.2 Black–Scholes Model

163

where 𝑟=

1 𝑇 − 𝑡 ∫𝑡

𝑇

𝑟𝑢 𝑑𝑢,

𝐷=

1 𝑇 − 𝑡 ∫𝑡

𝑇

𝐷𝑢 𝑑𝑢,

𝜎2 =

1 𝑇 − 𝑡 ∫𝑡

𝑇

𝜎𝑢2 𝑑𝑢.

By analogy with the risk-neutral pricing methodology (see Problem 2.2.2.5, page 101), we can substitute the constants 𝑟, 𝐷 and 𝜎 of the Black–Scholes formula with 𝑟, 𝐷 and 𝜎, respectively so that the price of European call 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and put 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) options with strike 𝐾 are 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑− ) 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑− ) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑+ ) such that log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝑡) 𝑑± = . √ 𝜎 𝑇 −𝑡 Substituting the values of 𝑟, 𝐷 and 𝜎 2 we finally have 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒− ∫𝑡

𝑇

𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒− ∫𝑡

𝑇

where

𝐷𝑢 𝑑𝑢 𝐷𝑢 𝑑𝑢

Φ(𝑑+ ) − 𝐾𝑒− ∫𝑡

𝑇

𝑟𝑢 𝑑 𝑢

Φ(−𝑑+ ) − 𝐾𝑒− ∫𝑡

𝑇

Φ(𝑑− )

𝑟𝑢 𝑑 𝑢

Φ(−𝑑− )

𝑇

log(𝑆𝑡 ∕𝐾) + ∫𝑡 (𝑟𝑢 − 𝐷𝑢 ± 12 𝜎𝑢2 ) 𝑑𝑢 𝑑± = . √ 𝑇 ∫𝑡 𝜎𝑢2 𝑑𝑢

32. Non-Dividend-Paying Asset Price as Num´eraire. Let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 have the following diffusion process 𝑑𝑆𝑡 = (𝜇𝑡 − 𝐷𝑡 )𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡 𝑆𝑡 where 𝜇𝑡 , 𝐷𝑡 and 𝜎𝑡 are time-dependent drift, continuous dividend yield and volatility, functions, and let 𝑟𝑡 be the time-dependent risk-free interest rate from a money-market account. 𝑡 Show that under the risk-neutral measure ℚ, 𝑆𝑡 𝑒∫0 𝐷𝑢 𝑑 𝑢 follows a non-dividend diffusion process of the form ) ( 𝑡 𝑡 𝑡 𝑑 𝑆𝑡 𝑒∫0 𝐷𝑢 𝑑𝑢 = 𝑟𝑆𝑡 𝑒∫0 𝐷𝑢 𝑑𝑢 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑒∫0 𝐷𝑢 𝑑𝑢 𝑑𝑊𝑡ℚ where 𝑊𝑡ℚ = 𝑊𝑡 +

𝑡

∫0

𝜇𝑢 − 𝑟 𝑢 𝑑𝑢 is a ℚ-standard Wiener process. 𝜎𝑢

164

2.2.2 Black–Scholes Model

By definition the price of a European-style option with payoff Ψ(𝑆𝑇 ) at time 𝑡 < 𝑇 is ] [ | 𝑇 𝑉 (𝑆𝑡 , 𝑡) = 𝔼ℚ 𝑒− ∫𝑡 𝑟𝑢 𝑑𝑢 Ψ(𝑆𝑇 )|| ℱ𝑡 | where under the risk-neutral measure ℚ, the risk-free money-market account acts as the num´eraire. 𝑡 By denoting ℚ𝑆 as the equivalent martingale measure where the asset price 𝑆𝑡 𝑒∫0 𝐷𝑢 𝑑𝑢 is used as the num´eraire and using )−1 𝑡 ( 𝑡 𝑒∫0 𝑟𝑢 𝑑𝑢 𝑋𝑡 = 𝑆𝑡 𝑒∫0 𝐷𝑢 𝑑𝑢 as the discounted money-market account, show that under ℚ𝑆 the asset price 𝑆𝑡 now evolves according to the diffusion process 𝑑𝑆𝑡 ℚ = (𝑟𝑡 − 𝐷𝑡 + 𝜎𝑡2 )𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡 𝑆 𝑆𝑡 ℚ𝑆

where 𝑊𝑡

= 𝑊𝑡ℚ −

𝑡

𝜎 𝑑𝑢 is a ℚ𝑆 -standard Wiener process. ∫0 𝑢 Hence, show that for a strike price 𝐾 the price 𝐶𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) of a European asset-ornothing call with terminal payoff ⎧𝑆 ⎪ 𝑇 Ψ(𝑆𝑇 ) = ⎨ ⎪0 ⎩

if 𝑆𝑇 > 𝐾 if 𝑆𝑇 ≤ 𝐾

is given by 𝐶𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒− ∫𝑡

𝑇

𝐷𝑢 𝑑𝑢

Φ(𝑑+ )

where 𝑇

𝑥 log(𝑆𝑡 ∕𝐾) + ∫𝑡 (𝑟𝑢 − 𝐷𝑢 + 12 𝜎𝑢2 ) 𝑑𝑢 1 − 1 𝑢2 𝑑+ = and Φ(𝑥) = √ √ 𝑒 2 𝑑𝑢. ∫ 𝑇 −∞ 2𝜋 ∫𝑡 𝜎𝑢2 𝑑𝑢

Solution: From Girsanov’s theorem, under the risk-neutral measure ℚ, 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟𝑡 − 𝐷𝑡 )𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡ℚ 𝑆𝑡

2.2.2 Black–Scholes Model

where 𝑊𝑡ℚ = 𝑊𝑡 +

165 𝑡

∫0

𝜇𝑡 − 𝑟 𝑡 𝑑𝑢 is a ℚ-standard Wiener process. From It¯o’s lemma 𝜎𝑡

) ( 𝑡 𝑡 𝑡 𝑑 𝑆𝑡 𝑒∫0 𝐷𝑢 𝑑𝑢 = 𝐷𝑡 𝑆𝑡 𝑒∫0 𝐷𝑢 𝑑𝑢 𝑑𝑡 + 𝑒∫0 𝐷𝑢 𝑑𝑢 𝑑𝑆𝑡 [ ] 𝑡 𝑡 = 𝐷𝑡 𝑆𝑡 𝑒∫0 𝐷𝑢 𝑑𝑢 𝑑𝑡 + 𝑒∫0 𝐷𝑢 𝑑𝑢 (𝑟𝑡 − 𝐷𝑡 )𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡ℚ 𝑡

𝑡

= 𝑟𝑡 𝑆𝑡 𝑒∫0 𝐷𝑢 𝑑𝑢 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑒∫0 𝐷𝑢 𝑑𝑊𝑡ℚ . 𝑡

Given that 𝑆𝑡 𝑒∫0 𝐷𝑢 𝑑𝑢 is a non-dividend-paying asset and strictly positive, thus it can be used as a num´eraire. Under the change of num´eraire for a payoff Ψ(𝑆𝑇 ) [ (1) 𝑁𝑡(1) 𝔼ℚ

[ | ] | ] Ψ(𝑆𝑇 ) || Ψ(𝑆𝑇 ) || (2) ℚ(2) | ℱ𝑡 = 𝑁𝑡 𝔼 | ℱ𝑡 𝑁𝑇(1) || 𝑁𝑇(2) ||

where for 𝑖 = 1, 2, 𝑁 (𝑖) is a num´eraire and ℚ(𝑖) is the measure under which the asset prices discounted by 𝑁 (𝑖) are ℚ(𝑖) -martingales. Under the risk-neutral measure ℚ we have 𝑡

𝑇

𝑁𝑡(1) = 𝑒∫0 𝑟𝑢 𝑑𝑢 and 𝑁𝑇(1) = 𝑒∫0

𝑟𝑢 𝑑𝑢

and under the ℚ𝑆 measure 𝑡

𝑇

𝑁𝑡(2) = 𝑆𝑡 𝑒∫0 𝐷𝑢 𝑑𝑢 and 𝑁𝑇(2) = 𝑆𝑇 𝑒∫0

𝐷𝑢 𝑑𝑢

.

Hence, ] | | Ψ(𝑆𝑇 )| ℱ𝑡 𝑉 (𝑆𝑡 , 𝑡) = 𝔼 𝑒 | [ ] | 𝑡 Ψ(𝑆𝑇 ) | ∫0 𝑟𝑢 𝑑𝑢 ℚ 𝔼 =𝑒 | ℱ𝑡 𝑇 | 𝑒∫0 𝑟𝑢 𝑑𝑢 | [ | ] 𝑡 Ψ(𝑆𝑇 ) || ∫0 𝐷𝑢 𝑑𝑢 ℚ𝑆 = 𝑆𝑡 𝑒 𝔼 | ℱ𝑡 . 𝑇 𝑆𝑇 𝑒∫0 𝐷𝑢 𝑑𝑢 || ℚ

[

𝑇

− ∫𝑡 𝑟𝑢 𝑑𝑢

Given ⎧𝑆 ⎪ 𝑇 Ψ(𝑆𝑇 ) = ⎨ ⎪0 ⎩

if 𝑆𝑇 > 𝐾 if 𝑆𝑇 ≤ 𝐾

= 𝑆𝑇 1I{𝑆𝑇 >𝐾}

166

2.2.2 Black–Scholes Model

the asset-or-nothing call price under ℚ𝑆 is [

] 𝑆𝑇 1I{𝑆𝑇 >𝐾} || |ℱ 𝐶𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒 𝔼ℚ𝑆 | 𝑡 𝑇 ∫0 𝐷𝑢 𝑑𝑢 | 𝑆𝑇 𝑒 | ] [ 𝑇 | − ∫𝑡 𝐷𝑢 𝑑𝑢 ℚ𝑆 1I{𝑆𝑇 >𝐾} | ℱ𝑡 = 𝑆𝑡 𝑒 𝔼 | ( ) 𝑇 − ∫𝑡 𝐷𝑢 𝑑𝑢 | ℚ𝑆 𝑆𝑇 > 𝐾 | ℱ𝑡 . = 𝑆𝑡 𝑒 𝑡 ∫0 𝐷𝑢 𝑑𝑢

Under ℚ𝑆 the discounted money-market account is defined as )−1 𝑡 ( 𝑡 𝑒∫0 𝑟𝑢 𝑑𝑢 𝑋𝑡 = 𝑆𝑡 𝑒∫0 𝐷𝑢 𝑑𝑢 where 𝑋𝑡 is a ℚ𝑆 -martingale. Using It¯o’s lemma ) ( 𝑡 𝑑𝑋𝑡 = 𝑑 𝑆𝑡−1 𝑒∫0 (𝑟𝑢 −𝐷𝑢 )𝑑𝑢 𝑡

𝑡

= (𝑟𝑡 − 𝐷𝑡 )𝑆𝑡−1 𝑒∫0 (𝑟𝑢 −𝐷𝑢 )𝑑𝑢 𝑑𝑡 + 𝑒∫0 (𝑟𝑢 −𝐷𝑢 )𝑑𝑢 𝑑(𝑆𝑡−1 ) ( ( ) ) 𝑡 𝑡 𝑑𝑆𝑡 𝑑𝑆𝑡 −1 ∫0 (𝑟𝑢 −𝐷𝑢 )𝑑𝑢 −1 ∫0 (𝑟𝑢 −𝐷𝑢 )𝑑𝑢 𝑑𝑡 + 𝑆𝑡 𝑒 + − +… = (𝑟𝑡 − 𝐷𝑡 )𝑆𝑡 𝑒 𝑆𝑡 𝑆𝑡 [ ] = (𝑟𝑡 − 𝐷𝑡 )𝑋𝑡 𝑑𝑡 − 𝑋𝑡 (𝑟𝑡 − 𝐷𝑡 )𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡ℚ + 𝜎 2 𝑋𝑡 𝑑𝑡 = −𝜎𝑡 𝑋𝑡 𝑑𝑊𝑡ℚ + 𝜎𝑡2 𝑋𝑡 𝑑𝑡 ℚ𝑆

= −𝜎𝑡 𝑋𝑡 𝑑𝑊𝑡 ℚ𝑆

where 𝑊𝑡

= 𝑊𝑡ℚ −

𝑡

𝜎 𝑑𝑢 is a ℚ𝑆 -standard Wiener process. ∫0 𝑢 ℚ By substituting 𝑑𝑊𝑡ℚ = 𝑑𝑊𝑡 𝑆 + 𝜎𝑡 𝑑𝑡 into 𝑑𝑆𝑡 = (𝑟𝑡 − 𝐷𝑡 )𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑𝑊𝑡ℚ the asset price diffusion process under ℚ𝑆 becomes 𝑑𝑆𝑡 ℚ = (𝑟𝑡 − 𝐷𝑡 + 𝜎𝑡2 )𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡 𝑆 . 𝑆𝑡 To find the distribution of 𝑆𝑡 we note that 𝑑𝑆𝑡 1 − 𝑑(log 𝑆𝑡 ) = 𝑆𝑡 2

(

𝑑𝑆𝑡 𝑆𝑡

)2 +…

1 ℚ = (𝑟𝑡 − 𝐷𝑡 + 𝜎𝑡2 )𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡 𝑆 − 𝜎𝑡2 𝑑𝑡 2 ) ( 1 2 ℚ𝑆 = 𝑟𝑡 − 𝐷𝑡 + 𝜎𝑡 𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡 2 and taking integrals log 𝑆𝑇 = log 𝑆𝑡 +

𝑇

∫𝑡

(

) 1 𝑟𝑢 − 𝐷𝑢 + 𝜎𝑢2 𝑑𝑢 + ∫𝑡 2

𝑇

ℚ

𝜎𝑢 𝑑𝑊𝑢 𝑆 .

2.2.2 Black–Scholes Model ℚ𝑆

Since 𝑊𝑡

167

is a ℚ𝑆 -martingale we have 𝑇

ℚ 𝜎𝑢 𝑑𝑊𝑢 𝑆

∫𝑡

( ∼

𝑇

0,

∫𝑡

) 𝜎𝑢2 𝑑𝑢

and hence ( log 𝑆𝑇 ∼ 

log 𝑆𝑡 +

𝑇

∫𝑡

(

) 1 𝑟𝑢 − 𝐷𝑢 + 𝜎𝑢2 𝑑𝑢, ∫𝑡 2

𝑇

) 𝜎𝑢2 𝑑𝑢

.

Therefore, ( ) ℚ𝑆 𝑆𝑇 > 𝐾 || ℱ𝑡 ( ) 𝑇 = 𝑆𝑡 𝑒− ∫𝑡 𝐷𝑢 𝑑 𝑢 ℚ𝑆 log 𝑆𝑇 > log 𝐾 || ℱ𝑡

𝐶𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒− ∫𝑡

𝑇

𝐷𝑢 𝑑 𝑢

= 𝑆𝑡 𝑒− ∫𝑡

𝑇

𝐷𝑢 𝑑 𝑢

( ) | 𝑇 ⎛ ⎞ 1 2 ∫ 𝑟 𝑑𝑢 || ⎟ − − 𝐷 + 𝜎 log 𝐾 − log 𝑆 𝑡 𝑢 𝑢 𝑢 𝑡 ⎜ 2 |ℱ⎟ × ℚ𝑆 ⎜ 𝑍 > √ | 𝑡 𝑇 2 | ⎟ ⎜ ∫ 𝜎 𝑑𝑢 | ⎠ 𝑢 𝑡 ⎝ |

where 𝑍 ∼  (0, 1). Thus, 𝐶𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒− ∫𝑡

𝑇

𝐷𝑢 𝑑𝑢

Φ(𝑑+ )

where 𝑇

𝑥 log(𝑆𝑡 ∕𝐾) + ∫𝑡 (𝑟𝑢 − 𝐷𝑢 + 12 𝜎𝑢2 ) 𝑑𝑢 1 − 1 𝑢2 and Φ(𝑥) = 𝑑+ = √ √ 𝑒 2 𝑑𝑢. ∫−∞ 2𝜋 𝑇 ∫𝑡 𝜎𝑢2 𝑑𝑢

33. European Option Price Under Stochastic Interest Rate I. Let {𝑊𝑡𝑆 : 𝑡 ≥ 0} and {𝑊𝑡𝑟 : 𝑡 ≥ 0} be ℙ-standard Wiener processes on the probability space (Ω, ℱ , ℙ) and let the asset price 𝑆𝑡 have the following diffusion process 𝑑𝑆𝑡 = (𝜇𝑡 − 𝐷𝑡 )𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡𝑆 𝑆𝑡 where 𝜇𝑡 , 𝐷𝑡 and 𝜎𝑡 are time-dependent drift, continuous dividend yield and volatility functions and the risk-free interest rate 𝑟𝑡 is assumed to follow an Ornstein-Uhlenbeck process (or Vasicek process in the interest-rate modelling world) 𝑑𝑟𝑡 = 𝜅(𝜃 − 𝑟𝑡 )𝑑𝑡 + 𝛼𝑑𝑊𝑡𝑟

168

2.2.2 Black–Scholes Model

where 𝜅, 𝜃 and 𝛼 are constant parameters. In addition, we assume 𝑊𝑡𝑆 and 𝑊𝑡𝑟 are correlated with coefficient 𝜌 ∈ (−1, 1), 𝑑𝑊𝑡𝑆 ⋅ 𝑑𝑊𝑡𝑟 = 𝜌𝑑𝑡. Assume in the market there is a family of tradeable, risk-free zero-coupon bonds whose price at time 𝑡 is 𝑃 (𝑡, 𝑇 ) where they deliver a unit of currency at maturity 𝑇 , 𝑃 (𝑇 , 𝑇 ) = 1. The zero-coupon bond has the process 𝑑𝑃 (𝑡, 𝑇 ) = 𝜇𝑝 (𝑡, 𝑇 )𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 )𝑑𝑊𝑡𝑟 𝑃 (𝑡, 𝑇 ) where 𝜇𝑝 (𝑡, 𝑇 ) is the drift and 𝜎𝑝 (𝑡, 𝑇 ) is the volatility. ⟂ 𝑊𝑡𝑟 , show that (a) By defining {𝑌𝑡 : 𝑡 ≥ 0} as a ℙ-standard Wiener process where 𝑌𝑡 ⟂ we can write 𝑊𝑡𝑆 = 𝜌𝑊𝑡𝑟 +

√ 1 − 𝜌2 𝑌𝑡 .

(b) At time 𝑡, we consider a trader who has a portfolio valued at Π𝑡 holding 𝜙𝑡 units of risky asset 𝑆𝑡 and 𝜓𝑡 units invested in zero-coupon bonds. Using the two-dimensional Girsanov’s theorem, show that under the risk-neutral measure ℚ, 𝑑𝑆𝑡 ̃𝑆 = (𝑟𝑡 − 𝐷𝑡 )𝑑𝑡 + 𝜎𝑡 𝑑 𝑊 𝑆𝑡 𝑑𝑃 (𝑡, 𝑇 ) ̃𝑟 = 𝑟𝑡 𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 )𝑑 𝑊 𝑡 𝑃 (𝑡, 𝑇 ) ) ( ̃𝑟 𝑑𝑟𝑡 = 𝜅(𝜃 − 𝑟𝑡 ) − 𝛼𝛾𝑡 𝑑𝑡 + 𝛼𝑑 𝑊 𝑡 ̃ 𝑆 = 𝜌𝑊 ̃𝑟 + where 𝑊 𝑡

√ ̃𝑟 = 𝑊 𝑟 + 1 − 𝜌2 𝑌̃𝑡 , 𝑊 𝑡 𝑡

𝑡

∫0

𝛾𝑢 𝑑𝑢, 𝑌̃𝑡 = 𝑌𝑡 +

𝑡

∫0

𝜆𝑢 𝑑𝑢 are ℚ-

𝜇𝑝 (𝑡, 𝑇 ) − 𝑟𝑡 standard Wiener processes, 𝛾𝑡 = is the market price of interest risk and 𝜎𝑝 (𝑡, 𝑇 ) √ 𝜇 − 𝑟𝑡 . 𝜌𝛾𝑡 + 1 − 𝜌2 𝜆𝑡 = 𝑡 𝜎𝑡 (c) Using It¯o’s formula, show that the zero-coupon bond price 𝑃 (𝑡, 𝑇 ) satisfies the following PDE ) 𝜕𝑃 1 𝜕2𝑃 ( 𝜕𝑃 − 𝑟𝑡 𝑃 (𝑡, 𝑇 ) = 0 + 𝛼 2 2 + 𝜅(𝜃 − 𝑟𝑡 ) − 𝛼𝛾𝑡 𝜕𝑡 2 𝜕𝑟𝑡 𝜕𝑟𝑡 𝑃 (𝑇 , 𝑇 ) = 1.

2.2.2 Black–Scholes Model

169

(d) Using the Feynman-Kac formula, state the price of the zero-coupon bond 𝑃 (𝑡, 𝑇 ) at time 𝑡 as an expectation. 𝛼𝛾 , Assuming the market price of risk 𝛾𝑡 is a constant value 𝛾 and by setting 𝜃̃ = 𝜃 − 𝜅 show that the solution of 𝑃 (𝑡, 𝑇 ) can be written in the form 𝑃 (𝑡, 𝑇 ) = 𝑒𝐴(𝑡,𝑇 )−𝑟𝑡 𝐵(𝑡,𝑇 )

where ( 𝐴(𝑡, 𝑇 ) =

𝐵(𝑡, 𝑇 ) =

( ) ) 1 𝛼 2 𝛼2 (𝐵(𝑡, 𝑇 ) − 𝑇 + 𝑡) − 𝜃̃ − 𝐵(𝑡, 𝑇 )2 , 2 𝜅 4𝜅

) 1( 1 − 𝑒−𝜅(𝑇 −𝑡) 𝜅

𝜎𝑝 (𝑡, 𝑇 ) = −

and

) 𝛼( 1 − 𝑒−𝜅(𝑇 −𝑡) . 𝜅

(e) By denoting ℚ𝑇 as the 𝑇 -forward measure where the zero-coupon bond price is used as the num´eraire, show that under the ℚ𝑇 measure, the risky asset 𝑆𝑡 , zero-coupon bond 𝑃 (𝑡, 𝑇 ) and the interest rate 𝑟𝑡 follow the diffusions ) 𝑑𝑆𝑡 ( ̂𝑆 = 𝑟𝑡 − 𝐷𝑡 + 𝜌𝜎𝑡 𝜎𝑝 (𝑡, 𝑇 ) 𝑑𝑡 + 𝜎𝑡 𝑑 𝑊 𝑡 𝑆𝑡 ) 𝑑𝑃 (𝑡, 𝑇 ) ( ̂𝑟 = 𝑟𝑡 + 𝜎𝑝 (𝑡, 𝑇 )2 𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 )𝑑 𝑊 𝑡 𝑃 (𝑡, 𝑇 ) ( ) ̂𝑟 𝑑𝑟𝑡 = 𝜅(𝜃 − 𝑟𝑡 ) + 𝛼(𝜎(𝑡, 𝑇 ) − 𝛾𝑡 ) 𝑑𝑡 + 𝛼𝑑 𝑊 𝑡

̂𝑆 = 𝑊 ̃𝑆 − where 𝑊 𝑡 𝑡

𝑡

̂𝑟 = 𝑊 ̃𝑟 − 𝜌𝜎𝑝 (𝑢, 𝑇 ) 𝑑𝑢 and 𝑊 𝑡 𝑡

∫0 standard Wiener processes. (f) Finally, under ℚ𝑇 show that for a payoff

𝑡

∫0

𝜎𝑝 (𝑢, 𝑇 ) 𝑑𝑢 are ℚ𝑇 -

{ } Ψ(𝑆𝑇 ) = max 𝑆𝑇 − 𝐾, 0 the European call option price with strike price 𝐾 under stochastic interest rate is 𝐶stochIR (𝑆𝑡 , 𝑟𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒− ∫𝑡

𝑇

𝐷𝑢 𝑑𝑢

Φ(𝑑+ ) − 𝐾𝑃 (𝑡, 𝑇 )Φ(𝑑− )

170

2.2.2 Black–Scholes Model

where

𝑑± =

𝑇

( ) log 𝑆𝑡 ∕(𝑃 (𝑡, 𝑇 )𝐾) −

(

) 1 𝐷𝑢 ∓ 𝜎(𝑢, 𝑇 )2 𝑑𝑢 2

∫𝑡 √ 𝑇 ∫𝑡 𝜎(𝑢, 𝑇 )2 𝑑𝑢

√

𝜎𝑡2 + 𝜎𝑝 (𝑡, 𝑇 )2 − 2𝜌𝜎𝑡 𝜎𝑝 (𝑡, 𝑇 ) ) 𝛼( 𝜎𝑝 (𝑡, 𝑇 ) = − 1 − 𝑒−𝜅(𝑇 −𝑡) 𝜅 𝑥 1 − 21 𝑢2 𝑒 Φ(𝑥) = 𝑑𝑢. ∫−∞ √2𝜋 𝜎(𝑡, 𝑇 ) =

Solution: √ (a) By writing 𝑊𝑡𝑆 = 𝜌𝑊𝑡𝑟 + 1 − 𝜌2 𝑌𝑡 we have the following properties ) ( √ √ 𝔼(𝑊𝑡𝑆 ) = 𝔼 𝜌𝑊𝑡𝑟 + 1 − 𝜌2 𝑌𝑡 = 𝜌𝔼(𝑊𝑡𝑟 ) + 1 − 𝜌2 𝔼(𝑌𝑡 ) = 0 and Var(𝑊𝑡𝑆 ) = Var(𝜌𝑊𝑡𝑟 +

√ 1 − 𝜌2 𝑌𝑡 ) = 𝜌2 Var(𝑊𝑡𝑟 ) + (1 − 𝜌2 )Var(𝑌𝑡 ) = 𝑡

since 𝑊𝑡𝑟 ⟂ ⟂ 𝑌𝑡 Given both 𝑊𝑡𝑟 ∼  (0, 𝑡), 𝑌𝑡 ∼  (0, 𝑡) and 𝑊𝑡𝑟 ⟂ ⟂ 𝑌𝑡 therefore 𝜌𝑊𝑡𝑟 +

√ 1 − 𝜌2 𝑌𝑡 ∼  (0, 𝑡).

From It¯o’s formula and taking note that 𝑊𝑡𝑟 ⟂ ⟂ 𝑊𝑡 √ 1 − 𝜌2 𝑌𝑡 ) ⋅ 𝑑𝑊𝑡𝑟 √ = (𝜌𝑑𝑊𝑡𝑟 + 1 − 𝜌2 𝑑𝑌𝑡 ) ⋅ 𝑑𝑊𝑡𝑟 √ = 𝜌(𝑑𝑊𝑡𝑟 )2 + 1 − 𝜌2 𝑑𝑌𝑡 ⋅ 𝑑𝑊𝑡𝑟

𝑑𝑊𝑡𝑆 ⋅ 𝑑𝑊𝑡𝑟 = 𝑑(𝜌𝑊𝑡𝑟 +

= 𝜌𝑑𝑡. Thus, we can write 𝑊𝑡𝑆 = 𝜌𝑊𝑡𝑟 + (b) We first define,

√ 1 − 𝜌2 𝑌𝑡 .

𝑌̃𝑡 = 𝑌𝑡 +

𝑡

∫0

̃𝑟 = 𝑊 𝑟 + 𝑊 𝑡 𝑡

𝜆𝑢 𝑑𝑢

∫0

𝑡

𝛾𝑢 𝑑𝑢

where 𝜆𝑡 is the market price of asset risk and 𝛾𝑡 is the market price of interest rate risk. ̃ 𝑟. Since 𝑌𝑡 ⟂ ⟂ 𝑊𝑡𝑟 therefore we can easily deduce 𝑌̃𝑡 ⟂ ⟂𝑊 𝑡

2.2.2 Black–Scholes Model

171

From the two-dimensional Girsanov’s theorem, there exist a risk-neutral measure ℚ on the filtration ℱ𝑠 , 0 ≤ 𝑠 ≤ 𝑡 defined by the Radon-Nikod´ym process, 𝑡 𝑡 1 𝑡 2 1 𝑡 2 𝑟 𝑑ℚ = 𝑍𝑡 = 𝑒− 2 ∫0 𝜆𝑢 𝑑𝑢−∫0 𝜆𝑢 𝑑𝑌𝑢 ⋅ 𝑒− 2 ∫0 𝛾𝑢 𝑑𝑢−∫0 𝛾𝑢 𝑑𝑊𝑢 𝑑ℙ

̃ 𝑟 are ℚ-standard Wiener processes and 𝑌̃𝑡 ⟂ ̃ 𝑟. so that 𝑌̃𝑡 and 𝑊 ⟂𝑊 𝑡 𝑡 Let the portfolio Π𝑡 be defined as Π𝑡 = 𝜙𝑡 𝑆𝑡 + 𝜓𝑡 𝑃 (𝑡, 𝑇 ) where 𝜙𝑡 units are invested in risky asset 𝑆𝑡 and 𝜓𝑡 units are invested in zero-coupon bond 𝑃 (𝑡, 𝑇 ). Given the holder of the portfolio will receive 𝐷𝑡 𝑆𝑡 𝑑𝑡 for every risky asset held, thus ( ) 𝑑Π𝑡 = 𝜙𝑡 𝑑𝑆𝑡 + 𝐷𝑡 𝑆𝑡 𝑑𝑡 + 𝜓𝑡 𝑑𝑃 (𝑡, 𝑇 ) ( ) = 𝜙𝑡 𝜇𝑡 𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑𝑊𝑡𝑆 ( ) +𝜓𝑡 𝜇𝑝 (𝑡, 𝑇 )𝑃 (𝑡, 𝑇 )𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 )𝑃 (𝑡, 𝑇 )𝑑𝑊𝑡𝑟 [ ( )] √ = 𝑟𝑡 Π𝑡 𝑑𝑡 + 𝜙𝑡 𝑆𝑡 (𝜇𝑡 − 𝑟𝑡 )𝑑𝑡 + 𝜎𝑡 𝜌𝑑𝑊𝑡𝑟 + 1 − 𝜌2 𝑑𝑌𝑡 ) ] [( +𝜓𝑡 𝑃 (𝑡, 𝑇 ) 𝜇𝑝 (𝑡, 𝑇 ) − 𝑟𝑡 𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 )𝑑𝑊𝑡𝑟 By substituting ̃ 𝑟 − 𝛾𝑡 𝑑𝑡 𝑑𝑊𝑡𝑟 = 𝑑 𝑊 𝑡 𝑑𝑌𝑡 = 𝑑 𝑌̃𝑡 − 𝜆𝑡 𝑑𝑡 into 𝑑Π𝑡 we have 𝑑Π𝑡 = 𝑟𝑡 Π𝑡 𝑑𝑡 [ +𝜙𝑡 𝑆𝑡 (𝜇𝑡 − 𝑟𝑡 )𝑑𝑡 ( )] √ √ ̃ 𝑟 − 𝜌𝛾𝑡 𝑑𝑡 + 1 − 𝜌2 𝑑 𝑌̃𝑡 − 1 − 𝜌2 𝜆𝑡 𝑑𝑡 +𝜎𝑡 𝜌𝑑 𝑊 𝑡 [( ( )] ) ̃ 𝑟 − 𝛾𝑡 𝑑𝑡 +𝜓𝑡 𝑃 (𝑡, 𝑇 ) 𝜇𝑝 (𝑡, 𝑇 ) − 𝑟𝑡 𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 ) 𝑑 𝑊 𝑡 = 𝑟𝑡 Π𝑡 𝑑𝑡

[( ) ) ( √ +𝜙𝑡 𝑆𝑡 𝜇𝑡 − 𝑟𝑡 − 𝜌𝛾𝑡 + 1 − 𝜌2 𝜆𝑡 𝜎𝑡 𝑑𝑡 ( )] √ ̃ 𝑟 + 1 − 𝜌2 𝑑 𝑌̃𝑡 +𝜎𝑡 𝜌𝑑 𝑊 𝑡 ] [( ) ̃𝑟 . +𝜓𝑡 𝑃 (𝑡, 𝑇 ) 𝜇𝑝 (𝑡, 𝑇 ) − 𝑟𝑡 − 𝛾𝑡 𝜎𝑝 (𝑡, 𝑇 ) 𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 )𝑑 𝑊 𝑡 𝑡

By writing 𝐵𝑡 = 𝑒∫0 𝑟𝑢 𝑑𝑢 so that 𝑑𝐵𝑡 = 𝑟𝑡 𝐵𝑡 𝑑𝑡, and under the risk-neutral measure ̃ 𝑟 and 𝑊 ̃𝑡 are ℚ-martingales. In order for the discounted 𝐵 −1 Π𝑡 to ℚ, both 𝑊 𝑡 𝑡

172

2.2.2 Black–Scholes Model

be a ℚ-martingale, 𝑑(𝐵𝑡−1 Π𝑡 ) = −𝑟𝑡 𝐵𝑡−1 Π𝑡 𝑑𝑡 + 𝐵𝑡−1 𝑑Π𝑡 [( ) ) ( √ = 𝐵𝑡−1 𝜙𝑡 𝑆𝑡 𝜇𝑡 − 𝑟𝑡 − 𝜌𝛾𝑡 + 1 − 𝜌2 𝜆𝑡 𝜎𝑡 𝑑𝑡 ( )] √ ̃ 𝑟 + 1 − 𝜌2 𝑑 𝑌̃𝑡 +𝜎𝑡 𝜌𝑑 𝑊 𝑡 ] [( ) −1 ̃𝑟 +𝐵𝑡 𝜓𝑡 𝑃 (𝑡, 𝑇 ) 𝜇𝑝 (𝑡, 𝑇 ) − 𝑟𝑡 − 𝛾𝑡 𝜎𝑝 (𝑡, 𝑇 ) 𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 )𝑑 𝑊 𝑡 we require 𝜌𝛾𝑡 +

√ 𝜇 − 𝑟𝑡 1 − 𝜌2 𝜆𝑡 = 𝑡 𝜎𝑡

and

𝛾𝑡 =

𝜇𝑝 (𝑡, 𝑇 ) − 𝑟𝑡 𝜎𝑝 (𝑡, 𝑇 )

.

Hence, by substituting ̃ 𝑟 − 𝛾𝑡 𝑑𝑡, 𝑑𝑊𝑡𝑟 = 𝑑 𝑊 𝑡 𝑑𝑌𝑡 = 𝑑 𝑌̃𝑡 − 𝜆𝑡 𝑑𝑡, √ 𝜇 − 𝑟𝑡 𝜌𝛾𝑡 + 1 − 𝜌2 𝜆𝑡 = 𝑡 𝜎𝑡 into ( ) √ 𝑑𝑆𝑡 = (𝜇𝑡 − 𝐷𝑡 )𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝜌𝑑𝑊𝑡𝑟 + 1 − 𝜌2 𝑑𝑌𝑡 we have 𝑑𝑆𝑡 = (𝜇𝑡 − 𝐷𝑡 )𝑆𝑡 𝑑𝑡 ( ) √ √ ̃ 𝑟 − 𝜌𝛾𝑡 𝑑𝑡 + 1 − 𝜌2 𝑑 𝑌̃𝑡 − 1 − 𝜌2 𝜆𝑡 𝑑𝑡 +𝜎𝑡 𝑆𝑡 𝜌𝑑 𝑊 𝑡 ( ) √ ̃ 𝑟 + 1 − 𝜌2 𝑑 𝑌̃𝑡 = (𝜇𝑡 − 𝐷𝑡 )𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝜌𝑑 𝑊 𝑡 ( ) √ −𝜎𝑡 𝑆𝑡 𝜌𝛾𝑡 + 1 − 𝜌2 𝜆𝑡 𝑑𝑡 ( ) √ ̃ 𝑟 + 1 − 𝜌2 𝑑 𝑌̃𝑡 = (𝜇𝑡 − 𝐷𝑡 )𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝜌𝑑 𝑊 𝑡 ( ) 𝜇𝑡 − 𝑟 𝑡 −𝜎𝑡 𝑆𝑡 𝑑𝑡 𝜎𝑡 ( ) √ ̃ 𝑟 + 1 − 𝜌2 𝑑 𝑌̃𝑡 = (𝑟𝑡 − 𝐷𝑡 )𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝜌𝑑 𝑊 𝑡 ̃𝑆 = (𝑟𝑡 − 𝐷𝑡 )𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑 𝑊 𝑡 ̃ 𝑆 = 𝜌𝑊 ̃𝑟 + where 𝑊 𝑡 𝑡

√ 1 − 𝜌2 𝑌̃𝑡 .

2.2.2 Black–Scholes Model

173

On the other hand, by substituting ̃ 𝑟 − 𝛾𝑡 𝑑𝑡, 𝑑𝑊𝑡𝑟 = 𝑑 𝑊 𝑡 𝜇𝑝 (𝑡, 𝑇 ) − 𝑟𝑡 𝛾𝑡 = 𝜎𝑝 (𝑡, 𝑇 ) into 𝑑𝑃 (𝑡, 𝑇 ) = 𝜇𝑝 (𝑡, 𝑇 )𝑃 (𝑡, 𝑇 )𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 )𝑃 (𝑡, 𝑇 )𝑑𝑊𝑡𝑟 , 𝑑𝑟𝑡 = 𝜅(𝜃 − 𝑟𝑡 )𝑑𝑡 + 𝛼𝑑𝑊𝑡𝑟 the zero-coupon bond price and the instantaneous interest rate under the ℚ-measure become ( ) ̃ 𝑟 − 𝛾𝑡 𝑑𝑡 𝑑𝑃 (𝑡, 𝑇 ) = 𝜇𝑝 (𝑡, 𝑇 )𝑃 (𝑡, 𝑇 )𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 )𝑃 (𝑡, 𝑇 ) 𝑑 𝑊 𝑡

=

̃𝑟 𝜇𝑝 (𝑡, 𝑇 )𝑃 (𝑡, 𝑇 )𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 )𝑃 (𝑡, 𝑇 )𝑑 𝑊 𝑡 (

−𝜎𝑝 (𝑡, 𝑇 )

𝜇𝑝 (𝑡, 𝑇 ) − 𝑟𝑡 𝜎𝑝 (𝑡, 𝑇 )

)

𝑑𝑡

̃𝑟 = 𝑟𝑡 𝑃 (𝑡, 𝑇 )𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 )𝑃 (𝑡, 𝑇 )𝑑 𝑊 𝑡 and ( ) ̃ 𝑟 − 𝛾𝑡 𝑑𝑡 𝑑𝑟𝑡 = 𝜅(𝜃 − 𝑟𝑡 )𝑑𝑡 + 𝛼 𝑑 𝑊 𝑡 ) ( ̃ 𝑟. = 𝜅(𝜃 − 𝑟𝑡 ) − 𝛼𝛾𝑡 𝑑𝑡 + 𝛼𝑑 𝑊 𝑡 (c) From It¯o’s formula, 𝜕𝑃 1 𝜕2𝑃 𝜕𝑃 𝑑𝑟𝑡 + (𝑑𝑟𝑡 )2 + … 𝑑𝑡 + 𝜕𝑡 𝜕𝑟𝑡 2 𝜕𝑟2𝑡 [ ] 2 ) 𝜕𝑃 ( 𝜕𝑃 ̃ 𝑟 + 1 𝛼 2 𝜕 𝑃 𝑑𝑡 𝜅(𝜃 − 𝑟𝑡 ) − 𝛼𝛾𝑡 𝑑𝑡 + 𝛼𝑑 𝑊 𝑑𝑡 + = 𝑡 𝜕𝑡 𝜕𝑟𝑡 2 𝜕𝑟2𝑡 [ ] ) 𝜕𝑃 𝜕𝑃 𝜕𝑃 ̃ 𝑟 1 𝜕2𝑃 ( = 𝑑 𝑊𝑡 . 𝑑𝑡 + 𝛼 + 𝛼 2 2 + 𝜅(𝜃 − 𝑟𝑡 ) − 𝛼𝛾𝑡 𝜕𝑡 2 𝜕𝑟𝑡 𝜕𝑟𝑡 𝜕𝑟𝑡

𝑑𝑃 (𝑡, 𝑇 ) =

By equating coefficients with ̃𝑟 𝑑𝑃 (𝑡, 𝑇 ) = 𝑟𝑡 𝑃 (𝑡, 𝑇 )𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 )𝑃 (𝑡, 𝑇 )𝑑 𝑊 𝑡 we will have ) 𝜕𝑃 1 𝜕2𝑃 ( 𝜕𝑃 − 𝑟𝑡 𝑃 (𝑡, 𝑇 ) = 0 + 𝛼 2 2 + 𝜅(𝜃 − 𝑟𝑡 ) − 𝛼𝛾𝑡 𝜕𝑡 2 𝜕𝑟𝑡 𝜕𝑟𝑡

174

2.2.2 Black–Scholes Model

and 𝜎𝑝 (𝑡, 𝑇 ) = 𝛼

𝜕𝑃 𝑃 (𝑡, 𝑇 )−1 𝜕𝑟𝑡

with boundary condition 𝑃 (𝑇 , 𝑇 ) = 1. (d) From the Feynman-Kac formula, under the risk-neutral measure ℚ, the solution of the PDE at time 𝑡 is given by 𝑃 (𝑡, 𝑇 ) = 𝔼

ℚ

[ 𝑒

𝑇

− ∫𝑡 𝑟𝑢 𝑑𝑢

] [ ] | | 𝑇 ℚ − ∫𝑡 𝑟𝑢 𝑑𝑢 | | 𝑃 (𝑇 , 𝑇 )| ℱ𝑡 = 𝔼 𝑒 | ℱ𝑡 . | |

𝛼𝛾 , we therefore By setting 𝛾𝑡 = 𝛾 where 𝛾 is a constant value and writing 𝜃̃ = 𝜃 − 𝜅 have ̃ 𝑟. 𝑑𝑟𝑡 = 𝜅(𝜃̃ − 𝑟𝑡 )𝑑𝑡 + 𝛼𝑑 𝑊 𝑡 Solving the SDE directly, 𝑇

∫𝑡

𝑑𝑟𝑢 =

𝑇

∫𝑡

(

) 𝜅(𝜃̃ − 𝑟𝑢 ) 𝑑𝑢 + 𝛼

̃ − 𝑡) − 𝜅 𝑟𝑇 = 𝑟𝑡 = 𝜅 𝜃(𝑇

𝑇

∫𝑡

𝑇

∫𝑡

𝑟𝑢 𝑑𝑢 + 𝛼

̃𝑟 𝑑𝑊 𝑢 𝑇

∫𝑡

̃𝑟 𝑑𝑊 𝑢

which we finally have 𝑇

−

∫𝑡

𝑟𝑢 𝑑𝑢 =

1 ̃ − 𝑡) − 𝛼 (𝑟𝑇 − 𝑟𝑡 ) − 𝜃(𝑇 𝜅 𝜅 ∫𝑡

𝑇

̃ 𝑟. 𝑑𝑊 𝑢

From Problem 3.2.2.10 of Problems and Solutions of Mathematical Finance 1: Stochastic Calculus, we can write ) ( 𝑟𝑇 = 𝑟𝑡 𝑒−𝜅(𝑇 −𝑡) + 𝜃̃ 1 − 𝑒−𝜅(𝑇 −𝑡) +

𝑇

∫𝑡

̃𝑟 𝛼𝑒−𝜅(𝑇 −𝑢) 𝑑 𝑊 𝑢

and substituting it to the above equation 𝑇

−

∫𝑡

) ( ) 1 − 𝑒−𝜅(𝑇 −𝑡) 1 − 𝑒−𝜅(𝑇 −𝑡) + 𝜃̃ −𝑇 +𝑡 𝜅 𝜅 𝑇 ( ) 𝛼 ̃𝑟 + 𝑒−𝜅(𝑇 −𝑢) − 1 𝑑 𝑊 𝑢 ∫ 𝑘 𝑡 (

𝑟𝑢 𝑑𝑢 = 𝑟𝑡

2.2.2 Black–Scholes Model

175

or 𝑇

−

∫𝑡

𝑟𝑢 𝑑𝑢 = −𝑟𝑡 𝐵(𝑡, 𝑇 ) + 𝜃̃ (𝐵(𝑡, 𝑇 ) − 𝑇 + 𝑡) +

𝛼 𝑘 ∫𝑡

𝑇

( −𝜅(𝑇 −𝑢) ) ̃𝑟 𝑒 − 1 𝑑𝑊 𝑢

) 1( 1 − 𝑒−𝜅(𝑇 −𝑡) . where 𝐵(𝑡, 𝑇 ) = 𝜅 Taking mean, we have [ 𝔼

ℚ

𝑇

−

∫𝑡

| | 𝑟𝑢 𝑑𝑢| ℱ𝑡 | |

] = −𝑟𝑡 𝐵(𝑡, 𝑇 ) + 𝜃̃ (𝐵(𝑡, 𝑇 ) − 𝑇 + 𝑡)

since [ 𝔼

ℚ

𝛼 𝑘 ∫𝑡

𝑇

| ( −𝜅(𝑇 −𝑢) ) ̃ 𝑟 || ℱ𝑡 𝑒 − 1 𝑑𝑊 𝑢| |

] =0

and variance, [ Var

ℚ

𝑇

−

∫𝑡

| | 𝑟𝑢 𝑑𝑢| ℱ𝑡 | |

( Since 𝑟𝑇 ∼ 

𝑟𝑡 𝑒

−𝜅(𝑇 −𝑡)

]

[[ ]2 | ] ( )2 𝑇 ( | ) 𝛼 ℚ −𝜅(𝑇 −𝑠) 𝑟 ̃ | ℱ𝑡 𝑒 𝔼 − 1 𝑑𝑊 = | 𝑠 ∫𝑡 𝑘 | [ ]| ( )2 𝑇 ( )2 || 𝛼 𝑒−𝜅(𝑇 −𝑢) − 1 𝑑𝑢| ℱ𝑡 = 𝔼ℚ | ∫𝑡 𝑘 | ) ( )2 ( −𝜅(𝑇 −𝑡) 𝛼 1−𝑒 =− −𝑇 +𝑡 𝑘 𝜅 ( ) 𝑒−2𝜅(𝑇 −𝑡) 2𝑒−𝜅(𝑇 −𝑡) 𝛼2 1 + − − 2𝜅 𝜅 2 𝜅2 𝜅2 ( )2 𝛼2 𝛼 (𝐵(𝑡, 𝑇 ) − 𝑇 + 𝑡) − 𝐵(𝑡, 𝑇 )2 . =− 𝑘 2𝜅

) ) 𝛼2 [ ] ( −𝜅(𝑇 −𝑡) −2𝜅(𝑇 −𝑡) ̃ 1−𝑒 , +𝜃 1−𝑒 , we can deduce 𝜅

that 𝑇

−

∫𝑡

( 𝑟𝑢 𝑑𝑢 ∼  −𝑟𝑡 𝐵(𝑡, 𝑇 ) + 𝜃̃ (𝐵(𝑡, 𝑇 ) − 𝑇 + 𝑡) ,

) ( )2 𝛼 𝛼2 2 (𝐵(𝑡, 𝑇 ) − 𝑇 + 𝑡) − − 𝐵(𝑡, 𝑇 ) . 𝑘 2𝜅

176

2.2.2 Black–Scholes Model

Thus, 𝑃 (𝑡, 𝑇 ) = 𝔼 =𝑒

ℚ

[ 𝑒

| | ℱ𝑇 |

𝑇

− ∫𝑡 𝑟𝑢 𝑑𝑢 |

]

( )2 2 ̃ −𝑟𝑡 𝐵(𝑡,𝑇 )+𝜃(𝐵(𝑡,𝑇 )−𝑇 +𝑡)− 12 𝛼𝑘 (𝐵(𝑡,𝑇 )−𝑇 +𝑡)− 𝛼4𝜅 𝐵(𝑡,𝑇 )2

= 𝑒𝐴(𝑡,𝑇 )−𝑟𝑡 𝐵(𝑡,𝑇 ) where ( 𝐴(𝑡, 𝑇 ) =

( ) ) 1 𝛼 2 𝛼2 ̃ (𝐵(𝑡, 𝑇 ) − 𝑇 + 𝑡) − 𝜃− 𝐵(𝑡, 𝑇 )2 2 𝜅 4𝜅

and 𝐵(𝑡, 𝑇 ) =

) 1( 1 − 𝑒−𝜅(𝑇 −𝑡) . 𝜅

Thus, the volatility of the zero-coupon bond is given as 𝜕𝑃 𝑃 (𝑡, 𝑇 )−1 𝜕𝑟𝑡 = −𝛼𝐵(𝑡, 𝑇 ) ) 𝛼( = − 1 − 𝑒−𝜅(𝑇 −𝑡) . 𝜅

𝜎𝑝 (𝑡, 𝑇 ) = 𝛼

(e) Under the risk-neutral measure ℚ, the discounted zero-coupon bond 𝐵𝑡−1 𝑃 (𝑡, 𝑇 ) is a ℚ-martingale. Hence, we can deduce ) ( 𝑑 𝐵𝑡−1 𝑃 (𝑡, 𝑇 ) 𝐵𝑡−1 𝑃 (𝑡, 𝑇 )

𝑑𝑃 (𝑡, 𝑇 ) 𝑑𝐵𝑡 𝑑𝑃 (𝑡, 𝑇 ) 𝑑𝐵𝑡 − + − = 𝑃 (𝑡, 𝑇 ) 𝐵𝑡 𝑃 (𝑡, 𝑇 ) 𝐵𝑡

(

𝑑𝐵𝑡 𝐵𝑡

)2

̃ 𝑟 − 𝑟𝑡 𝑑𝑡 = 𝑟𝑡 𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 )𝑑 𝑊 𝑡 ̃𝑟 = 𝜎𝑝 (𝑡, 𝑇 )𝑑 𝑊 𝑡

is a ℚ-martingale. From It¯o’s formula, ( ( ) ) )2 ( −1 )) 𝑑 𝐵𝑡−1 𝑃 (𝑡, 𝑇 ) ( ( −1 1 𝑑 𝐵𝑡 𝑃 (𝑡, 𝑇 ) +… − 𝑑 log 𝐵𝑡 𝑃 (𝑡, 𝑇 ) = 2 𝐵𝑡−1 𝑃 (𝑡, 𝑇 ) 𝐵𝑡−1 𝑃 (𝑡, 𝑇 ) ̃ 𝑟 − 1 𝜎𝑝 (𝑡, 𝑇 )2 𝑑𝑡 = 𝜎𝑝 (𝑡, 𝑇 )𝑑 𝑊 𝑡 2 and solving the differential equation we have ( log

𝐵𝑡−1 𝑃 (𝑡, 𝑇 ) 𝐵0−1 𝑃 (0, 𝑇 )

)

𝑡

=

∫0

𝑡

̃𝑟 − 1 𝜎𝑝 (𝑢, 𝑇 ) 𝑑 𝑊 𝜎 (𝑢, 𝑇 )2 𝑑𝑢 𝑡 2 ∫0 𝑝

2.2.2 Black–Scholes Model

177

or 𝐵𝑡−1 𝑃 (𝑡, 𝑇 )

𝑡

̃𝑟 1

𝑡

= 𝑒∫0 𝜎𝑝 (𝑢,𝑇 )𝑑 𝑊𝑡 − 2 ∫0 𝜎𝑝 (𝑢,𝑇 )

𝐵0−1 𝑃 (0, 𝑇 )

2 𝑑𝑢

where 𝐵0 = 1. From Girsanov’s theorem, we can define a 𝑇 -forward measure ℚ𝑇 , given by the Radon-Nikod´ym derivative 𝑑ℚ𝑇 || 𝑃 (𝑡, 𝑇 ) = 𝑑ℚ ||ℱ𝑡 𝑃 (0, 𝑇 ) =

/

𝐵𝑡 𝐵0

𝐵𝑡−1 𝑃 (𝑡, 𝑇 ) 𝐵0−1 𝑃 (0, 𝑇 ) 𝑡

̃𝑟 1

𝑡

= 𝑒− ∫0 (−𝜎𝑝 (𝑢,𝑇 ))𝑑 𝑊𝑢 − 2 ∫0 𝜎𝑝 (𝑢,𝑇 ) ̂𝑟 = 𝑊 ̃𝑟 − so that 𝑊 𝑡 𝑡

2 𝑑𝑢

𝑡

𝜎 (𝑢, 𝑇 ) 𝑑𝑢 is a ℚ𝑇 -standard Wiener process. ∫0 𝑝 In a similar vein, we can also set ̃ 𝑟 = 𝜌𝑊 ̃𝑆 + 𝑊 𝑡 𝑡

√ ̃𝑡 1 − 𝜌2 𝑍

̃ 𝑆 and 𝑍 ̃𝑡 are ℚ-standard Wiener processes and 𝑊 ̃𝑆 ⟂ ̃ where 𝑊 𝑡 𝑡 ⟂ 𝑍𝑡 . Thus, we can write ̃𝑟 𝑑𝑃 (𝑡, 𝑇 ) = 𝑟𝑡 𝑃 (𝑡, 𝑇 )𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 )𝑃 (𝑡, 𝑇 )𝑑 𝑊 𝑡 ) ( √ ̃ 𝑆 + 1 − 𝜌2 𝑑 𝑍 ̃𝑡 = 𝑟𝑡 𝑃 (𝑡, 𝑇 )𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 )𝑃 (𝑡, 𝑇 ) 𝜌𝑑 𝑊 𝑡 and we can deduce ( ( ) ) )2 ( −1 ( ( −1 )) 𝑑 𝐵𝑡−1 𝑃 (𝑡, 𝑇 ) 1 𝑑 𝐵𝑡 𝑃 (𝑡, 𝑇 ) 𝑑 log 𝐵𝑡 𝑃 (𝑡, 𝑇 ) = +… − 2 𝐵𝑡−1 𝑃 (𝑡, 𝑇 ) 𝐵𝑡−1 𝑃 (𝑡, 𝑇 ) ̃ 𝑆 − 1 𝜌2 𝜎𝑝 (𝑡, 𝑇 )2 𝑑𝑡 = 𝜌𝜎𝑝 (𝑡, 𝑇 )𝑑 𝑊 𝑡 2 √ 2 ̃𝑡 − 1 (1 − 𝜌2 )𝜎𝑝 (𝑡, 𝑇 )2 𝑑𝑡 + 1 − 𝜌 𝜎𝑝 (𝑡, 𝑇 )𝑑 𝑍 2 and hence 𝐵𝑡−1 𝑃 (𝑡, 𝑇 ) 𝐵0−1 𝑃 (0, 𝑇 )

𝑡

̃ 𝑆 − 1 ∫ 𝑡 𝜌2 𝜎𝑝 (𝑢,𝑇 )2 𝑑𝑢 2 0

= 𝑒∫0 𝜌𝜎𝑝 (𝑢,𝑇 )𝑑 𝑊𝑡 𝑡

×𝑒∫0

√

𝑡

̃𝑡 − 1 ∫ (1−𝜌2 )𝜎𝑝 (𝑢,𝑇 )2 𝑑𝑢 1−𝜌2 𝜎𝑝 (𝑢,𝑇 )𝑑 𝑍 2 0 .

178

2.2.2 Black–Scholes Model

From the two-dimensional Girsanov’s theorem, there exist a 𝑇 -forward measure ℚ𝑇 on the filtration ℱ𝑠 , 0 ≤ 𝑠 ≤ 𝑡 defined by the Radon-Nikod´ym process, 𝑡 2 𝑑ℚ𝑇 ̃𝑆 1 𝑡 2 = 𝑒− ∫0 (−𝜌𝜎𝑝 (𝑢,𝑇 ))𝑑 𝑊𝑢 − 2 ∫0 𝜌 𝜎𝑝 (𝑢,𝑇 ) 𝑑𝑢 𝑑ℚ 𝑡

√

×𝑒− ∫0 (− 𝑡

̂𝑆 = 𝑊 ̃𝑆 − so that 𝑊 𝑡 𝑡

𝑡

̃𝑢 − 1 ∫ (1−𝜌2 )𝜎𝑝 (𝑢,𝑇 )2 𝑑𝑢 1−𝜌2 𝜎𝑝 (𝑢,𝑇 ))𝑑 𝑍 2 0

̂𝑡 = 𝑍 ̃𝑡 − 𝜌𝜎𝑝 (𝑢, 𝑇 ) 𝑑𝑢 and 𝑍

∫0 ̂𝑆 ⟂ ̂ ℚ𝑇 -standard Wiener processes and 𝑊 𝑡 ⟂ 𝑍𝑡 . Hence, under the ℚ𝑇 measure,

𝑡√

∫0

1 − 𝜌2 𝜎𝑝 (𝑢, 𝑇 ) 𝑑𝑢 are

( ) 𝑑𝑆𝑡 ̂ 𝑆 + 𝜌𝜎𝑝 (𝑡, 𝑇 )𝑑𝑡 = (𝑟𝑡 − 𝐷𝑡 )𝑑𝑡 + 𝜎𝑡 𝑑 𝑊 𝑡 𝑆𝑡 ) ( ̂𝑆 , = 𝑟𝑡 − 𝐷𝑡 + 𝜌𝜎𝑡 𝜎𝑝 (𝑡, 𝑇 ) 𝑑𝑡 + 𝜎𝑡 𝑑 𝑊 𝑡 ( ) 𝑑𝑃 (𝑡, 𝑇 ) ̂ 𝑟 + 𝜎𝑝 (𝑡, 𝑇 )𝑑𝑡 = 𝑟𝑡 𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 ) 𝑑 𝑊 𝑡 𝑃 (𝑡, 𝑇 ) ) ( ̂𝑟 = 𝑟𝑡 + 𝜎𝑝 (𝑡, 𝑇 )2 𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 )𝑑 𝑊 𝑡 and ( ) ) ( ̂ 𝑟 + 𝜎𝑝 (𝑡, 𝑇 )𝑑𝑡 𝑑𝑟𝑡 = 𝜅(𝜃 − 𝑟𝑡 ) − 𝛼𝛾𝑡 𝑑𝑡 + 𝛼 𝑑 𝑊 𝑡 ) ( ̂ 𝑟. = 𝜅(𝜃 − 𝑟𝑡 ) + 𝛼(𝜎𝑝 (𝑡, 𝑇 ) − 𝛾𝑡 ) 𝑑𝑡 + 𝛼𝑑 𝑊 𝑡 (f) Under the change of num´eraire for a payoff Ψ(𝑆𝑇 ) [ (1) 𝑁𝑡(1) 𝔼ℚ

[ | ] | ] Ψ(𝑆𝑇 ) || Ψ(𝑆𝑇 ) || (2) ℚ(2) | ℱ𝑡 = 𝑁𝑡 𝔼 | ℱ𝑡 𝑁𝑇(1) || 𝑁𝑇(2) ||

where for 𝑖 = 1, 2, 𝑁 (𝑖) is a num´eraire and ℚ(𝑖) is the measure under which the asset prices discounted by 𝑁 (𝑖) are ℚ(𝑖) -martingales. Under the risk-neutral measure ℚ we have 𝑡

𝑁𝑡(1) = 𝑒∫0 𝑟𝑢 𝑑𝑢

and

𝑇

𝑁𝑇(1) = 𝑒∫0

𝑟𝑢 𝑑𝑢

and under the 𝑇 -forward measure ℚ𝑇 𝑁𝑡(2) = 𝑃 (𝑡, 𝑇 ) and

𝑁𝑇(2) = 𝑃 (𝑇 , 𝑇 ) = 1.

Hence, with the change of num´eraire from the risk-neutral measure ℚ to the 𝑇 forward measure ℚ𝑇 , the European call option price at time 𝑡 under stochastic

2.2.2 Black–Scholes Model

179

interest rate is ] | | max{𝑆𝑇 − 𝐾, 0}| ℱ𝑡 𝐶stochIR (𝑆𝑡 , 𝑟𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝔼 𝑒 | [ ] | 𝑡 max{𝑆 − 𝐾, 0} | 𝑇 = 𝑒∫0 𝑟𝑢 𝑑𝑢 𝔼ℚ | ℱ𝑡 𝑇 | | 𝑒∫0 𝑟𝑢 𝑑𝑢 [ ] max{𝑆𝑇 − 𝐾, 0} || ℱ = 𝑃 (𝑡, 𝑇 )𝔼ℚ𝑇 | 𝑡 𝑃 (𝑇 , 𝑇 ) | [ { }| ] 𝑆 | 𝑇 ℚ𝑇 = 𝑃 (𝑡, 𝑇 )𝔼 max − 𝐾, 0 | ℱ𝑡 . | 𝑃 (𝑇 , 𝑇 ) | ℚ

[

𝑇

− ∫𝑡 𝑟𝑢 𝑑𝑢

From It¯o’s formula, ) ( 𝑑 𝑃 (𝑡, 𝑇 )−1 𝑆𝑡 𝑃 (𝑡, 𝑇 )−1 𝑆𝑡

( ) ( )2 )( 𝑑𝑆𝑡 𝑑𝑃 (𝑡, 𝑇 ) 𝑑𝑆𝑡 𝑑𝑃 (𝑡, 𝑇 ) 𝑑𝑃 (𝑡, 𝑇 ) − − + 𝑆𝑡 𝑃 (𝑡, 𝑇 ) 𝑆𝑡 𝑃 (𝑡, 𝑇 ) 𝑃 (𝑡, 𝑇 ) ( ) 𝑆 ̂ = 𝑟𝑡 − 𝐷𝑡 + 𝜌𝜎𝑡 𝜎𝑝 (𝑡, 𝑇 ) 𝑑𝑡 + 𝜎𝑡 𝑑 𝑊 𝑡 ) ( 2 ̂ − 𝑟𝑡 + 𝜎𝑝 (𝑡, 𝑇 ) 𝑑𝑡 − 𝜎𝑝 (𝑡, 𝑇 )𝑑 𝑊𝑡𝑟 − 𝜌𝜎𝑡 𝜎𝑝 (𝑡, 𝑇 )𝑑𝑡 =

+𝜎𝑝 (𝑡, 𝑇 )2 𝑑𝑡 ̂ 𝑆 − 𝜎𝑝 (𝑡, 𝑇 )𝑑 𝑊 ̂𝑟 = −𝐷𝑡 𝑑𝑡 + 𝜎𝑡 𝑑 𝑊 𝑡 𝑡 = −𝐷𝑡 𝑑𝑡 + 𝜎(𝑡, 𝑇 )𝑑 𝑉̂𝑡

where 𝜎(𝑡, 𝑇 ) =

√

𝜎𝑡2 + 𝜎𝑝 (𝑡, 𝑇 )2 − 2𝜌𝜎𝑡 𝜎𝑝 (𝑡, 𝑇 )

and ̂ 𝑆 − 𝜎𝑝 (𝑡, 𝑇 )𝑑 𝑊 ̂𝑟 𝜎𝑡 𝑑 𝑊 𝑡 𝑡 𝑑 𝑉̂𝑡 = √ 𝜎𝑡2 + 𝜎𝑝 (𝑡, 𝑇 )2 − 2𝜌𝜎𝑡 𝜎𝑝 (𝑡, 𝑇 ) is a ℚ𝑇 -standard Wiener process. Hence, by solving the geometric Brownian motion process, ( log

𝑃 (𝑇 , 𝑇 )−1 𝑆𝑇 𝑃 (𝑡, 𝑇 )−1 𝑆𝑡

)

( ∼

𝑇

−

∫𝑡

(

) 1 𝐷𝑢 + 𝜎(𝑢, 𝑇 )2 𝑑𝑢, ∫𝑡 2

𝑇

) 𝜎(𝑢, 𝑇 ) 𝑑𝑢 . 2

Following Problem 1.2.2.7 of Problems and Solutions in Mathematical Finance, Stochastic Calculus, Volume 1, the European call option price at time 𝑡 under stochastic

180

2.2.2 Black–Scholes Model

interest rate is 𝐶stochIR (𝑆𝑡 , 𝑟𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒− ∫𝑡

𝑇

𝐷𝑢 𝑑𝑢

Φ(𝑑+ ) − 𝐾𝑃 (𝑡, 𝑇 )Φ(𝑑− )

where

𝑑± =

( ) log 𝑆𝑡 ∕(𝑃 (𝑡, 𝑇 )𝐾) −

𝑇

(

) 1 𝐷𝑢 ∓ 𝜎(𝑢, 𝑇 )2 𝑑𝑢 2

∫𝑡 √ 𝑇 ∫𝑡 𝜎(𝑢, 𝑇 )2 𝑑𝑢

𝜎(𝑡, 𝑇 ) =

√

,

𝜎𝑡2 + 𝜎𝑝 (𝑡, 𝑇 ) − 2𝜌𝜎𝑡 𝜎𝑝 (𝑡, 𝑇 )

and 𝜎𝑝 (𝑡, 𝑇 ) = −

) 𝛼( 1 − 𝑒−𝜅(𝑇 −𝑡) . 𝜅

34. European Option Price Under Stochastic Interest Rate II. Let {𝑊𝑡𝑆 : 𝑡 ≥ 0} and {𝑊𝑡𝑟 : 𝑡 ≥ 0} be ℙ-standard Wiener processes on the probability space (Ω, 𝐹 , ℙ) and let the asset price 𝑆𝑡 have the following diffusion process 𝑑𝑆𝑡 = (𝜇𝑡 − 𝐷𝑡 )𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡𝑆 𝑆𝑡 where 𝜇𝑡 , 𝐷𝑡 and 𝜎𝑡 are time-dependent drift, continuous dividend yield and volatility functions and the risk-free interest rate 𝑟𝑡 is assumed to follow an Ornstein-Uhlenbeck process (or Vasicek process in the interest-rate modelling world) 𝑑𝑟𝑡 = 𝜅(𝜃 − 𝑟𝑡 )𝑑𝑡 + 𝛼𝑑𝑊𝑡𝑟 where 𝜅, 𝜃 and 𝛼 are constant parameters. In addition, we assume 𝑊𝑡𝑆 and 𝑊𝑡𝑟 are correlated with coefficient 𝜌 ∈ (−1, 1), 𝑑𝑊𝑡𝑆 ⋅ 𝑑𝑊𝑡𝑟 = 𝜌𝑑𝑡. Assume that in the market there is a family of tradeable, risk-free zero-coupon bonds whose price at time 𝑡 is 𝑃 (𝑡, 𝑇 ) where they deliver a unit of currency at maturity 𝑇 , 𝑃 (𝑇 , 𝑇 ) = 1. The zero-coupon bond has the process 𝑑𝑃 (𝑡, 𝑇 ) = 𝜇𝑝 (𝑡, 𝑇 )𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 )𝑑𝑊𝑡𝑟 𝑃 (𝑡, 𝑇 ) where 𝜇𝑝 (𝑡, 𝑇 ) is the drift and 𝜎𝑝 (𝑡, 𝑇 ) is the volatility

2.2.2 Black–Scholes Model

181

(a) By considering a hedging portfolio involving an option 𝑉 (𝑆𝑡 , 𝑟𝑡 , 𝑡) under stochastic interest rate which can only be exercised on the option expiry time 𝑇 > 𝑡, a risky asset 𝑆𝑡 and a zero-coupon bond 𝑃 (𝑡, 𝑇 ) show that 𝑉 (𝑆𝑡 , 𝑟𝑡 , 𝑡) satisfies the following PDE 1 𝜕2𝑉 𝜕2𝑉 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 + (𝑟𝑡 − 𝐷𝑡 )𝑆𝑡 + 𝜎𝑡2 𝑆𝑡2 2 + 𝛼 2 2 + 𝜌𝛼𝜎𝑡 𝑆𝑡 𝜕𝑡 2 2 𝜕𝑆 𝜕𝑟 𝜕𝑆 𝜕𝑆𝑡 𝜕𝑟𝑡 𝑡 𝑡 𝑡 ) 𝜕𝑉 ( + 𝜅(𝜃 − 𝑟𝑡 ) − 𝛼𝛾𝑡 − 𝑟𝑡 𝑉 (𝑆𝑡 , 𝑟𝑡 , 𝑡) = 0 𝜕𝑟𝑡 with boundary condition 𝑉 (𝑆𝑇 , 𝑟𝑇 , 𝑇 ) = Ψ(𝑆𝑇 ) where 𝛾𝑡 =

𝜇𝑝 (𝑡, 𝑇 ) − 𝑟𝑡

is the market price of risk of the interest rate and Ψ(𝑆𝑇 ) is 𝜎𝑝 (𝑡, 𝑇 ) the option payoff. (b) Using It¯o’s formula, show that the zero-coupon bond price 𝑃 (𝑡, 𝑇 ) satisfies the following PDE ) 𝜕𝑃 1 𝜕2𝑃 ( 𝜕𝑃 − 𝑟𝑡 𝑃 (𝑡, 𝑇 ) = 0 + 𝛼 2 2 + 𝜅(𝜃 − 𝑟𝑡 ) − 𝛼𝛾𝑡 𝜕𝑡 2 𝜕𝑟𝑡 𝜕𝑟𝑡 with boundary condition 𝑃 (𝑇 , 𝑇 ) = 1. 𝛼𝛾 (c) Assuming 𝛾𝑡 = 𝛾 where 𝛾 is a constant value, 𝜃̃ = 𝜃 − and writing the price of a 𝜅 zero-coupon bond maturing at time 𝑇 in the affine function form 𝑃 (𝑡, 𝑇 ) = 𝑒𝐴(𝑡,𝑇 )−𝑟𝑡 𝐵(𝑡,𝑇 ) with boundary conditions 𝐴(𝑇 , 𝑇 ) = 0,

𝐵(𝑇 , 𝑇 ) = 0

find the functions of 𝐴(𝑡, 𝑇 ) and 𝐵(𝑡, 𝑇 ) by solving the PDE satisfied by 𝑃 (𝑡, 𝑇 ). Hence, show the volatility of the zero-coupon bond to be 𝜎𝑝 (𝑡, 𝑇 ) = −

) 𝛼( 1 − 𝑒−𝜅(𝑇 −𝑡) . 𝜅

(d) To reduce the dimensionality of the PDE satisfied by 𝑉 (𝑆𝑡 , 𝑟𝑡 , 𝑡), let 𝑆̂𝑡 =

𝑆𝑡 𝑃 (𝑡, 𝑇 )

and

𝑉̂ (𝑆̂𝑡 , 𝑡) =

𝑉 (𝑆𝑡 , 𝑟𝑡 , 𝑡) 𝑃 (𝑡, 𝑇 )

182

2.2.2 Black–Scholes Model

and using the zero-coupon bond price PDE, show that 𝑉̂ (𝑆̂𝑡 , 𝑡) satisfies 𝜕 𝑉̂ 𝜕 2 𝑉̂ 𝜕 𝑉̂ 1 − 𝐷𝑡 𝑆̂𝑡 𝑡 = 0 + 𝜎(𝑡, 𝑇 )2 𝑆̂𝑡 𝜕𝑡 2 𝜕 𝑆̂2 𝜕 𝑆̂𝑡 𝑡

with boundary condition 𝑉̂ (𝑆̂𝑡 , 𝑇 ) = Ψ(𝑆̂𝑇 ) where 𝜎(𝑡, 𝑇 ) =

√

𝜎𝑡2 + 𝜎𝑝 (𝑡, 𝑇 )2 − 2𝜌𝜎𝑡 𝜎𝑝 (𝑡, 𝑇 ) and Ψ(𝑆̂𝑇 ) =

(e) Finally, for a payoff

Ψ(𝑆𝑇 ) . 𝑃 (𝑇 , 𝑇 )

{ } Ψ(𝑆𝑇 ) = max 𝐾 − 𝑆𝑇 , 0 deduce that the European put option price with strike price 𝐾 under stochastic interest rate is 𝑃stochIR (𝑆𝑡 , 𝑟𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑃 (𝑡, 𝑇 )Φ(−𝑑− ) − 𝑆𝑡 𝑒− ∫𝑡

𝑇

𝐷𝑢 𝑑𝑢

Φ(−𝑑+ )

where

𝑑± =

( ) log 𝑆𝑡 ∕(𝑃 (𝑡, 𝑇 )𝐾) −

𝑇

(

) 1 𝐷𝑢 ∓ 𝜎(𝑢, 𝑇 )2 𝑑𝑢 2

∫𝑡 √ 𝑇 ∫𝑡 𝜎(𝑢, 𝑇 )2 𝑑𝑢

𝜎(𝑡, 𝑇 ) =

,

√ 𝜎𝑡2 + 𝜎𝑝 (𝑡, 𝑇 )2 − 2𝜌𝜎𝑡 𝜎𝑝 (𝑡, 𝑇 ),

𝜎𝑝 (𝑡, 𝑇 ) = −

) 𝛼( 1 − 𝑒−𝜅(𝑇 −𝑡) 𝜅

and 𝑥

Φ(𝑥) =

∫−∞

1 − 1 𝑢2 √ 𝑒 2 𝑑𝑢. 2𝜋

Solution: (a) We let the portfolio at time 𝑡 be Π𝑡 = 𝑉 (𝑆𝑡 , 𝑟𝑡 , 𝑡) − Δ1 𝑆𝑡 − Δ2 𝑃 (𝑡, 𝑇 ) where it consists of purchasing one unit of option, and selling Δ1 and Δ2 units of risky asset 𝑆𝑡 and zero-coupon bond 𝑃 (𝑡, 𝑇 ) respectively. Since the holder of the portfolio

2.2.2 Black–Scholes Model

183

receives 𝐷𝑡 𝑆𝑡 𝑑𝑡 for every asset held and because the investor holds −Δ1 𝑆𝑡 , the portfolio changes by an amount −Δ1 𝐷𝑡 𝑆𝑡 𝑑𝑡 and therefore the change in portfolio Π𝑡 is 𝑑Π𝑡 = 𝑑𝑉 − Δ1 (𝑑𝑆𝑡 + 𝐷𝑡 𝑆𝑡 𝑑𝑡) − Δ2 𝑑𝑃 (𝑡, 𝑇 ) = 𝑑𝑉 − Δ1 𝑑𝑆𝑡 − Δ1 𝐷𝑡 𝑆𝑡 𝑑𝑡 − Δ2 𝑑𝑃 (𝑡, 𝑇 ) ( ) = 𝑑𝑉 − Δ1 𝜇𝑡 𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑𝑊𝑡𝑆 ( ) −Δ2 𝜇𝑝 (𝑡, 𝑇 )𝑃 (𝑡, 𝑇 )𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 )𝑃 (𝑡, 𝑇 )𝑑𝑊𝑡𝑟 ) ( = 𝑑𝑉 − Δ1 𝜇𝑡 𝑆𝑡 + Δ2 𝜇𝑝 (𝑡, 𝑇 )𝑃 (𝑡, 𝑇 ) 𝑑𝑡 − Δ1 𝜎𝑡 𝑆𝑡 𝑑𝑊𝑡𝑆 −Δ2 𝜎𝑝 (𝑡, 𝑇 )𝑃 (𝑡, 𝑇 )𝑑𝑊𝑡𝑟 . Expanding 𝑉 (𝑆𝑡 , 𝑟𝑡 , 𝑡) using Taylor’s theorem 𝑑𝑉 =

𝜕𝑉 𝜕𝑉 1 𝜕2𝑉 1 𝜕2𝑉 𝜕𝑉 𝑑𝑆𝑡 + 𝑑𝑟𝑡 + (𝑑𝑡)2 + (𝑑𝑆𝑡 )2 𝑑𝑡 + 2 𝜕𝑡 𝜕𝑆𝑡 𝜕𝑟𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡2 +

1 𝜕2𝑉 𝜕2𝑉 𝜕2 𝑉 2 (𝑑𝑟 ) + (𝑑𝑡)(𝑑𝑆 ) + (𝑑𝑡)(𝑑𝑟𝑡 ) 𝑡 𝑡 2 𝜕𝑟2𝑡 𝜕𝑡𝜕𝑆𝑡 𝜕𝑡𝜕𝑟𝑡

+

𝜕2𝑉 (𝑑𝑆𝑡 )(𝑑𝑟𝑡 ) + … 𝜕𝑆𝑡 𝜕𝑟𝑡

and substituting 𝑑𝑆𝑡 = (𝜇𝑡 − 𝐷𝑡 )𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑𝑊𝑡𝑆 and 𝑑𝑟𝑡 = 𝜅(𝜃 − 𝑟𝑡 ) + 𝛼𝑑𝑊𝑡𝑟 , and applying It¯o’s lemma, ( 𝑑𝑉 =

1 𝜕2𝑉 𝜕2 𝑉 𝜕2𝑉 1 𝜕𝑉 + 𝜎𝑡2 𝑆𝑡2 2 + 𝛼 2 2 + 𝜌𝛼𝜎𝑡 𝑆𝑡 𝜕𝑡 2 2 𝜕𝑟𝑡 𝜕𝑆𝑡 𝜕𝑟𝑡 𝜕𝑆𝑡 ) 𝜕𝑉 𝜕𝑉 𝜕𝑉 𝜕𝑉 +(𝜇𝑡 − 𝐷𝑡 )𝑆𝑡 + 𝜅(𝜃 − 𝑟𝑡 ) 𝑑𝑊𝑡𝑆 + 𝛼 𝑑𝑊𝑡𝑟 . 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝜕𝑆𝑡 𝜕𝑟𝑡 𝜕𝑆𝑡 𝜕𝑟𝑡

Substituting back into 𝑑Π𝑡 and rearranging the terms, we have ( 𝑑Π𝑡 =

𝜕𝑉 1 𝜕2𝑉 𝜕2𝑉 𝜕2𝑉 1 + 𝜎𝑡2 𝑆𝑡2 2 + 𝛼 2 2 + 𝜌𝛼𝜎𝑡 𝑆𝑡 𝜕𝑡 2 2 𝜕𝑟𝑡 𝜕𝑆𝑡 𝜕𝑟𝑡 𝜕𝑆𝑡

) 𝜕𝑉 𝜕𝑉 +(𝜇𝑡 − 𝐷𝑡 )𝑆𝑡 + 𝜅(𝜃 − 𝑟𝑡 ) − Δ1 𝜇𝑡 𝑆𝑡 − Δ2 𝜇𝑝 (𝑡, 𝑇 )𝑃 (𝑡, 𝑇 ) 𝑑𝑡 𝜕𝑆𝑡 𝜕𝑟 ( 𝑡 ) ( ) 𝜕𝑉 𝜕𝑉 𝑆 +𝜎𝑡 𝑆𝑡 − Δ1 𝑑𝑊𝑡 + 𝛼 − Δ2 𝜎𝑝 (𝑡, 𝑇 )𝑃 (𝑡, 𝑇 ) 𝑑𝑊𝑡𝑟 . 𝜕𝑆𝑡 𝜕𝑟𝑡 To eliminate the random components, we set Δ1 =

𝜕𝑉 𝜕𝑆𝑡

and

Δ2 = 𝛼𝜎𝑝 (𝑡, 𝑇 )−1 𝑃 (𝑡, 𝑇 )−1

𝜕𝑉 𝜕𝑟𝑡

184

2.2.2 Black–Scholes Model

leading to [ 𝑑Π𝑡 =

𝜕𝑉 1 𝜕2𝑉 𝜕2𝑉 𝜕2𝑉 𝜕𝑉 1 − 𝐷𝑡 𝑆𝑡 + 𝜎𝑡2 𝑆𝑡2 2 + 𝛼 2 2 + 𝜌𝛼𝜎𝑡 𝑆𝑡 𝜕𝑡 2 2 𝜕𝑟𝑡 𝜕𝑆𝑡 𝜕𝑟𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 ( ) ] 𝜇𝑝 (𝑡, 𝑇 ) 𝜕𝑉 + 𝜅(𝜃 − 𝑟𝑡 ) − 𝛼 𝑑𝑡. 𝜎𝑝 (𝑡, 𝑇 ) 𝜕𝑟𝑡

Under no arbitrage condition, the return on the amount Π𝑡 invested in a interest rate would see a growth of 𝑑Π𝑡 = 𝑟𝑡 Π𝑡 𝑑𝑡 and hence we have ( ) 𝛼 𝜕𝑉 𝜕𝑉 − 𝑑𝑡 𝑟𝑡 𝑉 (𝑆𝑡 , 𝑟𝑡 , 𝑡) − 𝑆𝑡 𝜕𝑆𝑡 𝜎𝑝 (𝑡, 𝑇 ) 𝜕𝑟𝑡 [ 1 𝜕2 𝑉 𝜕2𝑉 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 − 𝐷𝑡 𝑆𝑡 + 𝜎𝑡2 𝑆𝑡2 2 + 𝛼 2 2 + 𝜌𝛼𝜎𝑡 𝑆𝑡 = 𝜕𝑡 2 2 𝜕𝑟𝑡 𝜕𝑆𝑡 𝜕𝑟𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 ( ) ] 𝜇𝑝 (𝑡, 𝑇 ) 𝜕𝑉 + 𝜅(𝜃 − 𝑟𝑡 ) − 𝛼 𝑑𝑡. 𝜎𝑝 (𝑡, 𝑇 ) 𝜕𝑟𝑡 Removing 𝑑𝑡 and rearranging the terms we finally have the Black-Scholes equation under stochastic interest rate 1 𝜕2𝑉 𝜕2𝑉 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 + (𝑟𝑡 − 𝐷𝑡 )𝑆𝑡 + 𝜎𝑡2 𝑆𝑡2 2 + 𝛼 2 2 + 𝜌𝛼𝜎𝑡 𝑆𝑡 𝜕𝑡 2 2 𝜕𝑟𝑡 𝜕𝑆𝑡 𝜕𝑟𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 ) 𝜕𝑉 ( + 𝜅(𝜃 − 𝑟𝑡 ) − 𝛼𝛾𝑡 − 𝑟𝑡 𝑉 (𝑆𝑡 , 𝑟𝑡 , 𝑡) = 0 𝜕𝑟𝑡 with boundary condition 𝑉 (𝑆𝑇 , 𝑟𝑇 , 𝑇 ) = Ψ(𝑆𝑇 ) where 𝛾𝑡 =

𝜇𝑝 (𝑡, 𝑇 ) − 𝑟𝑡

is the market price of risk of the interest rate. 𝜎𝑝 (𝑡, 𝑇 ) (b) To find the PDE satisfied by the zero-coupon bond 𝑃 (𝑡, 𝑇 ), from Taylor’s theorem 𝑑𝑃 (𝑡, 𝑇 ) =

1 𝜕2𝑃 𝜕𝑃 𝜕𝑃 𝑑𝑟𝑡 + (𝑑𝑟𝑡 )2 + … 𝑑𝑡 + 𝜕𝑡 𝜕𝑟𝑡 2 𝜕𝑟2𝑡

and substituting 𝑑𝑟𝑡 = 𝜅(𝜃 − 𝑟𝑡 )𝑑𝑡 + 𝛼𝑑𝑊𝑡𝑟 and using It¯o’s formula ( 𝑑𝑃 (𝑡, 𝑇 ) =

𝜕𝑃 𝜕𝑃 1 𝜕2𝑃 + 𝛼 2 2 + 𝜅(𝜃 − 𝑟𝑡 ) 𝜕𝑡 2 𝜕𝑟𝑡 𝜕𝑟𝑡

) 𝑑𝑡 + 𝛼

𝜕𝑃 𝑑𝑊𝑡𝑟 . 𝜕𝑟𝑡

2.2.2 Black–Scholes Model

185

By equating coefficients with 𝑑𝑃 (𝑡, 𝑇 ) = 𝜇𝑝 (𝑡, 𝑇 )𝑃 (𝑡, 𝑇 )𝑑𝑡 + 𝜎𝑝 (𝑡, 𝑇 )𝑃 (𝑡, 𝑇 )𝑑𝑊𝑡𝑟 we have ( ) 𝜕𝑃 𝜕𝑃 1 2 𝜕2𝑃 + 𝜅(𝜃 − 𝑟𝑡 ) + 𝛼 𝑃 (𝑡, 𝑇 )−1 𝜇𝑝 (𝑡, 𝑇 ) = 𝜕𝑡 2 𝜕𝑟2𝑡 𝜕𝑟𝑡 and 𝜎𝑝 (𝑡, 𝑇 ) = 𝛼𝑃 (𝑡, 𝑇 )−1

𝜕𝑃 . 𝜕𝑟𝑡

Substituting the expressions of 𝜇𝑝 (𝑡, 𝑇 ) and 𝜎𝑝 (𝑡, 𝑇 ) into the market price of risk formula 𝛾𝑡 =

𝜇𝑝 (𝑡, 𝑇 ) − 𝑟𝑡 𝜎𝑝 (𝑡, 𝑇 )

we will have ) 𝜕𝑃 1 𝜕2𝑃 ( 𝜕𝑃 − 𝑟𝑡 𝑃 (𝑡, 𝑇 ) = 0 + 𝛼 2 2 + 𝜅(𝜃 − 𝑟𝑡 ) − 𝛼𝛾𝑡 𝜕𝑡 2 𝜕𝑟𝑡 𝜕𝑟𝑡 with boundary condition 𝑃 (𝑇 , 𝑇 ) = 1. 𝛼𝛾 , we therefore (c) By setting 𝛾𝑡 = 𝛾 where 𝛾 is a constant value and writing 𝜃̃ = 𝜃 − 𝜅 have ( ) 𝜕𝑃 1 𝜕2𝑃 𝜕𝑃 − 𝑟𝑡 𝑃 (𝑡, 𝑇 ) = 0 + 𝛼 2 2 + 𝜅(𝜃̃ − 𝑟𝑡 ) 𝜕𝑡 2 𝜕𝑟𝑡 𝜕𝑟𝑡 with boundary condition 𝑃 (𝑇 , 𝑇 ) = 1. By substituting ) ( 𝜕𝐵 𝜕𝐴 𝜕𝑃 𝑃 (𝑡, 𝑇 ), = − 𝑟𝑡 𝜕𝑡 𝜕𝑡 𝜕𝑡

𝜕𝑃 = −𝐵(𝑡, 𝑇 )𝑃 (𝑡, 𝑇 ), 𝜕𝑟𝑡

into the PDE and equating coefficients we eventually have 𝜕𝐵 = 𝜅𝐵(𝑡, 𝑇 ) − 1 𝜕𝑡 1 𝜕𝐴 ̃ = 𝜅 𝜃𝐵(𝑡, 𝑇 ) − 𝛼 2 𝐵(𝑡, 𝑇 )2 . 𝜕𝑡 2

𝜕2𝑃 = 𝐵(𝑡, 𝑇 )2 𝑃 (𝑡, 𝑇 ) 𝜕𝑟2𝑡

186

2.2.2 Black–Scholes Model

Solving

𝜕𝐵 − 𝜅𝐵(𝑡, 𝑇 ) = −1 we let the integrating factor be 𝐼 = 𝑒−𝜅𝑡 , and thus 𝜕𝑡 ) 𝑑 ( −𝜅𝑡 𝑒 𝐵(𝑡, 𝑇 ) = −𝑒−𝜅𝑡 𝑑𝑡 𝑒−𝜅𝑡 𝑒−𝜅𝑡 𝐵(𝑡, 𝑇 ) = +𝐶 𝜅

where 𝐶 is a constant. At time 𝑡 = 𝑇 , 𝐵(𝑇 , 𝑇 ) = 0 and hence 𝐶 = − 𝐵(𝑡, 𝑇 ) =

Solving

𝑒−𝜅𝑇 . Thus, 𝜅

) 1( 1 − 𝑒−𝜅(𝑇 −𝑡) . 𝜅

𝜕𝐴 1 ̃ = 𝜅 𝜃𝐵(𝑡, 𝑇 ) − 𝛼 2 𝐵(𝑡, 𝑇 )2 and because 𝐴(𝑇 , 𝑇 ) = 0, we have 𝜕𝑡 2 𝐴(𝑡, 𝑇 ) =

1 2 𝛼 2 ∫𝑡

𝑇

𝐵(𝑢, 𝑇 ) 𝑑𝑢 − 𝜅 𝜃̃

𝑇

∫𝑡

𝐵(𝑢, 𝑇 ) 𝑑𝑢.

Since 𝑇

∫𝑡

1 𝜅 1 = 𝜅 1 = 𝜅

𝐵(𝑢, 𝑇 ) 𝑑𝑢 =

𝑇

( ) 1 − 𝑒−𝜅(𝑇 −𝑢) 𝑑𝑢 ∫𝑡 ( )) 1( 𝑇 −𝑡− 1 − 𝑒−𝜅(𝑇 −𝑡) 𝜅 (𝑇 − 𝑡 − 𝐵(𝑡, 𝑇 ))

and 𝑇

∫𝑡

1 𝜅2 1 = 2 𝜅 1 = 2 𝜅

𝐵(𝑢, 𝑇 )2 𝑑𝑢 =

𝑇

( ) 1 − 2𝑒−𝜅(𝑇 −𝑢) + 𝑒2𝜅(𝑇 −𝑢) 𝑑𝑢 ∫𝑡 ( ) 1 1 − 𝑒−2𝜅(𝑇 −𝑡) (𝑇 − 𝑡 − 2𝐵(𝑡, 𝑇 )) + 2𝜅 𝜅2 ( ) 𝜅 𝑇 − 𝑡 − 𝐵(𝑡, 𝑇 ) − 𝐵(𝑡, 𝑇 )2 2

we have ( ) ( ) 𝜅 1 𝛼 2 𝑇 − 𝑡 − 𝐵(𝑡, 𝑇 ) − 𝐵(𝑡, 𝑇 )2 − 𝜃̃ (𝑇 − 𝑡 − 𝐵(𝑡, 𝑇 )) 2 𝜅 2 ( ( )2 ) 𝛼 1 𝛼2 (𝐵(𝑡, 𝑇 ) − 𝑇 + 𝑡) − = 𝜃̃ − 𝐵(𝑡, 𝑇 )2 . 2 𝜅 4𝜅

𝐴(𝑡, 𝑇 ) =

2.2.2 Black–Scholes Model

187

Therefore, the bond price is 𝑃 (𝑡, 𝑇 ) = 𝑒𝐴(𝑡,𝑇 )−𝑟𝑡 𝐵(𝑡,𝑇 ) where ( ( ) ) 𝛼2 1 𝛼 2 ̃ (𝐵(𝑡, 𝑇 ) − 𝑇 + 𝑡) − 𝐵(𝑡, 𝑇 )2 𝐴(𝑡, 𝑇 ) = 𝜃 − 2 𝜅 4𝜅 and 𝐵(𝑡, 𝑇 ) =

) 1( 1 − 𝑒−𝜅(𝑇 −𝑡) . 𝜅

Furthermore, the volatility of the zero-coupon bond is given as 𝜎𝑝 (𝑡, 𝑇 ) = 𝛼

𝜕𝑃 𝑃 (𝑡, 𝑇 )−1 𝜕𝑟𝑡

= −𝛼𝐵(𝑡, 𝑇 ) =−

) 𝛼( 1 − 𝑒−𝜅(𝑇 −𝑡) . 𝜅

(d) By setting 𝑆̂𝑡 =

𝑆𝑡 𝑃 (𝑡, 𝑇 )

and

𝑉̂ (𝑆̂𝑡 , 𝑡) =

𝑉 (𝑆𝑡 , 𝑟𝑡 , 𝑡) 𝑃 (𝑡, 𝑇 )

we have the following 𝜕𝑉 𝜕𝑃 𝜕 𝑉̂ 𝜕 𝑉̂ 𝜕 𝑆̂𝑡 = 𝑉̂ (𝑆̂𝑡 , 𝑡) + 𝑃 (𝑡, 𝑇 ) + 𝑃 (𝑡, 𝑇 ) 𝜕𝑡 𝜕𝑡 𝜕𝑡 𝜕 𝑆̂𝑡 𝜕𝑡 𝑆𝑡 𝜕𝑃 𝜕 𝑉̂ 𝜕𝑃 𝜕 𝑉̂ = 𝑉̂ (𝑆̂𝑡 , 𝑡) + 𝑃 (𝑡, 𝑇 ) − 𝑃 (𝑡, 𝑇 ) 2 𝜕𝑡 𝜕𝑡 𝑃 (𝑡, 𝑇 ) 𝜕 𝑆̂ 𝜕𝑡 𝜕𝑃 𝜕 𝑉̂ 𝜕𝑃 𝜕 𝑉̂ = 𝑉̂ (𝑆̂𝑡 , 𝑡) + 𝑃 (𝑡, 𝑇 ) − 𝑆̂𝑡 , 𝜕𝑡 𝜕𝑡 𝜕 𝑆̂ 𝜕𝑡

𝑡

𝑡

𝜕 𝑉̂ 𝜕 𝑆̂𝑡 𝜕 𝑉̂ 𝜕𝑉 = 𝑃 (𝑡, 𝑇 ) = , 𝜕𝑆𝑡 𝜕𝑆 ̂ 𝜕 𝑆𝑡 𝑡 𝜕 𝑆̂𝑡 𝜕 𝜕2𝑉 = 𝜕𝑆𝑡 𝜕𝑆𝑡2

(

𝜕 𝑉̂ 𝜕 𝑆̂𝑡

) =

1 𝜕 2 𝑉̂ 𝜕 2 𝑉̂ 𝜕 𝑆̂𝑡 = , 𝑃 (𝑡, 𝑇 ) 𝜕 𝑆̂2 𝜕 𝑆̂2 𝜕𝑆𝑡 𝑡

𝑡

188

2.2.2 Black–Scholes Model

𝜕 𝑉̂ 𝜕 𝑆̂𝑡 ̂ ̂ 𝜕𝑃 𝜕𝑉 = 𝑃 (𝑡, 𝑇 ) + 𝑉 (𝑆𝑡 , 𝑡) 𝜕𝑟𝑡 𝜕𝑟𝑡 𝜕 𝑆̂𝑡 𝜕𝑟𝑡 = −𝑃 (𝑡, 𝑇 )

𝑆𝑡 𝜕𝑃 𝜕 𝑉̂ 𝜕𝑃 + 𝑉̂ (𝑆̂𝑡 , 𝑡) 2 𝜕𝑟 𝜕𝑟𝑡 ̂ 𝑃 (𝑡, 𝑇 ) 𝑡 𝜕 𝑆𝑡

𝜕𝑃 𝜕 𝑉̂ 𝜕𝑃 + 𝑉̂ (𝑆̂𝑡 , 𝑡) , 𝜕𝑟𝑡 𝜕 𝑆̂𝑡 𝜕𝑟𝑡 ( ) 𝑆̂𝑡 𝜕 2 𝑉̂ 𝜕𝑃 𝜕2𝑉 𝜕2𝑉 𝜕 𝜕 𝑉̂ 𝜕 2 𝑉̂ 𝜕 𝑆̂𝑡 = = =− , = 𝜕𝑆𝑡 𝜕𝑟𝑡 𝜕𝑟𝑡 𝜕𝑆𝑡 𝜕𝑟𝑡 𝜕 𝑆̂ 𝑃 (𝑡, 𝑇 ) 𝜕 𝑆̂2 𝜕𝑟𝑡 𝜕 𝑆̂2 𝜕𝑟𝑡 = −𝑆̂𝑡

𝑡

𝑡

𝑡

2 ̂ 𝜕 𝑆̂𝑡 𝜕𝑃 ̂ 2 𝜕 𝑆̂𝑡 𝜕 𝑉̂ 𝜕𝑃 𝜕2𝑉 ̂𝑡 𝜕 𝑉 𝜕𝑃 − 𝑆̂𝑡 𝜕 𝑉 𝜕 𝑃 + 𝜕 𝑉 = − − 𝑆 𝜕𝑟𝑡 𝜕 𝑆̂ 𝜕𝑟𝑡 𝜕𝑟2𝑡 𝜕𝑟𝑡 𝜕 𝑆̂𝑡 𝜕𝑟𝑡 𝜕 𝑆̂𝑡 𝜕𝑟2𝑡 𝜕 𝑆̂𝑡 𝜕𝑟𝑡 𝜕𝑟𝑡 𝑡

𝜕2𝑃 𝜕𝑟2𝑡 ( ) ( ) 𝑆̂𝑡2 𝜕 2 𝑉̂ 𝜕𝑃 2 𝜕 𝑆̂𝑡 𝜕𝑃 𝜕 𝑉̂ 𝜕𝑃 𝜕 𝑉̂ 𝜕 2 𝑃 =− + − 𝑆̂𝑡 𝜕𝑃 𝜕𝑟𝑡 𝜕 𝑆̂ 𝜕𝑟𝑡 𝑃 (𝑡, 𝑇 ) 𝜕 𝑆̂2 𝜕𝑟𝑡 𝜕 𝑆̂𝑡 𝜕𝑟2𝑡 𝑡 𝑡 ( ) 𝜕 𝑉̂ 𝜕 𝑆̂𝑡 𝜕𝑃 𝜕𝑃 𝜕2𝑃 + + 𝑉̂ (𝑆̂𝑡 , 𝑡) 2 𝜕𝑟𝑡 𝜕 𝑆̂𝑡 𝜕𝑃 𝜕𝑟𝑡 𝜕𝑟𝑡 ( ) 2 2 𝑆̂𝑡 𝜕 2 𝑉̂ 𝜕𝑃 𝜕2𝑃 𝜕 𝑉̂ 𝜕 2 𝑃 = − 𝑆̂𝑡 + 𝑉̂ (𝑆̂𝑡 , 𝑡) 2 . 2 𝑃 (𝑡, 𝑇 ) 𝜕 𝑆̂2 𝜕𝑟𝑡 𝜕𝑟𝑡 𝜕 𝑆̂𝑡 𝜕𝑟𝑡 +𝑉̂ (𝑆̂𝑡 , 𝑡)

𝑡

Substituting the above expressions into the Black-Scholes model under stochastic 𝜕𝑃 interest rate, dividing it by 𝑃 (𝑡, 𝑇 ) and taking note that 𝜎𝑝 (𝑡, 𝑇 ) = 𝛼𝑃 (𝑡, 𝑇 )−1 , we 𝜕𝑟𝑡 will eventually have ) 𝜕 2 𝑉̂ 𝜕 𝑉̂ 𝜕 𝑉̂ 1 ( 2 𝜎𝑡 + 𝜎𝑝 (𝑡, 𝑇 )2 − 2𝜌𝜎𝑡 𝜎𝑝 (𝑡, 𝑇 ) 𝑆̂𝑡2 − 𝐷𝑡 𝑆̂𝑡 + 𝜕𝑡 2 𝜕 𝑆̂2 𝜕 𝑆̂𝑡 𝑡

(

) ) 𝜕𝑃 𝑉̂ (𝑆̂𝑡 , 𝑡) 𝜕𝑃 1 2 𝜕2𝑃 ( + 𝜅(𝜃 − 𝑟 ) − 𝛼𝛾 − 𝑟 𝑃 (𝑡, 𝑇 ) + 𝛼 𝑡 𝑡 𝑡 𝜕𝑡 2 𝜕𝑟2𝑡 𝜕𝑟𝑡 𝑃 (𝑡, 𝑇 )

(

) ) 𝜕𝑃 𝑆̂𝑡 𝜕 𝑉̂ 1 2 𝜕2𝑃 ( 𝜕𝑃 + 𝜅(𝜃 − 𝑟 ) − 𝛼𝛾 − 𝑟 𝑃 (𝑡, 𝑇 ) = 0. + 𝛼 𝑡 𝑡 𝑡 𝜕𝑡 2 𝜕𝑟2𝑡 𝜕𝑟𝑡 𝑃 (𝑡, 𝑇 ) 𝜕 𝑆̂ 𝑡

+

−

Since ) 𝜕𝑃 𝜕𝑃 1 𝜕2𝑃 ( − 𝑟𝑡 𝑃 (𝑡, 𝑇 ) = 0 + 𝛼 2 2 + 𝜅(𝜃 − 𝑟𝑡 ) − 𝛼𝛾𝑡 𝜕𝑡 2 𝜕𝑟𝑡 𝜕𝑟𝑡

2.2.2 Black–Scholes Model

189

therefore we have the reduced Black-Scholes equation of the form 𝜕 𝑉̂ 1 𝜕 2 𝑉̂ 𝜕 𝑉̂ − 𝐷𝑡 𝑆̂𝑡 =0 + 𝜎(𝑡, 𝑇 )2 𝑆̂𝑡2 2 𝜕𝑡 2 𝜕 𝑆̂ 𝜕 𝑆̂ 𝑡

𝑡

with boundary condition 𝑉̂ (𝑆̂𝑇 , 𝑇 ) = Ψ(𝑆̂𝑇 ) where 𝜎(𝑡, 𝑇 )2 = 𝜎𝑡2 + 𝜎𝑝 (𝑡, 𝑇 )2 − 2𝜌𝜎𝑡 𝜎𝑝 (𝑡, 𝑇 )

and

Ψ(𝑆̂𝑇 ) =

Ψ(𝑆𝑇 ) . 𝑃 (𝑇 , 𝑇 )

(e) By writing 𝑃̂𝑏𝑠 (𝑆̂𝑡 , 𝑡; 𝐾, 𝑇 ) as the European put option price at time 𝑡 satisfying the reduced Black-Scholes equation with payoff 𝑃̂𝑏𝑠 (𝑆̂𝑇 , 𝑇 ; 𝐾, 𝑇 ) =

1 max{𝐾 − 𝑆𝑇 , 0} = max{𝐾 − 𝑆̂𝑇 } 𝑃 (𝑇 , 𝑇 )

𝑆𝑇 , 𝑃 (𝑇 , 𝑇 ) = 1, 𝐾 is the strike price and 𝑇 > 𝑡, then the solution 𝑃 (𝑇 , 𝑇 ) can be deduced as where 𝑆̂𝑇 =

𝑇

𝑃̂𝑏𝑠 (𝑆̂𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾Φ(−𝑑− ) − 𝑆̂𝑡 𝑒− ∫𝑡 𝐷𝑢 𝑑𝑢 Φ(−𝑑+ ) ) 𝑇 ( 1 𝐷𝑢 ∓ 𝜎(𝑢, 𝑇 )2 𝑑𝑢 log(𝑆̂𝑡 ∕𝐾) − ∫𝑡 2 𝑑± = . √ 𝑇 2 ∫𝑡 𝜎(𝑢, 𝑇 ) 𝑑𝑢 Hence, by setting back 𝑆̂𝑡 = 𝑃 (𝑡, 𝑇 )−1 𝑆𝑡 , the European put option price at time 𝑡 under stochastic interest rate, 𝑃stochIR (𝑆𝑡 , 𝑟𝑡 , 𝑡; 𝐾, 𝑇 ) is 𝑃stochIR (𝑆𝑡 , 𝑟𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑃 (𝑡, 𝑇 )𝑃̂𝑏𝑠 (𝑆̂𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑃 (𝑡, 𝑇 )Φ(−𝑑− ) − 𝑆̂𝑡 𝑃 (𝑡, 𝑇 )𝑒− ∫𝑡 = 𝐾𝑃 (𝑡, 𝑇 )Φ(−𝑑− ) − 𝑆𝑡

𝑇

𝑇

𝐷𝑢 𝑑𝑢

𝑒− ∫𝑡 𝐷𝑢 𝑑𝑢 Φ(−𝑑

Φ(−𝑑+ )

+)

where

𝑑± =

( ) log 𝑆𝑡 ∕(𝑃 (𝑡, 𝑇 )𝐾) − √

𝑇

𝑇

∫𝑡

(

) 1 𝐷𝑢 ∓ 𝜎(𝑢, 𝑇 )2 𝑑𝑢 2

∫𝑡 𝜎(𝑢, 𝑇 )2 𝑑𝑢

,

190

2.2.3 Tree-Based Methods

𝜎(𝑡, 𝑇 ) =

√

𝜎𝑡2 + 𝜎𝑝 (𝑡, 𝑇 ) − 2𝜌𝜎𝑡 𝜎𝑝 (𝑡, 𝑇 ),

and 𝜎𝑝 (𝑡, 𝑇 ) = −

2.2.3

) 𝛼( 1 − 𝑒−𝜅(𝑇 −𝑡) . 𝜅

Tree-Based Methods

1. Risk-Neutral Approach. At time 𝑡, the value of the stock is 𝑆𝑡 and at time 𝑇 > 𝑡, the price has moved to either 𝑢𝑆𝑡 or 𝑑𝑆𝑡 , 0 < 𝑑 < 𝑢. We assume under continuous compounding that there is a risk-free interest rate 𝑟 and the stock pays a continuous dividend yield 𝐷 where it follows a GBM process 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 such that 𝜇 is the drift, 𝜎 is the volatility and {𝑊𝑡 : 𝑡 ≥ 0} is the ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ). Suppose 0 < 𝑑 < 𝑒(𝑟−𝐷)(𝑇 −𝑡) < 𝑢, show that the risk-neutral probabilities for upward and downward movement of the stock price are 𝜋=

𝑒(𝑟−𝐷)(𝑇 −𝑡) − 𝑑 𝑢−𝑑

and

1−𝜋 =

𝑢 − 𝑒(𝑟−𝐷)(𝑇 −𝑡) 𝑢−𝑑

respectively. Prove that 0 < 𝜋 < 1. Hence, find the market price of a European call option 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) which expires at time 𝑇 with strike price 𝐾. Solution: We first need to find the risk-neutral probabilities 𝜋, 1 − 𝜋 for upward/downward movement of the stock price where, under the risk-neutral probability measure ℚ ( ) 𝔼ℚ 𝑆𝑇 || 𝑆𝑡 = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) . Since the stock price would grow to either 𝑢𝑆𝑡 with probability 𝜋 or 𝑑𝑆𝑡 with probability 1 − 𝜋, we therefore have 𝜋𝑢𝑆𝑡 + (1 − 𝜋)𝑑𝑆𝑡 = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) such that 𝜋=

𝑒(𝑟−𝐷)(𝑇 −𝑡) − 𝑑 𝑢−𝑑

and

1−𝜋 =

𝑢 − 𝑒(𝑟−𝐷)(𝑇 −𝑡) . 𝑢−𝑑

2.2.3 Tree-Based Methods

191

Since 0 < 𝑑 < 𝑒(𝑟−𝐷)(𝑇 −𝑡) < 𝑢 we have 𝑒(𝑟−𝐷)(𝑇 −𝑡) − 𝑑 > 0 and

𝑢−𝑑 >0

and hence, 𝜋 > 0. In addition, because 0 < 𝑒(𝑟−𝐷)(𝑇 −𝑡) − 𝑑 < 𝑢 − 𝑑, therefore 𝜋 < 1. Hence, the European call option price at time 𝑡 is [ { }| ] 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝑆𝑇 − 𝐾, 0 | 𝑆𝑡 | [ { } { }] −𝑟(𝑇 −𝑡) 𝜋 max 𝑆𝑇 − 𝐾, 0 + (1 − 𝜋) max 𝑆𝑇 − 𝐾, 0 =𝑒 ) [( (𝑟−𝐷)(𝑇 −𝑡) } { −𝑑 𝑒 max 𝑢 𝑆𝑡 − 𝐾, 0 = 𝑒−𝑟(𝑇 −𝑡) 𝑢−𝑑 ) ( ] (𝑟−𝐷)(𝑇 −𝑡) } { 𝑢−𝑒 max 𝑑𝑆𝑡 − 𝐾, 0 . + 𝑢−𝑑

2. Self-Financing Trading Strategy Approach. At time 𝑡, the value of the stock is 𝑆𝑡 and at time 𝑇 > 𝑡, the price has moved to either 𝑢𝑆𝑡 or 𝑑𝑆𝑡 , 0 < 𝑑 < 𝑢. We assume under continuous compounding that there is a risk-free interest rate 𝑟 and the stock pays a continuous dividend yield 𝐷. Suppose 0 < 𝑑 < 𝑒(𝑟−𝐷)(𝑇 −𝑡) < 𝑢. Denoting 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) as the market price of a European put option which expires at time 𝑇 with strike price 𝐾, show by setting up a portfolio 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝜙𝑡 𝑆𝑡 + 𝜓𝑡 where 𝜙𝑡 is the number of units of 𝑆𝑡 and 𝜓𝑡 is the amount of cash invested in the money market that [ 𝜙𝑡 = 𝑒

−𝐷(𝑇 −𝑡)

𝑉 (𝑢𝑆𝑡 , 𝑇 ) − 𝑉 (𝑑𝑆𝑡 , 𝑇 ) (𝑢 − 𝑑)𝑆𝑡

]

and [ 𝜓𝑡 = 𝑒

−𝑟(𝑇 −𝑡)

] 𝑢𝑉 (𝑑𝑆𝑡 , 𝑇 ) − 𝑑𝑉 (𝑢 𝑆𝑡 , 𝑇 ) . 𝑢−𝑑

Hence, find the price of a European put option 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) which expires at time 𝑇 with strike price 𝐾. Solution: At time 𝑡, the portfolio is worth 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝜙𝑡 𝑆𝑡 + 𝜓𝑡 where 𝜙𝑡 is the number of units of 𝑆𝑡 and 𝜓𝑡 is the amount of cash invested in the money market.

192

2.2.3 Tree-Based Methods

Given that 𝑆𝑡 pays a continuous dividend yield 𝐷, then at time 𝑇 the number of stocks and the amount of cash will grow to 𝜙𝑡 𝑒𝐷(𝑇 −𝑡) and 𝜓𝑒𝑟(𝑇 −𝑡) , respectively. Because 𝑆𝑇 can either be 𝑆𝑇 = 𝑢𝑆𝑡 or 𝑆𝑇 = 𝑑𝑆𝑡 , we can write 𝑉 (𝑢𝑆𝑡 , 𝑇 ; 𝐾, 𝑇 ) = 𝜙𝑡 𝑒𝐷(𝑇 −𝑡) 𝑢𝑆𝑡 + 𝜓𝑡 𝑒𝑟(𝑇 −𝑡) and 𝑉 (𝑑𝑆𝑡 , 𝑇 ; 𝐾, 𝑇 ) = 𝜙𝑡 𝑒𝐷(𝑇 −𝑡) 𝑑𝑆𝑡 + 𝜓𝑡 𝑒𝑟(𝑇 −𝑡) . Hence, by solving the two equations we have [ 𝜙𝑡 = 𝑒−𝐷(𝑇 −𝑡)

𝑉 (𝑢𝑆𝑡 , 𝑇 ; 𝐾, 𝑇 ) − 𝑉 (𝑑𝑆𝑡 , 𝑇 ; 𝐾, 𝑇 ) (𝑢 − 𝑑)𝑆𝑡

]

and 𝜓𝑡 = 𝑒−𝑟(𝑇 −𝑡) (𝑉 (𝑢𝑆𝑡 , 𝑇 ; 𝐾, 𝑇 ) − 𝜙𝑡 𝑒𝐷(𝑇 −𝑡) 𝑢𝑆𝑡 ) ] [ −𝑟(𝑇 −𝑡) 𝑢𝑉 (𝑑𝑆𝑡 , 𝑇 ; 𝐾, 𝑇 ) − 𝑑𝑉 (𝑢𝑆𝑡 , 𝑇 ; 𝐾, 𝑇 ) =𝑒 . 𝑢−𝑑 Finally, because 𝑉 (𝑢𝑆𝑡 , 𝑇 ; 𝐾, 𝑇 ) = max{𝐾 − 𝑢𝑆𝑡 , 0}and 𝑉 (𝑑𝑆𝑡 , 𝑇 ; 𝐾, 𝑇 ) = max{𝐾 − 𝑑𝑆𝑡 , 0} therefore 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝜙𝑡 𝑆𝑡 + 𝜓𝑡 ) [( (𝑟−𝐷)(𝑇 −𝑡) } { −𝑑 𝑒 = 𝑒−𝑟(𝑇 −𝑡) max 𝐾 − 𝑢𝑆𝑡 , 0 𝑢−𝑑 ) ( ] (𝑟−𝐷)(𝑇 −𝑡) } { 𝑢−𝑒 max 𝐾 − 𝑑𝑆𝑡 , 0 . + 𝑢−𝑑

3. By referring to Problems 2.2.3.1 and 2.2.3.2 explain why, when we drop the assumption 0 < 𝑑 < 𝑒(𝑟−𝐷)(𝑇 −𝑡) < 𝑢, there is an arbitrage opportunity. Solution: To show that there is an arbitrage opportunity if we drop the assumption 0 < 𝑑 < 𝑒(𝑟−𝐷)(𝑇 −𝑡) < 𝑢 we first assume 𝑑 ≥ 𝑒(𝑟−𝐷)(𝑇 −𝑡) then, at time 𝑡, a speculator can borrow from the money market in order to buy stock worth 𝑆𝑡 . At time 𝑇 , the stock price is worth either 𝑢𝑆𝑡 or 𝑑𝑆𝑡 which is enough to pay off the money-market debt since 𝑢 > 𝑑 ≥ 𝑒(𝑟−𝐷)(𝑇 −𝑡) . Hence, this provides an arbitrage. In contrast, if 𝑢 ≤ 𝑒(𝑟−𝐷)(𝑇 −𝑡) , a speculator can short sell the stock worth 𝑆𝑡 at time 𝑡 and invest the proceeds in the money market whilst paying dividends to the owner of the stock. By time 𝑇 , the money invested is worth 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) and by replacing the cost of

2.2.3 Tree-Based Methods

193

the stock, and since 𝑑 < 𝑢 ≤ 𝑒(𝑟−𝐷)(𝑇 −𝑡) , the profit would either be 𝑆𝑡 (𝑒(𝑟−𝐷)(𝑇 −𝑡) − 𝑢) ≥ 0 or 𝑆𝑡 (𝑒(𝑟−𝐷)(𝑇 −𝑡) − 𝑑) ≥ 0 which is also an arbitrage. 4. Cox–Ross–Rubinstein Method. At time 𝑡, the value of the asset is 𝑆𝑡 and at time 𝑡 + Δ𝑡, Δ𝑡 > 0 the asset price is either increased to 𝑢𝑆𝑡 or decreased to 𝑑𝑆𝑡 , 0 < 𝑑 < 𝑢. In addition, under continuous compounding there is a risk-free interest rate 𝑟 and the asset pays a continuous dividend yield 𝐷. By assuming the asset price follows the geometric Brownian motion such that under the risk-neutral measure ℚ ( log

𝑆𝑡+Δ𝑡 𝑆𝑡

) ∼

[(

) ] 1 𝑟 − 𝐷 − 𝜎 2 Δ𝑡, 𝜎 2 Δ𝑡 2

where 𝜎 is the asset price volatility, show by using the first two moments of 𝑆𝑡+Δ𝑡 given 𝑆𝑡 that 𝜋𝑢 + (1 − 𝜋)𝑑 = 𝑒(𝑟−𝐷)Δ𝑡 𝜋𝑢2 + (1 − 𝜋)𝑑 2 = 𝑒(2(𝑟−𝐷)+𝜎

2 )Δ𝑡

where 𝜋 and 1 − 𝜋 are the risk-neutral probabilities of upward and downward movement of the asset price, respectively. 1 By setting 𝑢 = show that 𝑑 𝑢=𝐴+

√

𝐴2 − 1,

𝑑 =𝐴−

√

𝐴2 − 1 and

𝜋=

𝑒(𝑟−𝐷)Δ𝑡 − 𝑑 𝑢−𝑑

[ ] 2 where 𝐴 = 12 𝑒−(𝑟−𝐷)Δ𝑡 + 𝑒(𝑟−𝐷+𝜎 )Δ𝑡 . By expanding 𝑢 and 𝑑 up to (Δ𝑡) show that 𝑢 = 𝑒𝜎

√ Δ𝑡

and

𝑑 = 𝑒−𝜎

√ Δ𝑡

and to ensure 𝜋, 1 − 𝜋 ∈ (0, 1) deduce that 0 < Δ𝑡
0 the asset price is increased to 𝑢𝑆𝑡 , unchanged 𝑆𝑡 or decreased to 𝑑𝑆𝑡 , 0 < 𝑑 < 𝑢. In addition, under continuous compounding there is a riskfree interest rate 𝑟 and the asset pays a continuous dividend yield 𝐷. By assuming the asset price follows the geometric Brownian motion such that under the risk-neutral measure ℚ ( log

𝑆𝑡+Δ𝑡 𝑆𝑡

) ∼

[(

) ] 1 𝑟 − 𝐷 − 𝜎 2 Δ𝑡, 𝜎 2 Δ𝑡 2

200

2.2.3 Tree-Based Methods

( where 𝜎 is the asset price volatility, show by using the first two moments of log given 𝑆𝑡 that

𝑆𝑡+Δ𝑡 𝑆𝑡

)

) ( 1 𝜋1 log 𝑢 + 𝜋3 log 𝑑 = 𝑟 − 𝐷 − 𝜎 2 Δ𝑡 2 )2 ( 1 2 2 2 𝜋1 (log 𝑢) + 𝜋3 (log 𝑑) = 𝜎 Δ𝑡 + 𝑟 − 𝐷 − 𝜎 2 (Δ𝑡)2 2 𝜋 1 + 𝜋2 + 𝜋 3 = 1 where 𝜋1 , 𝜋2 and 𝜋3 are the risk-neutral probabilities of upward, unchanged and downward movement of the asset price, respectively. √ 1 By setting 𝑢 = and 𝑢 = 𝑒𝜆𝜎 Δ𝑡 , 𝜆 > 0 show by taking an approximation up to (Δ𝑡) 𝑑 that ( 1 1 𝑟−𝐷− + 2𝜆2 2𝜆𝜎 1 𝜋2 = 1 − 2 𝜆 ( 1 1 𝑟−𝐷− 𝜋3 = 2 − 2𝜆𝜎 2𝜆 𝜋1 =

) 1 2 √ Δ𝑡 𝜎 2 ) 1 2 √ Δ𝑡. 𝜎 2

Finally, to ensure 𝜋1 , 𝜋2 , 𝜋3 ∈ (0, 1) deduce that 𝜆>1

0
0. Solution: By definition, Γ𝑡 =

𝜕Δ𝑡 which can be approximated by 𝜕𝑆𝑡 Γ𝑡 ≈

Δ𝑡+𝛿𝑡 − Δ𝑡 . 𝑆𝑡+𝛿𝑡 − 𝑆𝑡

Hence, Δ𝑡+𝛿𝑡 ≈ Δ𝑡 + Γ𝑡 (𝑆𝑡+𝛿𝑡 − 𝑆𝑡 ) = 0.4127 + 0.1134 × (42.75 − 40) = 0.7246.

15. Consider a stock trading at 𝑆𝑡 = $49 with volatility 𝜎𝑡 = 35% at time 𝑡. A European put option 𝑃 (𝑆𝑡 , 𝑡) on the stock is priced at $2.3217 with vega 𝑡 = 0.3796. If the volatility increases by 0.5% at time 𝑡 + 𝛿𝑡, 𝛿𝑡 > 0, find the new put option price. Solution: By definition, 𝑡 =

𝜕𝑃 which can be approximated by 𝜕𝜎𝑡

𝑡 ≈

𝑃 (𝑆𝑡+𝛿𝑡 , 𝑡 + 𝛿𝑡) − 𝑃 (𝑆𝑡 , 𝑡) . 𝜎𝑡+𝛿𝑡 − 𝜎𝑡

2.2.4 The Greeks

245

Hence, 𝑃 (𝑆𝑡+𝛿𝑡 , 𝑡 + 𝛿𝑡) ≈ 𝑃 (𝑆𝑡 , 𝑡) + 𝑡 (𝜎𝑡+𝛿𝑡 − 𝜎𝑡 ) = 2.3217 − 0.3796 × 0.05 = 2.3407.

16. At time 𝑡, we have an asset price 𝑆𝑡 = $37 which pays a continuous dividend yield 𝐷 = 1.2% with volatility 𝜎 = 30%. By setting the risk-free interest rate 𝑟 = 8.5% we let 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the European call option price having a strike price 𝐾 = $35 with time to expiry 𝑇 − 𝑡 = 90 days, where 𝑇 is the option expiry time. From the information given in Table 2.3 find the price of the European put option at time 𝑡. Table 2.3

European call option Greek values.

European Call Option Greeks Δ𝑡 Γ𝑡 𝑡 Θ𝑡 (per calendar day)

Value 0.7110 0.0620 0.0624 −0.0146

Solution: From the Black–Scholes equation 𝜕𝐶 1 2 2 𝜕 2 𝐶 𝜕𝐶 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 or 1 Θ𝑡 + 𝜎 2 𝑆𝑡2 Γ𝑡 + (𝑟 − 𝐷)𝑆𝑡 Δ𝑡 − 𝑟𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0. 2 Given that Θ𝑡 = 0.0146 is quoted per calendar day, then per year Θ𝑡 = −0.0146 × 365 = −5.3290. Therefore, the call option price is ( ) 1 1 Θ𝑡 + 𝜎 2 𝑆𝑡2 Γ𝑡 + (𝑟 − 𝐷)𝑆𝑡 Δ𝑡 𝑟 2 ( 1 −5.3290 + 0.5 × 0.32 × 372 × 0.0620 = 0.085

𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) =

+(0.085 − 0.012) × 37 × 0.7110) = $4.8344.

246

2.2.4 The Greeks

From put–call parity, the put option price is 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) + 𝐾𝑒−𝑟(𝑇 −𝑡) 90

90

= 4.8344 − 37𝑒−0.012× 365 + 35𝑒−0.085× 365 = $2.2178.

17. Let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the evolution of the asset price 𝑆𝑡 have the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the asset price volatility. By setting 𝑟 as the risk-free interest rate, a derivative security written on this asset has price 𝑉 (𝑆𝑡 , 𝑡) with payoff Ψ(𝑆𝑇 ) at expiry time 𝑇 . 𝜕𝑉 Show that the interest-rate sensitivity 𝜌(𝑆𝑡 , 𝑡) = satisfies the differential equation 𝜕𝑟 𝐵𝑆 [𝜌(𝑆𝑡 , 𝑡)] = 𝑉 (𝑆𝑡 , 𝑡) − 𝑆𝑡

𝜕𝑉 𝜕𝑆𝑡

with boundary condition 𝜌(𝑆𝑇 , 𝑇 ) = 0 where 𝐵𝑆 denotes the differential operator 𝐵𝑆 =

𝜕 𝜕2 𝜕 1 − 𝑟. + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡

Interpret this result financially. By writing the solution of 𝜌(𝑆𝑡 , 𝑡) in the form 𝜌(𝑆𝑡 , 𝑡) = 𝑔(𝑡)𝑈 (𝑆𝑡 , 𝑡) with 𝑈 (𝑆𝑇 , 𝑇 ) = 0, show that [ ] 𝜕𝑉 𝜌(𝑆𝑡 , 𝑡) = −(𝑇 − 𝑡) 𝑉 (𝑆𝑡 , 𝑡) − 𝑆𝑡 . 𝜕𝑆𝑡 Solution: Since 𝑉 (𝑆𝑡 , 𝑡) satisfies 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡

2.2.4 The Greeks

247

then by differentiating the Black–Scholes equation with respect to 𝑟 we have 𝜕 𝜕𝑟

(

) 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

or ( ) ) ) ( ( 1 𝜕𝑉 𝜕 𝜕𝑉 𝜕𝑉 𝜕 2 𝜕𝑉 𝜕𝑉 𝜕 + 𝜎 2 𝑆𝑡2 2 + 𝑆𝑡 − 𝑉 (𝑆𝑡 , 𝑡) − 𝑟 + (𝑟 − 𝐷)𝑆𝑡 = 0. 𝜕𝑡 𝜕𝑟 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑟 𝜕𝑟 𝜕𝑆𝑡 𝜕𝑟 By setting 𝜌(𝑆𝑡 , 𝑡) =

𝜕𝑉 we have 𝜕𝑟

𝜕𝜌 1 2 2 𝜕 2 𝜌 𝜕𝜌 𝜕𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝜌(𝑆𝑡 , 𝑡) = 𝑉 (𝑆𝑡 , 𝑡) − 𝑆𝑡 . + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑉 (𝑆 , 𝑇 ) = 𝜌(𝑆𝑇 , 𝑇 ) = 0. Because 𝑉 (𝑆𝑇 , 𝑇 ) = Ψ(𝑆𝑇 ), thus 𝜕𝑟 𝑇 Hence, 𝐵𝑆 [𝜌(𝑆𝑡 , 𝑡)] = 𝑉 (𝑆𝑡 , 𝑡) − 𝑆𝑡

𝜕𝑉 𝜕𝑆𝑡

with boundary condition 𝜌(𝑆𝑇 , 𝑇 ) = 0. 𝜕𝑉 If we set the delta of the option = Δ then 𝜕𝑆𝑡 𝐵𝑆 [𝜌(𝑆𝑡 , 𝑡)] = 𝑉 (𝑆𝑡 , 𝑡) − Δ𝑆𝑡 which is equivalent to a hedging portfolio used to construct the Black–Scholes equation. By letting the solution of 𝜌(𝑆𝑡 , 𝑡) be written as 𝜌(𝑆𝑡 , 𝑡) = 𝑔(𝑡)𝑈 (𝑆𝑡 , 𝑡) where 𝑔(𝑡) is a function depending on time 𝑡, then by substituting it into the Black–Scholes differential operator 𝐵𝑆 [𝜌(𝑆𝑡 , 𝑡)] = 𝐵𝑆 [𝑔(𝑡)𝑈 (𝑆𝑡 , 𝑡)] =

𝜕 1 𝜕2 [𝑔(𝑡)𝑈 (𝑆𝑡 , 𝑡)] + 𝜎 2 𝑆𝑡2 2 [𝑔(𝑡)𝑈 (𝑆𝑡 , 𝑡)] 𝜕𝑡 2 𝜕𝑆𝑡 +(𝑟 − 𝐷)𝑆𝑡

=

𝜕 [𝑔(𝑡)𝑈 (𝑆𝑡 , 𝑡)] − 𝑟[𝑔(𝑡)𝑈 (𝑆𝑡 , 𝑡)] 𝜕𝑆𝑡

𝑑𝑔 𝑈 (𝑆𝑡 , 𝑡) + 𝑔(𝑡)𝐵𝑆 [𝑈 (𝑆𝑡 , 𝑡)]. 𝑑𝑡

248

2.2.4 The Greeks

𝜕𝑉 𝜕𝑉 and since both 𝑉 (𝑆𝑡 , 𝑡) and satisfy the Black– 𝜕𝑆𝑡 𝜕𝑆𝑡 Scholes equation (see Problem 2.2.2.9, page 110), therefore

By setting 𝑈 (𝑆𝑡 , 𝑡) = 𝑉 (𝑆𝑡 , 𝑡) − 𝑆𝑡

[ 𝐵𝑆 [𝑈 (𝑆𝑡 , 𝑡)] = 𝐵𝑆 [𝑉 (𝑆𝑡 , 𝑡)] − 𝐵𝑆

] 𝜕𝑉 𝑆𝑡 = 0. 𝜕𝑆𝑡

Thus, we can set [ ] 𝑑𝑔 𝜕𝑉 𝑉 (𝑆𝑡 , 𝑡) − 𝑆𝑡 𝐵𝑆 [𝜌(𝑆𝑡 , 𝑡)] = 𝑑𝑡 𝜕𝑆𝑡 𝑑𝑔 where = 1 with boundary condition 𝑔(𝑇 ) = 0. 𝑑𝑡 Solving the first-order differential equation and taking note that 𝑔(𝑇 ) = 0, we have 𝑔(𝑡) = −(𝑇 − 𝑡) and hence [ ] 𝜕𝑉 𝜌(𝑆𝑡 , 𝑡) = −(𝑇 − 𝑡) 𝑉 (𝑆𝑡 , 𝑡) − 𝑆𝑡 . 𝜕𝑆𝑡

18. Let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow a GBM process 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the asset price volatility. By setting 𝑟 as the risk-free interest rate, a derivative security written on this asset has price 𝑉 (𝑆𝑡 , 𝑡) with payoff Ψ(𝑆𝑇 ) at expiry time 𝑇 . 𝜕𝑉 Show that the volatility sensitivity (𝑆𝑡 , 𝑡) = satisfies 𝜕𝜎 𝐵𝑆 [(𝑆𝑡 , 𝑡)] + 𝜎𝑆𝑡2 Γ(𝑆𝑡 , 𝑡) = 0 with boundary condition (𝑆𝑇 , 𝑇 ) = 0 where Γ(𝑆𝑡 , 𝑡) =

𝜕2𝑉 𝜕𝑆𝑡2

is the gamma of the option price and 𝐵𝑆 denotes the differential operator 𝐵𝑆 =

𝜕 𝜕2 𝜕 1 − 𝑟. + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡

2.2.4 The Greeks

249

Let 𝑈 (𝑆𝑡 , 𝑡) be the price of another option that receives or pays 𝐾(𝑆𝑡 , 𝑡) per unit time (𝐾(𝑆𝑡 , 𝑡) > 0 denotes receiving an income whilst 𝐾(𝑆𝑡 , 𝑡) < 0 corresponds to a payment). By setting up a hedging portfolio show that 𝑈 (𝑆𝑡 , 𝑡) satisfies 𝐵𝑆 [𝑈 (𝑆𝑡 , 𝑡)] + 𝐾(𝑆𝑡 , 𝑡) = 0. Further, show that if 𝑈 (𝑆𝑇 , 𝑇 ) = 0 and 𝐾(𝑆𝑡 , 𝑡) > 0 for 0 ≤ 𝑡 < 𝑇 then 𝑈 (𝑆𝑡 , 𝑡) > 0 for 0 ≤ 𝑡 < 𝑇. Hence, deduce that if Γ(𝑆𝑡 , 𝑡) > 0 then (𝑆𝑡 , 𝑡) > 0 and finally show that (𝑆𝑡 , 𝑡) = (𝑇 − 𝑡)𝜎𝑆𝑡2 Γ(𝑆𝑡 , 𝑡). Solution: Since 𝑉 (𝑆𝑡 , 𝑡) satisfies 𝜕2 𝑉 𝜕𝑉 1 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 and by differentiating the Black–Scholes equation with respect to 𝜎 we have 𝜕 𝜕𝜎

(

) 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

or ( ) ) ) ( ( 𝜕𝑉 𝜕𝑉 𝜕 𝜕𝑉 1 𝜕2𝑉 𝜕 2 𝜕𝑉 𝜕 + 𝜎𝑆𝑡2 2 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 −𝑟 = 0. 𝜕𝑡 𝜕𝜎 2 𝜕𝜎 𝜕𝑆 𝜕𝜎 𝜕𝜎 𝜕𝑆𝑡 𝜕𝑆𝑡 𝑡 By setting (𝑆𝑡 , 𝑡) =

𝜕𝑉 𝜕2𝑉 we can write and Γ(𝑆𝑡 , 𝑡) = 𝜕𝜎 𝜕𝑆𝑡2

𝜕 𝜕 1 2 2 𝜕 2  + (𝑟 − 𝐷)𝑆𝑡 − 𝑟(𝑆𝑡 , 𝑡) + 𝜎𝑆𝑡2 Γ(𝑆𝑡 , 𝑡) = 0 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 and using the differential operator 𝐵𝑆 we eventually have 𝐵𝑆 [(𝑆𝑡 , 𝑡)] + 𝜎𝑆𝑡2 Γ(𝑆𝑡 , 𝑡) = 0 with boundary condition (𝑆𝑇 , 𝑇 ) =

𝜕𝑉 𝜕Ψ (𝑆 , 𝑇 ) = (𝑆 ) = 0. 𝜕𝜎 𝑇 𝜕𝜎 𝑇

At time 𝑡 we let the hedging portfolio Π𝑡 be constructed by taking a long position on option 𝑈 (𝑆𝑡 , 𝑡) and short Δ units of 𝑆𝑡 such that Π𝑡 = 𝑈 (𝑆𝑡 , 𝑡) − Δ𝑆𝑡 .

250

2.2.4 The Greeks

Since the asset pays a continuous dividend yield 𝐷 and the option 𝑈 (𝑆𝑡 , 𝑡) receives/pays 𝐾(𝑆𝑡 , 𝑡) per unit time, the instantaneous change in the portfolio is 𝑑Π𝑡 = 𝑑𝑈 + 𝐾(𝑆𝑡 , 𝑡)𝑑𝑡 − Δ(𝑑𝑆𝑡 + 𝐷𝑆𝑡 𝑑𝑡) = 𝑑𝑈 − Δ𝑑𝑆𝑡 + (𝐾(𝑆𝑡 , 𝑡) − Δ𝐷 𝑆𝑡 )𝑑𝑡. Expanding 𝑈 (𝑆𝑡 , 𝑡) using Taylor’s theorem we have 𝑑𝑈 =

𝜕𝑈 1 𝜕2𝑈 𝜕𝑈 𝑑𝑆𝑡 + 𝑑𝑆 2 + … 𝑑𝑡 + 𝜕𝑡 𝜕𝑆𝑡 2 𝜕𝑆𝑡2 𝑡

Substituting 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 and applying It¯o’s lemma ( ) 𝜕𝑈 𝜕𝑈 𝜕𝑈 1 2 2 𝜕 2 𝑈 + (𝜇 − 𝐷)𝑆𝑡 𝑑𝑊𝑡 . 𝑑𝑈 = + 𝜎 𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆 𝜕𝑆 𝜕𝑆𝑡 𝑡 𝑡 Hence, (

𝜕𝑈 1 2 2 𝜕 2 𝑈 𝜕𝑈 𝑑Π𝑡 = + (𝜇 − 𝐷)𝑆𝑡 + 𝐾(𝑆𝑡 , 𝑡) − 𝜇Δ𝑆𝑡 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 ) ( 𝜕𝑈 +𝜎𝑆𝑡 − Δ 𝑑𝑊𝑡 𝜕𝑆𝑡

) 𝑑𝑡

and to eliminate the risk we set Δ=

𝜕𝑈 𝜕𝑆𝑡

so that ( 𝑑Π𝑡 =

𝜕𝑈 1 2 2 𝜕 2 𝑈 𝜕𝑈 + (𝜇 − 𝐷)𝑆𝑡 + 𝐾(𝑆𝑡 , 𝑡) − 𝜇Δ𝑆𝑡 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡

) 𝑑𝑡.

Under the no-arbitrage condition the return on the amount Π𝑡 invested in a risk-free interest rate 𝑟 would be 𝑑Π𝑡 = 𝑟Π𝑡 𝑑𝑡

2.2.4 The Greeks

251

and taking into account that Δ =

𝜕𝑉 we have 𝜕𝑆𝑡 (

𝜕𝑈 1 2 2 𝜕 2 𝑈 𝜕𝑈 + (𝜇 − 𝐷)𝑆𝑡 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 ) +𝐾(𝑆𝑡 , 𝑡) − 𝜇Δ𝑆𝑡 𝑑𝑡 ( 𝜕𝑈 1 2 2 𝜕 2 𝑈 𝜕𝑈 𝑟(𝑈 (𝑆𝑡 , 𝑡) − Δ𝑆𝑡 )𝑑𝑡 = + (𝜇 − 𝐷)𝑆𝑡 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 ) +𝐾(𝑆𝑡 , 𝑡) − 𝜇Δ𝑆𝑡 𝑑𝑡 ( ) 𝜕𝑈 𝜕𝑈 𝜕𝑈 1 2 2 𝜕 2 𝑈 + (𝜇 − 𝐷)𝑆𝑡 𝑟 𝑈 (𝑆𝑡 , 𝑡) − 𝑆𝑡 + 𝜎 𝑆𝑡 = 2 𝜕𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝑟Π𝑡 𝑑𝑡 =

+𝐾(𝑆𝑡 , 𝑡) − 𝜇𝑆𝑡

𝜕𝑈 𝜕𝑆𝑡

and finally we have 𝜕𝑈 𝜕𝑈 1 2 2 𝜕 2 𝑈 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑈 (𝑆𝑡 , 𝑡) + 𝐾(𝑆𝑡 , 𝑡) = 0 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 or 𝐵𝑆 [𝑈 (𝑆𝑡 , 𝑡)] + 𝐾(𝑆𝑡 , 𝑡) = 0. Assume that if 𝑈 (𝑆𝑇 , 𝑇 ) = 0 and 𝐾(𝑆𝑡 , 𝑡) > 0 for 0 ≤ 𝑡 < 𝑇 then 𝑈 (𝑆𝑡 , 𝑡) ≤ 0 for 0 ≤ 𝑡 < 𝑇 . Let the portfolio be ̃ 𝑡 = 𝑈 (𝑆𝑡 , 𝑡) ≤ 0 Π where the instantaneous change in the portfolio becomes ̃ 𝑡 = 𝑑𝑈 (𝑆𝑡 , 𝑡) + 𝐾(𝑆𝑡 , 𝑡)𝑑𝑡. 𝑑Π Taking integrals and because 𝑈 (𝑆𝑇 , 𝑇 ) = 0 and 𝐾(𝑆𝑡 , 𝑡) > 0 for 0 ≤ 𝑡 < 𝑇 ̃𝑇 − Π ̃ 𝑡 = 𝑈 (𝑆𝑇 , 𝑇 ) − 𝑈 (𝑆𝑡 , 𝑡) + Π ̃ 𝑇 = 𝑈 (𝑆𝑇 , 𝑇 ) + Π 𝑇

=

∫𝑡 >0

𝑇

∫𝑡

𝐾(𝑆𝑡 , 𝑡)𝑑𝑡

𝑇

∫𝑡

𝐾(𝑆𝑡 , 𝑡)𝑑𝑡

𝐾(𝑆𝑡 , 𝑡)𝑑𝑡

252

2.2.4 The Greeks

which is a contradiction under the no-arbitrage condition. Therefore, 𝑈 (𝑆𝑡 , 𝑡) > 0 for 0 ≤ 𝑡 < 𝑇. By comparing 𝐵𝑆 [(𝑆𝑡 , 𝑡)] + 𝜎𝑆𝑡2 Γ(𝑆𝑡 , 𝑡) = 0,

(𝑆𝑇 , 𝑇 ) = 0

and 𝐵𝑆 [𝑈 (𝑆𝑡 , 𝑡)] + 𝐾(𝑆𝑡 , 𝑡) = 0,

𝑈 (𝑆𝑇 , 𝑇 ) = 0

we can deduce that if 𝐾(𝑆𝑡 , 𝑡) = 𝜎𝑆𝑡2 Γ(𝑆𝑡 , 𝑡) > 0 then 𝑈 (𝑆𝑡 , 𝑡) = (𝑆𝑡 , 𝑡) > 0 for 0 ≤ 𝑡 < 𝑇 . Hence, for any option price, if the gamma is always positive then it has a positive vega. 2 ̃ 𝑡 , 𝑡) where 𝐾(𝑆 ̃ 𝑡 , 𝑡) = −𝜎𝑆 2 Γ(𝑆𝑡 , 𝑡) and since 𝑆 2 𝜕 𝑉 satisfies Let (𝑆𝑡 , 𝑡) = 𝑔(𝑡)𝐾(𝑆 𝑡 𝑡 𝜕𝑆𝑡2 the Black–Scholes equation we have ̃ 𝑡 , 𝑡)] 𝐵𝑆 [(𝑆𝑡 , 𝑡)] = 𝐵𝑆 [𝑔(𝑡)𝐾(𝑆 =

2 𝜕 ̃ 𝑡 , 𝑡)] ̃ 𝑡 , 𝑡)] + 1 𝜎 2 𝑆 2 𝜕 [𝑔(𝑡)𝐾(𝑆 [𝑔(𝑡)𝐾(𝑆 𝑡 𝜕𝑡 2 𝜕𝑆𝑡2

+(𝑟 − 𝐷)𝑆𝑡

𝜕 ̃ 𝑡 , 𝑡)] − 𝑟[𝑔(𝑡)𝐾(𝑆 ̃ 𝑡 , 𝑡)] [𝑔(𝑡)𝐾(𝑆 𝜕𝑆𝑡

𝑑𝑔 ̃ ̃ 𝑡 , 𝑡)] 𝐾(𝑆𝑡 , 𝑡) + 𝑔(𝑡)𝐵𝑆 [𝐾(𝑆 𝑑𝑡 𝑑𝑔 ̃ 𝐾(𝑆𝑡 , 𝑡). = 𝑑𝑡 =

̃ 𝑡 , 𝑡) then Since 𝐵𝑆 [(𝑆𝑡 , 𝑡)] = 𝐾(𝑆 𝑑𝑔 =1 𝑑𝑡 with boundary condition 𝑔(𝑇 ) = 1. Solving the first-order differential equation we have 𝑔(𝑡) = −(𝑇 − 𝑡) and hence ̃ 𝑡 , 𝑡) = (𝑇 − 𝑡)𝜎𝑆 2 Γ(𝑆𝑡 , 𝑡). (𝑆𝑡 , 𝑡) = −(𝑇 − 𝑡)𝐾(𝑆 𝑡

2.2.4 The Greeks

253

19. Let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and at time 𝑡 let the asset price 𝑆𝑡 follow a GBM process 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the asset price volatility. Let the European call option 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) satisfy the following PDE 𝜕𝐶 𝜕𝐶 1 2 2 𝜕 2 𝐶 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0} and the European digital call option price 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) satisfy 𝜕𝐶 𝜕𝐶𝑑 1 2 2 𝜕 2 𝐶𝑑 + (𝑟 − 𝐷)𝑆𝑡 𝑑 − 𝑟𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝐶𝑑 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) =

{ 1 if 𝑆𝑇 > 𝐾 0 if 𝑆𝑇 ≤ 𝐾

where 𝑟 is the risk-free interest rate, 𝐾 is the strike price and 𝑇 is the option expiry time. Explain why, given 𝜕 max{𝑆𝑇 − 𝐾, 0} = 𝜕𝑆𝑇

{ 1 0

if 𝑆𝑇 > 𝐾 if 𝑆𝑇 ≤ 𝐾

𝜕𝐶 . 𝜕𝑆𝑡 ̂ 𝑡 , 𝑡; 𝐾, 𝑇 ) is the price of another European call option with strike 𝐾 and Suppose 𝐶(𝑆 expiry time 𝑇 where in this case the risk-free interest rate is ̂ 𝑟, the continuous dividend ̂ ̂ yield 𝐷 and volatility 𝜎. By choosing appropriate ̂ 𝑟 and 𝐷 in terms of 𝑟, 𝐷 and 𝜎 show that ̂ 𝜕 𝐶 ̂ 𝐶 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and Δ satisfy exactly the same Black–Scholes equa𝜕𝑆𝑡 tion and payoff. Hence, deduce that we have 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≠ Δ𝐶 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) where Δ𝐶 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) =

𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) Φ(𝑑− ) where log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) 𝑑− = √ 𝜎 𝑇 −𝑡

𝑥

and

Φ(𝑥) =

∫−∞

1 − 1 𝑢2 √ 𝑒 2 𝑑𝑢. 2𝜋

254

2.2.4 The Greeks

Solution: From the Black–Scholes equation 𝜕𝐶 1 2 2 𝜕 2 𝐶 𝜕𝐶 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0} and by differentiating with respect to 𝑆𝑡 𝜕 𝜕𝑆𝑡

(

𝜕𝐶 𝜕𝐶 1 2 2 𝜕 2 𝐶 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡

) =0

or (

( ) 𝜕2𝐶 1 2 2 𝜕2 𝜕𝐶 + 𝑆 𝜎 𝑡 𝜕𝑆𝑡2 2 𝜕𝑆𝑡2 𝜕𝑆𝑡 ( ) 𝜕 𝜕𝐶 𝜕𝐶 𝜕𝐶 + (𝑟 − 𝐷)𝑆𝑡 = 0. +(𝑟 − 𝐷) −𝑟 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕 𝜕𝑡

𝜕𝐶 𝜕𝑆𝑡

By setting Δ𝐶 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) =

)

+ 𝜎 2 𝑆𝑡

𝜕𝐶 we have 𝜕𝑆𝑡

𝜕Δ 𝜕Δ𝐶 1 2 2 𝜕 2 Δ𝐶 + (𝑟 − 𝐷 + 𝜎 2 )𝑆𝑡 𝐶 − 𝐷Δ𝐶 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 { Δ𝐶 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) =

1 if 𝑆𝑇 > 𝐾 0 if 𝑆𝑇 ≤ 𝐾

which is not the same problem that 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) satisfies. Hence, 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≠ Δ𝐶 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). ̂ 𝑡 , 𝑡; 𝐾, 𝑇 ), it satisfies For the case 𝐶(𝑆 ̂ 𝜕 𝐶̂ 1 2 2 𝜕 2 𝐶̂ ̂ 𝑡 𝜕𝐶 − ̂ ̂ 𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + (̂ 𝑟 − 𝐷)𝑆 𝑟𝐶(𝑆 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 ̂ 𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0} 𝐶(𝑆 ̂ 𝐶 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = and its delta Δ

𝜕 𝐶̂ satisfies 𝜕𝑆𝑡

̂ ̂𝐶 1 ̂𝐶 𝜕2 Δ 𝜕Δ 𝜕Δ ̂ + 𝜎 2 )𝑆𝑡 𝐶 − 𝐷 ̂Δ ̂ 𝐶 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + (̂ 𝑟−𝐷 + 𝜎 2 𝑆𝑡2 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 ̂ 𝐶 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = Δ

{ 1 0

if 𝑆𝑇 > 𝐾 if 𝑆𝑇 ≤ 𝐾.

2.2.4 The Greeks

255

̂ 𝐶 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and By comparing the PDEs and payoffs satisfied by 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and Δ in order for ̂ 𝐶 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = Δ ̂ = 𝑟 and ̂ ̂ + 𝜎 2 = 𝑟 − 𝐷. Hence, we can set 𝐷 𝑟−𝐷 ̂ 𝑟 = 2𝑟 − 𝐷 − 𝜎 2

and

̂ = 𝑟. 𝐷

Following Problem 2.2.4.2 (page 220) we have ̂ 𝐶 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = Δ 𝜕 ̂ = 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) 𝜕𝑆𝑡 ̂ = 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑̂+ )

̂ + 1 𝜎 2 )(𝑇 − 𝑡) 𝑟−𝐷 log(𝑆𝑡 ∕𝐾) + (̂ 2 . √ 𝜎 𝑇 −𝑡 ̂ = 𝑟 we have By substituting ̂ 𝑟 = 2𝑟 − 𝐷 − 𝜎 2 and 𝐷 where 𝑑̂+ =

̂

𝑒−𝐷(𝑇 −𝑡) = 𝑒−𝑟(𝑇 −𝑡) and 𝑑̂+ =

log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) = 𝑑− . √ 𝜎 𝑇 −𝑡

Hence, 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) Φ(𝑑− ). 20. Let {𝑊𝑡 : 𝑡 ≥ 0} be the standard Wiener process on the probability space (Ω, ℱ, ℙ) where at time 𝑡 the stock price 𝑆𝑡 follows the following diffusion process 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 for constant drift 𝜇, continuous dividend yield 𝐷 and volatility 𝜎. The risk-free interest rate is a constant 𝑟 and a derivative security written on this asset has price 𝑉 (𝑆𝑡 , 𝑡) with payoff Ψ(𝑆𝑇 ) at expiry time 𝑇 . By defining the sensitivity of the option price with respect to the continuous dividend yield 𝜑(𝑆𝑡 , 𝑡) =

𝜕𝑉 𝜕𝐷

256

2.2.4 The Greeks

show that 𝐵𝑆 [𝜑(𝑆𝑡 , 𝑡)] − 𝑆𝑡

𝜕𝑉 =0 𝜕𝑆𝑡

with boundary condition 𝜑(𝑆𝑇 , 𝑇 ) = 0 where 𝐵𝑆 denotes the Black–Scholes differential operator 𝐵𝑆 =

𝜕 𝜕2 𝜕 1 − 𝑟. + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡

Deduce that 𝜑(𝑆𝑡 , 𝑡) = −(𝑇 − 𝑡)𝑆𝑡

𝜕𝑉 𝜕𝑆𝑡

and give a financial interpretation of this result. Solution: Given that 𝑉 (𝑆𝑡 , 𝑡) satisfies 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 and by differentiating the Black–Scholes equation with respect to 𝐷 we have 𝜕 𝜕𝐷

(

) 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

or ( ) ) ) ( ( 𝜕 𝜕𝑉 1 𝜕𝑉 𝜕𝑉 𝜕 2 𝜕𝑉 𝜕𝑉 𝜕 + 𝜎 2 𝑆𝑡2 2 − 𝑆𝑡 −𝑟 + (𝑟 − 𝐷)𝑆𝑡 = 0. 𝜕𝑡 𝜕𝐷 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝐷 𝜕𝐷 𝜕𝑆𝑡 𝜕𝐷 By substituting 𝜑(𝑆𝑡 , 𝑡) =

𝜕𝑉 we have 𝜕𝐷

𝜕𝜑 𝜕𝜑 1 2 2 𝜕 2 𝜑 𝜕𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝜑(𝑆𝑡 , 𝑡) − 𝑆𝑡 =0 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 or in terms of the differential operator 𝐵𝑆 [𝜑(𝑆𝑡 , 𝑡)] − 𝑆𝑡

𝜕𝑉 =0 𝜕𝑆𝑡

with boundary condition 𝜑(𝑆𝑇 , 𝑇 ) =

𝜕 𝜕 𝑉 (𝑆𝑇 , 𝑇 ) = Ψ(𝑆𝑇 ) = 0. 𝜕𝐷 𝜕𝐷

2.2.4 The Greeks

257

Let 𝜑(𝑆𝑡 , 𝑡) = 𝑔(𝑡)𝑈 (𝑆𝑡 , 𝑡), where 𝑈 (𝑆𝑡 , 𝑡) = 𝑆𝑡

𝜕𝑉 . At expiry 𝜕𝑆𝑡

𝜑(𝑆𝑇 , 𝑇 ) = 𝑔(𝑇 )𝑈 (𝑆𝑇 , 𝑇 ) = 𝑔(𝑇 )𝑆𝑇

𝜕𝑉 = 0. 𝜕𝑆𝑇

Hence, 𝑔(𝑇 ) = 0. In addition, 𝐵𝑆 [𝜑(𝑆𝑡 , 𝑡)] = 𝐵𝑆 [𝑔(𝑡)𝑈 (𝑆𝑡 , 𝑡)] 1 𝜕 𝜕2 [𝑔(𝑡)𝑈 (𝑆𝑡 , 𝑡)] + 𝜎 2 𝑆𝑡2 2 [𝑔(𝑡)𝑈 (𝑆𝑡 , 𝑡)] 𝜕𝑡 2 𝜕𝑆𝑡 𝜕 + (𝑟 − 𝐷)𝑆𝑡 [𝑔(𝑡)𝑈 (𝑆𝑡 , 𝑡)] − 𝑟𝑔(𝑡)𝑈 (𝑆𝑡 , 𝑡) 𝜕𝑆𝑡 𝑑𝑔 = 𝑈 (𝑆𝑡 , 𝑡) + 𝑔(𝑡)𝐵𝑆 [𝑈 (𝑆𝑡 , 𝑡)] 𝑑𝑡 [ ] 𝑑𝑔 𝜕𝑉 𝜕𝑉 = + 𝑔(𝑡)𝐵𝑆 𝑆𝑡 𝑆 𝑑𝑡 𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑉 . = 𝑆𝑡 𝜕𝑆𝑡 =

[ ] 𝜕𝑉 = 0 (see Problem 2.2.2.9, page 110) we have Since 𝐵𝑆 𝑆𝑡 𝜕𝑆𝑡 𝑑𝑔 = 1. 𝑑𝑡 Because 𝑔(𝑇 ) = 0 we have 𝑔(𝑡) = −(𝑇 − 𝑡) and therefore 𝜑(𝑆𝑡 , 𝑡) = −(𝑇 − 𝑡)𝑆𝑡

By setting the option delta

𝜕𝑉 . 𝜕𝑆𝑡

𝜕𝑉 = Δ𝑡 we have 𝜕𝑆𝑡 𝐵𝑆 [𝜑(𝑆𝑡 , 𝑡)] = Δ𝑡 𝑆𝑡

258

2.2.4 The Greeks

which is equivalent to the number of shares needed to be bought or sold in a delta-hedging portfolio. Furthermore, because 𝜑(𝑆𝑡 , 𝑡) = −(𝑇 − 𝑡)Δ𝑡 𝑆𝑡 then as the number of shares increases (decreases), the sensitivity of the option with respect to the continuous dividend yield also decreases (increases). In addition, because of the presence of the time to expiration 𝜏 = 𝑇 − 𝑡, we can see that any change in the dividend will have a bigger effect on longer-term options than shorter-term options. 21. Let 𝑆𝑡 be the price of an asset following a GBM process 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility and 𝑊𝑡 is the standard Wiener process on the probability space (Ω, ℱ, ℙ). Consider a digital (or cash-or-nothing) put option written on 𝑆𝑡 with strike 𝐾 having a terminal payoff Ψ(𝑆𝑇 ) = 1I{𝑆𝑇 𝑡. By considering a hedging portfolio consisting of long 1∕𝜀 European put option with strike 𝐾 + 𝜀 and short 1∕𝜀 European put option with strike 𝐾, show that as 𝜀 → 0 the digital put option 𝑃𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is 𝑃𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) =

𝜕𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) 𝜕𝐾

where 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑− ) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑+ ) such that log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝑡) 𝑑± = √ 𝜎 𝑇 −𝑡

𝑥

and

Φ(𝑥) =

∫−∞

1 − 1 𝑢2 √ 𝑒 2 𝑑𝑢. 2𝜋

Hence, deduce that the asset-or-nothing put option 𝑃𝑎 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) with terminal payoff Ψ(𝑆𝑇 ) = 𝑆𝑇 1I{𝑆𝑇 0 (see Problem 2.2.4.3, page 221). Given that the market behaviour is volatile, Var(𝜀𝑡 |𝑆𝑡 ) would increase since 𝜎 increases in value. Hence, 𝑉 (𝑆𝑡 , 𝑡) would also increase in value as the option price increases with volatility. Then the hedge portfolio would also rise because of the increase in the variance estimate of the hedging error. 23. At time 𝑡 a contract was written to sell 100 units of “strip” consisting of one European call option 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and two European put options 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) written on a nondividend-paying stock 𝑆𝑡 with the same strike 𝐾 and time to expiry 𝑇 − 𝑡 = 270 days, where 𝑇 is the expiry time. The writer delta-hedged the contract at time 𝑡 and since then has not rebalanced the portfolio. At time 𝜏 (𝑡 < 𝜏) the writer decided to close out the position with the information given in Table 2.4. Table 2.4

European option prices and deltas.

Time Stock Price Call Price Put Price Call Option Delta Put Option Delta

𝑡

𝜏

$60 $6.4884 $13.9466 0.4565 −0.5435

$65 $7.3351 $10.4433

By assuming the risk-free interest rate 𝑟 as a constant value, calculate the profit of the writer at time 𝜏.

2.2.4 The Greeks

263

Solution: At time 𝑡 we denote one unit of short strip as 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 2𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) where 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) are the European call and put option prices written on 𝑆𝑡 with the same strike 𝐾 and expiry time 𝑇 (𝑡 < 𝑇 ). Let the portfolio consist of short one unit of 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and long Δ𝑡 units of stock 𝑆𝑡 so that Π𝑡 = −𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + Δ𝑡 𝑆𝑡 . By differentiating the portfolio Π𝑡 with respect to 𝑆𝑡 𝜕Π𝑡 𝜕𝑉 =− + Δ𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 and for the portfolio to be delta neutral we set

𝜕Π𝑡 = 0 and hence 𝜕𝑆𝑡

𝜕𝑉 𝜕𝑆𝑡 𝜕𝐶 𝜕𝑃 = +2 𝜕𝑆𝑡 𝜕𝑆𝑡 = 0.4565 − 2 × 0.5435 = −0.6305.

Δ𝑡 =

In order to replicate the option payoff 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) we let 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = Δ𝑡 𝑆𝑡 + Ψ𝑡 where Ψ𝑡 is the amount of cash invested in a risk-free money market. Hence, Ψ𝑡 = 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − Δ𝑡 𝑆𝑡 = 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 2𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − Δ𝑡 𝑆𝑡 = 6.4884 + 2 × 13.9466 + 0.6305 × 60 = 72.2116. Thus, to maintain a delta-hedged portfolio the writer needs to invest 100 × 72.2116 = $7,221.16 into the money market and sell 100 × 0.6304 = 63.04 units of 𝑆𝑡 = $60. In order to calculate the interest earned between time 𝑡 and 𝜏, from the put–call parity 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 − 𝐾𝑒−𝑟(𝑇 −𝑡) 𝐶(𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) − 𝑃 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) = 𝑆𝜏 − 𝐾𝑒−𝑟(𝑇 −𝜏) and by substituting 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = $6.4884, 𝐶(𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) = $7.3351, 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = $13.9466, 𝑃 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) = $10.4433, 𝑆𝑡 = $60 and 𝑆𝜏 = $65 we have 𝐾𝑒−𝑟(𝑇 −𝑡) = $67.4582

and

𝐾𝑒−𝑟(𝑇 −𝜏) = $68.1082

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2.2.4 The Greeks

such that 𝑒𝑟(𝜏−𝑡) =

68.1082 . 67.4582

Hence, Gains on “strip” sold = 100 × [6.4884 + 2 × 13.9466 − 7.3351 −2 × 10.4432] = $615.99 Gains on share price purchased = −100 × 0.6305 × (65 − 60) = −$315.25 Interest earned = 7,221.16 × (𝑒𝑟(𝜏−𝑡) − 1) ) ( 68.1082 −1 = 7,221.16 67.4582 = $69.58 so that Total profit at time 𝜏 = 615.99 − 315.25 + 69.58 = $370.32.

24. At time 𝑡 an investor has sold 1000 units of 33-strike call option on a non-dividend-paying stock 𝑆𝑡 . From the information given in Table 2.5, calculate how many units of 35-strike call and stock the investor should buy/sell in order to maintain a delta and gamma-neutral portfolio. Table 2.5 European call option Greek values for different strikes.

Delta Gamma

33-Strike Call

35-Strike Call

0.6986 0.07584

0.5217 0.06245

Solution: Let 𝐾1 = 33 and 𝐾2 = 35 denote the strike prices for 33-strike call 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) and 35-strike call 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) option prices, respectively, where 𝑇 is the option expiry time. Let the portfolios be constructed as follows: (1) Π(1) 𝑡 = −𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + Δ𝑡 𝑆𝑡 (2) Π(2) 𝑡 = 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − Δ𝑡 𝑆𝑡 (2) where Δ(1) 𝑡 and Δ𝑡 are the number of shares needed to be bought/sold for each of the (2) portfolios Π(1) 𝑡 and Π𝑡 , respectively.

2.2.4 The Greeks

265

In order to maintain gamma neutrality using 35-strike calls, we let the combined portfolios be (2) Π𝑡 = 𝑛1 Π(1) 𝑡 + 𝑛2 Π𝑡

where 𝑛1 and 𝑛2 are the numbers of 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) and 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) options needed to set the gamma of the portfolio Π𝑡 to zero. By differentiating Π𝑡 twice with respect to 𝑆𝑡 we have 𝜕 2 Π𝑡 𝜕𝑆𝑡2 and by setting

𝜕 2 Π𝑡 𝜕𝑆𝑡2

= −𝑛1

𝜕2𝐶 𝜕2𝐶 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝑛2 2 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) 2 𝜕𝑆𝑡 𝜕𝑆𝑡

=0

−𝑛1

𝜕2𝐶 𝜕2𝐶 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝑛2 2 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) = 0. 2 𝜕𝑆𝑡 𝜕𝑆𝑡

𝜕2𝐶 𝜕2𝐶 (𝑆 , 𝑡; 𝐾 ) = 0.07584 and (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) = 0.06245, we have 𝑡 1 𝜕𝑆𝑡2 𝜕𝑆𝑡2 𝑛2 = 1214.4115. Thus, to ensure that portfolio Π𝑡 is gamma neutral the investor needs to buy 1214.4115 units of 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ). Finally, to delta-neutralise portfolio Π𝑡 we note that Given 𝑛1 = 1000,

𝜕Π(1) 𝜕Π(2) 𝜕Π𝑡 = 𝑛1 𝑡 + 𝑛2 𝑡 𝜕𝑆𝑡 𝜕𝑆 𝜕𝑆𝑡 [ 𝑡 ] [ ] 𝜕𝐶 𝜕𝐶 (1) (2) = 𝑛1 − (𝑆 , 𝑡; 𝐾1 , 𝑇 ) + Δ𝑡 + 𝑛2 (𝑆 , 𝑡; 𝐾2 , 𝑇 ) − Δ𝑡 𝜕𝑆𝑡 𝑡 𝜕𝑆𝑡 𝑡 =0 or 𝜕𝐶 𝜕𝐶 (𝑆 , 𝑡; 𝐾1 , 𝑇 ) − 𝑛2 (𝑆 , 𝑡; 𝐾2 , 𝑇 ) 𝜕𝑆𝑡 𝑡 𝜕𝑆𝑡 𝑡 = 1000 × 0.6986 − 1214.4115 × 0.5217 = 65.04152.

(2) 𝑛1 Δ(1) 𝑡 − 𝑛2 Δ𝑡 = 𝑛1

Hence, in order to delta-hedge the portfolio the investor needs to buy 65.04152 shares of 𝑆𝑡 .

3 American Options In the previous chapter we concentrated solely on European options, but in most exchangetraded options nearly all stock and equity options are American options whilst stock indices are mainly European options. For a European option, the option price is only dependent on the underlying asset price at expiry and is independent of the path that the underlying asset price follows over the life of the option. Hence, the payoff is path independent and it follows that the discounted expected value of the payoff is also path independent. In contrast, an American option gives the holder of the contract the right but not the obligation to enter into a future transaction at any time until the expiry date of the contract. Therefore, American options belong to path-dependent options since they depend on the path of the underlying asset price prior to the option’s expiry time. Given that investors have the freedom to exercise their American options any time during the life of the contract, naturally they are more expensive than European options which can only be exercised at expiry.

3.1 INTRODUCTION As an American option value is greater than its equivalent European counterpart, it does not satisfy the same problem specification as the European option. Hence, the American option cannot satisfy the Black–Scholes equation. However, both have the same value at the option expiry time. By setting 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) to be the American call and put option prices, respectively whilst 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) are the European call and put option prices, respectively with 𝑆𝑡 being the common dividend-paying stock price at time 𝑡, 𝑇 > 𝑡 being the option’s expiry time, 𝐷 being the continuous dividend yield, 𝐾 being the common strike and 𝑟 being the risk-free interest rate, we have the following properties for the call options 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) ≤ 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑆𝑡 and for the put options 𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) ≤ 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐾. For European options, the put–call parity is 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) whilst for American options, the put–call parity is 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾 ≤ 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑆𝑡 − 𝐾𝑒−𝑟(𝑇 −𝑡) .

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3.1 INTRODUCTION

Conceptually, the pricing formulations for the American option can be framed as follows: s optimal stopping time formulation s free boundary formulation s linear complementarity formulation. Optimal Stopping Time Formulation Consider an American option with terminal payoff Ψ(𝑆𝑇 ) where 𝑇 is the option expiry time and suppose the option is exercised at time 𝑢 ≤ 𝑇 . Under the risk-neutral measure ℚ, conditional on the filtration ℱ𝑡 , the discounted price of the option at time 𝑡 ≤ 𝑢 with payoff at time 𝑢 is given by [ ] | 𝔼ℚ 𝑒−𝑟(𝑢−𝑡) Ψ(𝑆𝑢 )| ℱ𝑡 | where 𝔼ℚ denotes the expectation under the risk-neutral measure ℚ, and under which the stock 𝑆𝑡 follows the diffusion process 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡ℚ where 𝑟 is the risk-free interest rate, 𝐷 is the continuous dividend yield, 𝜎 is the stock volatility and 𝑊𝑡ℚ is a ℚ-standard Wiener process. As the holder of the American option can exercise at any time during the life of the option contract, the American option price at time 𝑡 ≤ 𝑢 ≤ 𝑇 is [ ] | 𝑉 (𝑆𝑡 , 𝑡) = sup 𝔼ℚ 𝑒−𝑟(𝑢−𝑡) Ψ(𝑆𝑢 )| ℱ𝑡 | 𝑡≤𝑢≤𝑇 where the supremum is taken over all possible stopping times 𝑢. The supremum is reached at the optimal stopping time 𝜏 such that { } 𝜏 = inf 𝑡 ≤ 𝑢 ≤ 𝑇 : 𝑉 (𝑆𝑢 , 𝑢) = Ψ(𝑆𝑢 ) . 𝑢

Note that the optimal stopping time problem is useful for theoretical arguments when formulating the problem via a probabilistic approach. However, it is impractical when it comes to numerical computation, except for special types of American options like immediate-touch options.

Free Boundary Formulation The Black–Scholes equation which is satisfied by European options is no longer valid where early exercise is permitted. As the American option gives its holder much more flexibility over when to exercise it, it therefore has a higher value. When early exercise is permitted, the American option satisfies the Black–Scholes inequality 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡) ≤ 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡

3.1 INTRODUCTION

269

with constraints 𝑉 (𝑆𝑡 , 𝑡) ≥ Ψ(𝑆𝑡 ),

𝑉 (𝑆𝑇 , 𝑇 ) = Ψ(𝑆𝑇 )

where Ψ(𝑆𝑡 ) is the intrinsic value of the American option payoff at time 𝑡. Thus, the problem of finding the option price value is known as a free boundary problem since the optimal exercise price is not known a priori. To decide when to exercise rather than holding the option, there are two criteria to adhere to: s The option price can never be worth less than its intrinsic value; otherwise there will be an arbitrage opportunity. s The holder of the option must exercise so as to give the option its maximum value. At the optimal exercise boundary 𝑆𝑡∞ the following hold: s The option price and the intrinsic function must be continuous as functions of the stock price, 𝑆𝑡 . s The sensitivity of the option price with respect to the stock price, 𝜕𝑉 || is contin𝜕𝑆𝑡 ||𝑆𝑡 =𝑆 ∞ 𝑡 uous as a function of the stock price, 𝑆𝑡 . This condition is known as the smooth pasting condition. When trying to understand the local behaviour of American options, the free boundary formulation is an extremely useful tool. However, it is not a very practical method, since with the presence of free boundaries they have to be determined via expensive numerical strategies which are non-trivial. Linear Complementarity Formulation From the free boundary problem, the American option price 𝑉 (𝑆𝑡 , 𝑡) can be formulated further as a linear complementarity problem. In general, by writing 𝐵𝑆 to denote the Black–Scholes operator 𝐵𝑆 =

𝜕 𝜕2 𝜕 1 −𝑟 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡

the linear complementarity formulation to determine the American option 𝑉 (𝑆𝑡 , 𝑡) is given as 𝐵𝑆 [𝑉 (𝑆𝑡 , 𝑡)] ≤ 0,

𝑉 (𝑆𝑡 , 𝑡) ≥ Ψ(𝑆𝑡 ),

[𝑉 (𝑆𝑡 , 𝑡) − Ψ(𝑆𝑡 )] ⋅ 𝐵𝑆 [𝑉 (𝑆𝑡 , 𝑡)] = 0

such that 𝑉 (𝑆𝑇 , 𝑇 ) = Ψ(𝑆𝑇 )

and

𝜕𝑉 is continuous in 𝑆𝑡 . 𝜕𝑆𝑡

270

3.1 INTRODUCTION

There are several advantages to this formulation: s s s s

The formulation does not explicitly contain a free boundary. The formulation is independent of the number of free boundaries. The free boundary is recovered from the solution. The problem can be solved numerically, such as using finite-difference approximations or numerical optimisation. However, for high-dimensional (multi-asset) American options, the numerical scheme is impractical. An alternative is to use the least-squares Monte Carlo method.

Binomial Option Pricing The binomial formula can also be modified to evaluate American options. Like its European counterparts, the binomial formula for American options also consists of a lattice representing the movements of the stock price where the option is priced by working backwards through the lattice. For the case of American options, since there is a possibility of early exercise, at each node we take the maximum of the value of the option if it is held to expiration and the gain that could be realised with immediate exercise. Thus, for an American call the value of the option at a node is given by { 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = max 𝑆𝑡 − 𝐾, [ ]} 𝑒−𝑟Δ𝑡 𝜋𝐶𝑎𝑚 (𝑢𝑆𝑡 , 𝑡 + Δ𝑡; 𝐾, 𝑇 ) + (1 − 𝜋)𝐶𝑎𝑚 (𝑑𝑆𝑡 , 𝑡 + Δ𝑡; 𝐾, 𝑇 ) and for an American put it is given by { 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = max 𝐾 − 𝑆𝑡 , [ ]} 𝑒−𝑟Δ𝑡 𝜋𝑃𝑎𝑚 (𝑢𝑆𝑡 , 𝑡 + Δ𝑡; 𝐾, 𝑇 ) + (1 − 𝜋)𝑃𝑎𝑚 (𝑑𝑆𝑡 , 𝑡 + Δ𝑡; 𝐾, 𝑇 ) where 𝑆𝑡 is the stock price at time 𝑡 (equivalent to a node in a tree), 𝑟 is the risk-free interest rate, 𝐾 is the strike price, 𝑇 > 𝑡 is the option expiry time, Δ𝑡 is the binomial time step, 𝜋 and 1 − 𝜋 are the risk-neutral probabilities for the increase and decrease in the stock price, respectively. In contrast, for the case of Bermudan options, since early exercise is only allowed on predetermined dates, the value of a Bermudan call option, 𝐶𝑏𝑒𝑟𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) at a node becomes 𝐶𝑏𝑒𝑟𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) { ⎧ max 𝑆𝑡 − 𝐾, ]} early exercise ⎪ 𝑒−𝑟Δ𝑡 [𝜋𝐶 (𝑢𝑆 , 𝑡 + Δ𝑡; 𝐾, 𝑇 ) + (1 − 𝜋)𝐶 (𝑑𝑆 , 𝑡 + Δ𝑡; 𝐾, 𝑇 ) is allowed 𝑏𝑒𝑟𝑚 𝑡 𝑏𝑒𝑟𝑚 𝑡 ⎪ =⎨ [ ] ⎪ 𝑒−𝑟Δ𝑡 𝜋𝐶𝑏𝑒𝑟𝑚 (𝑢𝑆𝑡 , 𝑡 + Δ𝑡; 𝐾, 𝑇 ) + (1 − 𝜋)𝐶𝑏𝑒𝑟𝑚 (𝑑𝑆𝑡 , 𝑡 + Δ𝑡; 𝐾, 𝑇 ) early exercise ⎪ is not allowed ⎩

3.2.1 Basic Properties

271

and the corresponding value of a Bermudan put option, 𝑃𝑏𝑒𝑟𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is 𝑃𝑏𝑒𝑟𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) { ⎧ max 𝐾 − 𝑆𝑡 , ]} early exercise ⎪ 𝑒−𝑟Δ𝑡 [𝜋𝑃 (𝑢𝑆 , 𝑡 + Δ𝑡; 𝐾, 𝑇 ) + (1 − 𝜋)𝑃 (𝑑𝑆 , 𝑡 + Δ𝑡; 𝐾, 𝑇 ) is allowed 𝑏𝑒𝑟𝑚 𝑡 𝑏𝑒𝑟𝑚 𝑡 ⎪ =⎨ [ ] ⎪ 𝑒−𝑟Δ𝑡 𝜋𝑃𝑏𝑒𝑟𝑚 (𝑢𝑆𝑡 , 𝑡 + Δ𝑡; 𝐾, 𝑇 ) + (1 − 𝜋)𝑃𝑏𝑒𝑟𝑚 (𝑑𝑆𝑡 , 𝑡 + Δ𝑡; 𝐾, 𝑇 ) early exercise ⎪ is not allowed. ⎩

3.2 PROBLEMS AND SOLUTIONS 3.2.1

Basic Properties

1. Consider the American and European call options with prices 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ), respectively where 𝑆𝑡 is the dividend-paying asset price at time 𝑡 < 𝑇 , 𝑇 is the expiry time of the options and 𝐾 is the common strike price. Show that 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑆𝑡 (1 − 𝑒−𝐷(𝑇 −𝑡) ) where 𝐷 is the continuous dividend yield. Solution: Assume 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) < 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) for all 𝑡. Therefore, at time 𝑡 we can set up a portfolio with long one American call and short one European call options Π𝑡 = 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) < 0. Assume at time 𝜏, 𝑡 ≤ 𝜏 ≤ 𝑇 , that the American call option is exercised, giving a payoff 𝑆𝜏 − 𝐾 > 0 and by exercising the American call option it gives us Net profit at time 𝜏 = 𝑆𝜏 − 𝐾 − 𝐶𝑎𝑚 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) + 𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) > 0. At expiry time 𝑇 , and because we are holding the asset now worth 𝑆𝑇 (bought at strike price 𝐾 at time 𝜏), then irrespective of whether the holder of the European call option exercises the option or not ⎧𝑆 − 𝐾 ⎪ 𝑇 Net profit at time 𝑇 = ⎨ ⎪0 ⎩ ≥ 0.

if 𝑆𝑇 > 𝐾 if 𝑆𝑇 ≤ 𝐾

Hence, we always have a guaranteed positive return for the case when 𝑆𝑡 > 𝐾, which implies the existence of an arbitrage opportunity.

272

3.2.1 Basic Properties

In contrast, if the American call option is not exercised then at terminal time 𝑇 Π𝑇 = 𝐶𝑎𝑚 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) − 𝐶𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0} − max{𝑆𝑇 − 𝐾, 0} = 0. Hence, we also have an arbitrage opportunity. Therefore, we conclude that 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). For the case 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑆𝑡 (1 − 𝑒−𝐷(𝑇 −𝑡) ) we assume 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) > 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑆𝑡 (1 − 𝑒−𝐷(𝑇 −𝑡) ) and set up the following portfolio at time 𝑡 Π𝑡 = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑆𝑡 (1 − 𝑒−𝐷(𝑇 −𝑡) ) − 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) < 0. Assume at time 𝜏, 𝑡 ≤ 𝜏 ≤ 𝑇 , that the American call option is exercised with payoff 𝑆𝜏 − 𝐾 > 0. Thus, the value of the portfolio at time 𝜏 is Π𝜏 = 𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) + 𝑆𝜏 (1 − 𝑒−𝐷(𝑇 −𝜏) ) − (𝑆𝜏 − 𝐾) = 𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) − 𝑆𝜏 𝑒−𝐷(𝑇 −𝜏) + 𝐾 ≥ 𝑆𝜏 − 𝐾 − 𝑆𝜏 𝑒−𝐷(𝑇 −𝜏) + 𝐾 = 𝑆𝜏 (1 − 𝑒−𝐷(𝑇 −𝜏) ) >0 which is an arbitrage opportunity. In addition, if the American call option is not exercised then at terminal time 𝑇 , the portfolio is now worth Π𝑇 = 𝐶𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) − 𝐶𝑎𝑚 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0} − max{𝑆𝑇 − 𝐾, 0} =0 which is also an arbitrage opportunity. Hence, 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑆𝑡 (1 − 𝑒−𝐷(𝑇 −𝑡) ). 2. Consider the American and European call options with prices 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ), respectively where 𝑆𝑡 is the dividend-paying asset price at time 𝑡 < 𝑇 , 𝑇 is the expiry time of the options and 𝐾 is the common strike price. Show that 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) +

𝑛 ∑ 𝑖=1

𝛿𝑖 𝑒−𝑟(𝑡𝑖 −𝑡)

3.2.1 Basic Properties

273

where 𝛿1 , 𝛿2 , … , 𝛿𝑛 is a sequence of discrete dividends paid at time 𝑡1 , 𝑡2 , … , 𝑡𝑛 , respectively where 𝑡 < 𝑡1 < 𝑡2 < ⋯ < 𝑡𝑛 < 𝑇 . Solution: For the proof of the inequality 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) see Problem 3.2.1.1 (page 271). ∑ For the case 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑛𝑖=1 𝛿𝑖 𝑒−𝑟(𝑡𝑖 −𝑡) we assume 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) > 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) +

𝑛 ∑ 𝑖=1

𝛿𝑖 𝑒−𝑟(𝑡𝑖 −𝑡)

and set up the following portfolio at time 𝑡, Π𝑡 = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) +

𝑛 ∑ 𝑖=1

𝛿𝑖 𝑒−𝑟(𝑡𝑖 −𝑡) − 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) < 0.

Assume at time 𝜏, 𝑡 < 𝑡1 < 𝑡2 < ⋯ < 𝑡𝑘−1 < 𝜏 < 𝑡𝑘 < 𝑡𝑘+1 < ⋯ < 𝑡𝑛 < 𝑇 , that the American call option is exercised by the owner with payoff 𝑆𝜏 − 𝐾 > 0. Thus the value of the portfolio at time 𝜏 is Π𝜏 = 𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) + = 𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) + ≥ 𝑆𝜏 − 𝐾 + =

𝑛 ∑ 𝑖=𝑘

𝑛 ∑ 𝑖=𝑘

𝑛 ∑ 𝑖=𝑘

𝑛 ∑ 𝑖=𝑘

𝛿𝑖 𝑒−𝑟(𝑡𝑖 −𝜏) − (𝑆𝜏 − 𝐾) 𝛿𝑖 𝑒−𝑟(𝑡𝑖 −𝜏) − 𝑆𝜏 + 𝐾

𝛿𝑖 𝑒−𝑟(𝑡𝑖 −𝜏) − 𝑆𝜏 + 𝐾

𝛿𝑖 𝑒−𝑟(𝑡𝑖 −𝜏)

>0

which is an arbitrage opportunity. In addition, if the American call option is not exercised then at terminal time 𝑇 , the portfolio is now worth Π𝑇 = 𝐶𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) − 𝐶𝑎𝑚 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0} − max{𝑆𝑇 − 𝐾, 0} =0 which is also an arbitrage opportunity. Hence, 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + ∑𝑛 −𝑟(𝑡𝑖 −𝑡) . 𝑖=1 𝛿𝑖 𝑒

274

3.2.1 Basic Properties

3. Consider the American and European call options with prices 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ), respectively where 𝑆𝑡 is the spot price at time 𝑡 < 𝑇 , 𝑇 is the expiry time of the options and 𝐾 is the common strike price. Let 𝑟 be the risk-free interest rate. Show that if the stock does not pay dividends, then it is never optimal to exercise an American call option before expiry, i.e. 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) but this is not true for an American put option. Solution: Given that 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝑆𝑡 − 𝐾, 0}, the price of a European call option is always greater than the intrinsic value of an American call option. Therefore, it is inadvisable to exercise the American call option early since there is no incentive to hold onto the stock which pays no dividends, and hence 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). By delaying the exercise of an American call option until expiry time, the holder will gain interest on the strike. However, by delaying the exercise of an American put option the holder would lose the interest gained on the strike. N.B. Note that by setting the continuous dividend yield or discrete dividends to zero, from Problems 3.2.1.1 and 3.2.1.2 (see pages 271–273) we have 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) or 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ).

4. Consider the American and European put options with prices 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ), respectively where 𝑆𝑡 is the spot price at time 𝑡 < 𝑇 , 𝑇 is the expiry time of the options, 𝐾 is the common strike price and 𝑟 is the constant risk-free interest rate. Show that 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝐾(1 − 𝑒−𝑟(𝑇 −𝑡) ). Solution: Assume that 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) < 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) for all 𝑡. Therefore, at time 𝑡 we can set up a portfolio with long one American put and short one European put options, Π𝑡 = 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) < 0. Assume at time 𝜏, 𝑡 ≤ 𝜏 ≤ 𝑇 , that the American put option is exercised giving a payoff 𝐾 − 𝑆𝜏 > 0. Thus, by exercising the American put option it gives us Net profit at time 𝜏 = 𝐾 − 𝑆𝜏 − 𝑃𝑎𝑚 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) + 𝑃𝑏𝑠 (𝑆𝜏 , 𝑡; 𝐾, 𝑇 ) > 0 and we can invest 𝐾 in the bank that will give us 𝐾𝑒𝑟(𝑇 −𝜏) at time 𝑇 .

3.2.1 Basic Properties

275

At expiry time 𝑇 , and because we are holding 𝐾𝑒𝑟(𝑇 −𝜏) worth of cash, then irrespective of whether the holder of the European put option exercises the option or not, we are always guaranteed to have a positive return 𝐾𝑒𝑟(𝑇 −𝜏) − 𝐾 ≥ 0 which implies an arbitrage opportunity. Therefore, 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). In contrast, if the American put option is not exercised then at terminal time 𝑇 Π𝑇 = 𝑃𝑎𝑚 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) − 𝑃𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝐾 − 𝑆𝑇 , 0} − max{𝐾 − 𝑆𝑇 , 0} = 0. Hence, we also have an arbitrage opportunity. Therefore, we can conclude that 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). For the case 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝐾(1 − 𝑒−𝑟(𝑇 −𝑡) ) we assume 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) > 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝐾(1 − 𝑒−𝑟(𝑇 −𝑡) ) and set up the following portfolio at time 𝑡 Π𝑡 = 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝐾(1 − 𝑒−𝑟(𝑇 −𝑡) ) − 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) < 0. Assume at time 𝜏, 𝑡 ≤ 𝜏 ≤ 𝑇 , that the American put option is exercised with the payoff 𝐾 − 𝑆𝜏 where 𝐾 > 𝑆𝜏 . Thus, the value of the portfolio at time 𝜏 is Π𝜏 = 𝑃𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) + 𝐾(1 − 𝑒−𝑟(𝑇 −𝜏) ) − (𝐾 − 𝑆𝜏 ) = 𝑃𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) − 𝐾𝑒−𝑟(𝑇 −𝜏) + 𝑆𝜏 ≥ 𝐾 − 𝑆𝜏 − 𝐾𝑒−𝑟(𝑇 −𝜏) + 𝑆𝜏 = 𝐾(1 − 𝑒−𝑟(𝑇 −𝜏) ) >0 which is an arbitrage opportunity. In addition, if the American put option is not exercised then at terminal time 𝑇 , the portfolio is now worth Π𝑇 = 𝑃𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) − 𝑃𝑎𝑚 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝐾 − 𝑆𝑇 , 0} − max{𝐾 − 𝑆𝑇 , 0} =0 which is also an arbitrage opportunity. Hence, 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝐾(1 − 𝑒−𝑟(𝑇 −𝑡) ).

276

3.2.1 Basic Properties

5. Consider the American and European put options with prices 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ), respectively where 𝑆𝑡 is a dividend-paying asset price at time 𝑡 < 𝑇 , 𝑇 is the expiry time of the options and 𝐾 is the common strike price. Show that in the absence of a risk-free interest rate, 𝑟 = 0 it is never optimal to exercise an American put option before expiry time, i.e. 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) but this is not the case for an American call option. Solution: Since 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝐾 − 𝑆𝑡 , 0}, which is the intrinsic value of an American put option, and if 𝑟 = 0 then by exercising the option early there is no financial gain in holding 𝐾 in the money market. Therefore, it is never optimal to exercise an American put option before expiry time. By delaying the exercise of an American put option until expiry time, the holder will gain dividend on the stock. However, by delaying the exercise of an American call option the holder would lose dividend gained on the stock. N.B. Note that by setting the interest rate to zero, from Problem 3.2.1.4 (see page 274) we have 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) or 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ).

6. Consider an American call option with price 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) where 𝑆𝑡 is the spot price at time 𝑡 < 𝑇 , 𝑇 is the expiry date of the option and 𝐾 is the strike price. Show that 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝑆𝑡 − 𝐾, 0}. Solution: We assume 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) < max{𝑆𝑡 − 𝐾, 0} and at time 𝑡 we set up the following portfolio Π𝑡 = 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − max{𝑆𝑡 − 𝐾, 0} < 0 At expiry time 𝑇 Π𝑇 = 𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) − max{𝑆𝑇 − 𝐾, 0} = max{𝑆𝑇 − 𝐾, 0} − max{𝑆𝑇 − 𝐾, 0} =0 which is an arbitrage opportunity. Therefore, 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝑆𝑡 − 𝐾, 0}. N.B. We can also prove the following. Assume 𝑆𝑡 > 𝐾 and let 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) < 𝑆𝑡 − 𝐾. Because 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0 an investor can then buy the option at time 𝑡 and exercise it immediately, leading to a profit 𝑆𝑡 − 𝐾 > 0. The net profit is therefore 𝑆𝑡 − 𝐾 −

3.2.1 Basic Properties

277

𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) > 0, which in turn provides an arbitrage opportunity to the investor. Thus, 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝑆𝑡 − 𝐾, 0}. 7. Let 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of an American call option where 𝑆𝑡 is the spot price at time 𝑡 < 𝑇 , 𝑇 is the expiry time of the option and 𝐾 is the strike price. Show that 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑆𝑡 . Solution: Assume that 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) > 𝑆𝑡 and hence at time 𝑡 we can sell an American call option 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and buy a stock 𝑆𝑡 , giving us a profit of 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑆𝑡 > 0. Given that we own the stock, we can always cover the delivery of the stock should the option be exercised by the holder at any time until time 𝑇 . Therefore, we are always guaranteed to have a positive return max{𝑆𝑡 − 𝐾, 0} ≥ 0 which implies an arbitrage opportunity. Therefore, 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑆𝑡 . 8. For a non-dividend-paying asset, let 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of an American call option where 𝑆𝑡 is the asset price at time 𝑡 < 𝑇 , 𝑇 is the expiry time of the option, 𝐾 is the strike price and 𝑟 is a constant interest rate. Show that 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝑆𝑡 − 𝐾𝑒−𝑟(𝑇 −𝑡) , 0}. Solution: From the put–call parity for European options 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 − 𝐾𝑒−𝑟(𝑇 −𝑡) where 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) are the European call and put option prices at time 𝑡 written on 𝑆𝑡 with common strike 𝐾 and expiry time 𝑇 > 𝑡. We then have 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑆𝑡 − 𝐾𝑒−𝑟(𝑇 −𝑡) ≥ 𝑆𝑡 − 𝐾𝑒−𝑟(𝑇 −𝑡) since 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0. Because 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) we therefore have 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝑆𝑡 − 𝐾𝑒−𝑟(𝑇 −𝑡) . Since 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0, then 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝑆𝑡 − 𝐾𝑒−𝑟(𝑇 −𝑡) , 0}. 9. Let 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of an American call option where 𝑆𝑡 is the asset price at time 𝑡 < 𝑇 , 𝑇 is the expiry time of the option, 𝐾 is the strike price and 𝑟 is a constant interest rate. Assume that the asset pays a continuous dividend yield 𝐷. Show that 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) , 0}.

278

3.2.1 Basic Properties

Solution: From the put–call parity for European options 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) where 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) are the European call and put option prices at time 𝑡 written on 𝑆𝑡 with common strike 𝐾 and expiry time 𝑇 > 𝑡. We then have 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) ≥ 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) since 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0. Because 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) we therefore have 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) . Since 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0, then 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) , 0}. 10. Let 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of an American call option where 𝑆𝑡 is the asset price at time 𝑡 < 𝑇 , 𝑇 is the expiry time of the option, 𝐾 is the strike price and 𝑟 is a constant interest rate. Assume that the asset pays a sequence of discrete dividends 𝛿𝑖 at time 𝑡𝑖 , 𝑖 = 1, 2, … , 𝑛 where 𝑡 < 𝑡1 < 𝑡2 < ⋯ < 𝑡𝑛 < 𝑇 . Show that { 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max

𝑆𝑡 −

𝑛 ∑ 𝑖=1

} 𝛿𝑖 𝑒

−𝑟(𝑡𝑖 −𝑡)

− 𝐾𝑒

−𝑟(𝑇 −𝑡)

,0

.

Solution: From the put–call parity for European options 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 −

𝑛 ∑ 𝑖=1

𝛿𝑖 𝑒−𝑟(𝑡𝑖 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡)

where 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) are the European call and put option prices at time 𝑡 written on 𝑆𝑡 with common strike 𝐾 and expiry time 𝑇 > 𝑡. We then have 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑆𝑡 − ≥ 𝑆𝑡 − since 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0.

𝑛 ∑ 𝑖=1

𝑛 ∑ 𝑖=1

𝛿𝑖 𝑒−𝑟(𝑡𝑖 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡)

𝛿𝑖 𝑒−𝑟(𝑡𝑖 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡)

3.2.1 Basic Properties

279

Because 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) we therefore have 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝑆𝑡 −

𝑛 ∑ 𝑖=1

𝛿𝑖 𝑒−𝑟(𝑡𝑖 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) .

Since 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0, then { 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max

𝑆𝑡 −

𝑛 ∑ 𝑖=1

} 𝛿𝑖 𝑒−𝑟(𝑡𝑖 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) , 0

.

11. Consider an American call option with price 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) where 𝑆𝑡 is the spot price at time 𝑡 < 𝑇 , 𝑇 is the expiry time of the option and 𝐾 is the strike price. By assuming the stock does not pay dividends, should the holder of an American call option exercise the option if 𝑆𝑡 > 𝐾 whilst believing that the future stock price will go below 𝐾? Solution: If the option is sold at time 𝑡 then Net profit = Profit of option sold − Cost of option bought = 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0. On the contrary, if the holder does nothing at time 𝑡 then Net profit = −Cost of option bought = −𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) < 0. Since 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝑆𝑡 − 𝐾, 0} = 𝑆𝑡 − 𝐾 and if the option is exercised at time 𝑡 then Net profit = Payoff at time 𝑡 − Cost of option bought = 𝑆𝑡 − 𝐾 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 0. Therefore, the holder should sell the option rather than exercise it. 12. Consider an American put option with price 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) where 𝑆𝑡 is the spot price at time 𝑡 < 𝑇 , 𝑇 is the expiry time of the option and 𝐾 is the strike price. Show that 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝐾 − 𝑆𝑡 , 0}.

280

3.2.1 Basic Properties

Solution: We assume 𝐾 > 𝑆𝑡 and let 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) < 𝐾 − 𝑆𝑡 . Because 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0 an investor can buy a put option at time 𝑡 and exercise it immediately, leading to a profit 𝐾 − 𝑆𝑡 > 0. The net profit is therefore 𝐾 − 𝑆𝑡 − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) > 0, which in turn provides an arbitrage opportunity to the investor. Hence, 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝐾 − 𝑆𝑡 , 0}. 13. Let 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of an American put option where 𝑆𝑡 is the spot price at time 𝑡 < 𝑇 , 𝑇 is the expiry time of the option and 𝐾 is the strike price. Show that 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐾. Solution: Assume 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) > 𝐾 and hence at time 𝑡 we can sell an American put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and deposit the money in the bank, earning a risk-free interest rate 𝑟 which will give us at least 𝐾𝑒𝑟(𝜏−𝑡) for 𝑡 ≤ 𝜏 ≤ 𝑇 . Given that we own the cash, we can always cover the purchase of the stock for the price of 𝐾, should the option be exercised by the holder at any time until time 𝑇 . Therefore, we are always guaranteed to have a positive return max{𝐾 − 𝑆𝑡 , 0} ≥ 0, which implies an arbitrage opportunity. Therefore, 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐾. 14. For a non-dividend-paying asset, let 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of an American put option where 𝑆𝑡 is the asset price at time 𝑡 < 𝑇 , 𝑇 is the expiry time of the option, 𝐾 is the strike price and 𝑟 is a constant interest rate. Show that 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 , 0}. Solution: From the put–call parity for European options 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 − 𝐾𝑒−𝑟(𝑇 −𝑡) where 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) are the European call and put option prices at time 𝑡 written on 𝑆𝑡 with common strike 𝐾 and expiry time 𝑇 > 𝑡. We then have 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 ≥ 𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 since 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0. Because 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) we have 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 . Since 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0 therefore 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 , 0}.

3.2.1 Basic Properties

281

15. Let 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of an American put option where 𝑆𝑡 is the asset price at time 𝑡 < 𝑇 , 𝑇 is the expiry time of the option, 𝐾 is the strike price and 𝑟 is a constant interest rate. Assume that 𝑆𝑡 pays a continuous dividend yield 𝐷. Show that 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) , 0}. Solution: From the put–call parity for European options we have 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) where 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) are the European call and put option prices at time 𝑡 written on 𝑆𝑡 with common strike 𝐾 and expiry time 𝑇 > 𝑡. We then express 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) ≥ 𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) since 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0. Because 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) we have 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) . Since 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0 therefore 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) , 0}. 16. Let 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of an American put option where 𝑆𝑡 is the asset price at time 𝑡 < 𝑇 , 𝑇 is the expiry time of the option, 𝐾 is the strike price and 𝑟 is a constant interest rate. Assume that the asset pays a sequence of discrete dividends 𝛿𝑖 at time 𝑡𝑖 , 𝑖 = 1, 2, … , 𝑛 where 𝑡 < 𝑡1 < 𝑡2 < ⋯ < 𝑡𝑛 < 𝑇 . Show that { } 𝑛 ∑ −𝑟(𝑇 −𝑡) −𝑟(𝑡𝑖 −𝑡) − 𝑆𝑡 + 𝛿𝑖 𝑒 ,0 . 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max 𝐾𝑒 𝑖=1

Solution: From the put–call parity for European options we have 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 −

𝑛 ∑ 𝑖=1

𝛿𝑖 𝑒−𝑟(𝑡𝑖 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡)

where 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) are the European call and put option prices at time 𝑡 written on 𝑆𝑡 with common strike 𝐾 and expiry time 𝑇 > 𝑡. We then have 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 + ≥ 𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 + since 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0.

𝑛 ∑ 𝑖=1

𝛿𝑖 𝑒−𝑟(𝑡𝑖 −𝑡)

𝑛 ∑ 𝑖=1

𝛿𝑖 𝑒−𝑟(𝑡𝑖 −𝑡)

282

3.2.1 Basic Properties

Because 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) we have 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 +

𝑛 ∑ 𝑖=1

𝛿𝑖 𝑒−𝑟(𝑡𝑖 −𝑡) .

Since 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 0 therefore { 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max

𝐾𝑒

−𝑟(𝑇 −𝑡)

− 𝑆𝑡 +

𝑛 ∑ 𝑖=1

} 𝛿𝑖 𝑒

−𝑟(𝑡𝑖 −𝑡)

,0

.

17. Put–Call Parity I. Consider the American call and put options with prices 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ), respectively where 𝑆𝑡 is the spot price at time 𝑡 < 𝑇 , 𝑇 is the expiry time of the options, 𝐾 is the common strike price and 𝑟 is a constant risk-free interest rate. Show that 𝑆𝑡 − 𝐾 ≤ 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑆𝑡 − 𝐾𝑒−𝑟(𝑇 −𝑡) . Solution: Since 𝑆𝑡 does not pay any dividends, then 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) where 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is the European call option price written on 𝑆𝑡 at time 𝑡 with strike price 𝐾 and option expiry 𝑇 . In addition, from Problem 3.2.1.4 (page 274) we have 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝐾(1 − 𝑒−𝑟(𝑇 −𝑡) ) where 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is the European put option price written on 𝑆𝑡 at time 𝑡 with strike price 𝐾 and option expiry 𝑇 . Thus, 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) −𝐾(1 − 𝑒−𝑟(𝑇 −𝑡) ) = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) −𝐾(1 − 𝑒−𝑟(𝑇 −𝑡) ) = 𝑆𝑡 − 𝐾𝑒−𝑟(𝑇 −𝑡) − 𝐾(1 − 𝑒−𝑟(𝑇 −𝑡) ) = 𝑆𝑡 − 𝐾 since 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 − 𝐾 −𝑟(𝑇 −𝑡) .

3.2.1 Basic Properties

283

In contrast, using the put–call parity for European options again 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 − 𝐾𝑒−𝑟(𝑇 −𝑡) .

18. Put–Call Parity II. Consider the American call and put options with prices 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ), respectively where 𝑆𝑡 is the spot price at time 𝑡 < 𝑇 which pays a continuous dividend yield 𝐷, 𝑇 is the expiry time of the options, 𝐾 is the common strike price and 𝑟 is a constant risk-free interest rate. Show that 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾 ≤ 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑆𝑡 − 𝐾𝑒−𝑟(𝑇 −𝑡) . Solution: From Problems 3.2.1.1 and 3.2.1.4 (pages 271–274) we have 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑆𝑡 (1 − 𝑒−𝐷(𝑇 −𝑡) ) 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝐾(1 − 𝑒−𝑟(𝑇 −𝑡) ) where 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) are the European call and put option prices, respectively written on 𝑆𝑡 at time 𝑡 with strike price 𝐾 and option expiry 𝑇 . Thus, 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) −𝐾(1 − 𝑒−𝑟(𝑇 −𝑡) ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) − 𝐾(1 − 𝑒−𝑟(𝑇 −𝑡) ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾 since 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾 −𝑟(𝑇 −𝑡) . In contrast, using the put–call parity for European options again 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑆𝑡 (1 − 𝑒−𝐷(𝑇 −𝑡) ) −𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) +𝑆𝑡 (1 − 𝑒−𝐷(𝑇 −𝑡) ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) + 𝑆𝑡 (1 − 𝑒−𝐷(𝑇 −𝑡) ) = 𝑆𝑡 − 𝐾𝑒−𝑟(𝑇 −𝑡) .

19. Put–Call Parity III. Consider the American call and put options with prices 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ), respectively where 𝑆𝑡 is the spot price at time 𝑡 < 𝑇 which pays a sequence of discrete dividends 𝛿1 , 𝛿2 , … , 𝛿𝑛 at times 𝑡1 , 𝑡2 , … , 𝑡𝑛 , respectively, 𝑡 < 𝑡1
100 the trader can exercise the call and pay $100 to own the stock. Therefore, he makes an extra profit of $2.53 compared with exercising the option now. In contrast, if 𝑆𝑇 ≤ 100 the trader would not exercise the option and would have $102.53 in the bank. Had the trader exercised the option at time 𝑡 = 0, he would own stock worth 110 and, by selling the stock and putting the proceeds in the bank, it will grow to $110 × 𝑒0.05×0.5 = $112.78 in 6 months’ time. Thus, if the trader only exercises the option at the expiry time and if 𝑆𝑇 ≤ 100 then he will be making a loss of $112.78 − $102.53 = $10.25.

286

3.2.1 Basic Properties

(c) If the trader exercises the call option now, he will pay $100 for the stock and can sell it for $110 and deposit the profit of $10 in the bank. At time 𝑇 = 0.5 the trader will earn $10 × 𝑒0.05×0.5 = $10.25. Conversely, if he waits until expiry time, he can short sell the stock now and deposit $110 in the bank, thus earning $110 × 𝑒0.05×0.5 = $112.78 in 6 months’ time. To cover his short-selling activity he can exercise the call option at expiry time by paying $100 for 𝑆𝑇 . Thus, he makes a profit of $12.78, which is more than the $10.25 if he exercised immediately. 23. Consider an American put option on a non-dividend-paying stock with strike price $12 with expiry time 8 months from now. The current stock price is $10 and the continuously compounded interest rate is 5% per annum. Given that the price of the American put option is $2.50, calculate the range of values for the corresponding American call option with the same strike price and expiry time. 8 = 23 years, 𝑟 = 0.05 Solution: Let the current time be 𝑡 = 0, so that 𝑆0 = 10, 𝑇 = 12 and 𝐾 = 12. By denoting 𝑃 (𝑆0 , 0; 𝐾, 𝑇 ) = 2.50 as the price of an American put option then, from the put–call parity, the corresponding American call option 𝐶(𝑆0 , 0; 𝐾, 𝑇 ) satisfies

𝑃 (𝑆0 , 0; 𝐾, 𝑇 ) + 𝑆0 − 𝐾 ≤ 𝐶(𝑆0 , 0; 𝐾, 𝑇 ) ≤ 𝑃 (𝑆0 , 0; 𝐾, 𝑇 ) + 𝑆0 − 𝐾𝑒−𝑟𝑇 . Thus, 2

2.5 + 10 − 12 ≤ 𝐶(𝑆0 , 0; 𝐾, 𝑇 ) ≤ 2.5 + 10 − 12 × 𝑒−0.05× 3 or $0.50 ≤ 𝐶(𝑆0 , 0; 𝐾, 𝑇 ) ≤ $0.89.

24. Consider two identical American call options 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇1 ) and 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇2 ) having expiry times 𝑇1 and 𝑇2 with 𝑇1 < 𝑇2 , 𝑆𝑡 is the spot price at time 𝑡 and 𝐾 is the common strike price. Show that 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇1 ) ≤ 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇2 ) for 𝑇1 < 𝑇2 . Solution: We assume 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇1 ) > 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇2 ) where at time 𝑡 we can set up a portfolio Π𝑡 = 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇2 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇1 ) < 0.

3.2.1 Basic Properties

287

At time 𝑇1 , the portfolio is now worth Π𝑇1 = 𝐶(𝑆𝑇1 , 𝑇1 ; 𝐾, 𝑇2 ) − max{𝑆𝑇1 − 𝐾, 0} ⎧ 𝐶(𝑆 , 𝑇 ; 𝐾, 𝑇 ) 𝑇1 1 2 ⎪ =⎨ ⎪ 𝐶(𝑆𝑇 , 𝑇1 ; 𝐾, 𝑇2 ) − 𝑆𝑇 + 𝐾 1 1 ⎩

if 𝑆𝑇1 ≤ 𝐾 if 𝑆𝑇1 > 𝐾.

Since 𝐶(𝑆𝑇1 , 𝑇1 ; 𝐾, 𝑇2 ) ≥ 𝑆𝑇1 − 𝐾 then Π𝑇1 ≥ 0, which is a contradiction. Therefore, 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇1 ) ≤ 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇2 ) for 𝑇1 < 𝑇2 . 25. Consider two identical American put options 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇1 ) and 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇2 ) having expiry times 𝑇1 and 𝑇2 with 𝑇1 < 𝑇2 , 𝑆𝑡 is the spot price at time 𝑡 and 𝐾 is the common strike price. Show that 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇1 ) ≤ 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇2 ) for 𝑇1 < 𝑇2 . Solution: We assume 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇1 ) > 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇2 ) where at time 𝑡 we can set up a portfolio Π𝑡 = 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇2 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇1 ) < 0. At time 𝑇1 , the portfolio is now worth Π𝑇1 = 𝑃 (𝑆𝑇1 , 𝑇1 ; 𝐾, 𝑇2 ) − max{𝐾 − 𝑆𝑇1 , 0} ⎧ 𝑃 (𝑆 , 𝑇 ; 𝐾, 𝑇 ) 𝑇1 1 2 ⎪ =⎨ ⎪ 𝑃 (𝑆𝑇 , 𝑇1 ; 𝐾, 𝑇2 ) − 𝐾 + 𝑆𝑇 1 1 ⎩

if 𝑆𝑇1 ≥ 𝐾 if 𝑆𝑇1 < 𝐾.

Since 𝑃 (𝑆𝑇1 , 𝑇1 ; 𝐾, 𝑇2 ) ≥ 𝐾 − 𝑆𝑇1 then Π𝑇1 ≥ 0, which is a contradiction. Therefore, 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇1 ) ≤ 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇2 ) for 𝑇1 < 𝑇2 .

288

3.2.1 Basic Properties

26. Consider two identical American call options 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) and 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) having strike prices 𝐾1 and 𝐾2 with 𝐾1 < 𝐾2 , 𝑆𝑡 is the spot price at time 𝑡 and 𝑇 > 𝑡 is the expiry time. Show that 0 ≤ 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≤ 𝐾2 − 𝐾1 for 𝐾1 < 𝐾2 . Solution: We first assume 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) < 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) where at time 𝑡 we can set up a portfolio Π𝑡 = 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) < 0. At expiry time 𝑇 Π𝑇 = 𝐶(𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) = max{𝑆𝑇 − 𝐾1 , 0} − max{𝑆𝑇 − 𝐾2 , 0} ⎧0 ⎪ ⎪ = ⎨ 𝑆𝑇 − 𝐾1 ⎪ ⎪𝐾 − 𝐾 1 ⎩ 2

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2 .

Hence, Π𝑇 ≥ 0 which is a contradiction. We next assume 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) > 𝐾2 − 𝐾1 where at time 𝑡 we set up the following portfolio Π𝑡 = 𝐾2 − 𝐾1 − 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) < 0. At expiry time 𝑇 Π𝑇 = 𝐾2 − 𝐾1 − 𝐶(𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) + 𝐶(𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) = 𝐾2 − 𝐾1 − max{𝑆𝑇 − 𝐾1 , 0} + max{𝑆𝑇 − 𝐾2 , 0} ⎧ 𝐾 2 − 𝐾1 ⎪ ⎪ = ⎨ 𝐾2 − 𝑆𝑇 ⎪ ⎪0 ⎩

if 𝑆𝑇 ≤ 𝐾1 if 𝐾1 < 𝑆𝑇 ≤ 𝐾2 if 𝑆𝑇 > 𝐾2 .

Hence, Π𝑇 ≥ 0 which is also a contradiction. Therefore, 0 ≤ 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) ≤ 𝐾2 − 𝐾1 for 𝐾1 < 𝐾2 .

3.2.1 Basic Properties

289

27. Consider two identical American put options 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) and 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) having strike prices 𝐾1 and 𝐾2 with 𝐾1 < 𝐾2 , 𝑆𝑡 is the spot price at time 𝑡 and 𝑇 > 𝑡 is the expiry time. Show that 0 ≤ 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) ≤ 𝐾2 − 𝐾1 for 𝐾1 < 𝐾2 . Solution: We first assume 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) < 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) where at time 𝑡 we can set up a portfolio Π𝑡 = 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) < 0. At expiry time 𝑇 Π𝑇 = 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) = max{𝐾2 − 𝑆𝑇 , 0} − max{𝐾1 − 𝑆𝑇 , 0} ⎧ 𝐾2 − 𝐾1 ⎪ ⎪ = ⎨ 𝐾 2 − 𝑆𝑇 ⎪ ⎪0 ⎩

if 𝑆𝑇 < 𝐾1 if 𝐾1 ≤ 𝑆𝑇 < 𝐾2 if 𝑆𝑇 ≥ 𝐾2 .

Hence, Π𝑇 ≥ 0 which is a contradiction. We next assume 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) > 𝐾2 − 𝐾1 where at time 𝑡 we set up the following portfolio Π𝑡 = 𝐾2 − 𝐾1 − 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) + 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) < 0. At expiry time 𝑇 Π𝑇 = 𝐾2 − 𝐾1 − 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) + 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 ) = 𝐾2 − 𝐾1 − max{𝐾2 − 𝑆𝑇 , 0} + max{𝐾1 − 𝑆𝑇 , 0} ⎧0 ⎪ ⎪ = ⎨ 𝑆𝑇 − 𝐾1 ⎪ ⎪𝐾 − 𝐾 1 ⎩ 2

if 𝑆𝑇 < 𝐾1 if 𝐾1 ≤ 𝑆𝑇 < 𝐾2 if 𝑆𝑇 ≥ 𝐾2 .

Hence, Π𝑇 ≥ 0 which is also a contradiction. Therefore, 0 ≤ 𝑃 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) − 𝑃 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) ≤ 𝐾2 − 𝐾1 for 𝐾1 < 𝐾2 .

290

3.2.1 Basic Properties

28. Let the prices of two American call options with strikes $50 and $60 be $2.50 and $3.00, respectively where both options have the same time to expiry. (a) Is the no-arbitrage condition violated? (b) Suggest a spread position so that the portfolio will ensure an arbitrage opportunity. Solution: We assume the options are priced at time 𝑡 with spot price 𝑆𝑡 and we let 𝐾1 = $50, 𝐾2 = $60 and 𝑇 > 𝑡 is the expiry time so that 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) = $2.50 and 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) = $3.00. (a) Since 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) < 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) for 𝐾1 < 𝐾2 , this violates the condition 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) ≥ 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ). (b) We let the portfolio at time 𝑡 be Π𝑡 = 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) < 0 where at time 𝑇 the portfolio will be worth Π𝑇 = max{𝑆𝑇 − 50, 0} − max{𝑆𝑇 − 60, 0} ⎧0 ⎪ ⎪ = ⎨ 𝑆𝑇 − 50 ⎪ ⎪ 10 ⎩

if 𝑆𝑇 ≤ 50 if 50 < 𝑆𝑇 ≤ 60 if 𝑆𝑇 > 60.

Thus, Π𝑇 ≥ 0 which shows that the portfolio Π𝑡 = 𝐶(𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶(𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) provides an arbitrage opportunity. 29. At time 𝑡 we consider an American put option on a non-dividend-paying stock 𝑆𝑡 with strike price 𝐾 > 𝑆𝑡 and 𝑟 is the risk-free interest rate. The stock price is forecast to fall in value where at the expiry time 𝑇 , 𝑆𝑇 < 𝑆𝑡 . Is it advisable to exercise the option early? If the stock pays a dividend 𝛿 > 0 at intermediate time 𝜏, 𝑡 < 𝜏 < 𝑇 , should we exercise the option early? Solution: If the option is exercised early at time 𝑡, the payoff is Ψ(𝑆𝑡 ) = max{𝐾 − 𝑆𝑡 , 0} = 𝐾 − 𝑆𝑡 . If the option is exercised at expiry time 𝑇 , the payoff is Ψ(𝑆𝑇 ) = max{𝐾 − 𝑆𝑇 , 0} = 𝐾 − 𝑆𝑇 . Hence, the present value of the payoff at time 𝑡 is 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 < 𝐾 − 𝑆𝑡 . Therefore, it is advisable to exercise early if the stock price is forecast to fall in value.

3.2.1 Basic Properties

291

If the stock pays a dividend at time 𝜏 then the present value of the payoff at time 𝑡 is 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑒−𝑟(𝑇 −𝑡) − (𝑆𝑡 − 𝛿𝑒−𝑟(𝜏−𝑡) ) = 𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 + 𝛿𝑒−𝑟(𝜏−𝑡) . Thus, early exercise is advisable if 𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 + 𝛿𝑒−𝑟(𝜏−𝑡) < 𝐾 − 𝑆𝑡 or 𝛿𝑒−𝑟(𝜏−𝑡) < 𝐾(1 − 𝑒−𝑟(𝑇 −𝑡) ).

30. Consider an investor holding one American call option and one European call option on the same non-dividend-paying stock with strike price 𝐾 = $50. The price of the American call option is $5 with time to expiry 8 months, whilst the European call option is priced at $7 with time to expiry 6 months. The continuously compounded risk-free interest rate is 𝑟 = 5%. Show that the investor can construct an arbitrage portfolio. Solution: At time 𝑡 = 0 with spot price 𝑆0 we denote the price of an American call option by 𝐶𝑎𝑚 (𝑆0 , 0; 𝐾, 𝑇1 ) = $5

for 𝑇1 =

8 12

=

2 3

years

and the European call option price by 𝐶𝑏𝑠 (𝑆0 , 0; 𝐾, 𝑇2 ) = $7

for 𝑇2 =

6 12

=

1 2

years.

We first set up a portfolio at time 𝑡 = 0 Π0 = 𝐶𝑎𝑚 (𝑆0 , 0; 𝐾, 𝑇1 ) − 𝐶𝑏𝑠 (𝑆0 , 0; 𝐾, 𝑇2 ) < 0. At time 𝑡 = 𝑇2 we have Π𝑇2 = 𝐶𝑎𝑚 (𝑆𝑇2 , 𝑇2 ; 𝐾, 𝑇1 ) − 𝐶𝑏𝑠 (𝑆𝑇2 , 𝑇2 ; 𝐾, 𝑇2 ) = 𝐶𝑎𝑚 (𝑆𝑇2 , 𝑇2 ; 𝐾, 𝑇1 ) − max{𝑆𝑇2 − 𝐾, 0} ⎧ 𝐶 (𝑆 , 𝑇 ; 𝐾, 𝑇 ) 1 ⎪ 𝑎𝑚 𝑇2 2 =⎨ ⎪ 𝐶𝑎𝑚 (𝑆𝑇 , 𝑇2 ; 𝐾, 𝑇1 ) − 𝑆𝑇 + 𝐾 2 2 ⎩ ⎧ 𝐶 (𝑆 , 𝑇 ; 𝐾, 𝑇 ) 1 ⎪ 𝑎𝑚 𝑇2 2 ≥⎨ ⎪ 𝐾(1 − 𝑒−𝑟(𝑇1 −𝑇2 ) ) ⎩

if 𝑆𝑇2 ≤ 𝐾 if 𝑆𝑇2 > 𝐾

if 𝑆𝑇2 ≤ 𝐾 if 𝑆𝑇2 > 𝐾

292

3.2.2 Time-Independent Options

since 𝐶𝑎𝑚 (𝑆𝑇2 , 𝑇2 ; 𝐾, 𝑇1 ) ≥ 𝑆𝑇2 − 𝐾𝑒−𝑟(𝑇1 −𝑇2 ) . Further, because 𝐶𝑎𝑚 (𝑆𝑇2 , 𝑇2 ; 𝐾, 𝑇1 ) ≥ 0, then Π𝑇2 > 0 which constitutes an arbitrage opportunity.

3.2.2

Time-Independent Options

1. Generalised Perpetual American Option. Consider an economy which consists of a riskfree asset and a stock, whose values at time 𝑡 are 𝐵𝑡 and 𝑆𝑡 , respectively. Assume that these values evolve according to the following diffusion processes 𝑑𝐵𝑡 = 𝑟𝐵𝑡 𝑑𝑡,

𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡

where 𝐷 is the continuous dividend yield, 𝑟 is the risk-free rate, 𝜇 is the stock price growth rate and 𝜎𝑡 > 0 is the stock price volatility. In addition, {𝑊𝑡 : 0 ≤ 𝑡 ≤ 𝑇 } is the ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and ℱ𝑡 , 0 ≤ 𝑡 ≤ 𝑇 is the filtration generated by the standard Wiener process. Show that a perpetual American option 𝑉 (𝑆𝑡 ) (which has not yet been exercised) satisfies the following second-order ODE 𝑑𝑉 1 2 2 𝑑2𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 ) = 0. 𝜎 𝑆𝑡 2 2 𝑑𝑆 𝑑𝑆𝑡 𝑡 Show also that the general solution of the above equation is 𝛼

𝛼

𝑉 (𝑆𝑡 ) = 𝐴𝑆𝑡 + + 𝐵𝑆𝑡 − where 𝐴 and 𝐵 are unknown constants, 𝛼+ > 0 and 𝛼− < 0. Solution: We consider a perpetual American option with an arbitrary payoff and because it is time independent we can denote the option price at time 𝑡 as 𝑉 (𝑆𝑡 ). By setting a Δ-hedged portfolio Π𝑡 = 𝑉 (𝑆𝑡 ) − Δ𝑆𝑡 and because during time 𝑑𝑡 the stock pays out a continuous dividend 𝐷𝑆𝑡 𝑑𝑡 we have 𝑑Π𝑡 = 𝑑𝑉 − Δ(𝑑𝑆𝑡 + 𝐷𝑆𝑡 𝑑𝑡). By Taylor’s expansion and subsequently using It̄o’s lemma we can write 𝑑𝑉 1 𝑑2𝑉 1 𝑑3𝑉 𝑑𝑆𝑡 + (𝑑𝑆𝑡 )2 + (𝑑𝑆𝑡 )3 + ⋯ 2 𝑑𝑆𝑡 2 𝑑𝑆𝑡 3! 𝑑𝑆𝑡3 ] 1 𝑑2𝑉 𝑑𝑉 [ (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑑𝑊𝑡 + 𝜎 2 𝑆𝑡2 𝑑𝑡 = 𝑑𝑆𝑡 2 𝑑𝑆𝑡2 [ ] 1 2 2 𝑑2𝑉 𝑑𝑉 𝑑𝑉 = + (𝜇 − 𝐷)𝑆𝑡 𝑑𝑊𝑡 𝜎 𝑆𝑡 𝑑𝑡 + 𝜎 2 𝑑𝑆𝑡 𝑑𝑆𝑡 𝑑𝑆𝑡2

𝑑𝑉 =

3.2.2 Time-Independent Options

293

and 𝑑𝑆𝑡 + 𝐷𝑆𝑡 𝑑𝑡 = 𝜇𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 . By substituting the above expressions into 𝑑Π𝑡 we have [ 𝑑Π𝑡 =

] ] [ 1 2 2 𝑑2𝑉 𝑑𝑉 𝑑𝑉 + (𝜇 − 𝐷)𝑆𝑡 − Δ𝜇𝑆𝑡 𝑑𝑡 + 𝜎 − Δ 𝑑𝑊𝑡 . 𝜎 𝑆𝑡 2 𝑑𝑆𝑡 𝑑𝑆𝑡 𝑑𝑆𝑡2

To eliminate risk we set Δ=

𝑑𝑉 𝑑𝑆𝑡

and hence [ 𝑑Π𝑡 =

] 1 2 2 𝑑2𝑉 𝑑𝑉 − 𝐷𝑆𝑡 𝑑𝑡. 𝜎 𝑆𝑡 2 𝑑𝑆𝑡 𝑑𝑆𝑡2

Given that there is no time limit to exercise the option, to avoid arbitrage we set 𝑑Π𝑡 = 𝑟Π𝑑𝑡 or [

] [ ] 𝑑𝑉 𝑑𝑉 1 2 2 𝑑2𝑉 − 𝐷𝑆 ) − 𝑆 𝜎 𝑆𝑡 𝑑𝑡 = 𝑟 𝑉 (𝑆 𝑑𝑡. 𝑡 𝑡 2 𝑑𝑆𝑡 𝑑𝑆𝑡 𝑡 𝑑𝑆𝑡2

Hence, 1 2 2 𝑑2𝑉 𝑑𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 ) = 0. 𝜎 𝑆𝑡 2 𝑑𝑆𝑡 𝑑𝑆𝑡2 To solve the second-order ODE we look for solutions of the form 𝑉 (𝑆𝑡 ) = 𝐶𝑆𝑡𝑚 where 𝐶 is a constant. By substituting 𝑉 (𝑆𝑡 ) = 𝐶𝑆𝑡𝑚 ,

𝑑𝑉 = 𝑚𝐶𝑆𝑡𝑚−1 , 𝑑𝑆𝑡

𝑑2𝑉 = 𝑚(𝑚 − 1)𝐶𝑆𝑡𝑚−2 𝑑𝑆𝑡2

into the ODE we have 1 2 𝜎 𝑚(𝑚 − 1) + (𝑟 − 𝐷)𝑚 − 𝑟 = 0 2

294

3.2.2 Time-Independent Options

or ) ( 1 1 2 2 𝜎 𝑚 + 𝑟 − 𝐷 − 𝜎 2 𝑚 − 𝑟 = 0. 2 2 Hence,

𝑚=

−(𝑟 − 𝐷 − 12 𝜎 2 ) ±

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟 𝜎2

.

Since √

1 1 (𝑟 − 𝐷 − 𝜎 2 )2 + 2𝜎 2 𝑟 > (𝑟 − 𝐷 − 𝜎 2 ) 2 2

the solution of the ODE must be of the form 𝛼

𝛼

𝑉 (𝑆𝑡 ) = 𝐴𝑆𝑡 + + 𝐵𝑆𝑡 − where 𝐴 and 𝐵 are unknown constants,

𝛼+ =

−(𝑟 − 𝐷 − 12 𝜎 2 ) +

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟 𝜎2

>0

and

𝛼− =

−(𝑟 − 𝐷 − 12 𝜎 2 ) −

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟 𝜎2

< 0.

2. Perpetual American Call Option. By definition a perpetual American call option gives the holder the right but not the obligation to buy an underlying asset (where the current price at time 𝑡 is 𝑆𝑡 ) for a specified strike price 𝐾 at any time in the future with no expiry time. Assume that the price of the perpetual American call option 𝐶(𝑆𝑡 ) satisfies 1 2 2 𝑑2𝐶 𝑑𝐶 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝐶(𝑆𝑡 ) = 0, 𝜎 𝑆𝑡 2 2 𝑑𝑆𝑡 𝑑𝑆𝑡

0 < 𝑆𝑡 < 𝑆 ∞

with boundary conditions 𝐶(0) = 0,

𝐶(𝑆 ∞ ) = 𝑆 ∞ − 𝐾

where 𝑟 is the risk-free interest rate, 𝐷 is the continuous dividend yield, 𝜎 is the constant volatility and 𝑆 ∞ > 𝐾 denotes the unknown optimal exercise boundary such that for 𝑆𝑡 ≥ 𝑆 ∞ the option should be exercised whilst for 𝑆𝑡 < 𝑆 ∞ the option should be held.

3.2.2 Time-Independent Options

295

Show that ⎧ (𝑆 ∞ − 𝐾) ⎪ 𝐶(𝑆𝑡 ) = ⎨ ⎪ ⎩ 𝑆𝑡 − 𝐾

(

𝑆𝑡 𝑆∞

)𝛼+

if 𝑆𝑡 < 𝑆 ∞ if 𝑆𝑡 ≥ 𝑆 ∞

where

𝛼+ =

−(𝑟 − 𝐷 − 12 𝜎 2 ) +

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟 𝜎2

> 0.

Hence, show that the optimal exercise boundary is 𝑆∞ =

𝛼+ 𝐾 𝛼+ − 1

and that for this optimal value the “smooth-pasting” condition Δ=

𝑑𝐶 || =1 𝑑𝑆𝑡 ||𝑆𝑡 =𝑆 ∞

is satisfied. Deduce that if 𝐷 = 0, it is not optimal to exercise the perpetual American call option. Solution: Given that 𝑆 ∞ > 𝐾 is the optimal exercise boundary then, for 𝑆𝑡 ≥ 𝑆 ∞ , we can deduce that the perpetual American call option is equal to its intrinsic value, that is 𝐶(𝑆𝑡 ) = 𝑆𝑡 − 𝐾,

𝑆𝑡 ≥ 𝑆 ∞ .

For 𝑆𝑡 < 𝑆 ∞ , 𝐶(𝑆𝑡 ) satisfies the second-order ODE and from Problem 3.2.2.1 (page 292) the general solution is 𝛼

𝛼

𝐶(𝑆𝑡 ) = 𝐴𝑆𝑡 + + 𝐵𝑆𝑡 − where 𝐴 and 𝐵 are unknown constants,

𝛼+ =

−(𝑟 − 𝐷 − 12 𝜎 2 ) +

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟 𝜎2

>0

and

𝛼− =

−(𝑟 − 𝐷 − 12 𝜎 2 ) −

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟 𝜎2

< 0.

296

3.2.2 Time-Independent Options

From the initial conditions, for 𝑆𝑡 → 0, 𝐶(𝑆𝑡 ) → 0 and because 𝛼− < 0 we can set 𝐵 = 0. Thus, 𝛼

𝐶(𝑆𝑡 ) = 𝐴𝑆𝑡 + . Because 𝑆 ∞ > 𝐾 therefore 𝐶(𝑆 ∞ ) = 𝑆 ∞ − 𝐾. By continuity 𝐴(𝑆 ∞ )𝛼+ = 𝑆 ∞ − 𝐾 or 𝐴 = (𝑆 ∞ − 𝐾)

(

1 𝑆∞

)𝛼+

.

Therefore, for 𝑆𝑡 < 𝑆 ∞ ( 𝐶(𝑆𝑡 ) = (𝑆 ∞ − 𝐾)

𝑆𝑡 𝑆∞

)𝛼+

.

In order to find the optimal exercise boundary we differentiate 𝐶(𝑆𝑡 ) with respect to 𝑆 ∞ ( ) ) [ ] 𝑆𝑡 𝛼+ 𝑆𝑡 𝛼+ 𝛼+ (𝑆 ∞ − 𝐾) − 𝑆∞ 𝑆∞ 𝑆∞ ( )𝛼+ [ ] ∞ 𝛼+ (𝑆 − 𝐾) 𝑆𝑡 = 1− ∞ 𝑆 𝑆∞

𝑑𝐶 = 𝑑𝑆 ∞

and by setting

(

𝑑𝐶 = 0 we have 𝑑𝑆 ∞ 𝑆∞ =

𝛼+ 𝐾 . 𝛼+ − 1

𝑑𝐶 once again with respect to 𝑆 ∞ 𝑑𝑆 ∞ ( ) [ ] ( ) 𝛼+ 𝛼+ (𝑆 ∞ − 𝐾) 𝑆𝑡 𝛼+ 𝑆𝑡 𝛼+ 𝛼+ 𝐾 𝑑2𝐶 = − 1 − − 𝑆∞ 𝑆∞ 𝑆∞ 𝑆∞ 𝑑(𝑆 ∞ )2 (𝑆 ∞ )2

By differentiating

and by substituting 𝑆 ∞ =

𝛼+ 𝐾 we have 𝛼+ − 1 𝑑 2 𝐶 || < 0. 𝑑(𝑆 ∞ )2 ||𝑆 ∞ = 𝛼+ 𝐾 𝛼+ −1

Hence, 𝑆 ∞ =

𝛼+ 𝐾 is a local maximum point. 𝛼+ − 1

3.2.2 Time-Independent Options

297

( For 𝑆𝑡
𝑆 ∗

with boundary conditions 𝑃 (∞) = 0,

𝑃 (𝑆 ∗ ) = 𝐾 − 𝑆 ∗

where 𝑟 is the risk-free interest rate, 𝐷 is the continuous dividend yield, 𝜎 is the constant volatility and 𝑆 ∗ < 𝐾 denotes the unknown optimal exercise boundary such that for 𝑆𝑡 ≤ 𝑆 ∗ the option should be exercised whilst for 𝑆𝑡 > 𝑆 ∗ the option should be held. Show that ⎧ 𝐾 − 𝑆𝑡 ⎪ ( )𝛼− 𝑃 (𝑆𝑡 ) = ⎨ 𝑆𝑡 ∗ ⎪ (𝐾 − 𝑆 ) ⎩ 𝑆∗

if 𝑆𝑡 ≤ 𝑆 ∗ if 𝑆𝑡 > 𝑆 ∗

where (

𝛼− =

− 𝑟−𝐷−

1 2 𝜎 2

√(

) −

𝑟 − 𝐷 − 12 𝜎 2

𝜎2

)2

+ 2𝜎 2 𝑟

< 0.

Hence, show that the optimal exercise boundary is 𝑆∗ =

𝛼− 𝐾 𝛼− − 1

and for this optimal value the “smooth-pasting” condition Δ=

𝑑𝑃 || = −1 𝑑𝑆𝑡 ||𝑆𝑡 =𝑆 ∗

is satisfied. Deduce that if 𝑟 = 0 then it is not optimal to exercise the perpetual put option.

3.2.2 Time-Independent Options

299

Solution: Given that 𝑆 ∗ < 𝐾 is the optimal exercise boundary then, for 𝑆𝑡 ≤ 𝑆 ∗ , the perpetual American put option can be deduced to be equal to its intrinsic value, that is 𝑃 (𝑆𝑡 ) = 𝐾 − 𝑆𝑡 ,

𝑆𝑡 < 𝑆 ∗ .

For 𝑆𝑡 > 𝑆 ∗ , 𝑃 (𝑆𝑡 ) satisfies the second-order ODE and from Problem 3.2.2.1 (page 292) the general solution is 𝛼

𝛼

𝑃 (𝑆𝑡 ) = 𝐴𝑆𝑡 + + 𝐵𝑆𝑡 − where 𝐴 and 𝐵 are unknown constants, (

𝛼+ =

1 2 𝜎 2

− 𝑟−𝐷−

√(

)

𝑟 − 𝐷 − 12 𝜎 2

+

)2

+ 2𝜎 2 𝑟

𝜎2

>0

and (

𝛼− =

− 𝑟−𝐷−

1 2 𝜎 2

√(

) −

𝑟 − 𝐷 − 12 𝜎 2

)2

+ 2𝜎 2 𝑟

𝜎2

< 0.

From the initial conditions, for 𝑆𝑡 → ∞, 𝑃 (𝑆𝑡 ) → 0 and because 𝛼+ > 0 we can set 𝐴 = 0. Thus, 𝛼

𝑃 (𝑆𝑡 ) = 𝐵𝑆𝑡 − . Because 𝑃 (𝑆 ∗ ) = 𝐾 − 𝑆 ∗ by continuity 𝐵(𝑆 ∗ )𝛼− = 𝐾 − 𝑆 ∗ or 𝐵 = (𝐾 − 𝑆 ∗ )

(

1 𝑆∗

)𝛼−

.

Therefore, for 𝑆𝑡 > 𝑆 ∗ ( 𝑃 (𝑆𝑡 ) = (𝐾 − 𝑆 ∗ )

𝑆𝑡 𝑆∗

)𝛼−

.

To find the optimal exercise boundary we differentiate 𝑃 (𝑆𝑡 ) with respect to 𝑆 ∗ (

) ( ) 𝑆𝑡 𝛼− 𝛼− (𝐾 − 𝑆 ∗ ) 𝑆𝑡 𝛼− − 𝑆∗ 𝑆∗ 𝑆∗ ( )𝛼− [ ] ∗ 𝑆𝑡 𝛼− (𝐾 − 𝑆 ) =− 1+ 𝑆∗ 𝑆∗

𝑑𝑃 =− 𝑑𝑆 ∗

300

3.2.2 Time-Independent Options

and by setting

𝑑𝑃 = 0 we have 𝑑𝑆 ∗ 𝑆∗ =

By differentiating

𝛼− 𝐾 . 𝛼− − 1

𝑑𝑃 once again with respect to 𝑆 ∗ 𝑑𝑆 ∗

𝛼 𝑑2𝑃 = −∗ 𝑆 𝑑(𝑆 ∗ )2 and by substituting 𝑆 ∗ =

(

𝑆𝑡 𝑆∗

)𝛼− [

] ( )𝛼− 𝑆𝑡 𝛼− 𝐾 𝛼− (𝐾 − 𝑆 ∗ ) 1+ + 𝑆∗ 𝑆∗ (𝑆 ∗ )2

𝛼− 𝐾 we have 𝛼− − 1 𝑑 2 𝑃 || < 0. 𝑑(𝑆 ∗ )2 ||𝑆 ∗ = 𝛼− 𝐾 𝛼− −1

𝛼− 𝐾 is a local maximum point. 𝛼− − 1 ( )𝛼− 𝑆𝑡 and by differentiating 𝑃 (𝑆𝑡 ) with respect to For 𝑆𝑡 > 𝑆 ∗ , 𝑃 (𝑆𝑡 ) = (𝐾 − 𝑆 ∗ ) 𝑆∗

Hence, 𝑆 ∗ =

𝑆𝑡

𝛼 (𝐾 − 𝑆 ∗ ) 𝑑𝑃 = − 𝑑𝑆𝑡 𝑆∗

(

𝑆𝑡 𝑆∗

)𝛼− −1

.

By taking limits 𝑆𝑡 → (𝑆 ∗ )+

lim

𝑆𝑡 →(𝑆 ∗ )+

) ( 𝛼 𝐾 𝛼− 𝐾 − − 𝛼− − 1 𝑑𝑃 = 𝛼− 𝐾 𝑑𝑆𝑡 𝛼− − 1 = −1.

For 𝑆𝑡 ≤ 𝑆 ∗ , 𝑃 (𝑆𝑡 ) = 𝐾 − 𝑆𝑡 and by differentiating 𝑃 (𝑆𝑡 ) with respect to 𝑆𝑡 we have 𝑑𝑃 = −1 𝑑𝑆𝑡 and by taking limits lim

𝑆𝑡 →(𝑆 ∗ )−

𝑑𝑃 = −1. 𝑑𝑆𝑡

3.2.2 Time-Independent Options

Since

lim

𝑆𝑡 →(𝑆 ∗ )−

301

𝑑𝑃 𝑑𝑃 = lim = −1 therefore the “smooth-pasting” condition Δ = 𝑑𝑆𝑡 𝑆𝑡 →(𝑆 ∗ )+ 𝑑𝑆𝑡

𝑑𝑃 || = −1 is satisfied. 𝑑𝑆𝑡 ||𝑆𝑡 =𝑆 ∗ If 𝑟 = 0 then

𝛼− =

𝐷+

1 2 𝜎 2

√(

𝐷 + 12 𝜎 2

−

)2

𝜎2

= 0. Therefore, the optimal exercise boundary 𝑆 ∗ becomes 𝛼− 𝐾 𝑟→0 𝛼− − 1 = 0.

lim 𝑆 ∗ = lim

𝑟→0

Hence, if the risk-free interest rate is zero then it is not optimal to exercise the perpetual put option. 4. Let (Ω, ℱ, ℙ) be a probability space and let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ). Suppose at time 𝑡 that the stock price 𝑆𝑡 follows the GBM process with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 and a risk-free asset 𝐵𝑡 follows 𝑑𝐵𝑡 = 𝑟𝐵𝑡 𝑑𝑡 where 𝜇 is the drift rate, 𝐷 is the continuous dividend yield, 𝜎 is the asset price volatility and 𝑟 is the risk-free interest rate. 1 2 For a constant 𝜆 show that 𝑋𝑡 = 𝑒𝜆𝑊𝑡 − 2 𝜆 𝑡 is a martingale. By setting 𝑇 = min{𝑡 ≥ 0 : 𝑊𝑡 = 𝑎 + 𝑏𝑡} as the first-passage time of hitting the slope line 𝑎 + 𝑏𝑡 where 𝑎 and 𝑏 are constants, using the optimal stopping theorem show that the Laplace transform of its distribution is given by √ ) ( 2 𝔼 𝑒−𝜃𝑇 = 𝑒−𝑎(𝑏+ 𝑏 +2𝜃) ,

𝜃 > 0.

We consider the perpetual American call and put options denoted by 𝐶(𝑆𝑡 ) and 𝑃 (𝑆𝑡 ), respectively for a specified strike price 𝐾. Let 𝑆 ∞ > 𝐾 and 𝑆 ∗ < 𝐾 denote the unknown optimal exercise boundaries for the perpetual American call and put options, respectively. Here, for 𝑆𝑡 ≥ 𝑆 ∞ (𝑆𝑡 > 𝑆 ∗ ) the perpetual American call (put) option should be exercised (held) and for 𝑆𝑡 < 𝑆 ∞ (𝑆𝑡 ≤ 𝑆 ∗ ) the perpetual American call (put) option should be held (exercised).

302

3.2.2 Time-Independent Options

Under the risk-neutral measure ℚ and using the Laplace transform for the distribution of the time to hit the optimal exercise boundaries show that ⎧ (𝑆 ∞ − 𝐾) ⎪ 𝐶(𝑆𝑡 ) = ⎨ ⎪ ⎩ 𝑆𝑡 − 𝐾

(

𝑆𝑡 𝑆∞

)𝛼+

if 𝑆𝑡 < 𝑆 ∞ if 𝑆𝑡 ≥ 𝑆 ∞

and ⎧ 𝐾 − 𝑆𝑡 ⎪ ( )𝛼− 𝑃 (𝑆𝑡 ) = ⎨ 𝑆𝑡 ⎪ (𝐾 − 𝑆 ∗ ) ⎩ 𝑆∗

if 𝑆𝑡 ≤ 𝑆 ∗ if 𝑆𝑡 > 𝑆 ∗

where (

𝛼+ =

− 𝑟−𝐷−

1 2 𝜎 2

√(

)

𝑟 − 𝐷 − 12 𝜎 2

+

)2

+ 2𝜎 2 𝑟

𝜎2

>0

and (

𝛼− =

− 𝑟−𝐷−

1 2 𝜎 2

√(

) −

𝑟 − 𝐷 − 12 𝜎 2

)2

+ 2𝜎 2 𝑟

𝜎2 1 2

< 0.

Solution: To show that 𝑋𝑡 = 𝑒𝜆𝑊𝑡 − 2 𝜆 𝑡 is a martingale and the Laplace transform of the first-passage time distribution see Problems 2.2.3.3 and 2.2.4.7, respectively of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus. We first consider the perpetual American call option case where we let 𝜏 ∞ be the time to hit the unknown optimal exercise boundary 𝑆 ∞ defined as 𝜏 ∞ = min{𝑢 − 𝑡 ≥ 0 : 𝑆𝑢 = 𝑆 ∞ }. From Girsanov’s theorem, under the risk-neutral measure ℚ we have 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡 where 𝑊𝑡ℚ = 𝑊𝑡 +

(𝜇 − 𝑟) 𝜎

𝑡 is a ℚ-standard Wiener process.

3.2.2 Time-Independent Options

303

Hence, the perpetual American call option price at time 𝑡 is defined as [ ] ∞ | 𝐶(𝑆𝑡 ) = 𝔼ℚ 𝑒−𝑟𝜏 max{𝑆𝜏 ∞ − 𝐾, 0}| ℱ𝑡 | [ ] ⎧ (𝑆 ∞ − 𝐾)𝔼ℚ 𝑒−𝑟𝜏 ∞ || ℱ if 𝑆𝑡 < 𝑆 ∞ ⎪ | 𝑡 =⎨ ⎪𝑆 − 𝐾 if 𝑆𝑡 ≥ 𝑆 ∞ . ⎩ 𝑡 By solving the SDE, for 𝑢 > 𝑡 we have 𝑆𝑢 = 𝑆𝑡 exp

[(

) ] 1 ℚ 𝑟 − 𝐷 − 𝜎 2 (𝑢 − 𝑡) + 𝜎𝑊𝑢−𝑡 2

and hence the hitting time 𝜏 ∞ becomes ) ] } { [( 1 ℚ 𝜏 ∞ = min 𝑢 − 𝑡 ≥ 0 : 𝑆𝑡 exp 𝑟 − 𝐷 − 𝜎 2 (𝑢 − 𝑡) + 𝜎𝑊𝑢−𝑡 = 𝑆∞ { } ( 2∞ ) ( ) 1 2 1 𝑆 1 ℚ 𝑟 − 𝐷 − 𝜎 (𝑢 − 𝑡) = min 𝑢 − 𝑡 ≥ 0 : 𝑊𝑢−𝑡 = log − 𝜎 𝑆𝑡 𝜎 2 { ( ∞) ( ) } 1 1 𝑆 1 𝑟 − 𝐷 − 𝜎2 𝑣 = min 𝑣 ≥ 0 : 𝑊𝑣ℚ = log − 𝜎 𝑆𝑡 𝜎 2 where 𝑣 = 𝑢 − 𝑡.

( ∞) ( ) 1 1 𝑆 1 𝑟 − 𝐷 − 𝜎 2 , the Laplace log and 𝑏 = − 𝜎 𝑆𝑡 𝜎 2 transform of the hitting-time distribution becomes Therefore, by setting 𝑎 =

√ [ ] ∞| 2 𝔼ℚ 𝑒−𝑟𝜏 | ℱ𝑡 = 𝑒−𝑎(𝑏+ 𝑏 +2𝑟) | { ( ∞)[ ( ) 1 1 1 𝑆 𝑟 − 𝐷 − 𝜎2 − = exp − log 𝜎 𝑆𝑡 𝜎 2 √ ⎤⎫ ( ) ⎪ 1 2 2 1 𝑟−𝐷− 𝜎 + + 2𝑟⎥⎬ 2 ⎥⎪ 2 𝜎 ⎦⎭ ( )𝛼+ 𝑆𝑡 = 𝑆∞

where (

𝛼+ =

− 𝑟−𝐷−

1 2 𝜎 2

√(

) +

𝑟 − 𝐷 − 12 𝜎 2

𝜎2

)2

+ 2𝜎 2 𝑟

> 0.

304

3.2.2 Time-Independent Options

Thus, ⎧ (𝑆 ∞ − 𝐾) ⎪ 𝐶(𝑆𝑡 ) = ⎨ ⎪ ⎩ 𝑆𝑡 − 𝐾

(

𝑆𝑡 𝑆∞

)𝛼+

if 𝑆𝑡 < 𝑆 ∞ if 𝑆𝑡 ≥ 𝑆 ∞

(

) 𝑆𝑡 𝛼+ = 0. 𝑆𝑡 →0 𝑆𝑡 →0 𝑆 ∞ For the case of a perpetual American put option, let 𝜏 ∗ be the time to hit the unknown optimal exercise boundary 𝑆 ∗ defined as such that lim 𝐶(𝑆𝑡 ) = (𝑆 ∞ − 𝐾) lim

𝜏 ∗ = min{𝑢 − 𝑡 ≥ 0 : 𝑆𝑢 = 𝑆 ∗ } and the option price is defined as [ ] ∗ | 𝑃 (𝑆𝑡 ) = 𝔼ℚ 𝑒−𝑟𝜏 max{𝐾 − 𝑆𝜏 ∗ , 0}| ℱ𝑡 | ⎧ 𝐾 − 𝑆𝑡 if 𝑆𝑡 ≤ 𝑆 ∗ ⎪ =⎨ [ ] ∗ ⎪ (𝐾 − 𝑆 ∗ )𝔼ℚ 𝑒−𝑟𝜏 || ℱ𝑡 if 𝑆𝑡 > 𝑆 ∗. | ⎩ ) ] [( ℚ Since 𝑆𝑢 = 𝑆𝑡 exp 𝑟 − 12 𝜎 2 (𝑢 − 𝑡) + 𝜎𝑊𝑢−𝑡 for 𝑢 > 𝑡, and using the reflection princi∗ ple of the Wiener process, the hitting time 𝜏 can be written as } ) ( ) 1 2 𝑆∞ 1 𝑟 − 𝐷 − 𝜎 (𝑢 − 𝑡) 𝜏 = min 𝑢 − 𝑡 ≥ 0 : − 𝑆𝑡 𝜎 2 { } ( ∞) ( ) 1 1 𝑆 1 ℚ 𝑟 − 𝐷 − 𝜎 2 (𝑢 − 𝑡) = min 𝑢 − 𝑡 ≥ 0 : −𝑊𝑢−𝑡 = − log + 𝜎 𝑆𝑡 𝜎 2 { } ( ∞) ( ) 1 2 1 ̃ ℚ = − 1 log 𝑆 𝑟 − 𝐷 − = min 𝑢 − 𝑡 ≥ 0 : 𝑊 (𝑢 − 𝑡) 𝜎 + 𝑢−𝑡 𝜎 𝑆 𝜎 2 { ( ∞) 𝑡 ( ) } 1 1 ̃ ℚ = − 1 log 𝑆 𝑟 − 𝐷 − 𝜎2 𝑣 = min 𝑣 ≥ 0 : 𝑊 + 𝑣 𝜎 𝑆𝑡 𝜎 2 {

∗

ℚ 𝑊𝑢−𝑡

1 = log 𝜎

(

̃ ℚ is a ℚ-standard Wiener process. where 𝑣 = 𝑢 − 𝑡 and 𝑊 𝑣 ( ) ( ) 1 𝑆∞ 1 1 𝑟 − 𝐷 − 𝜎 2 , the Laplace transform of By setting 𝑎 = − log and 𝑏 = 𝜎 𝑆𝑡 𝜎 2 the hitting-time distribution becomes √ [ ] ∗| 2 𝔼ℚ 𝑒−𝑟𝜏 | ℱ𝑡 = 𝑒−𝑎(𝑏+ 𝑏 +2𝑟) | { ( ∞)[ ( ) 1 1 1 𝑆 𝑟 − 𝐷 − 𝜎2 = exp log 𝜎 𝑆𝑡 𝜎 2

3.2.3 Time-Dependent Options

305

√ + ( =

𝑆𝑡 𝑆∗

⎤⎫ ( ) ⎪ 1 2 2 1 𝑟−𝐷− 𝜎 + 2𝑟⎥⎬ ⎥⎪ 2 𝜎2 ⎦⎭ )𝛼−

where (

𝛼− =

− 𝑟−𝐷−

1 2 𝜎 2

√(

) −

𝑟 − 𝐷 − 12 𝜎 2

)2

+ 2𝜎 2 𝑟

𝜎2

< 0.

Hence, ⎧ 𝐾 − 𝑆𝑡 ⎪ ( )𝛼− 𝑃 (𝑆𝑡 ) = ⎨ 𝑆𝑡 ∗ ⎪ (𝐾 − 𝑆 ) ⎩ 𝑆∗ ( where lim 𝑃 (𝑆𝑡 ) = (𝐾 − 𝑆 ) lim ∗

𝑆𝑡 →∞

3.2.3

𝑆𝑡 →∞

𝑆𝑡 𝑆∗

if 𝑆𝑡 ≤ 𝑆 ∗ if 𝑆𝑡 > 𝑆 ∗

)𝛼− = 0.

Time-Dependent Options

1. Consider an economy which consists of a risk-free asset and a stock, whose values at time 𝑡 are 𝐵𝑡 and 𝑆𝑡 , respectively. Assume that these values evolve according to the following diffusion processes 𝑑𝐵𝑡 = 𝑟𝐵𝑡 𝑑𝑡,

𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡

such that 𝐷 is the continuous dividend yield, 𝑟 is the risk-free rate, 𝜇 is the stock price growth rate and 𝜎𝑡 is the stock price volatility. In addition, {𝑊𝑡 : 𝑡 ≥ 0} is the ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ). Show that the American option 𝑉 (𝑆𝑡 , 𝑡) satisfies the following inequality 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 − 𝑟𝑉 (𝑆𝑡 , 𝑡) ≤ 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 with constraint 𝑉 (𝑆𝑡 , 𝑡) ≥ Ψ(𝑆𝑡 ) where Ψ(𝑆𝑡 ) is the intrinsic value of the American option at time 𝑡.

306

3.2.3 Time-Dependent Options

Solution: We first set up a Δ-hedged portfolio Π𝑡 = 𝑉 (𝑆𝑡 , 𝑡) − Δ𝑆𝑡 and because during time 𝑑𝑡 the stock pays out a continuous dividend 𝐷𝑆𝑡 𝑑𝑡 we have 𝑑Π𝑡 = 𝑑𝑉 − Δ(𝑑𝑆𝑡 + 𝐷𝑆𝑡 𝑑𝑡). By Taylor’s expansion and subsequently using It̄o’s lemma we can write 1 𝜕2𝑉 1 𝜕3𝑉 𝜕𝑉 𝜕𝑉 2 𝑑𝑆𝑡 + (𝑑𝑆 ) + (𝑑𝑆𝑡 )3 + ⋯ 𝑑𝑡 + 𝑡 𝜕𝑡 𝜕𝑆𝑡 2 𝜕𝑆𝑡2 3! 𝜕𝑆𝑡3 ] 1 𝜕2𝑉 𝜕𝑉 [ 𝜕𝑉 (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑑𝑊𝑡 + 𝜎 2 𝑆𝑡2 2 𝑑𝑡 𝑑𝑡 + = 𝜕𝑡 𝜕𝑆𝑡 2 𝜕𝑆𝑡 [ ] 𝜕𝑉 𝜕2 𝑉 𝜕𝑉 1 𝜕𝑉 = 𝑑𝑊𝑡 + 𝜎 2 𝑆𝑡2 2 + (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎 𝜕𝑡 2 𝜕𝑆 𝜕𝑆 𝜕𝑆𝑡 𝑡 𝑡

𝑑𝑉 =

and 𝑑𝑆𝑡 + 𝐷𝑆𝑡 𝑑𝑡 = 𝜇𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 . By substituting the above expressions into 𝑑Π𝑡 we have [

] 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 + (𝜇 − 𝐷)𝑆𝑡 − Δ𝜇𝑆𝑡 𝑑𝑡 𝑑Π𝑡 = + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 [ ] 𝜕𝑉 +𝜎 − Δ 𝑑𝑊𝑡 . 𝜕𝑆𝑡 To eliminate the random component we set Δ=

𝜕𝑉 𝜕𝑆𝑡

and hence [ 𝑑Π𝑡 =

] 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 − 𝐷𝑆𝑡 + 𝜎 𝑆𝑡 𝑑𝑡. 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

If we assume 𝑑Π𝑡 > 𝑟Π𝑡 𝑑𝑡 then we would experience an arbitrage since a trader can borrow money at the risk-free rate and then set up a Δ-hedged portfolio such that the return from the portfolio is greater than the return from a bank.

3.2.3 Time-Dependent Options

307

Because there are times to exercise the option, the simple arbitrage argument used in the European options is not valid here. Hence, we can conclude that for an American option 𝑑Π𝑡 ≤ 𝑟Π𝑡 𝑑𝑡 or

[

] [ ] 𝜕𝑉 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 − 𝐷𝑆𝑡 𝑆 𝑑𝑡. + 𝜎 𝑆𝑡 𝑑𝑡 ≤ 𝑟 𝑉 (𝑆𝑡 , 𝑡) − 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝑡 𝜕𝑆𝑡2

Therefore, 𝜕2 𝑉 𝜕𝑉 1 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡) ≤ 0. + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 Here, for a European option, the inequality would become an equality. Finally, from Problems 3.2.1.6 (page 276) and 3.2.1.12 (page 279) we can deduce that 𝑉 (𝑆𝑡 , 𝑡) ≥ Ψ(𝑆𝑡 ). 2. Linear Complementarity Problem. We consider the price of an American option 𝑉 (𝑆𝑡 , 𝑡) written on an underlying asset 𝑆𝑡 at time 𝑡 which satisfies the following inequality 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 − 𝑟𝑉 (𝑆𝑡 , 𝑡) ≤ 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 with constraint 𝑉 (𝑆𝑡 , 𝑡) ≥ Ψ(𝑆𝑡 ) where Ψ(𝑆𝑡 ) is the option’s intrinsic value, 𝑟 is the risk-free interest rate, 𝐷 is the continuous dividend yield and 𝜎 is the constant volatility. Show that 𝑉 (𝑆𝑡 , 𝑡) with its corresponding intrinsic value Ψ(𝑆𝑡 ) satisfies the linear complementarity problem [ ] 𝑉 (𝑆𝑡 , 𝑡) − Ψ(𝑆𝑡 ) ⋅ 𝐵𝑆 [𝑉 (𝑆𝑡 , 𝑡)] = 0 𝐵𝑆 [𝑉 (𝑆𝑡 , 𝑡)] ≤ 0, 𝑉 (𝑆𝑡 , 𝑡) ≥ Ψ(𝑆𝑡 ), where 𝐵𝑆 denotes the differential operator 𝐵𝑆 =

𝜕 𝜕2 𝜕 1 − 𝑟. + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡

Solution: Suppose that 𝑉 (𝑆𝑡 , 𝑡) = Ψ(𝑆𝑡 ). Then early exercise is optimal and by substituting 𝑉 (𝑆𝑡 , 𝑡) = Ψ(𝑆𝑡 ) into the inequality we have 𝜕Ψ 1 2 2 𝜕 2 Ψ 𝜕Ψ + (𝑟 − 𝐷)𝑆𝑡 − 𝑟Ψ(𝑆𝑡 ) < 0 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 since the option’s intrinsic value does not satisfy the Black–Scholes equation.

308

3.2.3 Time-Dependent Options

Hence, 𝑉 (𝑆𝑡 , 𝑡) = Ψ(𝑆𝑡 ),

𝐵𝑆 [𝑉 (𝑆𝑡 , 𝑡)] < 0.

In contrast, if 𝑉 (𝑆𝑡 , 𝑡) > Ψ(𝑆𝑡 ) then early exercise is not optimal and therefore 𝑉 (𝑆𝑡 , 𝑡) > Ψ(𝑆𝑡 ),

𝐵𝑆 [𝑉 (𝑆𝑡 , 𝑡)] = 0.

Collectively we can write 𝐵𝑆 [𝑉 (𝑆𝑡 , 𝑡)] ≤ 0,

𝑉 (𝑆𝑡 , 𝑡) ≥ Ψ(𝑆𝑡 ),

[

] 𝑉 (𝑆𝑡 , 𝑡) − Ψ(𝑆𝑡 ) ⋅ 𝐵𝑆 [𝑉 (𝑆𝑡 , 𝑡)] = 0.

3. Continuous Limit of Binomial Model for American Options. In a discrete-time Black– Scholes world, we consider the binomial tree model to calculate the price of an American option. At time 𝑡, where the current spot price is 𝑆𝑡 , we build a tree of possible scenarios of future stock prices 𝑆𝑖(𝑗) = 𝑢𝑗 𝑑 𝑖−𝑗 𝑆𝑡 ,

𝑖 = 0, 1, … , 𝑁,

𝑗 = 0, 1, 2, … , 𝑖 √

√

such that 𝑆0(0) = 𝑆𝑡 , 𝑁 is the total number of time periods, 𝑢 = 𝑒𝜎 Δ𝑡 , 𝑑 = 𝑒−𝜎 Δ𝑡 , Δ𝑡 = (𝑇 − 𝑡)∕𝑁 is the binomial time step, 𝑇 is the expiry time and 𝜎 is the stock price volatility. The intermediate American option price 𝑉 (𝑆𝑛(𝑚) , 𝑡 + 𝑛Δ𝑡) calculated at the 𝑚-th possible tree value and at time step 𝑡 + 𝑛Δ𝑡, where 𝑚 ≤ 𝑛, 𝑛 ≤ 𝑁 is defined as , 𝑡 + 𝑛Δ𝑡) 𝑉 (𝑆𝑛(𝑚){

[ ]} = max Ψ(𝑆𝑛(𝑚) ), 𝑒−𝑟Δ𝑡 𝜋𝑉 (𝑢𝑆𝑛(𝑚) , 𝑡 + (𝑛 + 1)Δ𝑡) + (1 − 𝜋)𝑉 (𝑑𝑆𝑛(𝑚) , 𝑡 + (𝑛 + 1)Δ𝑡)

𝑒(𝑟−𝐷)Δ𝑡 − 𝑑 is the risk-neutral probawhere Ψ(𝑆𝑛(𝑚) ) is the option’s intrinsic value, 𝜋 = 𝑢−𝑑 bility, 𝑟 is the risk-free interest rate and 𝐷 is the continuous dividend yield. Using Taylor’s expansion up to 𝑂(Δ𝑡) show that in the limit Δ𝑡 → 0 the binomial method approximates the continuous-time linear complementarity problem of the form 𝐵𝑆 [𝑉 (𝑆, 𝑡′ )] ≤ 0,

𝑉 (𝑆, 𝑡′ ) ≥ Ψ(𝑆),

[

] 𝑉 (𝑆, 𝑡′ ) − Ψ(𝑆) ⋅ 𝐵𝑆 [𝑉 (𝑆, 𝑡′ )] = 0

where 𝑆 = 𝑆𝑛(𝑚) , 𝑡′ = 𝑡 + 𝑛Δ𝑡 and 𝐵𝑆 denotes the differential operator 𝐵𝑆 =

𝜕 1 2 2 𝜕2 𝜕 + + (𝑟 − 𝐷)𝑆 𝜎 𝑆 − 𝑟. 𝜕𝑡′ 2 𝜕𝑆 𝜕𝑆 2

Solution: By setting 𝑆 = 𝑆𝑛(𝑚) and 𝑡′ = 𝑡 + 𝑛Δ𝑡, the intermediate American option price can be written as [ { ]} 𝑉 (𝑆, 𝑡′ ) = max Ψ(𝑆), 𝑒−𝑟Δ𝑡 𝜋𝑉 (𝑢𝑆, 𝑡′ + Δ𝑡) + (1 − 𝜋)𝑉 (𝑑𝑆, 𝑡′ + Δ𝑡)

3.2.3 Time-Dependent Options

309

where we either have [ ] 𝑉 (𝑆, 𝑡′ ) = Ψ(𝑆) and 𝑉 (𝑆, 𝑡′ ) > 𝑒−𝑟Δ𝑡 𝜋𝑉 (𝑢𝑆, 𝑡′ + Δ𝑡) + (1 − 𝜋)𝑉 (𝑑𝑆, 𝑡′ + Δ𝑡) or [ ] 𝑉 (𝑆, 𝑡′ ) > Ψ(𝑆) and 𝑉 (𝑆, 𝑡′ ) = 𝑒−𝑟Δ𝑡 𝜋𝑉 (𝑢𝑆, 𝑡′ + Δ𝑡) + (1 − 𝜋)𝑉 (𝑑𝑆, 𝑡′ + Δ𝑡) . By expanding 𝑉 (𝑢𝑆, 𝑡′ + Δ𝑡) and 𝑉 (𝑑𝑆, 𝑡′ + Δ𝑡) using Taylor’s theorem and using the same steps as discussed in Problem 2.2.3.8 (page 202), we can write 𝜋𝑉 (𝑢𝑆, 𝑡′ + Δ𝑡) + (1 − 𝜋)𝑉 (𝑑𝑆, 𝑡′ + Δ𝑡) − 𝑒𝑟Δ𝑡 𝑉 (𝑆, 𝑡′ ) ] [ 1 2 2 𝜕2𝑉 𝜕𝑉 𝜕𝑉 ′ + 𝑆 + (𝑟 − 𝐷)𝑆 ) + 𝑂(Δ𝑡) Δ𝑡. 𝜎 − 𝑟𝑉 (𝑆, 𝑡 = 𝜕𝑡′ 2 𝜕𝑆 𝜕𝑆 2 Since 𝜋𝑉 (𝑢𝑆, 𝑡′ + Δ𝑡) + (1 − 𝜋)𝑉 (𝑑𝑆, 𝑡′ + Δ𝑡) − 𝑒𝑟Δ𝑡 𝑉 (𝑆, 𝑡′ ) = 0 and 𝑉 (𝑆, 𝑡′ ) > Ψ(𝑆) or 𝜋𝑉 (𝑢𝑆, 𝑡′ + Δ𝑡) + (1 − 𝜋)𝑉 (𝑑𝑆, 𝑡′ + Δ𝑡) − 𝑒𝑟Δ𝑡 𝑉 (𝑆, 𝑡′ ) < 0 and 𝑉 (𝑆, 𝑡′ ) = Ψ(𝑆) we have 1 𝜕𝑉 𝜕2𝑉 𝜕𝑉 + 𝜎 2 𝑆 2 2 + (𝑟 − 𝐷)𝑆 − 𝑟𝑉 (𝑆, 𝑡′ ) + 𝑂(Δ𝑡) = 0 and 𝑉 (𝑆, 𝑡′ ) > Ψ(𝑆) ′ 𝜕𝑡 2 𝜕𝑆 𝜕𝑆 or 1 2 2 𝜕2𝑉 𝜕𝑉 𝜕𝑉 + + (𝑟 − 𝐷)𝑆 𝜎 𝑆 − 𝑟𝑉 (𝑆, 𝑡′ ) + 𝑂(Δ𝑡) < 0 and 𝑉 (𝑆, 𝑡′ ) = Ψ(𝑆). 𝜕𝑡′ 2 𝜕𝑆 𝜕𝑆 2 Hence, by taking limits Δ𝑡 → 0 𝐵𝑆 [𝑉 (𝑆, 𝑡′ )] = 0 and 𝑉 (𝑆, 𝑡′ ) > Ψ(𝑆) or 𝐵𝑆 [𝑉 (𝑆, 𝑡′ )] < 0 and 𝑉 (𝑆, 𝑡′ ) = Ψ(𝑆). Then we have 𝐵𝑆 [𝑉 (𝑆, 𝑡′ )] ≤ 0,

𝑉 (𝑆, 𝑡′ ) ≥ Ψ(𝑆),

[

] 𝑉 (𝑆, 𝑡′ ) − Ψ(𝑆) ⋅ 𝐵𝑆 [𝑉 (𝑆, 𝑡′ )] = 0.

310

3.2.3 Time-Dependent Options

4. Consider a binomial tree model for an underlying asset process {𝑆𝑛 : 0 ≤ 𝑛 ≤ 3} where 𝑆0 = 100. Let

𝑆𝑛+1

⎧ 𝑢𝑆 ⎪ 𝑛 =⎨ ⎪ 𝑑𝑆𝑛 ⎩

with probability 𝜋 with probability 1 − 𝜋

√

where 𝑢 = 𝑒𝜎 Δ𝑡 and 𝑑 = 1∕𝑢 where 𝜎 is the volatility and Δ𝑡 is the binomial time step. By assuming the risk-neutral interest rate 𝑟 = 5%, continuous dividend yield 𝐷 = 1% and volatility 𝜎 = 10%, we wish to price an American put option with strike 𝐾 = 105 and expiry time 𝑇 = 1 year in a 3-period binomial tree model. (a) Find the risk-neutral probabilities 𝜋 and 1 − 𝜋. (b) Find the price of the American put option. (c) What is the trading strategy to hedge this option at the initial time period? Solution: (a) Given 𝑆0 = 100, 𝐾 = 105, 𝑟 = 0.05, 𝐷 = 0.01, 𝜎 = 0.1 and time step Δ𝑡 = 13 , therefore 𝑢 = 𝑒𝜎

√ Δ𝑡

=𝑒

0.1 √ 3

= 1.0594

and 𝑑 = 𝑒−𝜎

√ Δ𝑡

=𝑒

0.1 −√

3

= 0.9439.

The risk-neutral probabilities are 𝜋=

𝑒(𝑟−𝐷)Δ𝑡 − 𝑑 = 0.6019 𝑢−𝑑

and 1 − 𝜋 = 1 − 0.6019 = 0.3981. (b) The binomial tree in Figure 3.1 shows the price movement of 𝑆0 in a 3-period binomial model. By setting 𝑆𝑖(𝑗) = 𝑢𝑗 𝑑 𝑖−𝑗 𝑆0 ,

𝑖 = 0, 1, … , 𝑛,

𝑗 = 0, 1, … , 𝑖

the American put option price at each of the lattice points is [ ]} { (𝑗+1) (𝑗) + (1 − 𝜋)𝑉𝑖+1 𝑉𝑖(𝑗) = max Ψ(𝑆𝑖(𝑗) ), 𝑒−𝑟Δ𝑡 𝜋𝑉𝑖+1

3.2.3 Time-Dependent Options

311 ∙ ∙ ∙

2

∙ ∙

∙

such that

0

2

∙ 2

0

0

∙

Figure 3.1

2

0

0

∙

0

0

0

0∙

3

3

A 3-period binomial tree model.

{ } Ψ(𝑆𝑖(𝑗) ) = max 𝐾 − 𝑆𝑖(𝑗) , 0

where 𝑖 = 0, 1, … , 𝑛 and 𝑗 = 0, 1, … , 𝑖. Hence, at time period 𝑛 = 3 𝑉3(0) = Ψ(𝑆3(0) ) = max{𝐾 − 𝑑 3 𝑆0 , 0} = max{105 − 0.94393 × 100, 0} = 20.9035 𝑉3(1) = Ψ(𝑆3(1) ) = max{𝐾 − 𝑢𝑑 2 𝑆0 , 0} = max{105 − 1.0594 × 0.94392 × 100, 0} = 10.6131 𝑉3(2) = Ψ(𝑆3(2) ) = max{𝐾 − 𝑢2 𝑑𝑆0 , 0} = max{105 − 1.05942 × 0.9439 × 100, 0} =0 𝑉3(3) = Ψ(𝑆3(3) ) = max{𝐾 − 𝑢3 𝑆0 , 0} = max{105 − 1.05943 × 100, 0} = 0.

0

312

3.2.3 Time-Dependent Options

At time period 𝑛 = 2 [ ]} { 𝑉2(0) = max Ψ(𝑆2(0) , 𝑒−𝑟Δ𝑡 𝜋𝑉3(1) + (1 − 𝜋)𝑉3(0) [ ]} { } { = max max 𝐾 − 𝑑 2 𝑆0 , 0 , 𝑒−𝑟Δ𝑡 𝜋𝑉3(1) + (1 − 𝜋)𝑉3(0) } { { = max max 105 − 0.94392 × 100, 0 , } 0.05 𝑒− 3 [0.6019 × 10.6131 + 0.3981 × 20.9035]

𝑉2(1)

= max{15.9053, 14.4666} = 15.9053 [ ]} { = max Ψ(𝑆2(1) , 𝑒−𝑟Δ𝑡 𝜋𝑉3(2) + (1 − 𝜋)𝑉3(1) [ ]} { } { = max max 𝐾 − 𝑢𝑑𝑆0 , 0 , 𝑒−𝑟Δ𝑡 𝜋𝑉3(2) + (1 − 𝜋)𝑉3(1) = max {max {105 − 1.0594 × 0.9439 × 100, 0} , } 0.05 𝑒− 3 [0.6019 × 0 + 0.3981 × 10.6131]

𝑉2(2)

= max{5.0032, 4.1552} = 5.0032 [ ]} { = max Ψ(𝑆2(2) , 𝑒−𝑟Δ𝑡 𝜋𝑉3(3) + (1 − 𝜋)𝑉3(2) [ ]} { } { = max max 𝐾 − 𝑢2 𝑆0 , 0 , 𝑒−𝑟Δ𝑡 𝜋𝑉3(3) + (1 − 𝜋)𝑉3(2) } { { = max max 105 − 1.05942 × 100, 0 , } 0.05 𝑒− 3 [0.6019 × 0 + 0.3981 × 0] = max{0, 0} = 0.

At time period 𝑛 = 1 [ ]} { 𝑉1(0) = max Ψ(𝑆1(0) , 𝑒−𝑟Δ𝑡 𝜋𝑉2(1) + (1 − 𝜋)𝑉2(0) [ ]} { } { = max max 𝐾 − 𝑑𝑆0 , 0 , 𝑒−𝑟Δ𝑡 𝜋𝑉2(1) + (1 − 𝜋)𝑉2(0) = max {max {105 − 0.9439 × 100, 0} , 𝑒−

0.05 3

[0.6019 × 5.0032 + 0.3981 × 15.9053]

= max{10.61, 9.1889} = 10.61

}

3.2.3 Time-Dependent Options

313

[ ]} { 𝑉1(1) = max Ψ(𝑆1(1) , 𝑒−𝑟Δ𝑡 𝜋𝑉2(2) + (1 − 𝜋)𝑉2(1) [ ]} { } { = max max 𝐾 − 𝑢𝑆0 , 0 , 𝑒−𝑟Δ𝑡 𝜋𝑉2(2) + (1 − 𝜋)𝑉2(1) = max {max {105 − 1.0594 × 100, 0} , 𝑒−

0.05 3

} [0.6019 × 0 + 0.3981 × 5.0032]

= max{0, 1.9589} = 1.9589 and finally at time period 𝑛 = 0 [ ]} { 𝑉0(0) = max Ψ(𝑆0(0) , 𝑒−𝑟Δ𝑡 𝜋𝑉1(1) + (1 − 𝜋)𝑉1(0) [ ]} { } { = max max 𝐾 − 𝑆0 , 0 , 𝑒−𝑟Δ𝑡 𝜋𝑉1(1) + (1 − 𝜋)𝑉1(0) = max {max {105 − 100, 0} , 𝑒−

0.05 3

}

[0.6019 × 1.9589 + 0.3981 × 10.61]

= max{5, 5.3136} = 5.3136. Therefore, the price of the American put based on a 3-period binomial model is 𝑉0(0) = 5.3136. (c) At time period 𝑛 = 0, let 𝜙0 and 𝜓0 be the unit of the underlying asset 𝑆0 and the amount of cash invested in the money market, respectively. At time period 𝑛 = 1, the asset price can either be 𝑆1(0) = 𝑑𝑆0 or 𝑆11 = 𝑢𝑆0 and 𝜓0 will grow to 𝜓0 𝑒𝑟Δ𝑡 . Thus, we can write 𝑉1(0) = 𝜙0 𝑆1(0) + 𝜓0 𝑒𝑟Δ𝑡 𝑉1(1) = 𝜙0 𝑆1(1) + 𝜓0 𝑒𝑟Δ𝑡 . Hence, 𝜙0 =

𝑉1(1) − 𝑉1(0) 𝑆1(1) − 𝑆1(0)

=

𝑉1(1) − 𝑉1(0) 𝑢𝑆0 − 𝑑𝑆0

and 𝜓0 = 𝑒−𝑟Δ𝑡 (𝑉1(0) − 𝜙0 𝑆1(0) ). By substituting 𝑉1(0) = 10.61, 𝑉1(1) = 1.9589, 𝑆1(0) = 𝑑𝑆0 = 94.39, 𝑆1(1) = 𝑢𝑆0 = 105.94, 𝑟 = 0.05 and Δ𝑡 = 13 we have 𝜙0 = −0.7490

and 𝜓0 = 79.9642

314

3.2.3 Time-Dependent Options

which implies we need to sell −0.7778 units of the underlying asset 𝑆0 and put $79.9642 into the money market at 5% interest rate.

5. Smooth-Pasting Condition for an American Call Option. We consider an American call option, 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) with strike 𝐾 written on an underlying asset priced at 𝑆𝑡 at time 𝑡 and 𝑇 > 𝑡 is the option expiry time. Here the option has an unknown optimal exercise boundary 𝑆𝑡∞ such that the option should be exercised if 𝑆𝑡 ≥ 𝑆𝑡∞ and held if 𝑆𝑡 < 𝑆𝑡∞ . By assuming 𝑆𝑡∞ > 𝐾 show that the American call option satisfies the “smooth-pasting” condition 𝜕𝐶 || =1 𝜕𝑆𝑡 ||𝑆𝑡 =𝑆 ∞ 𝑡

at the optimal exercise boundary 𝑆𝑡∞ . Solution: We prove this result via contradiction. From definition, the value of an American call option can be written as ⎧ 𝑓 (𝑆 , 𝑡; 𝐾, 𝑇 ) 𝑡 ⎪ 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ ⎪ 𝑆𝑡 − 𝐾 ⎩

if 𝑆𝑡 < 𝑆𝑡∞ if 𝑆𝑡 ≥ 𝑆𝑡∞

𝜕𝐶 = 1 for 𝑆𝑡 ≥ 𝑆𝑡∞ . For a 𝜕𝑆𝑡 graphical interpretation of the American call option see Figure 3.2.

where 𝑓 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) solves the Black–Scholes equation and

(

;

)

−

(

;

)

∞

Figure 3.2

American call option price.

3.2.3 Time-Dependent Options

For 𝑆𝑡 < 𝑆 ∞ , suppose

315

𝜕𝐶 || < 1 then 𝜕𝑆𝑡 ||𝑆𝑡 =𝑆 ∞ 𝑡

𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) < max{𝑆𝑡 − 𝐾, 0} which contradicts the fact that 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝑆𝑡 − 𝐾, 0}. 𝜕𝐶 || Conversely if > 1 then as the underlying asset price gets closer to 𝑆𝑡∞ the 𝜕𝑆𝑡 ||𝑆𝑡 =𝑆 ∞ 𝑡 value of the American call option can be increased by choosing a larger 𝑆𝑡∞ , which is a contradiction to the fact that 𝑆𝑡∞ is the optimal exercise boundary. Hence, in order to satisfy the optimal exercise strategy of an American call option we have 𝜕𝐶 || =1 𝜕𝑆𝑡 ||𝑆𝑡 =𝑆 ∞ 𝑡

where the two parts of 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) are joined smoothly without any discontinuity at 𝑆𝑡 = 𝑆𝑡∞ . 6. Smooth-Pasting Condition for an American Put Option. We consider an American put option 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) with strike 𝐾 written on an underlying asset priced at 𝑆𝑡 at time 𝑡 where 𝑇 > 𝑡 is the option expiry time. Here the option has an unknown optimal exercise boundary 𝑆𝑡∗ such that the option should be exercised if 𝑆𝑡 ≤ 𝑆 ∗ and held if 𝑆𝑡 > 𝑆𝑡∗ . By assuming 𝑆𝑡∗ < 𝐾 show that the American put option satisfies the “smooth-pasting” condition 𝜕𝑃 || = −1 𝜕𝑆𝑡 ||𝑆𝑡 =𝑆 ∗ 𝑡

at the optimal exercise boundary 𝑆𝑡∗ . Solution: We prove this result via contradiction. From definition, the value of an American put option can be written as ⎧𝐾 − 𝑆 𝑡 ⎪ 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ ⎪ 𝑔(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎩

if 𝑆𝑡 ≤ 𝑆𝑡∗ if 𝑆𝑡 > 𝑆𝑡∗

𝜕𝑃 where 𝑔(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) solves the Black–Scholes equation and = −1 for 𝑆𝑡 ≤ 𝑆𝑡∗ . For a 𝜕𝑆𝑡 graphical interpretation see Figure 3.3. 𝜕𝑃 || < −1 then for 𝑆𝑡 > 𝑆𝑡∗ Suppose 𝜕𝑆𝑡 ||𝑆𝑡 =𝑆 ∗ 𝑡 𝑃 (𝑆𝑡 , 𝑡) < max{𝐾 − 𝑆𝑡 , 0} which contradicts the fact that 𝑃 (𝑆𝑡 , 𝑡) ≥ max{𝐾 − 𝑆𝑡 , 0}.

316

3.2.3 Time-Dependent Options (

;

)

−

(

;

)

∗

Figure 3.3

American put option price.

𝜕𝑃 || > −1 then as the underlying asset price gets closer to 𝑆𝑡∗ the 𝜕𝑆𝑡 ||𝑆𝑡 =𝑆 ∗ 𝑡 value of the American put option can be increased by choosing a smaller 𝑆𝑡∗ , which is a contradiction to the fact that 𝑆𝑡∗ is the optimal exercise boundary. Hence, in order to satisfy the optimal exercise strategy of an American put option we have Conversely, if

𝜕𝑃 || = −1 𝜕𝑆𝑡 ||𝑆𝑡 =𝑆 ∗ 𝑡

where the two parts of 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) are joined smoothly without any discontinuity at 𝑆𝑡 = 𝑆𝑡∗ . 7. American Call Option Asymptotic Optimal Exercise Boundary. We consider an American call option price 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) written on an underlying asset priced at 𝑆𝑡 at time 𝑡 satisfying the following inequality 𝜕𝐶 𝜕𝐶 1 2 2 𝜕 2 𝐶 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 0 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 with constraints 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝑆𝑡 − 𝐾, 0},

𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0}

where 𝑟 is the risk-free interest rate, 𝐷 is the continuous dividend yield, 𝜎 is the constant volatility, 𝐾 is the strike price and 𝑇 > 𝑡 is the option expiry time. The option has an

3.2.3 Time-Dependent Options

317

unknown optimal exercise boundary 𝑆𝑡∞ ≥ 𝐾 where the option should be exercised if 𝑆𝑡 ≥ 𝑆𝑡∞ and held if 𝑆𝑡 < 𝑆𝑡∞ . Show that for 𝐷 < 𝑟 lim

𝑡→𝑇

𝜕𝐶 ≤ 0, 𝜕𝑡

for 𝐾 < lim 𝑆𝑡 ≤ 𝑡→𝑇

𝑟𝐾 𝐷

and lim

𝑡→𝑇

𝜕𝐶 > 0, 𝜕𝑡

and hence deduce that lim 𝑆𝑡∞ = 𝑡→𝑇

For 𝐷 ≥ 𝑟 show that

for lim 𝑆𝑡 > 𝑡→𝑇

𝑟𝐾 𝐷

𝑟𝐾 . 𝐷 lim 𝑆𝑡∞ = 𝐾.

𝑡→𝑇

{

} 𝑟 ,1 . 𝑡→𝑇 𝐷 Explain the financial implications if we set 𝐷 = 0 or 𝑟 = 0.

Finally, deduce that lim 𝑆𝑡∞ = 𝐾 max

Solution: By considering the case when the American call option is not optimal to exercise, then the option price satisfies the Black–Scholes equation 𝜕𝐶 𝜕𝐶 1 2 2 𝜕 2 𝐶 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 with constraint 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) > max{𝑆𝑡 − 𝐾, 0}. For case 𝐷 < 𝑟 we only consider lim 𝑆𝑡 > 𝐾 such that 𝑡→𝑇

lim 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = lim 𝑆𝑡 − 𝐾.

𝑡→𝑇

𝑡→𝑇

We then have lim

𝑡→𝑇

𝜕𝐶 1 𝜕2𝐶 𝜕𝐶 + lim 𝑟𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = − lim 𝜎 2 𝑆𝑡2 2 − lim (𝑟 − 𝐷)𝑆𝑡 𝑡→𝑇 2 𝑡→𝑇 𝜕𝑡 𝜕𝑆𝑡 𝑡→𝑇 𝜕𝑆𝑡 = −(𝑟 − 𝐷) lim 𝑆𝑡 + 𝑟(lim 𝑆𝑡 − 𝐾) 𝑡→𝑇

= 𝐷 lim 𝑆𝑡 − 𝑟𝐾. 𝑡→𝑇

𝑡→𝑇

318

3.2.3 Time-Dependent Options

Suppose lim gives

𝑡→𝑇

𝜕𝐶 > 0, then expanding 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) using Taylor’s theorem for 𝑡 → 𝑇 𝜕𝑡

lim 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) + lim

𝑡→𝑇

𝑡→𝑇

< 𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 𝑆𝑇 − 𝐾

𝜕𝐶 (𝑡 − 𝑇 ) + 𝑂((𝑡 − 𝑇 )2 ) 𝜕𝑡

which is a contradiction. Therefore, in order to hold the option we have lim

𝑡→𝑇

𝜕𝐶 ≤ 0, 𝜕𝑡

for 𝐾 < lim 𝑆𝑡 ≤ 𝑡→𝑇

𝑟𝐾 𝐷

and conversely lim

𝑡→𝑇

𝜕𝐶 > 0, 𝜕𝑡

for lim 𝑆𝑡 > 𝑡→𝑇

𝑟𝐾 𝐷

where the option should be exercised. Hence, the optimal exercise boundary is obtained by setting lim 𝑆𝑡∞ =

𝑡→𝑇

𝑟𝐾 . 𝐷

For the case 𝐷 ≥ 𝑟 we assume lim 𝑆𝑡∞ > 𝐾 where early exercise is not optimal for 𝐾 < 𝑡→𝑇

lim 𝑆𝑡 < lim 𝑆𝑡∞ . Given 𝐷 ≥ 𝑟 and lim 𝑆𝑡 > 𝐾 therefore, by not exercising the option the

𝑡→𝑇

𝑡→𝑇

𝑡→𝑇

interest earned on the strike is greater than the dividend lost from holding the asset, which is a contradiction. Hence, lim 𝑆𝑡∞ ≤ 𝐾.

𝑡→𝑇

Because 𝑆𝑡∞ ≥ 𝐾, 𝑡 < 𝑇 we can deduce that lim 𝑆𝑡∞ = 𝐾.

𝑡→𝑇

In general we can write

lim 𝑆 ∞ 𝑡→𝑇 𝑡

⎧ 𝑟𝐾 ⎪ =⎨ 𝐷 ⎪𝐾 ⎩

if 𝐷 < 𝑟

if 𝐷 ≥ 𝑟 } { 𝑟 ,1 . = 𝐾 max 𝐷

3.2.3 Time-Dependent Options

319

If 𝐷 = 0 then lim 𝑆𝑡∞ = ∞

𝑡→𝑇

which shows that it is never optimal to exercise an American call option before the expiry time. In contrast, if 𝑟 = 0 then lim 𝑆𝑡∞ = 𝐾

𝑡→𝑇

which implies that it is always optimal to exercise an American call option whenever the option is deep in the money. 8. American Put Option Asymptotic Optimal Exercise Boundary. We consider an American put option price 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) written on an underlying asset priced at 𝑆𝑡 at time 𝑡 satisfying the following inequality 𝜕2𝑃 𝜕𝑃 𝜕𝑃 1 − 𝑟𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≤ 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 with constraints 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max{𝐾 − 𝑆𝑡 , 0},

𝑃 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝐾 − 𝑆𝑇 , 0}

where 𝑟 is the risk-free interest rate, 𝐷 is the continuous dividend yield, 𝜎 is the constant volatility, 𝐾 is the strike price and 𝑇 > 𝑡 is the option expiry time. The option has an unknown optimal exercise boundary 𝑆𝑡∗ ≤ 𝐾 where the option should be exercised if 𝑆𝑡 ≤ 𝑆𝑡∗ and held if 𝑆𝑡 > 𝑆𝑡∞ . Show that for 𝐷 > 𝑟 lim

𝑡→𝑇

𝜕𝑃 ≤ 0, 𝜕𝑡

for

𝑟𝐾 ≤ lim 𝑆𝑡 < 𝐾 𝑡→𝑇 𝐷

and lim

𝑡→𝑇

𝜕𝑃 > 0, 𝜕𝑡

and hence deduce that lim 𝑆𝑡∗ = 𝑡→𝑇

For 𝐷 ≤ 𝑟 show that

for lim 𝑆𝑡 < 𝑡→𝑇

𝑟𝐾 𝐷

𝑟𝐾 . 𝐷 lim 𝑆𝑡∗ = 𝐾.

𝑡→𝑇

{

} 𝑟 ,1 . 𝑡→𝑇 𝐷 Explain the financial implications if we set 𝐷 = 0 or 𝑟 = 0.

Finally, deduce that lim 𝑆𝑡∗ = 𝐾 min

320

3.2.3 Time-Dependent Options

Solution: We only consider the case when the American put option is not optimal to exercise where the option price satisfies the Black–Scholes equation 𝜕2𝑃 𝜕𝑃 1 𝜕𝑃 − 𝑟𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 with constraint 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) > max{𝐾 − 𝑆𝑡 , 0}. For case 𝐷 ≤ 𝑟 we consider lim 𝑆𝑡 < 𝐾 so that 𝑡→𝑇

lim 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾 − lim 𝑆𝑡 .

𝑡→𝑇

𝑡→𝑇

We then have lim

𝑡→𝑇

𝜕𝑃 1 𝜕2 𝑃 𝜕𝑃 + lim 𝑟𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = − lim 𝜎 2 𝑆𝑡2 2 − lim (𝑟 − 𝐷)𝑆𝑡 𝑡→𝑇 𝑡→𝑇 𝜕𝑡 2 𝜕𝑆𝑡 𝑡→𝑇 𝜕𝑆𝑡 = (𝑟 − 𝐷) lim 𝑆𝑡 − 𝑟(lim 𝑆𝑡 − 𝐾) 𝑡→𝑇

𝑡→𝑇

= 𝑟𝐾 − 𝐷 lim 𝑆𝑡 . 𝑡→𝑇

Suppose lim gives

𝑡→𝑇

𝜕𝑃 > 0, then expanding 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) using Taylor’s theorem for 𝑡 → 𝑇 𝜕𝑡

lim 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) + lim

𝑡→𝑇

𝑡→𝑇

< 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 𝐾 − 𝑆𝑇

𝜕𝑃 (𝑡 − 𝑇 ) + 𝑂((𝑡 − 𝑇 )2 ) 𝜕𝑡

which is a contradiction. Therefore, in order to hold the option we have lim

𝑡→𝑇

𝜕𝑃 ≤ 0, 𝜕𝑡

for

𝑟𝐾 ≤ lim 𝑆𝑡 < 𝐾 𝑡→𝑇 𝐷

and conversely the option should be exercised when lim

𝑡→𝑇

𝜕𝑃 > 0, 𝜕𝑡

for lim 𝑆𝑡 < 𝑡→𝑇

𝑟𝐾 . 𝐷

Hence, the optimal exercise boundary is obtained by setting lim 𝑆𝑡∗ =

𝑡→𝑇

𝑟𝐾 . 𝐷

3.2.3 Time-Dependent Options

321

For the case 𝐷 ≤ 𝑟 we assume lim 𝑆𝑡∗ < 𝐾 where early exercise is not optimal for 𝑡→𝑇

lim 𝑆𝑡∗ < lim 𝑆𝑡 < 𝐾. Given 𝐷 ≤ 𝑟 and lim 𝑆𝑡 < 𝐾 therefore, by not exercising the option

𝑡→𝑇

𝑡→𝑇

𝑡→𝑇

the holder assumes the interest lost on the strike is lower than the dividend gained from holding the asset, which is a contradiction. Hence, lim 𝑆𝑡∗ ≥ 𝐾.

𝑡→𝑇

Because 𝑆𝑡∞ ≤ 𝐾, 𝑡 < 𝑇 we can deduce that lim 𝑆𝑡∗ = 𝐾.

𝑡→𝑇

In general, we can write ⎧ 𝑟𝐾 ⎪ lim 𝑆 ∗ = ⎨ 𝐷 𝑡→𝑇 𝑡 ⎪𝐾 ⎩

if 𝐷 > 𝑟

if 𝐷 ≤ 𝑟 } { 𝑟 ,1 . = 𝐾 min 𝐷

If 𝐷 = 0 then lim 𝑆𝑡∗ = 𝐾

𝑡→𝑇

which implies it is always optimal to exercise an American put option whenever the option is deep in the money. If 𝑟 = 0 then lim 𝑆𝑡∗ = 0

𝑡→𝑇

which shows that when there is zero interest rate there is no financial gain in holding 𝐾 from early exercise and therefore it is never optimal to exercise an American put option until expiry time. Thus, it is better to hold on to the option whilst earning dividend 𝐷 from 𝑆𝑡 till the expiry time. 9. Upper Bound of American Option Price. Let 𝑉 (𝑆𝑡 , 𝑡) be the price of an American option on an underlying asset 𝑆𝑡 at time 𝑡 with terminal payoff Ψ(𝑆𝑇 ) where 𝑇 is the option expiry time. In addition, let 𝑟 be the risk-free interest rate. Under the risk-neutral measure ℚ and conditional on the filtration ℱ𝑡 , we can express the option price as a primal problem [ ] | 𝑉 (𝑆𝑡 , 𝑡) = sup 𝔼ℚ 𝑒−𝑟(𝑇 −𝜏) Ψ(𝑆𝜏 )| ℱ𝑡 | 𝑡≤𝜏≤𝑇 where the supremum is taken over all possible stopping times 𝜏, 𝜏 ∈ [𝑡, 𝑇 ].

322

3.2.3 Time-Dependent Options

For arbitrary martingales 𝑀𝑡 , show that 𝑉 (𝑆𝑡 , 𝑡) ≤ inf 𝔼

ℚ

[

{

max 𝑒

𝑀

−𝑟(𝑇 −𝜏)

𝑡≤𝜏≤𝑇

] }| | Ψ(𝑆𝜏 ) − 𝑀𝜏 | ℱ𝑡 + 𝑀𝑡 |

where the right-hand side of the inequality is the dual problem. Hence, show that 𝑉 (𝑆𝑡 , 𝑡) ≤ 𝔼ℚ

[

] | max 𝑒−𝑟(𝑇 −𝜏) Ψ(𝑆𝜏 )|| ℱ𝑡 . 𝑡≤𝜏≤𝑇 |

Solution: For an arbitrary martingale 𝑀𝑡 [ ] | 𝑉 (𝑆𝑡 , 𝑡) = sup 𝔼ℚ 𝑒−𝑟(𝑇 −𝜏) Ψ(𝑆𝜏 )| ℱ𝑡 | 𝑡≤𝜏≤𝑇 [ ] | = sup 𝔼ℚ 𝑒−𝑟(𝑇 −𝜏) Ψ(𝑆𝜏 ) + 𝑀𝜏 − 𝑀𝜏 | ℱ𝑡 | 𝑡≤𝜏≤𝑇 { [ ] [ ]} | = sup 𝔼ℚ 𝑒−𝑟(𝑇 −𝜏) Ψ(𝑆𝜏 ) − 𝑀𝜏 | ℱ𝑡 + 𝔼ℚ 𝑀𝜏 || ℱ𝑡 | 𝑡≤𝜏≤𝑇 [ ] | = sup 𝔼ℚ 𝑒−𝑟(𝑇 −𝜏) Ψ(𝑆𝜏 ) − 𝑀𝜏 | ℱ𝑡 + 𝑀𝑡 | 𝑡≤𝜏≤𝑇 [ ] { −𝑟(𝑇 −𝜏) }| ℚ | ≤𝔼 Ψ(𝑆𝜏 ) − 𝑀𝜏 | ℱ𝑡 + 𝑀𝑡 . max 𝑒 𝑡≤𝜏≤𝑇 | Thus, taking the infimum 𝑉 (𝑆𝑡 , 𝑡) ≤ inf 𝔼 𝑀

ℚ

[

{

max 𝑒

𝑡≤𝜏≤𝑇

−𝑟(𝑇 −𝜏)

] }| | Ψ(𝑆𝜏 ) − 𝑀𝜏 | ℱ𝑡 + 𝑀𝑡 . |

Setting 𝑀𝑡 = 0, we have 𝑉 (𝑆𝑡 , 𝑡) ≤ 𝔼ℚ

[

] | max 𝑒−𝑟(𝑇 −𝜏) Ψ(𝑆𝜏 )|| ℱ𝑡 . 𝑡≤𝜏≤𝑇 |

10. Black Approximation. Consider an American call option written on an underlying asset 𝑆𝑡 at time 𝑡 with strike price 𝐾 and option expiry time 𝑇 > 𝑡. The asset pays one discrete dividend 𝛿 at time 𝜏, 𝑡 < 𝜏 < 𝑇 . In addition, let 𝜎 be the asset volatility and 𝑟 be the riskfree interest rate. The Black approximation to price an American call option with a single dividend is given as { } 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≈ max 𝐶𝑏𝑠 (𝑆𝑡 − 𝛿𝑒−𝑟(𝜏−𝑡) , 𝑡; 𝐾, 𝑇 ), 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝜏)

3.2.3 Time-Dependent Options

323

where it is set as the maximum of two European options that expire at times 𝑇 and 𝜏 such that 𝐶𝑏𝑠 (𝑋, 𝑡; 𝑌 , 𝑇 ) = 𝑋Φ(𝑑+ ) − 𝑌 𝑒−𝑟(𝑇 −𝑡) Φ(𝑑− ) log(𝑋∕𝑌 ) + (𝑟 ± 12 𝜎 2 )(𝑇 − 𝑡) 𝑑± = √ 𝜎 𝑇 −𝑡 𝑥 1 − 1 𝑢2 Φ(𝑥) = √ 𝑒 2 𝑑𝑢. ∫−∞ 2𝜋 What do the first and second terms of the approximation symbolise? Is the approximation value an upper or lower bound of the American call option price? Finally, show that it is never optimal to exercise the American call option with a single discrete dividend when ] [ 𝛿 ≤ 𝐾 1 − 𝑒−𝑟(𝑇 −𝜏) . Solution: The first term provides the European call option price value when the probability of early exercise is zero, whilst the second term assumes that the probability of early exercise before the ex-dividend date 𝜏 is one. Since the two terms are sub-optimal values, therefore { } 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ≥ max 𝐶𝑏𝑠 (𝑆𝑡 − 𝛿𝑒−𝑟(𝜏−𝑡) , 𝑡; 𝐾, 𝑇 ), 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝜏) . To show that it is never optimal to exercise the American call option with one discrete dividend when ] [ 𝛿 ≤ 𝐾 1 − 𝑒−𝑟(𝑇 −𝜏) we first consider the possibility of early exercise prior to the dividend date 𝜏. If the option is exercised at time 𝜏, the buyer of the call option receives 𝑆𝜏 − 𝐾. If the option is not exercised before 𝜏, the asset price will drop to 𝑆𝜏 − 𝛿. Because the value of the call option 𝐶𝑎𝑚 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) ≥ 𝑆𝜏 − 𝛿 − 𝐾 −𝑟(𝑇 −𝜏) therefore it is not optimal to exercise at time 𝜏 if 𝑆𝜏 − 𝛿 − 𝐾𝑒−𝑟(𝑇 −𝜏) ≥ 𝑆𝜏 − 𝐾

324

3.2.3 Time-Dependent Options

or ] [ 𝛿 ≤ 𝐾 1 − 𝑒−𝑟(𝑇 −𝜏) .

11. Barone-Adesi and Whaley Formula. Let 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of an American put option on an underlying asset 𝑆𝑡 at time 𝑡 such that it satisfies the Black–Scholes equation for 𝑆𝑡 > 𝑆𝑡∗ 𝜕𝑃𝑎𝑚 1 2 2 𝜕 2 𝑃𝑎𝑚 𝜕𝑃 + (𝑟 − 𝐷)𝑆𝑡 𝑎𝑚 − 𝑟𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) > max{𝐾 − 𝑆𝑡 , 0} where 𝑆𝑡∗ ≤ 𝐾 is the unknown optimal exercise boundary, 𝐾 is the strike price, 𝑟 is the risk-free interest rate, 𝐷 is the continuous dividend yield, 𝜎 is the constant volatility and 𝑇 > 𝑡 is the expiry time. By writing the American put option price as 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝜀(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) where 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is the price of a European put option and 𝜀(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is the earlyexercise premium, show that for 𝑆𝑡 > 𝑆 ∗ , 𝜀(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) satisfies 𝜕𝜀 𝜕𝜀 1 2 2 𝜕 2 𝜀 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝜀(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 lim 𝜀(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾(1 − 𝑒−𝑟(𝑇 −𝑡) ),

𝑆𝑡 →0

lim 𝜀(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0,

𝑆𝑡 →∞

lim 𝜀(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0.

𝑡→𝑇

By setting 𝜀(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = (1 − 𝑒−𝑟(𝑇 −𝑡) )𝜐(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) show that for 𝑆𝑡 > 𝑆𝑡∗ , 𝜐(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) satisfies 𝑟 𝜕𝜐 𝜕𝜐 1 2 2 𝜕 2 𝜐 + (𝑟 − 𝐷)𝑆𝑡 − 𝜐(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 1 − 𝑒−𝑟(𝑇 −𝑡) 𝜕𝑆𝑡2 lim 𝜐(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾,

𝑆𝑡 →0

By assuming

lim 𝜐(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0,

𝑆𝑡 →∞

lim 𝜐(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0.

𝑡→𝑇

𝜕𝜐 = 0 show that the American put option can be approximated by 𝜕𝑡 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝐴𝑆𝑡𝛼

3.2.3 Time-Dependent Options

325

where 𝐴 is a constant,

𝛼=

−(𝑟 − 𝐷 −

1 2 𝜎 )− 2

√( )2 𝑟 − 𝐷 − 12 + 2𝜎 𝑟̃ 𝜎2

and 𝑟̃ =

𝑟 1 − 𝑒−𝑟(𝑇 −𝑡)

.

Finally, using the “smooth-pasting” condition and the solution of 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) at 𝑆𝑡 = 𝑆𝑡∗ , show that the constant 𝐴 is [ 𝐴=−

where Δ∗𝐸 =

Δ∗𝐸 + 1

]

𝛼(𝑆𝑡∗ )𝛼−1

𝜕𝑃𝑏𝑠 || with 𝑆𝑡∗ satisfying the following equation 𝜕𝑆𝑡 ||𝑆𝑡 =𝑆 ∗ 𝑡

(

Δ∗𝐸 + 1 − 𝛼 𝛼

) 𝑆𝑡∗ + 𝑃𝑏𝑠 (𝑆𝑡∗ , 𝑡; 𝐾, 𝑇 ) − 𝐾 = 0.

Solution: For 𝑆𝑡 > 𝑆𝑡∗ , both 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) satisfy the Black– Scholes equation. Therefore, 𝜀(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) also satisfies 𝜕𝜀 𝜕𝜀 1 2 2 𝜕 2 𝜀 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝜀(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0. + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 Given that 𝜀(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) as 𝑆𝑡 → 0, we have 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) → 𝐾 and 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) → 𝐾𝑒−𝑟(𝑇 −𝑡) . Thus, lim 𝜀(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾(1 − 𝑒−𝑟(𝑇 −𝑡) ).

𝑆𝑡 →0

In contrast, for the case when 𝑆𝑡 → ∞, we have 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) → 0 and 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) → 0. Therefore, lim 𝜀(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0.

𝑆𝑡 →∞

326

3.2.3 Time-Dependent Options

Finally, for 𝑡 → 𝑇 , both 𝑃𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) → max{𝐾 − 𝑆𝑇 , 0} and 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) → max{𝐾 − 𝑆𝑇 , 0}, and so lim 𝜀(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0.

𝑡→𝑇

By setting 𝜀(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = (1 − 𝑒−𝑟(𝑇 −𝑡) )𝜐(𝑆𝑡 , 𝑡) we have 𝜕𝜀 𝜕𝜐 = −𝑟𝑒−𝑟(𝑇 −𝑡) 𝜐(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + (1 − 𝑒−𝑟(𝑇 −𝑡) ) 𝜕𝑡 𝜕𝑡 2 𝜕𝜀 𝜕𝜐 𝜕 2 𝜀 −𝑟(𝑇 −𝑡) 𝜕 𝜐 = (1 − 𝑒−𝑟(𝑇 −𝑡) ) , = (1 − 𝑒 ) 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡2 𝜕𝑆𝑡2

and substituting them into the Black–Scholes equation we eventually have 𝜕𝜐 1 2 2 𝜕 2 𝜐 𝑟 𝜕𝜐 + (𝑟 − 𝐷)𝑆𝑡 − 𝜐(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 1 − 𝑒−𝑟(𝑇 −𝑡) 𝜕𝑆𝑡2 such that lim 𝜐(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = lim 𝜀(𝑆𝑡 , 𝑡; 𝐾, 𝑇 )(1 − 𝑒−𝑟(𝑇 −𝑡) )−1 = 𝐾

𝑆𝑡 →0

𝑆𝑡 →0

lim 𝜐(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = lim 𝜀(𝑆𝑡 , 𝑡; 𝐾, 𝑇 )(1 − 𝑒−𝑟(𝑇 −𝑡) )−1 = 0

𝑆𝑡 →∞

𝑆𝑡 →0

and lim 𝜐(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = lim 𝜀(𝑆𝑡 , 𝑡; 𝐾, 𝑇 )(1 − 𝑒−𝑟(𝑇 −𝑡) )−1 = 0.

𝑡→𝑇

By setting

𝑡→𝑇

𝜕𝜐 = 0, the equation becomes 𝜕𝑡 𝑟 𝜕𝜐 1 2 2 𝜕2𝜐 + (𝑟 − 𝐷)𝑆𝑡 − 𝜐(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 𝜎 𝑆𝑡 2 𝜕𝑆𝑡 1 − 𝑒−𝑟(𝑇 −𝑡) 𝜕𝑆𝑡2

and following the workings of the perpetual American options (see Problem 3.2.2.1, page 292) we can write the general solution as 𝜐(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐴𝑆𝑡𝛼 + 𝐵𝑆𝑡𝛽 where 𝐴 and 𝐵 are constants,

𝛼=

−(𝑟 − 𝐷 −

1 2 𝜎 )− 2

√(

𝑟−𝐷−

𝜎2

1 2

)2

+ 2𝜎 𝑟̃

0

𝑟 . From the initial condition, for 𝑆𝑡 → ∞, 𝜐(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) → 0 and 1 − 𝑒−𝑟(𝑇 −𝑡) because 𝛽 > 0 therefore 𝐵 = 0. Hence, such that 𝑟̃ =

𝜐(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐴𝑆𝑡𝛼 . Finally, at 𝑆𝑡 = 𝑆𝑡∗ (see Problem 3.2.3.6, page 315) we have 𝑃𝑎𝑚 (𝑆𝑡∗ , 𝑡; 𝐾, 𝑇 ) = 𝐾 − 𝑆𝑡∗ and

𝜕𝑃𝑎𝑚 || = −1 𝜕𝑆𝑡 ||𝑆𝑡 =𝑆 ∗ 𝑡

and by writing Δ∗𝑏𝑠 =

𝜕𝑃𝑏𝑠 || we therefore have two equations 𝜕𝑆𝑡 ||𝑆𝑡 =𝑆 ∗ 𝑡

𝑃𝑏𝑠 (𝑆𝑡∗ , 𝑡; 𝐾, 𝑇 ) + 𝐴(𝑆𝑡∗ )𝛼 = 𝐾 − 𝑆𝑡∗ Δ∗𝑏𝑠 + 𝛼𝐴(𝑆𝑡∗ )𝛼−1 = −1

and by solving them simultaneously [ 𝐴=−

Δ∗𝑏𝑠 + 1

]

𝛼(𝑆𝑡∗ )𝛼−1

and 𝑆𝑡∗ satisfies (

Δ∗𝑏𝑠 + 1 − 𝛼 𝛼

) 𝑆𝑡∗ + 𝑃𝑏𝑠 (𝑆𝑡∗ , 𝑡; 𝐾, 𝑇 ) − 𝐾 = 0.

12. Consider a binomial tree model for an underlying asset process {𝑆𝑛 : 0 ≤ 𝑛 ≤ 3} following the Cox–Ross–Rubinstein model. With the initial asset price 𝑆0 = 100, let

𝑆𝑛+1 √

⎧ 𝑢𝑆 ⎪ 𝑛 =⎨ ⎪ 𝑑𝑆𝑛 ⎩

with probability 𝜋 with probability 1 − 𝜋

where 𝑢 = 𝑒𝜎 Δ𝑡 and 𝑑 = 1∕𝑢, with 𝜎 the volatility and Δ𝑡 the binomial time step. Assuming the risk-free interest rate 𝑟 = 1%, continuous dividend yield 𝐷 = 0.5% and volatility 𝜎 = 8%, we wish to price a Bermudan call option with strike 𝐾 = 95 and expiry time 𝑇 = 1

328

3.2.3 Time-Dependent Options

year in a 3-period binomial tree model. For the Bermudan call option, early exercise is only allowed in the fourth month of the contract. (a) Find the risk-neutral probabilities 𝜋 and 1 − 𝜋. (b) Find the price of the Bermudan call option. Solution: (a) Given 𝑆0 = 100, 𝐾 = 95, 𝑟 = 0.01, 𝐷 = 0.005, 𝜎 = 0.08 and time step Δ𝑡 = 13 , then 𝑢 = 𝑒𝜎

√ Δ𝑡

=𝑒

0.08 √ 3

= 1.0472

and 𝑑 = 𝑒−𝜎

√ Δ𝑡

=

1 1 = = 0.9549. 𝑢 1.0472

Therefore, the risk-neutral probabilities are 𝜋=

𝑒(𝑟−𝐷)Δ𝑡 − 𝑑 = 0.5065 𝑢−𝑑

and 1 − 𝜋 = 1 − 0.5065 = 0.4935. (b) The binomial tree in Figure 3.4 shows the price movement of 𝑆0 in a 3-period binomial model. ∙ ∙ ∙

2

∙ ∙

∙

0

2

∙ 2

0

0

∙ Figure 3.4

2

0

0

∙

0

0

0

0∙

3

A 3-period binomial tree model.

3

0

3.2.3 Time-Dependent Options

329

By setting 𝑆𝑖(𝑗) = 𝑢𝑗 𝑑 𝑖−𝑗 𝑆0 ,

𝑖 = 0, 1, … , 𝑛,

𝑗 = 0, 1, … , 𝑖

the Bermudan call option price at each of the lattice points is

𝑉𝑖(𝑗)

{ ⎧ max Ψ(𝑆𝑖(𝑗) ), [ ]} ⎪ ⎪ 𝑒−𝑟Δ𝑡 𝜋𝑉 (𝑗+1) + (1 − 𝜋)𝑉 (𝑗) early exercise is allowed 𝑖+1 𝑖+1 =⎨ ⎪ [ ] ⎪ 𝑒−𝑟Δ𝑡 𝜋𝑉 (𝑗+1) + (1 − 𝜋)𝑉 (𝑗) early exercise is not allowed ⎩ 𝑖+1 𝑖+1

such that { } Ψ(𝑆𝑖(𝑗) ) = max 𝑆𝑖(𝑗) − 𝐾, 0 where 𝑖 = 0, 1, … , 𝑛 and 𝑗 = 0, 1, … , 𝑖. Hence, at time period 𝑛 = 3 (i.e., option expiry time 𝑇 = 1 year) 𝑉3(0) = Ψ(𝑆3(0) ) = max{𝑑 3 𝑆0 − 𝐾, 0} = max{0.95493 × 100 − 95, 0} =0 𝑉3(1) = Ψ(𝑆3(1) ) = max{𝑢𝑑 2 𝑆0 − 𝐾, 0} = max{1.0472 × 0.95492 × 100 − 95, 0} = 0.4873 𝑉3(2) = Ψ(𝑆3(2) ) = max{𝑢2 𝑑𝑆0 − 𝐾, 0} = max{1.04722 × 0.9549 × 100 − 95, 0} = 9.7170 𝑉3(3) = Ψ(𝑆3(3) ) = max{𝑢3 𝑆0 − 𝐾, 0} = max{1.04723 × 100 − 95, 0} = 19.8389.

330

3.2.3 Time-Dependent Options

At time period 𝑛 = 2 (i.e., 𝑡 = 8 months) where early exercise is not allowed [ ] 𝑉2(0) = 𝑒−𝑟Δ𝑡 𝜋𝑉3(1) + (1 − 𝜋)𝑉3(0) = 𝑒−

0.01 3

[0.5065 × 0.4873 + 0.4935 × 0] = 0.2460

[ ] 𝑉2(1) = 𝑒−𝑟Δ𝑡 𝜋𝑉3(2) + (1 − 𝜋)𝑉3(1) = 𝑒−

0.01 3

[0.5065 × 9.7170 + 0.4935 × 0.4873] = 5.1450

[ ] 𝑉2(2) = 𝑒−𝑟Δ𝑡 𝜋𝑉3(3) + (1 − 𝜋)𝑉3(2) = 𝑒−

0.01 3

[0.5065 × 19.8389 + 0.4935 × 9.7170] = 14.7944.

At time period 𝑛 = 1 (i.e., 𝑡 = 4 months) where early exercise is allowed [ ]} { 𝑉1(0) = max Ψ(𝑆1(0) ), 𝑒−𝑟Δ𝑡 𝜋𝑉2(1) + (1 − 𝜋)𝑉2(0) [ ]} { } { = max max 𝑑𝑆0 − 𝐾, 0 , 𝑒−𝑟Δ𝑡 𝜋𝑉2(1) + (1 − 𝜋)𝑉2(0) = max{max{0.9549 × 100 − 95, 0}, 𝑒−

0.01 3

[0.5065 × 5.1450 + 0.4935 × 0.2460]} = max{0.4900, 2.7183} = 2.7183 [ ]} { 𝑉1(1) = max Ψ(𝑆1(1) ), 𝑒−𝑟Δ𝑡 𝜋𝑉2(2) + (1 − 𝜋)𝑉2(1) [ ]} { } { = max max 𝑢𝑆0 − 𝐾, 0 , 𝑒−𝑟Δ𝑡 𝜋𝑉2(2) + (1 − 𝜋)𝑉2(1) = max{max{1.0472 × 100 − 95, 0}, 𝑒−

0.01 3

[0.5065 × 14.7944 + 0.4935 × 5.1450]} = max{9.7200, 9.9990} = 9.9990 and finally at time period 𝑛 = 0 (i.e., 𝑡 = 0) where early exercise is not allowed [ ] 𝑉0(0) = 𝑒−𝑟Δ𝑡 𝜋𝑉1(1) + (1 − 𝜋)𝑉1(0) = 𝑒−

0.01 3

[0.5065 × 9.9990 + 0.4935 × 2.7183] = 6.3847.

3.2.3 Time-Dependent Options

331

Therefore, the price of the Bermudan call option based on a 3-period binomial model is 𝑉0(0) = 6.3847. { } 13. One-Touch Option (PDE Approach). Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and at time 𝑡, let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. 𝑑 (𝑆 , 𝑡; 𝐾, 𝑇 ) that pays $1 A one-touch call option is an American digital call option 𝐶𝑎𝑚 𝑡 at the expiry time 𝑇 if the underlying asset price 𝑆𝑢 , 𝑡 ≤ 𝑢 ≤ 𝑇 is above the strike value 𝐾. If the underlying asset value has not reached the strike price by 𝑇 then the option expires worthless. 𝑑 (𝑆 , 𝑡; 𝐾, 𝑇 ) satisfies the As long as the option is not exercised then for 0 < 𝑆𝑡 < 𝐾, 𝐶𝑎𝑚 𝑡 Black–Scholes equation 𝑑 𝜕𝐶𝑎𝑚

𝜕𝑡

𝑑 𝑑 𝜕 2 𝐶𝑎𝑚 𝜕𝐶𝑎𝑚 1 𝑑 + (𝑟 − 𝐷)𝑆 − 𝑟𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 2 𝑆𝑡2 𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

with boundary conditions 𝑑 (0, 𝑡; 𝐾, 𝑇 ) = 0, 𝐶𝑎𝑚

𝑑 𝐶𝑎𝑚 (𝐾, 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) ,

𝑑 𝐶𝑎𝑚 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 1I{𝑆𝑇 ≥𝐾} .

𝑑 (𝑆 , 𝑡; 𝐾, 𝑇 ) in the form By writing the solution of 𝐶𝑎𝑚 𝑡 𝑑 𝑑 𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) 𝐶𝑎𝑚 𝑑 (𝑆 , 𝑡; 𝐾, 𝑇 ) and 𝑃 𝑑 (𝑆 , 𝑡; 𝐾, 𝑇 ) are the European digital call and put options where 𝐶𝑏𝑠 𝑡 𝑏𝑠 𝑡 satisfying 𝑑 𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝐶𝑏𝑠

find the Black–Scholes equation for 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) together with its corresponding boundary conditions for 0 < 𝑆𝑡 < 𝐾. ( ) 𝑆𝑡 Writing 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒𝛼𝑥+𝛽𝜏 𝐵(𝑥, 𝜏) where 𝑥 = log and 𝜏 = 12 𝜎 2 (𝑇 − 𝑡), 𝐾 show that by setting 1 𝛼 = − 𝑘1 2

and

1 𝛽 = − 𝑘21 − 𝑘0 4

332

3.2.3 Time-Dependent Options

where 𝑘1 =

𝑟−𝐷 1 2 𝜎 2

− 1 and 𝑘0 =

𝑟

, the Black–Scholes equation for 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is

1 2 𝜎 2

reduced to a heat equation of the form 𝜕𝐵 𝜕2𝐵 = 𝜕𝜏 𝜕𝑥2 𝐵(𝑥, 0) = 𝑓 (𝑥),

𝐵(0, 𝜏) = 0, 𝑥 < 0.

Given that the solution of the heat equation is 0

2 1 1 𝐵(𝑥, 𝜏) = √ 𝑓 (𝑧)𝑒−(𝑥−𝑧) ∕4𝜏 𝑑𝑧 − √ ∫ 2 𝜋𝜏 −∞ 2 𝜋𝜏 ∫0

∞

2 ∕4𝜏

𝑓 (−𝑧)𝑒−(𝑥−𝑧)

𝑑𝑧

deduce that for 0 < 𝑆𝑡 < 𝐾 (

𝑑 𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝐶𝑎𝑚

𝑆𝑡 𝐾

)2𝛼

𝑑 𝑃𝑏𝑠

(

𝐾2 , 𝑡; 𝐾, 𝑇 𝑆𝑡

)

where (

(

)

𝑑 𝐶𝑏𝑠 𝑋𝑡 , 𝑡; 𝐾, 𝑇 = 𝑒−𝑟(𝑇 −𝑡) Φ

) ( ) log 𝑋𝑡 ∕𝐾 + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

and 𝑑 𝑃𝑏𝑠

(

)

𝑌𝑡 , 𝑡; 𝐾, 𝑇 = 𝑒

( −𝑟(𝑇 −𝑡)

Φ

) ( ) − log 𝑌𝑡 ∕𝐾 − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

respectively and 𝑥

1 2 1 Φ(𝑥) = √ 𝑒− 2 𝑢 𝑑𝑢 2𝜋 ∫−∞

is the standard normal cumulative density. 𝑑 (𝑆 , 𝑡; 𝐾, 𝑇 ) = 𝐶 𝑑 (𝑆 , 𝑡; 𝐾, 𝑇 ) + 𝑃 𝑑 (𝑆 , 𝑡; 𝐾, 𝑇 ) + Solution: Given that 𝐶𝑎𝑚 𝑡 𝑏𝑠 𝑡 𝑏𝑠 𝑡 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) we can therefore express 𝑑 𝜕𝐶𝑎𝑚

𝜕𝑡

𝑑 𝜕𝐶𝑎𝑚

𝜕𝑆𝑡

=

=

𝑑 𝜕𝐶𝑏𝑠

𝜕𝑡

𝑑 𝜕𝐶𝑏𝑠

𝜕𝑆𝑡

+

+

𝑑 𝜕𝑃𝑏𝑠

𝜕𝑡

𝑑 𝜕𝑃𝑏𝑠

𝜕𝑆𝑡

+

𝜕𝑉 𝜕𝑡

+

𝜕𝑉 𝜕𝑆𝑡

3.2.3 Time-Dependent Options

333

and 𝑑 𝜕 2 𝐶𝑎𝑚

𝜕𝑆𝑡2

=

𝑑 𝜕 2 𝐶𝑏𝑠

𝜕𝑆𝑡2

+

𝑑 𝜕 2 𝑃𝑏𝑠

𝜕𝑆𝑡2

+

𝜕2𝑉 . 𝜕𝑆𝑡2

𝑑 (𝑆 , 𝑡; 𝐾, 𝑇 ) By substituting the above expressions into the Black–Scholes equation for 𝐶𝑎𝑚 𝑡 and taking note that 𝑑 𝜕𝐶𝑏𝑠

𝑑 𝑑 𝜕 2 𝐶𝑏𝑠 𝜕𝐶𝑏𝑠 1 𝑑 + (𝑟 − 𝐷)𝑆 − 𝑟𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0, + 𝜎 2 𝑆𝑡2 𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

0 < 𝑆𝑡 < 𝐾

and 𝑑 𝜕𝑃𝑏𝑠

𝑑 𝑑 𝜕 2 𝑃𝑏𝑠 𝜕𝑃𝑏𝑠 1 𝑑 + (𝑟 − 𝐷)𝑆 − 𝑟𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0, + 𝜎 2 𝑆𝑡2 𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

0 < 𝑆𝑡 < 𝐾

we have 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 − 𝑟𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0, + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡

0 < 𝑆𝑡 < 𝐾

with boundary conditions 𝑉 (0, 𝑡; 𝐾, 𝑇 ) = 0,

𝑉 (𝐾, 𝑡; 𝐾, 𝑇 ) = 0,

𝑉 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = −1I{𝑆𝑇 𝐾} | ℱ𝑡 𝑢 | 𝑡≤𝑢≤𝑇 | ( ) | = 𝑒−𝑟(𝑇 −𝑡) ℚ max 𝑆𝑢 ≥ 𝐾 || ℱ𝑡 𝑡≤𝑢≤𝑇 |

]

−𝑟(𝑇 −𝑡) ℚ

such that 𝑑 𝐶𝑎𝑚 (𝐾, 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) .

From Girsanov’s theorem, under the risk-neutral measure ℚ we can easily show that 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡

338

3.2.3 Time-Dependent Options

where 𝑊𝑡ℚ = 𝑊𝑡 + show that for 𝑡 < 𝑇

(𝜇 − 𝑟) 𝜎

𝑡 is a ℚ-standard Wiener process. Using It̄o’s lemma we can 1 2 )(𝑇 −𝑡)+𝜎𝑊𝑇ℚ−𝑡

𝑆𝑇 = 𝑆𝑡 𝑒(𝑟−𝐷− 2 𝜎

̂

= 𝑆𝑡 𝑒𝜎 𝑊𝑇 −𝑡

̂𝑇 −𝑡 = 𝜈(𝑇 − 𝑡) + 𝑊 ℚ and 𝜈 = 1 (𝑟 − 𝐷 − 1 𝜎 2 ). where 𝑊 𝑇 −𝑡 𝜎 2 By writing ̂𝑢−𝑡 𝑀𝑇 −𝑡 = max 𝑊 𝑡≤𝑢≤𝑇

therefore max 𝑆𝑢 = 𝑆𝑡 𝑒𝜎𝑀𝑇 −𝑡 .

𝑡≤𝑢≤𝑇

By substituting the above expression into the option price we have ) ( | 𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) ℚ 𝑆𝑡 𝑒𝜎𝑀𝑇 −𝑡 ≥ 𝐾 | ℱ𝑡 𝐶𝑎𝑚 | [ ( )] | −𝑟(𝑇 −𝑡) 𝜎𝑀𝑇 −𝑡 1 − ℚ 𝑆𝑡 𝑒 ≤ 𝐾 | ℱ𝑡 =𝑒 | [ ( ( )| )] 1 𝐾 | −𝑟(𝑇 −𝑡) =𝑒 1 − ℚ 𝑀𝑇 −𝑡 ≤ log . |ℱ 𝜎 𝑆𝑡 || 𝑡 From Problem 4.2.2.15 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus we can show that ( ℚ

𝑀𝑇 −𝑡

1 𝐾 | ≤ log( )|| ℱ𝑡 𝜎 𝑆𝑡 |

)

⎛ 1 log(𝐾∕𝑆 ) − 𝜈(𝑇 − 𝑡) ⎞ 𝑡 ⎟ ⎜ = Φ⎜ 𝜎 √ ⎟ 𝑇 −𝑡 ⎟ ⎜ ⎠ ⎝ ( )− 2𝜈 ⎛ − 1 log(𝐾∕𝑆𝑡 ) − 𝜈(𝑇 − 𝑡) ⎞ 𝜎 𝑆𝑡 ⎟ ⎜ − Φ⎜ 𝜎 √ ⎟. 𝐾 𝑇 −𝑡 ⎟ ⎜ ⎠ ⎝

Hence, )] log(𝐾∕𝑆𝑡 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) =𝑒 1−Φ √ 𝜎 𝑇 −𝑡 ⎛ − log(𝐾∕𝑆 ) − (𝑟 − 𝐷 − 1 𝜎 2 )(𝑇 − 𝑡) ⎞ ( )2𝛼 𝑡 𝑆𝑡 ⎟ 2 −𝑟(𝑇 −𝑡) ⎜ + 𝑒 Φ⎜ √ ⎟ 𝐾 𝜎 𝑇 −𝑡 ⎜ ⎟ ⎝ ⎠ [

𝑑 𝐶𝑎𝑚 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 )

−𝑟(𝑇 −𝑡)

(

3.2.3 Time-Dependent Options

339

⎛ log(𝑆 ∕𝐾) + (𝑟 − 𝐷 − 1 𝜎 2 )(𝑇 − 𝑡) ⎞ 𝑡 ⎟ ⎜ 2 =𝑒 Φ⎜ √ ⎟ 𝜎 𝑇 −𝑡 ⎟ ⎜ ⎠ ⎝ ( ) ( )2𝛼 − log(𝐾∕𝑆𝑡 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) 𝑆𝑡 −𝑟(𝑇 −𝑡) + 𝑒 Φ √ 𝐾 𝜎 𝑇 −𝑡 ( )2𝛼 ) ( 2 𝑆𝑡 𝐾 𝑑 𝑑 = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑃𝑏𝑠 , 𝑡; 𝐾, 𝑇 𝐾 𝑆𝑡 −𝑟(𝑇 −𝑡)

( 1 where 𝛼 = 2

1−

𝑟−𝐷 1 2 𝜎 2

) .

{ } 15. Immediate-Touch Option (PDE Approach). Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and at time 𝑡, let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. An immediate-touch call option is an American digital call option 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) that immediately pays $1 if the underlying asset price 𝑆𝑢 , 𝑡 ≤ 𝑢 ≤ 𝑇 is above the strike value 𝐾 where 𝑇 is the expiry time. If the underlying asset value has not reached the strike price by 𝑇 then the option expires worthless, that is 𝐶𝑑 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 0, 0 < 𝑆𝑇 < 𝐾. As long as the option is not exercised then for 0 < 𝑆𝑡 < 𝐾, 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) satisfies the Black–Scholes equation 𝜕𝐶 𝜕𝐶𝑑 1 2 2 𝜕 2 𝐶𝑑 + (𝑟 − 𝐷)𝑆𝑡 𝑑 − 𝑟𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 with boundary conditions 𝐶𝑑 (0, 𝑡; 𝐾, 𝑇 ) = 0,

𝐶𝑑 (𝐾, 𝑡; 𝐾, 𝑇 ) = 1.

By writing the solution of 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) in the form 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶 ∞ (𝑆𝑡 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) where 𝐶 ∞ (𝑆𝑡 ) satisfies the perpetual call problem 1 2 2 𝑑2𝐶 ∞ 𝑑𝐶 ∞ + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝐶 ∞ (𝑆𝑡 ) = 0 𝜎 𝑆𝑡 2 2 𝑑𝑆𝑡 𝑑𝑆𝑡

340

3.2.3 Time-Dependent Options

with boundary conditions 𝐶 ∞ (0) = 0,

𝐶 ∞ (𝐾) = 1

find 𝐶 ∞ (𝑆𝑡 ) by solving the second-order ODE. For 0 < 𝑆𝑡 < 𝐾 find the Black–Scholes equation for 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) together with its corresponding boundary conditions. ( ) 𝑆𝑡 and 𝜏 = 12 𝜎 2 (𝑇 − 𝑡) show Writing 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒𝛼𝑥+𝛽𝜏 𝐵(𝑥, 𝜏) where 𝑥 = log 𝐾 that by setting 1 𝛼 = − (𝑘 − 1) 2 where 𝑘 =

𝑟−𝐷 1 2 𝜎 2

𝐷

and 𝑘′ =

1 2 𝜎 2

1 𝛽 = − (𝑘 + 1)2 − 𝑘′ 4

and

, the Black–Scholes equation for 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is reduced

to a heat equation of the form

𝜕𝐵 𝜕2𝐵 = 𝜕𝜏 𝜕𝑥2 𝐵(𝑥, 0) = 𝑓 (𝑥),

𝐵(0, 𝜏) = 0,

𝑥 < 0.

Given that the solution of the heat equation with the above boundary conditions is 0

2 1 1 𝐵(𝑥, 𝜏) = √ 𝑓 (𝑦)𝑒−(𝑥−𝑦) ∕4𝜏 𝑑𝑦 − √ ∫ 2 𝜋𝜏 −∞ 2 𝜋𝜏 ∫0

∞

𝑓 (−𝑦)𝑒−(𝑥−𝑦)

deduce that for 0 < 𝑆𝑡 < 𝐾 ( 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) =

𝑆𝑡 𝐾

(

)𝜆+ Φ(𝑑+ ) +

𝑆𝑡 𝐾

)𝜆− Φ(𝑑− )

where

𝜆± =

−(𝑟 − 𝐷 − 12 𝜎 2 ) ±

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟

𝜎2 √ log(𝑆𝑡 ∕𝐾) ± (𝑇 − 𝑡) (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟 𝑑± = √ 𝜎 𝑇 −𝑡 and 𝑥

1 2 1 𝑒− 2 𝑢 𝑑𝑢 Φ(𝑥) = √ 2𝜋 ∫−∞

is the standard normal cdf.

2 ∕4𝜏

𝑑𝑦

3.2.3 Time-Dependent Options

341

Solution: Given that 𝐶 ∞ (𝑆𝑡 ) satisfies the perpetual call problem then from Problem 3.2.2.2 (page 294), the solution of the ODE is 𝜆

𝐶(𝑆𝑡 ) = 𝐴𝑆𝑡 +

where 𝜆+ = Since

−(𝑟 − 𝐷 − 12 𝜎 2 ) +

𝐶 ∞ (𝐾)

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟

> 0.

𝜎2

=1

𝐴 = 𝐾 −𝜆+ and hence ( 𝐶 (𝑆𝑡 ) = ∞

𝑆𝑡 𝐾

)𝜆+

.

Since 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶 ∞ (𝑆𝑡 ) + 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) we can write 𝜕𝐶𝑑 𝜕𝐶 , = 𝜕𝑡 𝜕𝑆𝑡

𝜕𝐶𝑑 𝑑𝐶 ∞ 𝜕𝐶 = + , 𝜕𝑆𝑡 𝑑𝑆𝑡 𝜕𝑆𝑡

𝜕 2 𝐶𝑑 𝜕𝑆𝑡2

=

𝑑 2𝐶 ∞ 𝜕2𝐶 + 𝑑𝑆𝑡2 𝜕𝑆𝑡2

and by substituting them into the Black–Scholes equation for 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and taking note that 𝑑𝐶 ∞ 1 2 2 𝑑2𝐶 ∞ + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝐶 ∞ (𝑆𝑡 ) = 0 𝜎 𝑆𝑡 2 2 𝑑𝑆𝑡 𝑑𝑆𝑡 we have 𝜕𝐶 𝜕𝐶 1 2 2 𝜕 2 𝐶 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0, + 𝜎 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

0 < 𝑆𝑡 < 𝐾

with boundary conditions ( 𝐶(0, 𝑡; 𝐾, 𝑇 ) = 0,

𝐶(𝐾, 𝑡; 𝐾, 𝑇 ) = 0,

𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = − (

Given that 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) =

𝑒𝛼𝑥+𝛽𝜏 𝐵(𝑥, 𝜏)

where 𝑥 = log

𝑆𝑡 𝐾

𝑆𝑇 𝐾

)𝜆+ ,

0 < 𝑆𝑇 < 𝐾.

) and 𝜏 = 12 𝜎 2 (𝑇 − 𝑡) we

have [ ] 𝜕𝐶 𝜕𝐵 𝜕𝐶 𝜕𝜏 1 = ⋅ = − 𝜎 2 𝑒𝛼𝑥+𝛽𝜏 + 𝛽𝐵(𝑥, 𝜏) 𝜕𝑡 𝜕𝜏 𝜕𝑡 2 𝜕𝜏 [ ] 𝜕𝐶 𝜕𝑥 𝑒𝛼𝑥+𝛽𝜏 𝜕𝐵 𝜕𝐶 = = ⋅ + 𝛼𝐵(𝑥, 𝜏) 𝜕𝑆𝑡 𝜕𝑥 𝜕𝑆𝑡 𝑆𝑡 𝜕𝑥

342

3.2.3 Time-Dependent Options

and

{

[ ]} 𝜕𝐵 𝑒𝛼𝑥+𝛽𝜏 𝛼𝐵(𝑥, 𝜏) + 𝑆𝑡 𝜕𝑥 ] [ 2 𝛼𝑥+𝛽𝜏 𝜕𝐵 𝜕 𝐵 𝑒 + (2𝛼 − 1) + 𝛼(𝛼 − 1)𝐵(𝑥, 𝜏) . = 𝜕𝑥 𝜕𝑥2 𝑆𝑡2

𝜕 𝜕2 𝐶 = 𝜕𝑆𝑡 𝜕𝑆𝑡2

By substituting the above expressions into the Black–Scholes equation for 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) we eventually have ) [ ( ] 𝜕𝐵 1 2 𝜕𝐵 1 2 𝜕𝐵 + − (2𝛼 − 1) + 𝑟 − 𝐷 𝜎 𝜎 2 𝜕𝜏 𝜕𝑥 2 𝜕𝑥2 ] [ 1 2 +𝐵 𝜎 [𝛼(𝛼 − 1) − 𝛽] + 𝛼(𝑟 − 𝐷) − 𝑟 = 0. 2 To eliminate the terms

𝜕𝐵 and 𝐵 we set 𝜕𝑥 1 2 𝜎 (2𝛼 − 1) + 𝑟 − 𝐷 = 0 2 1 2 𝜎 [𝛼(𝛼 − 1) − 𝛽] + 𝛼(𝑟 − 𝐷) − 𝑟 = 0. 2

By solving the equations simultaneously we have 1 𝛼 = − (𝑘 − 1) 2 where 𝑘 = Hence,

𝑟−𝐷 1 2 𝜎 2

𝐷

and 𝑘′ =

1 2 𝜎 2

and

1 𝛽 = − (𝑘 + 1)2 − 𝑘′ 4

.

1

1

𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒− 2 (𝑘−1)𝑥− 4 (𝑘+1)

2 𝜏−𝑘′ 𝜏

𝐵(𝑥, 𝜏)

with boundary conditions 1

1

𝐵(𝑥, 0) = 𝑒 2 (𝑘−1)𝑥 𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = −𝑒 2 (𝑘−1)𝑥

(

𝐾𝑒𝑥 𝐾

)𝜆+

1

= −𝑒 2 (𝑘−1+2𝜆+ )𝑥 ,

𝑥 𝐾. { } 16. Immediate-Touch Option (Probabilistic Approach). Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and at time 𝑡, let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. By setting 𝜏 = inf {𝑡 ≥ 0 : 𝑊𝑡 = 𝑎 + 𝑏𝑡} as the first-passage time of hitting the slope 𝑎 + 𝑏𝑡 where 𝑎, 𝑏 ∈ ℝ, show that the pdf is |𝑎| − 1 (𝑎+𝑏𝑡)2 𝑒 2𝑡 𝑓𝜏 (𝑡) = √ . 𝑡 2𝜋𝑡 An immediate-touch call option is an American digital call option 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) that immediately pays $1 if the underlying asset price 𝑆𝑢 , 𝑡 ≤ 𝑢 ≤ 𝑇 is above the strike value 𝐾 where 𝑇 is the expiry time. If the underlying asset value has not reached the strike price by 𝑇 then the option expires, worthless, that is 𝐶𝑑 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 0, 0 < 𝑆𝑇 < 𝐾. Under the risk-neutral measure ℚ show that 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) =

𝑇 −𝑡

∫0

𝑒−𝑟𝑣 𝑓𝜏ℚ (𝑣) 𝑑𝑣

(𝜇 − 𝑟) } { 𝑡 is a ℚ-standard Wiener where 𝜏 = inf 𝑡 ≥ 0 : 𝑊𝑡ℚ = 𝑤 − 𝜃𝑡 , 𝑊𝑡ℚ = 𝑊𝑡 + 𝜎 ( ) ( ) 1 1 𝐾 1 𝑟 − 𝐷 − 𝜎 2 and 𝑓𝜏ℚ (𝑡) is the pdf of 𝜏 under the ℚ process, 𝑤 = log ,𝜃= 𝜎 𝑆𝑡 𝜎 2 measure.

346

3.2.3 Time-Dependent Options

By setting 𝛼 = 𝜎𝜃 and 𝛽 2 = 𝛼 2 + 2𝜎 2 𝑟 show that we can write ( 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) =

𝐾 𝑆𝑡

) 𝛼±𝛽

𝑇 −𝑡

𝜎2

∫0

log(𝐾∕𝑆𝑡 ) − 12 𝑒 2𝜎 𝑣 √ 𝜎𝑣 2𝜋𝑣

[

( )] log 𝑆𝐾 ±𝛽𝑣 𝑡

𝑑𝑣.

Taking partial fractions of the form ) ) ) ( ( ( log 𝐾∕𝑆𝑡 log 𝐾∕𝑆𝑡 − 𝛽𝑡 log 𝐾∕𝑆𝑡 + 𝛽𝑡 = + √ √ √ 𝜎 𝑡 2𝜎 𝑡 2𝜎 𝑡 and by choosing appropriate integrands, show that for 0 < 𝑆𝑡 < 𝐾 ( 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) =

𝑆𝑡 𝐾

(

)𝜆+ Φ(𝑑+ ) +

𝑆𝑡 𝐾

)𝜆− Φ(𝑑− )

where

𝜆± =

−(𝑟 − 𝐷 − 12 𝜎 2 ) ±

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟 𝜎2

√ log(𝑆𝑡 ∕𝐾) ± (𝑇 − 𝑡) (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟 𝑑± = √ 𝜎 𝑇 −𝑡 and 𝑥

1 2 1 𝑒− 2 𝑢 𝑑𝑢 Φ(𝑥) = √ ∫ 2𝜋 −∞

is the standard normal cumulative density. Solution: In order to show that |𝑎| − 1 (𝑎+𝑏𝑡)2 𝑒 2𝑡 𝑓𝜏 (𝑡) = √ 𝑡 2𝜋𝑡 which is the first-passage time density of a standard Wiener process hitting a slope, see Problem 4.2.2.16 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus. From Girsanov’s theorem, under the risk-neutral measure ℚ the SDE for the asset price is 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡

3.2.3 Time-Dependent Options

347

(𝜇 − 𝑟) where 𝑊𝑡ℚ = 𝑊𝑡 + 𝑡 is a ℚ-standard Wiener process. Hence, using It¯o’s lemma 𝜎 for 𝑢 > 𝑡 we can write ̂

𝑆𝑢 = 𝑆𝑡 𝑒𝜎 𝑊𝑢−𝑡 ( ) ̂𝑢−𝑡 = 𝑊 ℚ + 𝜃(𝑢 − 𝑡), 𝜃 = 1 𝑟 − 𝐷 − 1 𝜎 2 . where 𝑊 𝑢−𝑡 𝜎 2 By definition the immediate-touch call option price is 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝔼ℚ ℚ

( (

max 𝑒

=𝔼

( =𝔼

| max 𝑒−𝑟(𝑢−𝑡) 1I{𝑆𝑢 ≥𝐾} || ℱ𝑡 𝑡≤𝑢≤𝑇 |

ℚ

−𝑟(𝑢−𝑡)

𝑡≤𝑢≤𝑇

max 𝑒

𝑡≤𝑢≤𝑇

)

| 1I{𝑆 𝑒𝜎 𝑊̂𝑢−𝑡 ≥𝐾} || ℱ𝑡 𝑡 |

−𝑟(𝑢−𝑡)

)

| | 1I ̂ 1 ( 𝐾 ) | ℱ𝑡 {𝑊𝑢−𝑡 ≥ 𝜎 log 𝑆 } | 𝑡 |

) .

Since }

{ max 𝑆𝑢 ≥ 𝐾

𝑡≤𝑢≤𝑇

{ ⇔

̂𝑢−𝑡 ≥ 1 log max 𝑊 𝑡≤𝑢≤𝑇 𝜎

(

𝐾 𝑆𝑡

)} ⇔ {𝜏 ≤ 𝑇 }

where { ( )} 1 𝐾 ̂ 𝜏 = inf 𝑢 ≥ 𝑡 : 𝑊𝑢−𝑡 = log 𝜎 𝑆𝑡 therefore ( ) | 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝔼ℚ 𝑒−𝑟(𝜏−𝑡) 1I{𝜏≤𝑇 } | ℱ𝑡 | 𝑇

=

∫𝑡

𝑒−𝑟(𝑢−𝑡) 𝑓𝜏ℚ (𝑢) 𝑑𝑢

where 𝑓𝜏ℚ (𝑢) =

𝑤 − 1 (𝑤−𝜃(𝑢−𝑡))2 𝑒 2(𝑢−𝑡) , √ (𝑢 − 𝑡) 2𝜋(𝑢 − 𝑡)

𝑤=

1 log 𝜎

and 𝜃=

( ) 1 1 𝑟 − 𝐷 − 𝜎2 . 𝜎 2

By letting 𝑣 = 𝑢 − 𝑡 we can write 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) =

𝑇 −𝑡

∫0

1 2 𝑤 𝑒− 2𝑣 (𝑤−𝜃𝑣) 𝑑𝑣. 𝑒−𝑟𝑣 √ 𝑣 2𝜋𝑣

(

𝐾 𝑆𝑡

)

348

3.2.3 Time-Dependent Options

1 Thus, by substituting 𝛼 = 𝜎𝜃 = 𝑟 − 𝐷 − 𝜎 2 , 𝜎𝑤 = log 2 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) =

𝑇 −𝑡

∫0 (

= ( = ( = ( =

𝐾 𝑆𝑡 𝐾 𝑆𝑡 𝐾 𝑆𝑡 𝐾 𝑆𝑡

(

) 𝐾 , 𝛽 2 = 𝛼 2 + 2𝑟𝜎 2 𝑆𝑡

[ ( ) ]2 1 𝐾 −𝑟𝑣 log(𝐾∕𝑆𝑡 ) − 2𝜎 2 𝑣 log 𝑆𝑡 −𝛼𝑣 𝑒 𝑒 𝑑𝑣 √

)

𝜎𝑣 2𝜋𝑣

𝛼 𝜎2

𝑇 −𝑡

log(𝐾∕𝑆𝑡 ) − 2𝜎12 𝑣 𝑒 √ 𝜎𝑣 2𝜋𝑣

𝑇 −𝑡

log(𝐾∕𝑆𝑡 ) − 2𝜎12 𝑣 𝑒 √ 𝜎𝑣 2𝜋𝑣

𝑇 −𝑡

log(𝐾∕𝑆𝑡 ) − 2𝜎12 𝑣 𝑒 √ 𝜎𝑣 2𝜋𝑣

∫0 )

𝛼 𝜎2

∫0 )

𝛼 𝜎2

∫0 ) 𝛼±𝛽

𝑇 −𝑡

𝜎2

∫0

[{

] ( )}2 log 𝑆𝐾 +(𝛼 2 +2𝑟𝜎 2 )𝑣2 𝑡

[{

] ( )}2 +𝛽 2 𝑣2 log 𝑆𝐾 𝑡

{[

log(𝐾∕𝑆𝑡 ) − 12 𝑒 2𝜎 𝑣 √ 𝜎𝑣 2𝜋𝑣

𝑑𝑣

𝑑𝑣

]2 ( ) ( )} log 𝑆𝐾 ±𝛽𝑣 ∓2𝛽𝑣 log 𝑆𝐾 𝑡

𝑡

[ ( ) ]2 log 𝑆𝐾 ±𝛽𝑣 𝑡

𝑑𝑣.

By setting log(𝐾∕𝑆𝑡 ) log(𝐾∕𝑆𝑡 ) − 𝛽𝑣 log(𝐾∕𝑆𝑡 ) + 𝛽𝑣 = + √ √ √ 𝜎 𝑣 2𝜎 𝑣 2𝜎 𝑣 we have ( 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) =

) 𝛼±𝛽

𝐾 𝑆𝑡 (

+

𝑇 −𝑡

𝜎2

∫0

𝐾 𝑆𝑡

) 𝛼±𝛽

log(𝐾∕𝑆𝑡 ) − 𝛽𝑣 − 12 𝑒 2𝜎 𝑣 √ 2𝜎 2𝜋𝑣

𝑇 −𝑡

𝜎2

∫0

[ ( ) ]2 log 𝑆𝐾 ±𝛽𝑣 𝑡

log(𝐾∕𝑆𝑡 ) + 𝛽𝑣 − 12 𝑒 2𝜎 𝑣 √ 2𝜎 2𝜋𝑣

[

𝑑𝑣

( ) ]2 log 𝑆𝐾 ±𝛽𝑣 𝑡

𝑑𝑣.

Hence, by selecting ( 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) =

𝐾 𝑆𝑡 (

+

) 𝛼+𝛽

𝐾 𝑆𝑡

𝑇 −𝑡

𝜎2

∫0 ) 𝛼−𝛽

log(𝐾∕𝑆𝑡 ) − 𝛽𝑣 − 12 𝑒 2𝜎 𝑣 √ 2𝜎 2𝜋𝑣

𝑇 −𝑡

𝜎2

∫0

[

( ) ]2 log 𝑆𝐾 +𝛽𝑣

log(𝐾∕𝑆𝑡 ) + 𝛽𝑣 − 12 𝑒 2𝜎 𝑣 √ 2𝜎 2𝜋𝑣

𝑡

[

( ) ]2 log 𝑆𝐾 −𝛽𝑣

and by setting 𝑥=

log(𝐾∕𝑆𝑡 ) + 𝛽𝑣 √ 2𝜎 2𝜋𝑣

and

𝑦=

𝑑𝑣

log(𝐾∕𝑆𝑡 ) − 𝛽𝑣 √ 2𝜎 2𝜋𝑣

𝑡

𝑑𝑣

𝑑𝑣

3.2.3 Time-Dependent Options

349

we eventually have ( 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) =

𝐾 𝑆𝑡 (

) 𝛼+𝛽

− log(𝐾∕𝑆𝑡 )−𝛽(𝑇 −𝑡) √ 𝜎 𝑇 −𝑡

𝜎2

∫−∞

𝐾 𝑆𝑡

) 𝛼−𝛽

1 − 1 𝑥2 √ 𝑒 2 𝑑𝑥 2𝜋

− log(𝐾∕𝑆𝑡 )+𝛽(𝑇 −𝑡) √ 𝜎 𝑇 −𝑡

𝜎2

1 − 1 𝑦2 √ 𝑒 2 𝑑𝑦 ∫−∞ 2𝜋 ) ( ) 𝛼+𝛽 ( log(𝑆𝑡 ∕𝐾) − 𝛽(𝑇 − 𝑡) 𝐾 𝜎2 Φ = √ 𝑆𝑡 𝜎 𝑇 −𝑡 ) ( ) 𝛼−𝛽 ( log(𝑆𝑡 ∕𝐾) + 𝛽(𝑇 − 𝑡) 𝐾 𝜎2 + Φ √ 𝑆𝑡 𝜎 𝑇 −𝑡 ) ( ) ( ( )− 𝛼−𝛽 𝜎2 𝑆𝑡 log(𝑆𝑡 ∕𝐾) + 𝛽(𝑇 − 𝑡) = Φ √ 𝐾 𝜎 𝑇 −𝑡 ) ( ( ) ( )− 𝛼+𝛽 𝜎2 𝑆𝑡 log(𝑆𝑡 ∕𝐾) − 𝛽(𝑇 − 𝑡) + Φ . √ 𝐾 𝜎 𝑇 −𝑡 +

√ By substituting 𝛼 = 𝑟 − 𝐷 −

1 2 𝜎 2

1 (𝑟 − 𝐷 − 𝜎 2 )2 + 2𝑟𝜎 2 we have, for 2

and 𝛽 =

0 < 𝑆𝑡 < 𝐾 ( 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) =

𝑆𝑡 𝐾

(

)𝜆+ Φ(𝑑+ ) +

𝑆𝑡 𝐾

)𝜆− Φ(𝑑− )

where

𝜆± =

−(𝑟 − 𝐷 − 12 𝜎 2 ) ±

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟 𝜎2

and √ log(𝑆𝑡 ∕𝐾) ± (𝑇 − 𝑡) (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟 𝑑± = . √ 𝜎 𝑇 −𝑡

4 Barrier Options In the last two chapters we have concentrated solely on European and American options which have a fixed strike and expiry date but different styles of exercise rights. In financial circles, these two types of derivatives are known as vanilla options, with the former also known as Black–Scholes options. In this chapter we will discuss barrier options, which are one of the most popular traded options that belong to a family of derivatives known as exotic options. Basically, an exotic option is a type of option which is neither a European nor an American option. Although they may have either European or American-style payoffs, exotic options have additional features that trigger the payoffs. The simplest exotic options are the digital and asset-or-nothing options which we have come across in Chapter 2. Another important difference between exotic options and vanilla options is that unlike vanilla options which are traded on major exchanges such as the Chicago Board Options Exchange, exotic options are usually traded OTC (i.e., without any supervision of an exchange). Hence these options are generally negotiated by brokers or dealers of the options’ sellers and buyers, as to how the trades are to be settled in the future. In finance, barrier options belong to a class of exotic options that can be terminated or activated when the threshold or barrier is crossed. These options are usually written on volatile stocks and given such a feature, barrier options are significantly cheaper than vanilla options. Hence, by trading such an option, it provides investors with an alternative hedging strategy without having to pay for the full vanilla price, which they believe is unlikely to occur in the lifetime of the option contract. By making the barrier be a function of time or paying a rebate when the barrier is triggered, there is a lot of flexibility for investors to express their view in the stock price movements within the contract. In addition, the option can also be easily extended to incorporate early-exercise features, but the solutions to such a problem usually depend on numerical methods. In this chapter, unless otherwise stated, we only consider European-style barrier options.

4.1 INTRODUCTION Conceptually, barrier options are conditional options which are dependent on some barriers being breached (or triggered) in their contractual lives. As these options are either activated or deactivated when the asset price crosses a barrier, these options are path-dependent. However, these options are also known as weakly path-dependent, since we only need to know whether or not the barrier is triggered and we are not concerned about the asset price path. Basically, there are two types of barrier options: knock-ins and knock-outs. For a knock-in: s If the option is an up-and-in barrier option, then the option is only active if the barrier is hit from below the barrier. Note that the option remains worthless if the asset price does not rise above the barrier. If at some point during the life of the option the barrier is hit, then the option will turn into a European option.

352

4.1 INTRODUCTION

s If the option is a down-and-in barrier option, then the option is only active if the barrier is hit from above the barrier. Note that the option remains worthless if the asset price does not fall below the barrier. If at some point during the life of the option the barrier is hit, then the option will turn into a European option. For a knock-out: s If the option is an up-and-out barrier option, then the option is only active if the asset price is below the barrier. Here, the option becomes worthless if the asset price rises above the barrier (or the barrier is hit from below). s If the option is a down-and-out barrier option, then the option is only active if the asset price is above the barrier. Here, the option becomes worthless if the asset price falls below the barrier (or the barrier is hit from above). In–Out Parity Just like the European and American options have their put–call parities, the barrier options have the in–out parity 𝑉𝑘∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑘∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) where the sum of the knock-in option 𝑉𝑘∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) and the knock-out option 𝑉𝑘∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) with the same asset price 𝑆𝑡 , strike 𝐾, barrier 𝐵 and expiry time 𝑇 is the same as the price of a European (Black–Scholes) option. From the above relationship, given that knock-ins and knock-outs have positive payoffs and hence positive option price values, we can immediately deduce that their option price values are less than the value of a European option. This is not unexpected, since barrier options have fewer rights than European options and thus they have a lower value. By incorporating a rebate 𝑅 payable at expiry 𝑇 , we have the following property 𝑅 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑘∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑅𝑒−𝑟(𝑇 −𝑡) 𝑉𝑘∕𝑖 𝑅 (𝑆 , 𝑡; 𝐾, 𝐵, 𝑇 ) and 𝑉 𝑅 (𝑆 , 𝑡; 𝐾, 𝐵, 𝑇 ) are knock-in and knock-out options with where 𝑉𝑘∕𝑖 𝑡 𝑘∕𝑜 𝑡 rebate 𝑅, respectively having the same asset price 𝑆𝑡 , strike 𝐾, barrier 𝐵, expiry time 𝑇 , and 𝑟 is the constant risk-free interest rate. In contrast, if the rebate 𝑅 is payable at knock-out or knock-in time, the corresponding in–out parity is

𝑅 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑘∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑅 𝑉𝑘∕𝑖

[(

𝑆𝑡 𝐵

where

𝜆± =

−(𝑟 − 𝐷 − 12 𝜎 2 ) ±

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟 𝜎2

.

(

)𝜆+ +

𝑆𝑡 𝐵

)𝜆− ]

4.1 INTRODUCTION

353

Barrier Options Pricing Under the Black–Scholes methodology we can solve the pricing issues of barrier options using both the probabilistic (martingale pricing) approach and partial differentiation equation (via the reflection principle) approach. Probabilistic Approach In the same vein as the martingale pricing framework of European options, we can also price barrier options via the risk-neutral expectation strategy. For instance, a knock-in barrier option payoff can be written as ⎧ Ψ(𝑆 )1I 𝑇 {max𝑡≤𝑢≤𝑇 𝑆𝑢 ≥𝐵 } ⎪ Ψ𝑘∕𝑖 (𝑆𝑇 ) = ⎨ ⎪ Ψ(𝑆𝑇 )1I{min 𝑡≤𝑢≤𝑇 𝑆𝑢 ≤𝐵 } ⎩

up-and-in option down-and-in option

whilst for a knock-out barrier option payoff ⎧ Ψ(𝑆 )1I 𝑇 {max𝑡≤𝑢≤𝑇 𝑆𝑢 𝐵 } ⎩

up-and-out option down-and-out option

where Ψ(𝑆𝑇 ) is the European option payoff at expiry time 𝑇 . In order to price this option we need to know the probability that the barrier will be triggered and also the probability that the option is exercised at expiry time. Hence, this requires the joint distribution of the Wiener process with drift (for the European option payoff) and its running maximum (for up-and-in and up-and-out options) or running minimum (for down-and-in and down-and-out options). Without loss of generality, let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process and if 𝑋𝑡 is a Wiener process with drift 𝜇 ∈ ℝ in the form 𝑋𝑡 = 𝜇𝑡 + 𝑊𝑡 and by defining 𝑀𝑡𝑋 = max 𝑋𝑠 0≤𝑠≤𝑡

to be the running maximum of the process 𝑋𝑡 up to time 𝑡, the cdf of the running maximum is ( ℙ(𝑀𝑡𝑋

≤ 𝑥) = Φ

𝑥 − 𝜇𝑡 √ 𝑡

)

( −𝑒

2𝜇𝑥

Φ

−𝑥 − 𝜇𝑡 √ 𝑡

) ,

𝑥≥0

354

4.1 INTRODUCTION

where Φ(⋅) is the standard normal cdf. The joint density of (𝑀𝑡𝑋 , 𝑋𝑡 ) of the running maximum and the Wiener process is 𝑓ℙ 𝑋

𝑀𝑡 ,𝑋𝑡

(𝑥, 𝑦) =

2(2𝑥 − 𝑦) 𝜇𝑦− 1 𝜇2 𝑡− 1 (2𝑥−𝑦)2 2𝑡 𝑒 2 , √ 𝑡 2𝜋𝑡

𝑥 ≥ 0, 𝑥 ≥ 𝑦.

As for the running minimum of the process 𝑋𝑡 up to time 𝑡, 𝑚𝑋 𝑡 = min 𝑋𝑠 0≤𝑠≤𝑡

the cdf of the running minimum is ( ℙ(𝑚𝑋 𝑡

≤ 𝑥) = Φ

𝑥 − 𝜇𝑡 √ 𝑡

)

( +𝑒

2𝜇𝑥

Φ

𝑥 + 𝜇𝑡 √ 𝑡

) ,

𝑥≤0

with the joint density of (𝑚𝑋 𝑡 , 𝑋𝑡 ) of the running minimum and the Wiener process 𝑓 ℙ𝑋

𝑚𝑡 ,𝑋𝑡

(𝑥, 𝑦) =

−2(2𝑥 − 𝑦) 𝜇𝑦− 1 𝜇2 𝑡− 1 (2𝑥−𝑦)2 2𝑡 𝑒 2 , √ 𝑡 2𝜋𝑡

𝑥 ≤ 0, 𝑥 ≤ 𝑦.

For the derivation of the above formulae, see Problem 4.2.2.14 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus. Partial Differentiation Equation Approach In order to find the closed-form solutions of barrier options via the PDE method, we first consider the reflected solution of the Black–Scholes equation. Theorem 4.1 Suppose 𝑉 (𝑆𝑡 , 𝑡) satisfies the Black–Scholes equation (see below) with the asset 𝑆𝑡 paying a continuous dividend yield 𝐷, 1 𝜕2𝑉 𝜕𝑉 𝜕𝑉 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0 𝜕𝑆𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 then for any constant 𝐵 (representing a barrier) ( 𝑈 (𝑆𝑡 , 𝑡) =

𝑆𝑡 𝐵

(

)2𝛼 𝑉

𝐵2 ,𝑡 𝑆𝑡

)

( 1 also satisfies the Black–Scholes equation provided that 𝛼 = 2

1−

𝑟−𝐷 1 2 𝜎 2

) . Note that 𝑆𝑡 and

𝐵 2 ∕𝑆𝑡 are on opposite sides of the barrier 𝐵 for 𝑆𝑡 ≠ 𝐵 and coincide when 𝑆𝑡 = 𝐵.

4.1 INTRODUCTION

355

Let us begin by focussing on the up-and-out barrier option and note that we can consider only the knock-out options as the knock-in options’ prices can be obtained easily from the in–out parity identity. If we consider the Black–Scholes equation for an asset 𝑆𝑡 paying a continuous dividend yield 𝐷, the up-and-out barrier option 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) with strike price 𝐾 and barrier 𝐵 satisfies the PDE 𝜕𝑉𝑢∕𝑜 𝜕𝑆𝑡

𝜕 2 𝑉𝑢∕𝑜 𝜕𝑉𝑢∕𝑜 1 + 𝜎 2 𝑆𝑡2 + (𝑟 − 𝐷)𝑆 − 𝑟𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 0 𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

provided the following boundary conditions are satisfied 0 < 𝑆𝑡 < 𝐵,

𝑡≤𝑇

𝑉𝑢∕𝑜 (𝐵, 𝑡; 𝐾, 𝐵, 𝑇 ) = 0 𝑉𝑢∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = Ψ(𝑆𝑇 ) where Ψ(𝑆𝑇 ) is the European (Black–Scholes) payoff at expiry time 𝑇 . Given that the up-and-out barrier option has a European payoff at expiry 𝑇 but becomes worthless on the barrier 𝐵, we can deduce that 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉̃𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) − 𝑉̂𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) so that 𝑉̃𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) satisfies the Black–Scholes equation 𝜕 𝑉̃𝑢∕𝑜 𝜕𝑆𝑡

𝜕 2 𝑉̃𝑢∕𝑜 𝜕 𝑉̃𝑢∕𝑜 1 + 𝜎 2 𝑆𝑡2 + (𝑟 − 𝐷)𝑆 − 𝑟𝑉̃𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 0 𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

with truncated payoff 0 < 𝑆𝑡 < 𝐵,

𝑡≤𝑇

𝑉̃𝑢∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = Ψ(𝑆𝑇 ). To account for the barrier, we therefore require 𝑉̂ (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) to satisfy 𝑉̂𝑢∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = 0 at expiry time 𝑇 and 𝑉̂𝑢∕𝑜 (𝐵, 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉̃𝑢∕𝑜 (𝐵, 𝑡; 𝐾, 𝐵, 𝑇 ) on the barrier 𝐵.

356

(

4.1 INTRODUCTION

To find the solution for 𝑉̂𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) we note that since ( ) ) )2𝛼 ( 2 𝑆𝑡 1 𝑟−𝐷 𝐵 , 𝑡; 𝐾, 𝐵, 𝑇 with 𝛼 = 1− 1 also satisfies the Black–Scholes 𝑉̃𝑢∕𝑜 𝐵 𝑆𝑡 2 𝜎2 2

equation, we can thus set the solution of an up-and-out barrier option as 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉̃𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) −

(

𝑆𝑡 𝐵

)2𝛼

𝑉̃𝑢∕𝑜

(

𝐵2 , 𝑡; 𝐾, 𝐵, 𝑇 𝑆𝑡

) .

Since 𝑆𝑡 and 𝐵 2 ∕𝑆𝑡 are on opposite sides of the barrier 𝐵 for 𝑆𝑡 ≠ 𝐵, therefore (

𝑆𝑇 𝐵

)2𝛼

𝑉̃

(

𝐵2 , 𝑡; 𝐾, 𝐵, 𝑇 𝑆𝑇

) =0

for all 0 < 𝑆𝑇 < 𝐵. Hence, one can easily check that the boundary conditions 𝑉𝑢∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = Ψ(𝑆𝑇 )

and

𝑉𝑢∕𝑜 (𝐵, 𝑡; 𝐾, 𝐵, 𝑇 ) = 0

are satisfied. Using the in–out parity property, the up-and-in barrier option price 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) at time 𝑡, 𝑡 ≤ 𝑇 is 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ). Following the same arguments, the down-and-out barrier option at time 𝑡 is 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉̃𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) −

(

𝑆𝑡 𝐵

)2𝛼

𝑉̃𝑑∕𝑜

(

𝐵2 , 𝑡; 𝐾, 𝐵, 𝑇 𝑆𝑡

)

where 𝑉̃𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) satisfies the following Black–Scholes equation 𝜕 𝑉̃𝑑∕𝑜 𝜕𝑆𝑡

𝜕 2 𝑉̃𝑑∕𝑜 𝜕 𝑉̃𝑑∕𝑜 1 + 𝜎 2 𝑆𝑡2 + (𝑟 − 𝐷)𝑆 − 𝑟𝑉̃𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 0 𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

with truncated payoff 𝑆𝑡 > 𝐵,

𝑡≤𝑇

𝑉̃𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = Ψ(𝑆𝑇 ). Finally, with the application of the in–out parity property, the down-and-in barrier option price 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) at time 𝑡, 𝑡 ≤ 𝑇 is 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ).

4.2.1 Probabilistic Approach

357

4.2 PROBLEMS AND SOLUTIONS 4.2.1

Probabilistic Approach

} { 1. Up-and-Out/Up-and-In Call Options. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. In addition, let 𝑟 be the risk-free interest rate. We consider a European up-and-out call option at time 𝑡, with expiry at time 𝑇 > 𝑡, strike 𝐾 and constant barrier 𝐵 > 𝐾. Explain why, for an up-and-out call option, we require 𝑆𝑡 < 𝐵 and 𝐵 > 𝐾, and show that the terminal payoff can be written as { } Ψ(𝑆𝑇 ) = max 𝑆𝑇 − 𝐾, 0 1I{ max 𝑆 < 𝐵} . 𝑢 𝑡≤𝑢≤𝑇

Using the identity 𝑈 𝑎𝑤− 1 ( √𝑤 )2 2 𝑇

1

𝑒 √ 2𝜋𝑇 ∫𝐿

[ ( 𝑑𝑤 = 𝑒

1 2 𝑎 𝑇 2

Φ

𝑈 − 𝑎𝑇 √ 𝑇

)

( −Φ

𝐿 − 𝑎𝑇 √ 𝑇

)]

𝑥

1 − 1 𝑢2 √ 𝑒 2 𝑑𝑢 is the cdf of a standard normal, show that under the ∫−∞ 2𝜋 risk-neutral measure ℚ, the up-and-out call option price at time 𝑡 is where Φ(𝑥) =

( ( ) ) 𝐶𝑢∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 ( ( ) )] [ − 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐵, 𝑇 + (𝐵 − 𝐾) 𝐶𝑑 𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) ( 2 ( )2𝛼 { 𝑆𝑡 𝐵 , 𝑡; 𝐾, 𝑇 𝐶𝑏𝑠 − 𝐵 𝑆𝑡 [ ) )]} ( 2 ( 2 𝐵 𝐵 − 𝐶𝑏𝑠 , 𝑡; 𝐵, 𝑇 + (𝐵 − 𝐾) 𝐶𝑑 , 𝑡; 𝐵, 𝑇 𝑆𝑡 𝑆𝑡 ( 1 where 𝛼 = 2

1−

𝑟−𝐷 1 2 𝜎 2

) , 𝐶𝑏𝑠 (𝑋, 𝑡; 𝑌 , 𝑇 ) and 𝐶𝑑 (𝑋, 𝑡; 𝑌 , 𝑇 ) are the vanilla and digital

358

4.2.1 Probabilistic Approach

call option prices defined as ) log(𝑋∕𝑌 ) + (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) 𝐶𝑏𝑠 (𝑋, 𝑡; 𝑌 , 𝑇 ) = 𝑋𝑒 Φ √ 𝜎 𝑇 −𝑡 ( ) log(𝑋∕𝑌 ) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) −𝑟(𝑇 −𝑡) −𝑌 𝑒 Φ √ 𝜎 𝑇 −𝑡 (

−𝐷(𝑇 −𝑡)

and ( 𝐶𝑑 (𝑋, 𝑡; 𝑌 , 𝑇 ) = 𝑒

−𝑟(𝑇 −𝑡)

Φ

log(𝑋∕𝑌 ) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

)

respectively. Hence, deduce that the European up-and-in call option price at time 𝑡 is ( ( ) ) 𝐶𝑢∕𝑖 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐵, 𝑇

) 𝐵2 𝐶𝑏𝑠 , 𝑡; 𝐾, 𝑇 𝑆𝑡 ( )2𝛼 [ ) )] ( 2 ( 2 𝑆𝑡 𝐵 𝐵 − , 𝑡; 𝐵, 𝑇 + (𝐵 − 𝐾) 𝐶𝑑 , 𝑡; 𝐵, 𝑇 . 𝐶𝑏𝑠 𝐵 𝑆𝑡 𝑆𝑡

( ) + (𝐵 − 𝐾) 𝐶𝑑 𝑆𝑡 , 𝑡; 𝐵, 𝑇 +

(

𝑆𝑡 𝐵

)2𝛼

(

Solution: Since the option is an up-and-out call we require 𝑆𝑡 for all 𝑡 to be below the barrier 𝐵, otherwise it knocks out. Furthermore, we require 𝐵 > 𝐾 for all time 𝑡, since if 𝐵 ≤ 𝐾 the option is worthless as no asset path would give a non-zero payoff. Therefore, the payoff of an up-and-out call option is { } Ψ(𝑆𝑇 ) = max 𝑆𝑇 − 𝐾, 0 1I{ max 𝑆 < 𝐵} . 𝑢 𝑡≤𝑢≤𝑇

Under the risk-neutral measure ℚ, 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡 (𝜇 − 𝑟) 𝑡 is a ℚ-standard Wiener process. Using It¯o’s lemma we can where 𝑊𝑡ℚ = 𝑊𝑡 + 𝜎 easily show that for 𝑡 < 𝑇 1 2 )(𝑇 −𝑡)+𝜎𝑊𝑇ℚ−𝑡

𝑆𝑇 = 𝑆𝑡 𝑒(𝑟−𝐷− 2 𝜎 ̂

= 𝑆𝑡 𝑒𝜎 𝑊𝑇 −𝑡 ̂𝑇 −𝑡 = 𝜈(𝑇 − 𝑡) + 𝑊 ℚ and 𝜈 = 1 (𝑟 − 𝐷 − 1 𝜎 2 ). By writing where 𝑊 𝑇 −𝑡 𝜎 2 ̂𝑢−𝑡 𝑀𝑇 −𝑡 = max 𝑊 𝑡≤𝑢≤𝑇

4.2.1 Probabilistic Approach

359

therefore max 𝑆𝑢 = 𝑆𝑡 𝑒𝜎𝑀𝑇 −𝑡

𝑡≤𝑢≤𝑇

and we can rewrite the payoff as { } Ψ(𝑆𝑇 ) = max 𝑆𝑇 − 𝐾, 0 1I{ max 𝑆 < 𝐵} 𝑢 𝑡≤𝑢≤𝑇 } { ̂ = max 𝑆𝑡 𝑒𝜎 𝑊𝑇 −𝑡 − 𝐾, 0 1I{𝑆 𝑒𝜎𝑀𝑇 −𝑡 𝜎 log 𝑆𝐾 𝑡 𝑡

Hence, the up-and-out call option price at time 𝑡 is ( ) 𝐶𝑢∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 [ ] = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ Ψ(𝑆𝑇 )|| ℱ𝑡 [ ( ) ̂ 𝑆𝑡 𝑒𝜎 𝑊𝑇 −𝑡 − 𝐾 1I{ = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ

= 𝑒−𝑟(𝑇 −𝑡)

( ) 𝑤= 𝜎1 log 𝑆𝐵 (

∫𝑤= 1 log 𝜎

𝑡

𝐾 𝑆𝑡

)

| ( ) ( )} || ℱ 1 𝐵 1 𝐾 ̂𝑇 −𝑡 > log 𝑀𝑇 −𝑡 < 𝜎 log 𝑆 ,𝑊 | 𝑡 𝜎 𝑆𝑡 𝑡 |

( ) 𝑚= 𝜎1 log 𝑆𝐵 ( 𝑡

∫𝑚=𝑤

) 𝑆𝑡 𝑒𝜎𝑤 − 𝐾 𝑓 ℚ

̂ 𝑀,𝑊

]

(𝑚, 𝑤) 𝑑𝑚𝑑𝑤

where ( )2 ⎧ √ 𝜈𝑤− 21 𝜈 2 (𝑇 −𝑡)− 12 2𝑚−𝑤 ⎪ 2(2𝑚−𝑤) 𝑇 −𝑡 √ 𝑒 ⎪ 𝑓 ℚ ̂ (𝑚, 𝑤) = ⎨ (𝑇 −𝑡) 2𝜋(𝑇 −𝑡) 𝑀,𝑊 ⎪ ⎪0 ⎩

𝑚 ≥ 0, 𝑚 ≥ 𝑤 otherwise.

Refer to Problem 4.2.2.14 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus for a derivation of the above density function.

360

4.2.1 Probabilistic Approach

Hence, ( ) 𝐶𝑢∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = −𝑒−𝑟(𝑇 −𝑡)

( ) 𝑤= 𝜎1 log 𝑆𝐵 ( (

∫𝑤= 1 log 𝜎

𝑡

𝐾 𝑆𝑡

)

𝑆𝑡 𝑒𝜎𝑤 − 𝐾 (

×√

𝜈𝑤− 21 𝜈 2 (𝑇 −𝑡)− 12

1 2𝜋(𝑇 − 𝑡)

𝑒

(

1 log 𝑆𝐵 𝜎 𝑡

)

1 =√ ( ) 2𝜋(𝑇 − 𝑡) ∫ 𝜎1 log 𝑆𝐾𝑡

−√

2𝑚−𝑤 √ 𝑇 −𝑡

)

( ) )2 |𝑚= 𝜎1 log 𝑆𝐵

| | | | | |𝑚=𝑤

(

2𝜋(𝑇 − 𝑡) ∫ 𝜎1 log

) 𝑆𝑡 𝑒𝜎𝑤 − 𝐾 𝑒

𝑡

𝐾 𝑆𝑡

𝑑𝑤

( 𝜎𝑤 ) −𝑟(𝑇 −𝑡)+𝜈𝑤− 21 𝜈 2 (𝑇 −𝑡)− 21 𝑆𝑡 𝑒 − 𝐾 𝑒

( ) 1 log 𝑆𝐵 ( 𝜎

1

𝑡

)

(

𝑤 √ 𝑇 −𝑡

)2

𝑑𝑤

( ) 2 ⎛ 2 log 𝑆𝐵 −𝑤 ⎞ 𝑡 1 2 1⎜ 𝜎 ⎟ √ −𝑟(𝑇 −𝑡)+𝜈𝑤− 2 𝜈 (𝑇 −𝑡)− 2 ⎜ ⎟ 𝑇 −𝑡

⎝

⎠

= 𝑆𝑡 𝐼1 − 𝐾𝐼2 − (𝑆𝑡 𝐼3 − 𝐾𝐼4 ) where (

𝐼1 = √

1 log 𝑆𝐵 𝜎 𝑡

1

(

2𝜋(𝑇 − 𝑡) ∫ 𝜎1 log

𝐾 𝑆𝑡

)

)

𝑒

(

)

1 log 𝑆𝐵 𝜎 𝑡

1

( −𝑟(𝑇 −𝑡)+𝜎𝑤+𝜈𝑤− 21 𝜈 2 (𝑇 −𝑡)− 12

𝐼2 = √ ( ) 𝑒 2𝜋(𝑇 − 𝑡) ∫ 𝜎1 log 𝑆𝐾𝑡 (

𝐼3 = √

1

1 log 𝑆𝐵 𝜎 𝑡

(

2𝜋(𝑇 − 𝑡) ∫ 𝜎1 log

𝐾 𝑆𝑡

)

)

𝑒

( −𝑟(𝑇 −𝑡)+𝜈𝑤− 21 𝜈 2 (𝑇 −𝑡)− 21

√

𝑤 √ 𝑇 −𝑡

)2

𝑑𝑤

)2

𝑑𝑤

( ) 2 ⎛ 2 log 𝑆𝐵 −𝑤 ⎞ 𝑡 1 2 1⎜ 𝜎 ⎟ √ −𝑟(𝑇 −𝑡)+𝜎𝑤+𝜈𝑤− 2 𝜈 (𝑇 −𝑡)− 2 ⎜ ⎟ 𝑇 −𝑡

⎝

⎠

(

1 𝐼4 = √ 2𝜋(𝑇 − 𝑡)

𝑤 𝑇 −𝑡

)

⎛ 𝜎2 log 𝑆𝐵 −𝑤 ⎞ ( ) 𝑡 ⎟ 1 𝐵 √ −𝑟(𝑇 −𝑡)+𝜈𝑤− 21 𝜈 2 (𝑇 −𝑡)− 12 ⎜ log ⎜ ⎟ 𝑇 −𝑡 𝜎 𝑆𝑡 ⎝ ⎠ ( ) 𝑒 ∫ 1 log 𝐾 𝜎 𝑆

𝑑𝑤

2

𝑑𝑤.

𝑡

Using the identity 1

𝑈 𝑎𝑤− 1 ( √𝑤 )2 2 𝑇

𝑒 √ 2𝜋𝑇 ∫𝐿

[ ( 𝑑𝑤 = 𝑒

1 2 𝑎 𝑇 2

Φ

𝑈 − 𝑎𝑇 √ 𝑇

)

( −Φ

𝐿 − 𝑎𝑇 √ 𝑇

)]

𝑑𝑤

4.2.1 Probabilistic Approach

361

we have (

𝐼1 = √

1

−𝑟(𝑇 −𝑡)− 12 𝜈 2 (𝑇 −𝑡)

2𝜋(𝑇 − 𝑡)

𝑒

1 log 𝜎

=𝑒

)

𝑒

∫ 1 log( 𝐾 ) 𝜎

−𝑟(𝑇 −𝑡)− 12 𝜈 2 (𝑇 −𝑡)+ 12 (𝜈+𝜎)2 (𝑇 −𝑡)

𝐵 𝑆𝑡

(𝜈+𝜎)𝑤− 12

( √

)2

𝑤 𝑇 −𝑡

𝑑𝑤

𝑆𝑡

⎡ ⎛ 1 log(𝐵∕𝑆 ) − (𝜈 + 𝜎)(𝑇 − 𝑡) ⎞ 𝑡 ⎢ ⎜𝜎 ⎟ √ ⎢Φ ⎜ ⎟ 𝑇 −𝑡 ⎢ ⎜ ⎟ ⎣ ⎝ ⎠

⎛ 1 log(𝐾∕𝑆 ) − (𝜈 + 𝜎)(𝑇 − 𝑡) ⎞⎤ 𝑡 ⎜ ⎟⎥ −Φ ⎜ 𝜎 √ ⎟⎥ 𝑇 −𝑡 ⎜ ⎟⎥ ⎝ ⎠⎦ 1 1 (𝑟 − 𝐷 − 𝜎 2 ), we have 𝜎 2 [ ( ) log(𝑆𝑡 ∕𝐵) + (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) −𝐷(𝑇 −𝑡) 𝐼1 = 𝑒 1−Φ √ 𝜎 𝑇 −𝑡 ( )] log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) −1 + Φ √ 𝜎 𝑇 −𝑡 [ ( ) log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) −𝐷(𝑇 −𝑡) Φ =𝑒 √ 𝜎 𝑇 −𝑡 ( )] log(𝑆𝑡 ∕𝐵) + (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) −Φ . √ 𝜎 𝑇 −𝑡

and knowing that 𝜈 =

Similarly we can deduce (

𝐼2 = √

1

−𝑟(𝑇 −𝑡)− 12 𝜈 2 (𝑇 −𝑡)

2𝜋(𝑇 − 𝑡) 1 2

𝑒

1 log 𝜎

∫ 1 log( 𝐾 ) 𝜎

1 2

𝐵 𝑆𝑡

)

𝑒

𝜈𝑤− 12

( √

𝑤 𝑇 −𝑡

)2

𝑑𝑤

𝑆𝑡

= 𝑒−𝑟(𝑇 −𝑡)− 2 𝜈 (𝑇 −𝑡)+ 2 𝜈 (𝑇 −𝑡) ⎡ ⎛ 1 log(𝐵∕𝑆 ) − 𝜈(𝑇 − 𝑡) ⎞ ⎛ 1 log(𝐾∕𝑆 ) − 𝜈(𝑇 − 𝑡) ⎞⎤ 𝑡 𝑡 ⎢ ⎜𝜎 ⎟ ⎜𝜎 ⎟⎥ √ √ ⎢Φ ⎜ ⎟ − Φ⎜ ⎟⎥ 𝑇 −𝑡 𝑇 −𝑡 ⎢ ⎜ ⎟ ⎜ ⎟⎥ ⎣ ⎝ ⎠ ⎝ ⎠⎦ [ ( ) 1 2 log(𝐵∕𝑆𝑡 ) − (𝑟 − 𝐷 − 2 𝜎 )(𝑇 − 𝑡) = 𝑒−𝑟(𝑇 −𝑡) Φ √ 𝜎 𝑇 −𝑡 ( )] log(𝐾∕𝑆𝑡 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) −Φ √ 𝜎 𝑇 −𝑡 [ ( ) log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) −𝑟(𝑇 −𝑡) Φ =𝑒 √ 𝜎 𝑇 −𝑡 ( )] log(𝑆𝑡 ∕𝐵) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) −Φ . √ 𝜎 𝑇 −𝑡

362

4.2.1 Probabilistic Approach

𝐼3 = √

1

−𝑟(𝑇 −𝑡)− 12 𝜈 2 (𝑇 −𝑡)−

2𝜋(𝑇 − 𝑡) (

×

1 log 𝜎

)

∫ 1 log( 𝐾 ) 𝜎

=𝑒

𝐵 𝑆𝑡

𝑒

2 𝜎 2 (𝑇 −𝑡)

(

( ))2 log 𝑆𝐵 𝑡

[

( )2 ( )] 𝜈+𝜎+ 𝜎(𝑇2−𝑡) log 𝑆𝐵 𝑤− 12 √𝑤

𝑒

𝑇 −𝑡

𝑡

𝑆𝑡

−𝑟(𝑇 −𝑡)− 12 𝜈 2 (𝑇 −𝑡)−

2 𝜎 2 (𝑇 −𝑡)

(

𝑑𝑤

( ))2 [ ( )]2 log 𝑆𝐵 + 12 𝜈+𝜎+ 𝜎(𝑇2−𝑡) log 𝑆𝐵 (𝑇 −𝑡) 𝑡

𝑡

[ ] ⎞ ⎡ ⎛1 2 log(𝐵∕𝑆 log(𝐵∕𝑆 ) − 𝜈 + 𝜎 + ) (𝑇 − 𝑡) ⎟ 𝑡 𝑡 ⎢ ⎜𝜎 𝜎(𝑇 − 𝑡) ⎟ × ⎢Φ ⎜ √ ⎟ ⎢ ⎜ 𝑇 − 𝑡 ⎟ ⎢ ⎜ ⎣ ⎝ ⎠ [ ] ⎛1 ⎞⎤ 2 ⎜ 𝜎 log(𝐾∕𝑆𝑡 ) − 𝜈 + 𝜎 + 𝜎(𝑇 − 𝑡) log(𝐵∕𝑆𝑡 ) (𝑇 − 𝑡) ⎟⎥ ⎟⎥ −Φ ⎜ √ ⎜ ⎟⎥ 𝑇 − 𝑡 ⎜ ⎟⎥ ⎝ ⎠⎦ [ ( ) 1 2 ( )−1− 2(𝑟−𝐷) 2 log(𝐵 ∕(𝑆𝑡 𝐾)) + (𝑟 − 𝐷 + 2 𝜎 )(𝑇 − 𝑡) 𝜎2 𝑆𝑡 −𝐷(𝑇 −𝑡) =𝑒 Φ √ 𝐵 𝜎 𝑇 −𝑡 ( )] log(𝐵∕𝑆𝑡 ) + (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) −Φ . √ 𝜎 𝑇 −𝑡

𝐼4 = √

1

−𝑟(𝑇 −𝑡)− 12 𝜈 2 (𝑇 −𝑡)−

2𝜋(𝑇 − 𝑡) (

×

1 log 𝜎

∫ 1 log( 𝐾 ) 𝜎

=𝑒

𝐵 𝑆𝑡

𝑒

)

2 𝜎 2 (𝑇 −𝑡)

(

( ))2 log 𝑆𝐵 𝑡

[

𝑒

( )2 ( )] 𝜈+ 𝜎(𝑇2−𝑡) log 𝑆𝐵 𝑤− 12 √𝑤

𝑆𝑡

−𝑟(𝑇 −𝑡)− 12 𝜈 2 (𝑇 −𝑡)−

𝑇 −𝑡

𝑡

2 𝜎 2 (𝑇 −𝑡)

𝑑𝑤

( ( ))2 [ ( )]2 log 𝑆𝐵 + 12 𝜈+ 𝜎(𝑇2−𝑡) log 𝑆𝐵 (𝑇 −𝑡) 𝑡

𝑡

[ ] ⎞ ⎡ ⎛1 2 log(𝐵∕𝑆 log(𝐵∕𝑆 ) − 𝜈 + ) (𝑇 − 𝑡) ⎟ 𝑡 𝑡 ⎢ ⎜𝜎 𝜎(𝑇 − 𝑡) ⎟ × ⎢Φ ⎜ √ ⎟ ⎢ ⎜ 𝑇 − 𝑡 ⎟ ⎢ ⎜ ⎣ ⎝ ⎠ [ ] ⎛1 ⎞⎤ 2 ⎜ 𝜎 log(𝐾∕𝑆𝑡 ) − 𝜈 + 𝜎(𝑇 − 𝑡) log(𝐵∕𝑆𝑡 ) (𝑇 − 𝑡) ⎟⎥ ⎟⎥ −Φ ⎜ √ ⎜ ⎟⎥ 𝑇 − 𝑡 ⎜ ⎟⎥ ⎝ ⎠⎦ [ ( ) 1 2 ( )1− 2(𝑟−𝐷) 2 log(𝐵 ∕(𝑆𝑡 𝐾)) + (𝑟 − 𝐷 − 2 𝜎 )(𝑇 − 𝑡) 𝜎2 𝑆𝑡 −𝑟(𝑇 −𝑡) =𝑒 Φ √ 𝐵 𝜎 𝑇 −𝑡 ( )] log(𝐵∕𝑆𝑡 ) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) −Φ . √ 𝜎 𝑇 −𝑡

4.2.1 Probabilistic Approach

363

Therefore, ( ) 𝐶𝑢∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝑆𝑡 𝐼1 − 𝐾𝐼2 − (𝑆𝑡 𝐼3 − 𝐾𝐼4 ) ( ) log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) −𝐷(𝑇 −𝑡) Φ = 𝑆𝑡 𝑒 √ 𝜎 𝑇 −𝑡 ( ) log(𝑆𝑡 ∕𝐵) + (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) −𝐷(𝑇 −𝑡) Φ −𝑆𝑡 𝑒 √ 𝜎 𝑇 −𝑡 ( ) log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) −𝐾𝑒−𝑟(𝑇 −𝑡) Φ √ 𝜎 𝑇 −𝑡 ( ) log(𝑆𝑡 ∕𝐵) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) −𝑟(𝑇 −𝑡) Φ +𝐾𝑒 √ 𝜎 𝑇 −𝑡 ( ) ( )−1− 2(𝑟−𝐷) log(𝐵 2 ∕(𝑆𝑡 𝐾)) + (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) 𝜎2 𝑆𝑡 −𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ √ 𝐵 𝜎 𝑇 −𝑡 ) ( ( )−1− 2(𝑟−𝐷) log(𝐵∕𝑆𝑡 ) + (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) 𝜎2 𝑆𝑡 −𝐷(𝑇 −𝑡) +𝑆𝑡 𝑒 Φ √ 𝐵 𝜎 𝑇 −𝑡 ( ) ( )1− 2(𝑟−𝐷) log(𝐵 2 ∕(𝑆𝑡 𝐾)) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) 𝜎2 𝑆𝑡 −𝑟(𝑇 −𝑡) +𝐾 𝑒 Φ √ 𝐵 𝜎 𝑇 −𝑡 ( ) ( )1− 2(𝑟−𝐷) log(𝐵∕𝑆𝑡 ) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) 𝜎2 𝑆𝑡 −𝐾 𝑒−𝑟(𝑇 −𝑡) Φ √ 𝐵 𝜎 𝑇 −𝑡 ( ( ( ) ) ) = 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 − 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐵, 𝑇 + (𝐾 − 𝐵)𝐶𝑑 𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) ( )1− 2(𝑟−𝐷) ) ( 2 𝜎2 𝑆𝑡 𝐵2 𝐵 𝐶𝑏𝑠 , 𝑡; 𝐾, 𝑇 − 𝐶𝑏𝑠 , 𝑡; 𝐵, 𝑇 − 𝑆𝑡 𝐵 𝑆𝑡 ) ( )1− 2(𝑟−𝐷) ( 2 𝜎2 𝑆𝑡 𝐵 (𝐾 − 𝐵)𝐶𝑑 , 𝑡; 𝐵, 𝑇 + 𝐵 𝑆𝑡 ( ( ( ) [ ) )] = 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 − 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐵, 𝑇 + (𝐵 − 𝐾)𝐶𝑑 𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) ( 2 ( )2𝛼 { 𝑆𝑡 𝐵 , 𝑡; 𝐾, 𝑇 𝐶𝑏𝑠 − 𝐵 𝑆𝑡 [ ) )]} ( 2 ( 2 𝐵 𝐵 − 𝐶𝑏𝑠 , 𝑡; 𝐵, 𝑇 + (𝐵 − 𝐾)𝐶𝑑 , 𝑡; 𝐵, 𝑇 𝑆𝑡 𝑆𝑡 ) ( 1 𝑟−𝐷 . where 𝛼 = 1− 1 2 𝜎2 (

𝑆𝑡 𝐵

)1− 2(𝑟−𝐷) 𝜎2

2

(

364

4.2.1 Probabilistic Approach

In contrast, for an up-and-in call option price its payoff at expiry 𝑇 can be written as { } Ψ(𝑆𝑇 ) = max 𝑆𝑇 − 𝐾, 0 1I{ max 𝑆 ≥ 𝐵} 𝑢 𝑡≤𝑢≤𝑇 ) ( { } = max 𝑆𝑇 − 𝐾, 0 1 − 1I{ max 𝑆 < 𝐵} 𝑢 𝑡≤𝑢≤𝑇 } { } { = max 𝑆𝑇 − 𝐾, 0 − max 𝑆𝑇 − 𝐾, 0 1I{ max 𝑆 < 𝐵} . 𝑢 𝑡≤𝑢≤𝑇

Hence, under the risk-neutral measure ℚ, the up-and-in call option price at time 𝑡 is [ ( ) { }| ] 𝐶𝑢∕𝑖 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝑒−𝑟(𝑇 −𝑡) 𝔼 max 𝑆𝑇 − 𝐾, 0 | ℱ𝑡 | [ ] | { } | −𝑟(𝑇 −𝑡) −𝑒 𝔼 max 𝑆𝑇 − 𝐾, 0 1I{ max 𝑆 < 𝐵} | ℱ𝑡 𝑢 | 𝑡≤𝑢≤𝑇 | ( ( ) ) = 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 − 𝐶𝑢∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ( ( ) ) = 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐵, 𝑇 + (𝐵 − 𝐾)𝐶𝑑 𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) ( )2𝛼 { ( 2 𝑆𝑡 𝐵 , 𝑡; 𝐾, 𝑇 + 𝐶𝑏𝑠 𝐵 𝑆𝑡 [ ) )]} ( 2 ( 2 𝐵 𝐵 − 𝐶𝑏𝑠 , 𝑡; 𝐵, 𝑇 + (𝐵 − 𝐾)𝐶𝑑 , 𝑡; 𝐵, 𝑇 . 𝑆𝑡 𝑆𝑡 { } 2. Down-and-Out/Down-and-In Call Options. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ , ℙ) and let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. We consider a European down-and-out call option at time 𝑡, with expiry at time 𝑇 > 𝑡, strike 𝐾 and constant barrier 𝐵. Explain why, for a down-and-out European call, 𝑆𝑡 > 𝐵, and show that the terminal payoff for 𝐵 ≤ 𝐾 or 𝐵 > 𝐾 can be written as { } Ψ(𝑆𝑇 ) = max 𝑆𝑇 − 𝐾, 0 1I{ min 𝑆 > 𝐵} . 𝑢 𝑡≤𝑢≤𝑇

Using the identity 1

𝑈 𝑎𝑤− 1 ( √𝑤 )2 2 𝑇

𝑒 √ 2𝜋𝑇 ∫𝐿

[ ( 𝑑𝑤 = 𝑒

1 2 𝑎 𝑇 2

Φ

𝑈 − 𝑎𝑇 √ 𝑇

)

( −Φ

𝐿 − 𝑎𝑇 √ 𝑇

)]

4.2.1 Probabilistic Approach

365

𝑥

1 − 1 𝑢2 √ 𝑒 2 𝑑𝑢 is the cdf of a standard normal, show that under the ∫−∞ 2𝜋 risk-neutral measure ℚ, the down-and-out call option price at time 𝑡 is where Φ(𝑥) =

𝐶𝑑∕𝑜

(

(

) 𝑆𝑡 2𝛼 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝑍, 𝑇 + (𝑍 − 𝐾) 𝐶𝑑 𝑆𝑡 , 𝑡; 𝑍, 𝑇 − 𝐵 [ ) )] ( 2 ( 2 𝐵 𝐵 × 𝐶𝑏𝑠 , 𝑡; 𝑍, 𝑇 + (𝑍 − 𝐾) 𝐶𝑑 , 𝑡; 𝑍, 𝑇 𝑆𝑡 𝑆𝑡 (

)

( where 𝑍 = max{𝐵, 𝐾}, 𝛼 =

(

)

1 2

1−

𝑟−𝐷 1 2 𝜎 2

)

) , 𝑟 is the risk-free interest rate,

𝐶𝑏𝑠 (𝑋, 𝑡; 𝑌 , 𝑇 ) and 𝐶𝑑 (𝑋, 𝑡; 𝑌 , 𝑇 ) are the vanilla and digital call option prices defined as ) log(𝑋∕𝑌 ) + (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) 𝐶𝑏𝑠 (𝑋, 𝑡; 𝑌 , 𝑇 ) = 𝑋𝑒 Φ √ 𝜎 𝑇 −𝑡 ( ) log(𝑋∕𝑌 ) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) −𝑟(𝑇 −𝑡) −𝑌 𝑒 Φ √ 𝜎 𝑇 −𝑡 (

−𝐷(𝑇 −𝑡)

and ( 𝐶𝑑 (𝑋, 𝑡; 𝑌 , 𝑇 ) = 𝑒

−𝑟(𝑇 −𝑡)

Φ

log(𝑋∕𝑌 ) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

)

respectively. Hence, deduce that the European down-and-in call option price at time 𝑡 is ( ( ) ) 𝐶𝑑∕𝑖 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇

( )2𝛼 ( ( ) ) 𝑆𝑡 −𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝑍, 𝑇 − (𝑍 − 𝐾) 𝐶𝑑 𝑆𝑡 , 𝑡; 𝑍, 𝑇 + 𝐵 [ ) )] ( 2 ( 2 𝐵 𝐵 × 𝐶𝑏𝑠 , 𝑡; 𝑍, 𝑇 + (𝑍 − 𝐾) 𝐶𝑑 , 𝑡; 𝑍, 𝑇 . 𝑆𝑡 𝑆𝑡

Solution: We consider two cases for the down-and-out call option price when 𝐵 ≤ 𝐾 and 𝐵 > 𝐾. We require 𝑆𝑡 > 𝐵 for all 𝑡 so as to ensure the option would not knock out at the starting time 𝑡. Hence, for either 𝐵 ≤ 𝐾 or 𝐵 > 𝐾, the payoff of a down-and-out call option is { } Ψ(𝑆𝑇 ) = max 𝑆𝑇 − 𝐾, 0 1I{ min 𝑆 > 𝐵} . 𝑢 𝑡≤𝑢≤𝑇

Under the risk-neutral measure ℚ, 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡

366

4.2.1 Probabilistic Approach

where 𝑊𝑡ℚ = 𝑊𝑡 + 𝑡 𝐵} 𝑢 𝑡≤𝑢≤𝑇 { } ̂ = max 𝑆𝑡 𝑒𝜎 𝑊𝑇 −𝑡 − 𝐾, 0 1I{𝑆 𝑒𝜎𝑚𝑇 −𝑡 >𝐵} ⎧ ( 𝜎𝑊 ̂ ⎪ 𝑆𝑡 𝑒 𝑇 −𝑡 ⎪ = ⎨( ̂𝑇 −𝑡 ⎪ 𝑆 𝑒𝜎 𝑊 ⎪ 𝑡 ⎩ ⎧ ( 𝜎𝑊 ̂ ⎪ 𝑆𝑡 𝑒 𝑇 −𝑡 ⎪ = ⎨( ̂𝑇 −𝑡 ⎪ 𝑆 𝑒𝜎 𝑊 ⎪ 𝑡 ⎩

𝑡

)

− 𝐾 1I{ ) − 𝐾 1I{ ) − 𝐾 1I{

}

if 𝐵 ≤ 𝐾

}

if 𝐵 > 𝐾

𝜎𝑚

̂ 𝑇 −𝑡 >𝐵,𝑆𝑡 𝑒𝜎 𝑊𝑇 −𝑡 >𝐾

𝜎𝑚

̂ 𝑇 −𝑡 >𝐵,𝑆𝑡 𝑒𝜎 𝑊𝑇 −𝑡 >𝐵

𝑆𝑡 𝑒

𝑆𝑡 𝑒

( ) ( )} ̂𝑇 −𝑡 > 1 log 𝐾 𝑚𝑇 −𝑡 > 𝜎1 log 𝑆𝐵 ,𝑊 𝜎 𝑆

if 𝐵 ≤ 𝐾

( ) ( )} ̂𝑇 −𝑡 > 1 log 𝐵 𝑚𝑇 −𝑡 > 𝜎1 log 𝑆𝐵 ,𝑊 𝜎 𝑆𝑡 𝑡

if 𝐵 > 𝐾.

𝑡

) − 𝐾 1I{

𝑡

Case 1: 𝐵 ≤ 𝐾 The down-and-out call option price at time 𝑡 is ( ) 𝐶𝑑∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 [ ] = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ Ψ(𝑆𝑇 )|| ℱ𝑡 [ ( ) ̂ −𝑟(𝑇 −𝑡) ℚ 𝑆𝑡 𝑒𝜎 𝑊𝑇 −𝑡 − 𝐾 1I{ 𝔼 =𝑒 = 𝑒−𝑟(𝑇 −𝑡)

𝑤=∞

𝑚=𝑤

( ) ( ) ∫𝑤= 1 log 𝐾 ∫𝑚= 1 log 𝐵 𝜎 𝑆 𝜎 𝑆 𝑡

𝑡

|

( ) ( )} || ℱ ̂𝑇 −𝑡 > 1 log 𝐾 𝑚𝑇 −𝑡 > 𝜎1 log 𝑆𝐵 ,𝑊 | 𝑡 𝜎 𝑆𝑡 𝑡

]

|

( 𝜎𝑤 ) 𝑆𝑡 𝑒 − 𝐾 𝑓 ℚ ̂ (𝑚, 𝑤) 𝑑𝑚𝑑𝑤 𝑚,𝑊

4.2.1 Probabilistic Approach

367

where ( )2 ⎧ √ ⎪ −2(2𝑚−𝑤) 𝜈𝑤− 21 𝜈 2 (𝑇 −𝑡)− 21 2𝑚−𝑤 𝑇 −𝑡 √ 𝑒 ⎪ 𝑓 ℚ ̂ (𝑚, 𝑤) = ⎨ (𝑇 −𝑡) 2𝜋(𝑇 −𝑡) 𝑚,𝑊 ⎪ ⎪0 ⎩

𝑚 ≤ 0, 𝑚 ≤ 𝑤 otherwise.

(See Problem 4.2.2.14 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus for the derivation of 𝑓 ℚ ̂ (𝑚, 𝑤).) 𝑚,𝑊

Thus, ( ) 𝐶𝑑∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = −𝑒−𝑟(𝑇 −𝑡)

𝑤=∞

∫

(

𝑤= 𝜎1 log 𝑆𝐾 𝑡

)

(

𝑆𝑡 𝑒𝜎𝑤 − 𝐾 (

𝜈𝑤− 21 𝜈 2 (𝑇 −𝑡)− 12

1

)

2𝑚−𝑤 √ 𝑇 −𝑡

𝑒 × √ 2𝜋(𝑇 − 𝑡)

)2 |𝑚=𝑤

| | 𝑑𝑤 | | | 1 (𝐵) |𝑚= 𝜎 log 𝑆 𝑡

= 𝑒−𝑟(𝑇 −𝑡)

𝑤=∞ (

∫ 1 log 𝜎

−𝑒−𝑟(𝑇 −𝑡)

𝐾 𝑆𝑡

)

(

1

(

𝜈𝑤− 21 𝜈 2 (𝑇 −𝑡)− 21

𝑆𝑡 𝑒𝜎𝑤 − 𝐾 √ 𝑒 2𝜋(𝑇 − 𝑡)

𝑤=∞ ( ) ∫ 1 log 𝐾 𝜎 𝑆 𝑡

)

𝑤 √ 𝑇 −𝑡

)2

𝑑𝑤

( ) 2 ⎛ 2 log 𝑆𝐵 −𝑤 ⎞ 𝑡 1 2 1⎜ 𝜎 ⎟ √ 𝜈𝑤− 2 𝜈 (𝑇 −𝑡)− 2 ⎜ ⎟ 𝑇 −𝑡

(

) 1 𝑆𝑡 𝑒𝜎𝑤 − 𝐾 √ 𝑒 2𝜋(𝑇 − 𝑡)

⎝

⎠

= 𝑆𝑡 𝐼1 − 𝐾𝐼2 − (𝑆𝑡 𝐼3 − 𝐾𝐼4 ) where

(

𝑤=∞

1 𝐼1 = √ ( )𝑒 2𝜋(𝑇 − 𝑡) ∫ 𝜎1 log 𝑆𝐾𝑡 𝑤=∞

−𝑟(𝑇 −𝑡)+𝜎𝑤+𝜈𝑤− 21 𝜈 2 (𝑇 −𝑡)− 12

(

1 𝐼2 = √ ( )𝑒 2𝜋(𝑇 − 𝑡) ∫ 𝜎1 log 𝑆𝐾𝑡

−𝑟(𝑇 −𝑡)+𝜈𝑤− 21 𝜈 2 (𝑇 −𝑡)− 21

√

𝑤 𝑇 −𝑡

𝑤 √ 𝑇 −𝑡

(

𝑤=∞

1 𝐼3 = √ ( ) 2𝜋(𝑇 − 𝑡) ∫ 𝜎1 log 𝑆𝐾𝑡

𝐼4 = √

1

𝑤=∞ (

2𝜋(𝑇 − 𝑡) ∫ 𝜎1 log

𝐾 𝑆𝑡

)2

𝑑𝑤

)2

𝑑𝑤 )

2 ⎛ 𝜎2 log 𝑆𝐵 −𝑤 ⎞ 𝑡 ⎟ √ −𝑟(𝑇 −𝑡)+𝜎𝑤+𝜈𝑤− 21 𝜈 2 (𝑇 −𝑡)− 12 ⎜ ⎜ ⎟ 𝑇 −𝑡 ⎝ ⎠ 𝑑𝑤 𝑒

)𝑒

( ) 2 ⎛ 2 log 𝑆𝐵 −𝑤 ⎞ 𝑡 1 2 1⎜ 𝜎 ⎟ √ −𝑟(𝑇 −𝑡)+𝜈𝑤− 2 𝜈 (𝑇 −𝑡)− 2 ⎜ ⎟ 𝑇 −𝑡

⎝

⎠

𝑑𝑤.

𝑑𝑤

368

4.2.1 Probabilistic Approach

Using the identity 𝑈 𝑎𝑤− 1 ( √𝑤 )2 2 𝑇

1

𝑒 √ 2𝜋𝑇 ∫𝐿

[ ( 𝑑𝑤 = 𝑒

1 2 𝑎 𝑇 2

Φ

𝑈 − 𝑎𝑇 √ 𝑇

)

( −Φ

𝐿 − 𝑎𝑇 √ 𝑇

)]

and following the same steps as described in Problem 4.2.1.1 (page 357), we have ⎡

⎛ 1 log(𝐾∕𝑆 ) + (𝜈 + 𝜎)(𝑇 − 𝑡) ⎞⎤ 𝑡 ⎜𝜎 ⎟⎥ 𝐼1 = 𝑒 √ ⎢1 − Φ ⎜ ⎟⎥ 𝑇 −𝑡 ⎢ ⎜ ⎟⎥ ⎣ ⎝ ⎠⎦ ) ( 1 2 log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 + 2 𝜎 )(𝑇 − 𝑡) = 𝑒−𝐷(𝑇 −𝑡) Φ √ 𝜎 𝑇 −𝑡 −𝐷(𝑇 −𝑡) ⎢

⎛ 1 log(𝐾∕𝑆 ) − 𝜈(𝑇 − 𝑡) ⎞⎤ ⎡ 𝑡 ⎜ ⎢ ⎟⎥ 𝐼2 = 𝑒−𝑟(𝑇 −𝑡) ⎢1 − Φ ⎜ 𝜎 √ ⎟⎥ 𝑇 −𝑡 ⎜ ⎢ ⎟⎥ ⎣ ⎝ ⎠⎦ ( ) 1 2 log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 − 2 𝜎 )(𝑇 − 𝑡) = 𝑒−𝑟(𝑇 −𝑡) Φ √ 𝜎 𝑇 −𝑡 ( 𝐼3 = 𝑒

−𝐷(𝑇 −𝑡)

𝑆𝑡 𝐵

)−1− 2(𝑟−𝐷) 𝜎2

( ) ⎛1 ⎡ ⎞⎤ 2 ⎜ 𝜎 log(𝐾∕𝑆𝑡 ) − 𝜈 + 𝜎 + 𝜎(𝑇 − 𝑡) log(𝐵∕𝑆𝑡 ) (𝑇 − 𝑡) ⎟⎥ ⎢ ⎟⎥ × ⎢1 − Φ ⎜ √ ⎜ ⎢ ⎟⎥ 𝑇 − 𝑡 ⎜ ⎢ ⎟⎥ ⎣ ⎝ ⎠⎦ ( )−1− 2(𝑟−𝐷) ( log(𝐵 2 ∕(𝑆 𝐾)) + (𝑟 − 𝐷 + 1 𝜎 2 )(𝑇 − 𝑡) ) 𝜎2 𝑆𝑡 𝑡 2 = 𝑒−𝐷(𝑇 −𝑡) Φ √ 𝐵 𝜎 𝑇 −𝑡 ( 𝐼4 = 𝑒

−𝑟(𝑇 −𝑡)

𝑆𝑡 𝐵

)1− 2(𝑟−𝐷) 𝜎2

( ) ⎞⎤ ⎡ ⎛1 2 ) − 𝜈 + ) (𝑇 − 𝑡) ⎟⎥ log(𝐾∕𝑆 log(𝐵∕𝑆 𝑡 𝑡 ⎢ ⎜𝜎 𝜎(𝑇 − 𝑡) ⎟⎥ × ⎢1 − Φ ⎜ √ ⎟⎥ ⎢ ⎜ 𝑇 −𝑡 ⎟⎥ ⎢ ⎜ ⎣ ⎠⎦ ⎝ ( )1− 2(𝑟−𝐷) ( log(𝐵 2 ∕(𝑆 𝐾)) + (𝑟 − 𝐷 − 1 𝜎 2 )(𝑇 − 𝑡) ) 𝜎2 𝑆𝑡 𝑡 2 . Φ = 𝑒−𝑟(𝑇 −𝑡) √ 𝐵 𝜎 𝑇 −𝑡

4.2.1 Probabilistic Approach

369

Therefore, ( ) 𝐶𝑑∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝑆𝑡 𝐼1 − 𝐾𝐼2 − (𝑆𝑡 𝐼3 − 𝐾𝐼4 ) ) log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) = 𝑆𝑡 √ 𝜎 𝑇 −𝑡 ( ) log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) −𝑟(𝑇 −𝑡) −𝐾𝑒 Φ √ 𝜎 𝑇 −𝑡 (

𝑒−𝐷(𝑇 −𝑡) Φ

( −𝑆𝑡

𝑒−𝐷(𝑇 −𝑡) (

−𝐾𝑒−𝑟(𝑇 −𝑡) (

𝑆𝑡 𝐵

𝑆𝑡 𝐵

𝜎2

)

( 1−

Φ (

)1− 2(𝑟−𝐷) 𝜎2

(

= 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 −

1 where 𝛼 = 2

(

)−1− 2(𝑟−𝐷)

𝑟−𝐷

𝑆𝑡 𝐵

Φ

log(𝐵 2 ∕(𝑆𝑡 𝐾)) + (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

log(𝐵 2 ∕(𝑆𝑡 𝐾)) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

)2𝛼

( 𝐶𝑏𝑠

𝐵2 , 𝑡; 𝐾, 𝑇 𝑆𝑡

)

)

)

1 2 𝜎 2

.

Case 2: 𝐵 > 𝐾 The down-and-out call option price at time 𝑡 is ( ) 𝐶𝑑∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 [ ] = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ Ψ(𝑆𝑇 )|| ℱ𝑡 [ ( ) ̂ 𝑆𝑡 𝑒𝜎 𝑊𝑇 −𝑡 − 𝐾 1I{ = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ

| ( ) ( )} || ℱ 1 𝐵 1 𝐵 ̂𝑇 −𝑡 > log 𝑚𝑇 −𝑡 > 𝜎 log 𝑆 ,𝑊 | 𝑡 𝜎 𝑆𝑡 𝑡 |

= 𝑒−𝑟(𝑇 −𝑡)

𝑤=∞

𝑚=𝑤

( ) ( ) ∫𝑤= 1 log 𝐵 ∫𝑚= 1 log 𝐵 𝜎 𝑆 𝜎 𝑆 𝑡

𝑡

]

( 𝜎𝑤 ) 𝑆𝑡 𝑒 − 𝐾 𝑓 ℚ ̂ (𝑚, 𝑤) 𝑑𝑚𝑑𝑤 𝑚,𝑊

)

370

4.2.1 Probabilistic Approach

where ( )2 ⎧ √ ⎪ −2(2𝑚−𝑤) 𝜈𝑤− 21 𝜈 2 (𝑇 −𝑡)− 12 2𝑚−𝑤 𝑇 −𝑡 √ 𝑒 ⎪ 𝑓 ℚ ̂ (𝑚, 𝑤) = ⎨ (𝑇 −𝑡) 2𝜋(𝑇 −𝑡) 𝑚,𝑊 ⎪ ⎪0 ⎩

𝑚 ≤ 0, 𝑚 ≤ 𝑤 otherwise.

Using the same steps as described in Case 1, we have ( ) 𝐶𝑑∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝑆𝑡 𝐼1 − 𝐾𝐼2 − (𝑆𝑡 𝐼3 − 𝐾𝐼4 ) where ⎡

⎛ 1 log(𝐵∕𝑆 ) + (𝜈 + 𝜎)(𝑇 − 𝑡) ⎞⎤ 𝑡 ⎜𝜎 ⎟⎥ 𝐼1 = 𝑒 √ ⎢1 − Φ ⎜ ⎟⎥ 𝑇 −𝑡 ⎢ ⎜ ⎟⎥ ⎣ ⎝ ⎠⎦ ( ) 1 2 log(𝑆𝑡 ∕𝐵) + (𝑟 − 𝐷 + 2 𝜎 )(𝑇 − 𝑡) = 𝑒−𝐷(𝑇 −𝑡) Φ √ 𝜎 𝑇 −𝑡 −𝐷(𝑇 −𝑡) ⎢

⎡ ⎛ 1 log(𝐵∕𝑆 ) − 𝜈(𝑇 − 𝑡) ⎞⎤ 𝑡 ⎢ ⎜ ⎟⎥ 𝐼2 = 𝑒−𝑟(𝑇 −𝑡) ⎢1 − Φ ⎜ 𝜎 √ ⎟⎥ 𝑇 −𝑡 ⎢ ⎜ ⎟⎥ ⎣ ⎝ ⎠⎦ ) ( 1 2 log(𝑆𝑡 ∕𝐵) + (𝑟 − 𝐷 − 2 𝜎 )(𝑇 − 𝑡) −𝑟(𝑇 −𝑡) Φ =𝑒 √ 𝜎 𝑇 −𝑡

( 𝐼3 = 𝑒

−𝐷(𝑇 −𝑡)

𝑆𝑡 𝐵

)−1− 2(𝑟−𝐷) 𝜎2

( ) ⎛1 ⎡ ⎞⎤ 2 ) − 𝜈 + 𝜎 + ) (𝑇 − 𝑡) ⎟⎥ log(𝐵∕𝑆 log(𝐵∕𝑆 𝑡 𝑡 ⎜𝜎 ⎢ 𝜎(𝑇 − 𝑡) ⎟⎥ × ⎢1 − Φ ⎜ √ ⎜ ⎢ ⎟⎥ 𝑇 − 𝑡 ⎜ ⎢ ⎟⎥ ⎣ ⎝ ⎠⎦ ( )−1− 2(𝑟−𝐷) ( log(𝐵∕𝑆 ) + (𝑟 − 𝐷 + 1 𝜎 2 )(𝑇 − 𝑡) ) 𝜎2 𝑆𝑡 𝑡 2 = 𝑒−𝐷(𝑇 −𝑡) Φ √ 𝐵 𝜎 𝑇 −𝑡

4.2.1 Probabilistic Approach

371

( 𝐼4 = 𝑒

−𝑟(𝑇 −𝑡)

𝑆𝑡 𝐵

)1− 2(𝑟−𝐷) 𝜎2

( ) ⎛1 ⎡ ⎞⎤ 2 ) − 𝜈 + ) (𝑇 − 𝑡) ⎟⎥ log(𝐵∕𝑆 log(𝐵∕𝑆 𝑡 𝑡 ⎜𝜎 ⎢ 𝜎(𝑇 − 𝑡) ⎟⎥ × ⎢1 − Φ ⎜ √ ⎜ ⎢ ⎟⎥ 𝑇 − 𝑡 ⎜ ⎢ ⎟⎥ ⎣ ⎝ ⎠⎦ ( )1− 2(𝑟−𝐷) ( log(𝐵∕𝑆 ) + (𝑟 − 𝐷 − 1 𝜎 2 )(𝑇 − 𝑡) ) 𝜎2 𝑆𝑡 𝑡 2 = 𝑒−𝑟(𝑇 −𝑡) . Φ √ 𝐵 𝜎 𝑇 −𝑡

Therefore,

𝐶𝑑∕𝑜

(

( )2𝛼 ( ( ) ) 𝑆𝑡 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐵, 𝑇 + (𝐵 − 𝐾) 𝐶𝑑 𝑆𝑡 , 𝑡; 𝐵, 𝑇 − 𝐵 [ ) )] ( 2 ( 2 𝐵 𝐵 × 𝐶𝑏𝑠 , 𝑡; 𝐵, 𝑇 + (𝐵 − 𝐾) 𝐶𝑑 , 𝑡; 𝐵, 𝑇 𝑆𝑡 𝑆𝑡 )

( where 𝛼 =

1 2

1−

𝑟−𝐷 1 2 𝜎 2

) .

Hence, by setting 𝑍 = max{𝐵, 𝐾} we can write ( )2𝛼 ( ( ( ) ) ) 𝑆𝑡 𝐶𝑑∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝑍, 𝑇 + (𝑍 − 𝐾) 𝐶𝑑 𝑆𝑡 , 𝑡; 𝑍, 𝑇 − 𝐵 [ ) )] ( 2 ( 2 𝐵 𝐵 × 𝐶𝑏𝑠 , 𝑡; 𝑍, 𝑇 + (𝑍 − 𝐾) 𝐶𝑑 , 𝑡; 𝑍, 𝑇 . 𝑆𝑡 𝑆𝑡 For the case of the down-and-in call option price, by definition its payoff at expiry time 𝑇 can be written as { } Ψ(𝑆𝑇 ) = max 𝑆𝑇 − 𝐾, 0 1I{ min 𝑆 ≤ 𝐵} 𝑢 𝑡≤𝑢≤𝑇 ) ( } { = max 𝑆𝑇 − 𝐾, 0 1 − 1I{ min 𝑆 > 𝐵} 𝑢 𝑡≤𝑢≤𝑇 } { } { = max 𝑆𝑇 − 𝐾, 0 − max 𝑆𝑇 − 𝐾, 0 1I{ min 𝑆 > 𝐵} . 𝑢 𝑡≤𝑢≤𝑇

372

4.2.1 Probabilistic Approach

Under the risk-neutral measure ℚ, the down-and-in call option price at time 𝑡 is ( ) 𝐶𝑑∕𝑖 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 [ { }| ] = 𝑒−𝑟(𝑇 −𝑡) 𝔼 max 𝑆𝑇 − 𝐾, 0 | ℱ𝑡 | [ ] | { } | −𝑒−𝑟(𝑇 −𝑡) 𝔼 max 𝑆𝑇 − 𝐾, 0 1I{ min 𝑆 > 𝐵} | ℱ𝑡 𝑢 | 𝑡≤𝑢≤𝑇 | ( ( ) ) = 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 − 𝐶𝑑∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) ( 2 ⎧ ( 𝑆 )2𝛼 𝐵 𝑡 ⎪ 𝐶𝑏𝑠 , 𝑡; 𝐾, 𝑇 𝑆𝑡 ⎪ 𝐵 ⎪ ( ( ( ) ) ) =⎨ 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 − 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐵, 𝑇 − (𝐵 − 𝐾) 𝐶𝑑 𝑆𝑡 , 𝑡; 𝐵, 𝑇 ⎪ ( )2𝛼 [ ) )] ( 2 ( 2 𝑆𝑡 ⎪ 𝐵 𝐵 (𝐵 , 𝑡; 𝐵, 𝑇 + − 𝐾) 𝐶 , 𝑡; 𝐵, 𝑇 𝐶 + 𝑏𝑠 𝑑 ⎪ 𝐵 𝑆𝑡 𝑆𝑡 ⎩

if 𝐵 ≤ 𝐾

if 𝐵 > 𝐾

( ( ( ) ) ) = 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 − 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝑍, 𝑇 − (𝑍 − 𝐾) 𝐶𝑑 𝑆𝑡 , 𝑡; 𝑍, 𝑇 ( +

𝑆𝑡 𝐵

)2𝛼 [

( 𝐶𝑏𝑠

𝐵2 , 𝑡; 𝑍, 𝑇 𝑆𝑡

)

( + (𝑍 − 𝐾) 𝐶𝑑

𝐵2 , 𝑡; 𝑍, 𝑇 𝑆𝑡

)]

where 𝑍 = max{𝐵, 𝐾}.

{ } 3. Up-and-Out/Up-and-In Put Options. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. We consider a European up-and-out put option at time 𝑡, with expiry at time 𝑇 > 𝑡, strike 𝐾 and constant barrier 𝐵. Explain why, for an up-and-out European put, 𝑆𝑡 < 𝐵, and show that the terminal payoff for 𝐵 ≥ 𝐾 or 𝐵 < 𝐾 can be written as { } Ψ(𝑆𝑇 ) = max 𝐾 − 𝑆𝑇 , 0 1I{ max 𝑆 < 𝐵} . 𝑢 𝑡≤𝑢≤𝑇

4.2.1 Probabilistic Approach

373

Using the identity 1

𝑈 𝑎𝑤− 1 ( √𝑤 )2 2 𝑇

𝑒 √ 2𝜋𝑇 ∫𝐿

[ ( 𝑑𝑤 = 𝑒

1 2 𝑎 𝑇 2

Φ

𝑈 − 𝑎𝑇 √ 𝑇

)

( −Φ

𝐿 − 𝑎𝑇 √ 𝑇

)]

𝑥

1 − 1 𝑢2 √ 𝑒 2 𝑑𝑢 is the cdf of a standard normal, show that under the ∫−∞ 2𝜋 risk-neutral measure ℚ, the up-and-out put option price at time 𝑡 is where Φ(𝑥) =

𝑃𝑢∕𝑜

(

(

) 𝑆𝑡 2𝛼 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝑃𝑏𝑠 𝑆𝑡 , 𝑡; 𝑍, 𝑇 + (𝐾 − 𝑍) 𝑃𝑑 𝑆𝑡 , 𝑡; 𝑍, 𝑇 − 𝐵 [ ( 2 ) )] ( 2 𝐵 𝐵 × 𝑃𝑏𝑠 , 𝑡; 𝑍, 𝑇 + (𝐾 − 𝑍) 𝑃𝑑 , 𝑡; 𝑍, 𝑇 𝑆𝑡 𝑆𝑡 (

)

( where 𝑍 = min{𝐵, 𝐾}, 𝛼 =

(

)

1 2

1−

𝑟−𝐷 1 2 𝜎 2

)

) , 𝑟 is the risk-free interest rate,

𝑃𝑏𝑠 (𝑋, 𝑡; 𝑌 , 𝑇 ) and 𝑃𝑑 (𝑋, 𝑡; 𝑌 , 𝑇 ) are the vanilla and digital put option prices defined as ) −log(𝑋∕𝑌 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) Φ 𝑃𝑏𝑠 (𝑋, 𝑡; 𝑌 , 𝑇 ) = 𝑌 𝑒 √ 𝜎 𝑇 −𝑡 ( ) −log(𝑋∕𝑌 ) − (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) −𝐷(𝑇 −𝑡) −𝑋𝑒 Φ √ 𝜎 𝑇 −𝑡 (

−𝑟(𝑇 −𝑡)

and ( 𝑃𝑑 (𝑋, 𝑡; 𝑌 , 𝑇 ) = 𝑒

−𝑟(𝑇 −𝑡)

Φ

−log(𝑋∕𝑌 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

)

respectively. Hence, deduce that the European up-and-in put option price at time 𝑡 is ( ( ) ) 𝑃𝑢∕𝑖 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝑃𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇

( )2𝛼 ( ( ) ) 𝑆𝑡 −𝑃𝑏𝑠 𝑆𝑡 , 𝑡; 𝑍, 𝑇 − (𝐾 − 𝑍) 𝑃𝑑 𝑆𝑡 , 𝑡; 𝑍, 𝑇 + 𝐵 [ ( 2 ) )] ( 2 𝐵 𝐵 × 𝑃𝑏𝑠 , 𝑡; 𝑍, 𝑇 + (𝐾 − 𝑍) 𝑃𝑑 , 𝑡; 𝑍, 𝑇 . 𝑆𝑡 𝑆𝑡

Solution: We require 𝑆𝑡 < 𝐵 so as to ensure the up-and-out put option would not knock out, especially at initiation of the contract.

374

4.2.1 Probabilistic Approach

For 𝐵 ≥ 𝐾 or 𝐵 < 𝐾, the payoff of an up-and-out put option is therefore { } Ψ(𝑆𝑇 ) = max 𝐾 − 𝑆𝑇 , 0 1I{ max 𝑆 < 𝐵} . 𝑢 𝑡≤𝑢≤𝑇

Under the risk-neutral measure ℚ, 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡 where 𝑊𝑡ℚ = 𝑊𝑡 + 𝑡 𝐵} 𝑢 𝑡≤𝑢≤𝑇 { } ̂ = max 𝐾 − 𝑆𝑡 𝑒𝜎 𝑊𝑇 −𝑡 , 0 1I{𝑆 𝑒𝜎𝑚𝑇 −𝑡 >𝐵} 𝑡 ) ( ̂𝑇 −𝑡 𝜎𝑊 { } 1I = 𝐾 − 𝑆𝑡 𝑒 𝜎𝑚 ̂ 𝑆𝑡 𝑒 𝑇 −𝑡 >𝐵,𝑆𝑡 𝑒𝜎 𝑊𝑇 −𝑡 𝜎 log 𝑆 ,𝑊𝑇 −𝑡 < 𝜎 log 𝑆𝐾 𝑡 𝑡

4.2.1 Probabilistic Approach

383

Hence, the down-and-out put option price at time 𝑡 is ( ) 𝑃𝑑∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 [ ] = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ Ψ(𝑆𝑇 )|| ℱ𝑡 [ ( ) ̂ −𝑟(𝑇 −𝑡) ℚ 𝐾 − 𝑆𝑡 𝑒𝜎 𝑊𝑇 −𝑡 1I{ 𝔼 =𝑒

|

( ) ( )} || ℱ ̂𝑇 −𝑡 < 1 log 𝐾 𝑚𝑇 −𝑡 > 𝜎1 log 𝑆𝐵 ,𝑊 | 𝑡 𝜎 𝑆𝑡 𝑡

= 𝑒−𝑟(𝑇 −𝑡)

( ) 𝑤= 𝜎1 log 𝑆𝐾 𝑡

|

𝑚=𝑤

𝑡

( ) ∫𝑤= 1 log 𝐵 𝜎 𝑆

(

∫𝑚= 1 log 𝜎

𝐵 𝑆𝑡

]

)

( ) 𝐾 − 𝑆𝑡 𝑒𝜎𝑤 𝑓 ℚ ̂ (𝑚, 𝑤) 𝑑𝑚𝑑𝑤 𝑚,𝑊

where ( )2 ⎧ √ ⎪ −2(2𝑚−𝑤) 𝜈𝑤− 12 𝜈 2 (𝑇 −𝑡)− 21 2𝑚−𝑤 𝑇 −𝑡 √ 𝑒 ⎪ 𝑓 ℚ ̂ (𝑚, 𝑤) = ⎨ (𝑇 −𝑡) 2𝜋(𝑇 −𝑡) 𝑚,𝑊 ⎪ ⎪0 ⎩

𝑚 ≤ 0, 𝑚 ≤ 𝑤 otherwise.

See Problem 4.2.2.14 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus to derive the expression 𝑓 ℚ ̂ (𝑚, 𝑤). 𝑀,𝑊

By switching the limits of the inner integral ( ) 𝑃𝑢∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝑒−𝑟(𝑇 −𝑡)

( ) 𝑤= 𝜎1 log 𝑆𝐾

∫

= −𝑒−𝑟(𝑇 −𝑡)

𝑡

(

𝑤= 𝜎1 log 𝑆𝐵 𝑡

𝑡

∫𝑚=𝑤

( ) 𝑤= 𝜎1 log 𝑆𝐾 ( (

∫𝑤= 1 log 𝜎

+𝑒−𝑟(𝑇 −𝑡)

)

( ) 𝑤= 𝜎1 log 𝑆𝐵 (

𝑡

𝐵 𝑆𝑡

)

) 𝑆𝑡 𝑒𝜎𝑤 − 𝐾 𝑓 ℚ ̂ (𝑚, 𝑤) 𝑑𝑚𝑑𝑤 𝑚,𝑊

)

(

𝜎

𝑡

𝐵 𝑆𝑡

)

𝜈𝑤− 21 𝜈 2 (𝑇 −𝑡)− 21

√

𝑆𝑡 𝑒𝜎𝑤 − 𝐾 √ 𝑒 2𝜋(𝑇 − 𝑡)

𝑤 𝑇 −𝑡

)2

𝑑𝑤

( ) 2 ⎛ 2 log 𝑆𝐵 −𝑤 ⎞ 𝑡 1 2 1⎜ 𝜎 ⎟ √ 𝜈𝑤− 2 𝜈 (𝑇 −𝑡)− 2 ⎜ ⎟ 𝑇 −𝑡

( ) 𝑤= 𝜎1 log 𝑆𝐾 (

∫𝑤= 1 log

(

1

) 1 𝑆𝑡 𝑒𝜎𝑤 − 𝐾 √ 𝑒 2𝜋(𝑇 − 𝑡)

⎝

⎠

= −𝑆𝑡 𝐼1 + 𝐾𝐼2 + (𝑆𝑡 𝐼3 − 𝐾𝐼4 ) where (

1

1 log 𝑆𝐾 𝜎 𝑡

)

𝐼1 = √ ( ) 𝑒 2𝜋(𝑇 − 𝑡) ∫ 𝜎1 log 𝑆𝐵𝑡

( −𝑟(𝑇 −𝑡)+𝜎𝑤+𝜈𝑤− 21 𝜈 2 (𝑇 −𝑡)− 21

𝑤 √ 𝑇 −𝑡

)2

𝑑𝑤

𝑑𝑤

384

4.2.1 Probabilistic Approach (

1 log 𝑆𝐾 𝜎 𝑡

)

1 𝐼2 = √ ( ) 𝑒 2𝜋(𝑇 − 𝑡) ∫ 𝜎1 log 𝑆𝐵𝑡

( −𝑟(𝑇 −𝑡)+𝜈𝑤− 21 𝜈 2 (𝑇 −𝑡)− 21

𝑤 √ 𝑇 −𝑡

)2

𝑑𝑤

(

𝐼3 = √

1 2𝜋(𝑇 − 𝑡)

)

2 ⎛ 2 log 𝑆𝐵 −𝑤 ⎞ ( ) 𝑡 1 2 1⎜ 𝜎 ⎟ 1 𝐾 √ −𝑟(𝑇 −𝑡)+𝜎𝑤+𝜈𝑤− 2 𝜈 (𝑇 −𝑡)− 2 log 𝑆 ⎜ ⎟ 𝑇 −𝑡 𝜎 𝑡 ⎝ ⎠ 𝑑𝑤 ( ) 𝑒 ∫ 1 log 𝐵 𝜎 𝑆 𝑡

(

1 𝐼4 = √ 2𝜋(𝑇 − 𝑡)

)

2 ⎛ 𝜎2 log 𝑆𝐵 −𝑤 ⎞ ( ) 𝑡 1 1 ⎟ 1 √ −𝑟(𝑇 −𝑡)+𝜈𝑤− 2 𝜈 2 (𝑇 −𝑡)− 2 ⎜ log 𝑆𝐾 ⎜ ⎟ 𝑇 −𝑡 𝜎 𝑡 ⎝ ⎠ 𝑑𝑤. ( ) 𝑒 ∫ 1 log 𝐵 𝜎 𝑆 𝑡

Using the identity 𝑈 𝑎𝑤− 1 ( √𝑤 )2 2 𝑇

1

𝑒 √ 2𝜋𝑇 ∫𝐿

[ ( 𝑑𝑤 = 𝑒

1 2 𝑎 𝑇 2

Φ

𝑈 − 𝑎𝑇 √ 𝑇

)

( −Φ

𝐿 − 𝑎𝑇 √ 𝑇

)]

and knowing that 𝜈 = 𝜎1 (𝑟 − 𝐷 − 12 𝜎 2 ), we have [ (

)

[ (

)

− log(𝑆𝑡 ∕𝐾) − (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) 𝐼1 = 𝑒−𝐷(𝑇 −𝑡) Φ √ 𝜎 𝑇 −𝑡 ( )] − log(𝑆𝑡 ∕𝐵) − (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) −Φ √ 𝜎 𝑇 −𝑡 − log(𝑆𝑡 ∕𝐾) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) −𝑟(𝑇 −𝑡) 𝐼2 = 𝑒 Φ √ 𝜎 𝑇 −𝑡 ( )] − log(𝑆𝑡 ∕𝐵) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) −Φ √ 𝜎 𝑇 −𝑡

)−1− 2(𝑟−𝐷) [ ( − log(𝐵 2 ∕(𝑆 𝐾)) − (𝑟 − 𝐷 + 1 𝜎 2 )(𝑇 − 𝑡) ) 𝜎2 𝑡 2 Φ 𝐼3 = 𝑒−𝐷(𝑇 −𝑡) √ 𝜎 𝑇 −𝑡 ( )] − log(𝐵∕𝑆𝑡 ) − (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) −Φ √ 𝜎 𝑇 −𝑡 (

𝑆𝑡 𝐵

)1− 2(𝑟−𝐷) [ ( − log(𝐵 2 ∕(𝑆 𝐾)) − (𝑟 − 𝐷 − 1 𝜎 2 )(𝑇 − 𝑡) ) 𝜎2 𝑡 2 Φ 𝐼4 = 𝑒 √ 𝜎 𝑇 −𝑡 ( )] 1 2 − log(𝐵∕𝑆𝑡 ) − (𝑟 − 𝐷 − 2 𝜎 )(𝑇 − 𝑡) −Φ . √ 𝜎 𝑇 −𝑡 (

−𝑟(𝑇 −𝑡)

𝑆𝑡 𝐵

4.2.1 Probabilistic Approach

385

Therefore, ( ) 𝑃𝑑∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = −𝑆𝑡 𝐼1 + 𝐾𝐼2 + (𝑆𝑡 𝐼3 − 𝐾𝐼4 ) ( = −𝑆𝑡

𝑒−𝐷(𝑇 −𝑡) Φ

+𝑆𝑡

𝑒−𝐷(𝑇 −𝑡) Φ

(

( +𝐾𝑒−𝑟(𝑇 −𝑡) Φ ( −𝐾𝑒−𝑟(𝑇 −𝑡) Φ ( +𝑆𝑡 ( −𝑆𝑡 ( −𝐾 ( +𝐾

𝑆𝑡 𝐵 𝑆𝑡 𝐵 𝑆𝑡 𝐵 𝑆𝑡 𝐵

− log(𝑆𝑡 ∕𝐾) − (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡 − log(𝑆𝑡 ∕𝐵) − (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

− log(𝑆𝑡 ∕𝐾) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡 − log(𝑆𝑡 ∕𝐵) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡 (

)−1− 2(𝑟−𝐷) 𝜎2

𝑒

−𝐷(𝑇 −𝑡)

𝑒

−𝐷(𝑇 −𝑡)

Φ (

)−1− 2(𝑟−𝐷) 𝜎2

(

)1− 2(𝑟−𝐷) 𝜎2

𝑒

−𝑟(𝑇 −𝑡)

𝑒

−𝑟(𝑇 −𝑡)

Φ (

)1− 2(𝑟−𝐷) 𝜎2

(

)

Φ

Φ

)

)

)

− log(𝐵 2 ∕(𝑆𝑡 𝐾)) − (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡 − log(𝐵∕𝑆𝑡 ) − (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

− log(𝐵∕𝑆𝑡 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡 (

)

= 𝑃𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 − 𝑃𝑏𝑠 𝑆𝑡 , 𝑡; 𝐵, 𝑇 + (𝐾 − 𝐵)𝑃𝑑 𝑆𝑡 , 𝑡; 𝐵, 𝑇 { ×

( 𝑃𝑏𝑠

𝐵2 , 𝑡; 𝐾, 𝑇 𝑆𝑡

( where 𝛼 =

1 2

1−

𝑟−𝐷 1 2 𝜎 2

)

[ − 𝑃𝑏𝑠

) .

(

𝐵2 , 𝑡; 𝐵, 𝑇 𝑆𝑡

)

)]

( + (𝐾 − 𝐵)𝑃𝑑

( −

)

)

− log(𝐵 2 ∕(𝑆𝑡 𝐾)) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

(

[

)

)

)

𝑆𝑡 𝐵

)2𝛼

𝐵2 , 𝑡; 𝐵, 𝑇 𝑆𝑡

)]}

386

4.2.2 Reflection Principle Approach

For the case of the down-and-in put option price, by definition its payoff at expiry time 𝑇 can be written as { } Ψ(𝑆𝑇 ) = max 𝐾 − 𝑆𝑇 , 0 1I{ min 𝑆 ≤ 𝐵} 𝑢 𝑡≤𝑢≤𝑇 ) ( } { = max 𝐾 − 𝑆𝑇 , 0 1 − 1I{ min 𝑆 > 𝐵} 𝑢 𝑡≤𝑢≤𝑇 } { } { = max 𝐾 − 𝑆𝑇 , 0 − max 𝐾 − 𝑆𝑇 , 0 1I{ min 𝑆 > 𝐵} . 𝑢 𝑡≤𝑢≤𝑇

Under the risk-neutral measure ℚ, the down-and-in put option price at time 𝑡 is [ ( ) { }| ] 𝑃𝑑∕𝑖 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝑒−𝑟(𝑇 −𝑡) 𝔼 max 𝐾 − 𝑆𝑇 , 0 | ℱ𝑡 | [ ] | { } | −𝑟(𝑇 −𝑡) −𝑒 𝔼 max 𝐾 − 𝑆𝑇 , 0 1I{ min 𝑆 > 𝐵} | ℱ𝑡 𝑢 | 𝑡≤𝑢≤𝑇 | ( ( ) ) = 𝑃𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 − 𝑃𝑑∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝑇 ( ( ) ) = 𝑃𝑏𝑠 𝑆𝑡 , 𝑡; 𝐵, 𝑇 + (𝐾 − 𝐵)𝑃𝑑 𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) ( )2𝛼 { ( 2 𝑆𝑡 𝐵 , 𝑡; 𝐾, 𝑇 𝑃𝑏𝑠 + 𝐵 𝑆𝑡 [ ( 2 ) )]} ( 2 𝐵 𝐵 − 𝑃𝑏𝑠 , 𝑡; 𝐵, 𝑇 + (𝐾 − 𝐵)𝑃𝑑 , 𝑡; 𝐵, 𝑇 . 𝑆𝑡 𝑆𝑡

4.2.2

Reflection Principle Approach

1. Reflection Principle for Black–Scholes Equation. Assume we are in the Black–Scholes world where at time 𝑡, the asset price 𝑆𝑡 follows the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 where 𝜇, 𝐷 and 𝜎 are the drift, continuous dividend yield and volatility, respectively, 𝑊𝑡 is the standard Wiener process and there is a risk-free asset which earns interest at a constant rate 𝑟. Let 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of a European option satisfying the Black–Scholes equation 𝜕𝑉 𝜕𝑉𝑏𝑠 1 2 2 𝜕 2 𝑉𝑏𝑠 + (𝑟 − 𝐷)𝑆𝑡 𝑏𝑠 − 𝑟𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡

4.2.2 Reflection Principle Approach

387

with 𝑉𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = Ψ(𝑆𝑇 ) where Ψ(𝑆𝑇 ) is the option payoff for a strike price 𝐾 at expiry time 𝑇 > 𝑡. Show that for a constant 𝐵, the function ( ( )2𝛼 ) 𝑉𝑏𝑠 𝐵 2 ∕𝑆𝑡 , 𝑡; 𝐾, 𝑇 𝑈 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 ∕𝐵 satisfies ] 𝜕𝑈 𝜕𝑈 1 2 2 𝜕 2 𝑈 [ 2 + 𝜎 (1 − 2𝛼) − (𝑟 − 𝐷) 𝑆𝑡 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 [ ] 2 2 − 𝑟 + 𝛼𝜎 − 2𝛼(𝑟 − 𝐷 + 𝛼𝜎 ) 𝑈 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0. ( 1 Explain the significance of the above PDE if we set 𝛼 = 2

1−

𝑟−𝐷 1 2 𝜎 2

) .

Solution: By setting 𝜉 = 𝐵 2 ∕𝑆𝑡 we can write ( )2𝛼 𝑈 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 ∕𝐵 𝑉𝑏𝑠 (𝜉, 𝑡; 𝐾, 𝑇 ) . Differentiating 𝑈 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) with respect to 𝑡 and 𝑆𝑡 we have 𝜕𝑈 = 𝜕𝑡

(

𝑆𝑡 𝐵

)2𝛼

𝜕𝑉𝑏𝑠 𝜕𝑡

(

𝑆𝑡 𝐵

)2𝛼

𝜕𝑉𝑏𝑠 𝜕𝜉 𝜕𝜉 𝜕𝑆𝑡 ( )2𝛼 ( ) 𝑆𝑡 𝜕𝑉𝑏𝑠 𝑆𝑡 2 2𝛼−1 −2𝛼 = 2𝛼𝑆𝑡 𝐵 𝑉𝑏𝑠 (𝜉, 𝑡; 𝐾, 𝑇 ) − 𝐵 𝜕𝜉 𝐵 𝜕𝑉 = 2𝛼𝑆𝑡2𝛼−1 𝐵 −2𝛼 𝑉𝑏𝑠 (𝜉, 𝑡; 𝐾, 𝑇 ) − 𝑆𝑡2𝛼−1 𝐵 −2𝛼 𝜉 𝑏𝑠 𝜕𝜉 ] ( )2𝛼 [ 𝜕𝑉 1 𝑆𝑡 = 2𝛼𝑉𝑏𝑠 (𝜉, 𝑡; 𝐾, 𝑇 ) − 𝜉 𝑏𝑠 𝑆𝑡 𝐵 𝜕𝜉

𝜕𝑈 = 2𝛼𝑆𝑡2𝛼−1 𝐵 −2𝛼 𝑉𝑏𝑠 (𝜉, 𝑡; 𝐾, 𝑇 ) + 𝜕𝑆𝑡

388

4.2.2 Reflection Principle Approach

𝜕𝑈 with respect to 𝑆𝑡 𝜕𝑆𝑡

and finally by differentiating 1 𝜕2𝑈 =− 2 𝜕𝑆𝑡2 𝑆𝑡

(

𝑆𝑡 𝐵

)2𝛼 [ 2𝛼𝑉𝑏𝑠 (𝜉, 𝑡; 𝐾, 𝑇 ) − 𝜉

𝜕𝑉𝑏𝑠 𝜕𝜉

]

) [ ] 𝜕𝑉 𝑆𝑡 2𝛼 2𝛼𝑉𝑏𝑠 (𝜉, 𝑡; 𝐾, 𝑇 ) − 𝜉 𝑏𝑠 𝐵 𝜕𝜉 ( )2𝛼 [ ] 𝜕 2 𝑉𝑏𝑠 𝜕𝜉 𝜕𝑉𝑏𝑠 𝜕𝜉 𝜕𝜉 𝜕𝑉𝑏𝑠 1 𝑆𝑡 − −𝜉 + 2𝛼 𝑆 𝐵 𝜕𝜉 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝜉 𝜕𝜉 2 𝜕𝑆𝑡 ( 𝑡 )( ) [ ] 2𝛼 𝜕𝑉𝑏𝑠 𝑆𝑡 2𝛼 − 1 = (𝜉, 𝑡; 𝐾, 𝑇 ) − 𝜉 2𝛼𝑉 𝑏𝑠 𝐵 𝜕𝜉 𝑆𝑡2 ( )2𝛼 [ ] 𝜕 2 𝑉𝑏𝑠 𝜕𝑉 1 𝑆𝑡 − 2 (2𝛼 − 1)𝜉 𝑏𝑠 − 𝜉 2 𝐵 𝜕𝜉 𝜕𝜉 2 𝑆𝑡 ( )2𝛼 [ ] 𝜕𝑉 𝜕 2 𝑉𝑏𝑠 1 𝑆𝑡 2𝛼(2𝛼 − 1)𝑉𝑏𝑠 (𝜉, 𝑡; 𝐾, 𝑇 ) − 2(2𝛼 − 1)𝜉 𝑏𝑠 + 𝜉 2 . = 2 𝐵 𝜕𝜉 𝜕𝜉 2 𝑆𝑡 +

2𝛼 𝑆𝑡2

By substituting

(

𝜕2𝑈 𝜕𝑈 𝜕𝑈 and into , 𝜕𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡2 ] 𝜕𝑈 𝜕𝑈 1 2 2 𝜕 2 𝑈 [ 2 + 𝜎 (1 − 2𝛼) − (𝑟 − 𝐷) 𝑆𝑡 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 [ ] 2 2 − 𝑟 + 𝛼𝜎 − 2𝛼(𝑟 − 𝐷 + 𝛼𝜎 ) 𝑈 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 )

we have ] 𝜕𝑈 𝜕𝑈 1 2 2 𝜕 2 𝑈 [ 2 + 𝜎 (1 − 2𝛼) − (𝑟 − 𝐷) 𝑆𝑡 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 [ ] 2 2 − 𝑟 + 𝛼𝜎 − 2𝛼(𝑟 − 𝐷 + 𝛼𝜎 ) 𝑈 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) } ( )2𝛼 { 𝜕𝑉𝑏𝑠 𝑆𝑡 𝜕𝑉𝑏𝑠 1 2 2 𝜕 2 𝑉𝑏𝑠 + (𝑟 − 𝐷)𝜉 (𝜉, 𝑡; 𝐾, 𝑇 ) + 𝜎 𝜉 − 𝑟𝑉 = 𝑏𝑠 𝐵 𝜕𝑡 2 𝜕𝜉 𝜕𝜉 2 =0 𝜕𝑉𝑏𝑠 1 2 2 𝜕 2 𝑉𝑏𝑠 𝜕𝑉 + (𝑟 − 𝐷)𝜉 𝑏𝑠 − 𝑟𝑉𝑏𝑠 (𝜉, 𝑡; 𝐾, 𝑇 ) = 0. + 𝜎 𝜉 2 𝜕𝑡 2 𝜕𝜉 (𝜕𝜉 ) ( ( )2𝛼 ) 1 𝑟−𝐷 𝑉𝑏𝑠 𝐵 2 ∕𝑆𝑡 , 𝑡; 𝐾, 𝑇 By setting 𝛼 = 1− 1 then 𝑈 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 ∕𝐵 2 𝜎2 2 satisfies since

𝜕𝑈 𝜕𝑈 1 2 2 𝜕 2 𝑈 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑈 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0. + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡

4.2.2 Reflection Principle Approach

389

2. In–Out Parity. Assume we are in the Black–Scholes world where at time 𝑡 the asset price 𝑆𝑡 follows the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 where 𝜇, 𝐷 and 𝜎 are the drift, continuous dividend yield and volatility, respectively, 𝑊𝑡 is the standard Wiener process and there is a risk-free asset which earns interest at a constant rate 𝑟. Let 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of a European option satisfying the Black–Scholes equation 𝜕𝑉𝑏𝑠 𝜕𝑉𝑏𝑠 1 2 2 𝜕 2 𝑉𝑏𝑠 + (𝑟 − 𝐷)𝑆 − 𝑟𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 with 𝑉𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = Ψ(𝑆𝑇 ) where Ψ(𝑆𝑇 ) is the option payoff for a strike price 𝐾 at expiry time 𝑇 > 𝑡. Let 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ), 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ), 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) and 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) be the European up-and-out/in and European down-and-out/in options with common barrier 𝐵, strike price 𝐾 and expiry time 𝑇 . By setting up independent portfolios, show that 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). Solution: At time 𝑡, we first set up the portfolios Π𝑢 (𝑆𝑡 , 𝑡) and Π𝑑 (𝑆𝑡 , 𝑡) with each having the following options Π𝑢 (𝑆𝑡 , 𝑡) = 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) Π𝑑 (𝑆𝑡 , 𝑡) = 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ). For the case of Π𝑢 (𝑆𝑡 , 𝑡), at expiry time 𝑇 only one of the two barrier options can be active. If barrier 𝐵 is triggered (𝑆𝑇 ≥ 𝐵), then 𝑉𝑢∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = 0 and 𝑉𝑢∕𝑖 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = Ψ(𝑆𝑇 ). However, if the barrier 𝐵 is not triggered (𝑆𝑇 < 𝐵), then 𝑉𝑢∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = Ψ(𝑆𝑇 ) and 𝑉𝑢∕𝑖 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = 0. In the same vein for Π𝑑 (𝑆𝑡 , 𝑡), at expiry time 𝑇 if the barrier 𝐵 is triggered (𝑆𝑇 ≤ 𝐵) then 𝑉𝑑∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = 0 and 𝑉𝑢∕𝑖 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = Ψ(𝑆𝑇 )

390

4.2.2 Reflection Principle Approach

while if 𝐵 is never triggered (𝑆𝑇 > 𝐵) then 𝑉𝑑∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = Ψ(𝑆𝑇 ) and 𝑉𝑑∕𝑖 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = 0. Hence, at time 𝑇 , the portfolios have the values 𝑉𝑢∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑢∕𝑖 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = Ψ(𝑆𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) 𝑉𝑑∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑑∕𝑖 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = Ψ(𝑆𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) and by discounting them back to time 𝑡 under the risk-neutral measure ℚ, we have 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ).

3. Up-and-Out/In Barrier Options. Assume we are in the Black–Scholes world where at time 𝑡 the asset price 𝑆𝑡 follows the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 where 𝜇, 𝐷 and 𝜎 are the drift, continuous dividend yield and volatility, respectively, 𝑊𝑡 is the standard Wiener process and there is a risk-free asset which earns interest at a constant rate 𝑟. Let 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of a European option satisfying the Black–Scholes equation 𝜕𝑉 𝜕𝑉𝑏𝑠 1 2 2 𝜕 2 𝑉𝑏𝑠 + (𝑟 − 𝐷)𝑆𝑡 𝑏𝑠 − 𝑟𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 with boundary condition 𝑉𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = Ψ(𝑆𝑇 ) where Ψ(𝑆𝑇 ) is the option payoff for a strike price 𝐾 at expiry time 𝑇 > 𝑡. Let 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) and 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) be the European up-and-out/in options, respectively with common barrier 𝐵, strike price 𝐾 and expiry time 𝑇 . Using the reflection principle, show that the formula for an up-and-out barrier option is 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉̂𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) −

(

𝑆𝑡 𝐵

)2𝛼

𝑉̂𝑢∕𝑜

(

𝐵2 , 𝑡; 𝐾, 𝐵, 𝑇 𝑆𝑡

)

4.2.2 Reflection Principle Approach

( 1 where 𝛼 = 2

1−

equation 𝜕 𝑉̂𝑢∕𝑜 𝜕𝑡

𝑟−𝐷

391

)

1 2 𝜎 2

and 𝑉̂𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) satisfies the following Black–Scholes

𝜕 2 𝑉̂𝑢∕𝑜 𝜕 𝑉̂𝑢∕𝑜 1 + (𝑟 − 𝐷)𝑆 − 𝑟𝑉̂𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 0 + 𝜎 2 𝑆𝑡2 𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

with discontinuous payoff 0 < 𝑆𝑡 < 𝐵,

𝑡≤𝑇

𝑉̂𝑢∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = Ψ(𝑆𝑇 ). Deduce the general solution for the up-and-in barrier option price. Solution: The up-and-out barrier option 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) with strike price 𝐾 and barrier 𝐵 satisfies the PDE 𝜕𝑉𝑢∕𝑜 𝜕𝑡

𝜕 2 𝑉𝑢∕𝑜 𝜕𝑉𝑢∕𝑜 1 + (𝑟 − 𝐷)𝑆 − 𝑟𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 0 + 𝜎 2 𝑆𝑡2 𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

with the following boundary conditions 0 < 𝑆𝑡 < 𝐵,

𝑡≤𝑇

𝑉𝑢∕𝑜 (𝐵, 𝑡; 𝐾, 𝐵, 𝑇 ) = 0 𝑉𝑢∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = Ψ(𝑆𝑇 ) where Ψ(𝑆𝑇 ) is the European option (Black–Scholes) payoff at expiry 𝑇 . As the up-andout barrier option has a European payoff at expiry time 𝑇 but becomes worthless on the barrier 𝐵, we can therefore write its solution as 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉̂𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) − 𝑉̃𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) so that 𝑉̂𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) satisfies the Black–Scholes equation 𝜕 𝑉̂𝑢∕𝑜 𝜕𝑡

𝜕 2 𝑉̂𝑢∕𝑜 𝜕 𝑉̂𝑢∕𝑜 1 + (𝑟 − 𝐷)𝑆 − 𝑟𝑉̂𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 0 + 𝜎 2 𝑆𝑡2 𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

with discontinuous payoff 0 < 𝑆𝑡 < 𝐵,

𝑡≤𝑇

𝑉̂𝑢∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = Ψ(𝑆𝑇 )

392

4.2.2 Reflection Principle Approach

whilst for 𝑉̃𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) we require on the barrier 𝐵 𝑉̃𝑢∕𝑜 (𝐵, 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉̂𝑢∕𝑜 (𝐵, 𝑡; 𝐾, 𝐵, 𝑇 ) and at expiry 𝑇 𝑉̃𝑢∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = 0. From the reflection principle (see Problem 4.2.2.1, page 386), by writing 𝜉𝑡 =

𝐵2 𝑆𝑡

) ) ( 2 𝑆𝑡 2𝛼 𝐵 , 𝑡; 𝐾, 𝐵, 𝑇 𝑉̂𝑢∕𝑜 𝐵 𝑆𝑡 ( )2𝛼 𝐵 = 𝑉̂𝑢∕𝑜 (𝜉𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) 𝜉𝑡

𝑉̃𝑢∕𝑜 (𝑆, 𝑡; 𝐾, 𝐵, 𝑇 ) =

( 1 with 𝛼 = 2

1−

𝑟−𝐷

(

)

1 2 𝜎 2

also satisfies the Black–Scholes equation

] ( )2𝛼 [ ̂ 𝜕 𝑉̂𝑢∕𝑜 𝜕 𝑉𝑢∕𝑜 1 2 2 𝜕 2 𝑉̂𝑢∕𝑜 𝐵 + (𝑟 − 𝐷)𝜉𝑡 − 𝑟𝑉̂𝑢∕𝑜 (𝜉𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 0 + 𝜎 𝜉𝑡 𝜉𝑡 𝜕𝑡 2 𝜕𝜉𝑡 𝜕𝜉𝑡2 with the following boundary conditions 0 < 𝜉𝑡 < 𝐵,

𝑡≤𝑇

( )2𝛼 ( )2𝛼 𝐵 𝐵 ̂ Ψ(𝜉𝑇 ) 𝑉𝑢∕𝑜 (𝜉𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = 𝜉𝑡 𝜉𝑡 or 𝜕 𝑉̂𝑢∕𝑜 𝜕𝑡

𝜕 2 𝑉̂𝑢∕𝑜 𝜕 𝑉̂𝑢∕𝑜 1 + (𝑟 − 𝐷)𝜉 − 𝑟𝑉̂𝑢∕𝑜 (𝜉𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 0 + 𝜎 2 𝜉𝑡2 𝑡 2 𝜕𝜉𝑡 𝜕𝜉𝑡2

with boundary conditions 0 < 𝜉𝑡 < 𝐵,

𝑡≤𝑇

𝑉̂𝑢∕𝑜 (𝜉𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = Ψ(𝜉𝑇 ).

4.2.2 Reflection Principle Approach

393

Thus, we can set (

𝑉̃𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) =

𝑆𝑡 𝐵

)2𝛼

𝑉̂𝑢∕𝑜

(

𝐵2 , 𝑡; 𝐾, 𝐵, 𝑇 𝑆𝑡

) .

To test the boundary conditions, we note that at the barrier 𝑆𝑡 = 𝐵 we have ) ( 2 ( )2𝛼 𝐵 𝐵 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝑉̂𝑢∕𝑜 (𝐵, 𝑡; 𝐾, 𝐵, 𝑇 ) 𝑉̂𝑢∕𝑜 𝑉̃𝑢∕𝑜 (𝐵, 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝐵 𝐵 whilst at expiry time 𝑇 , we have (

𝑆𝑇 𝐵

)2𝛼

(

𝐵2 , 𝑇 ; 𝐾, 𝐵, 𝑇 𝑆𝑇 ( )2𝛼 ( 2 ) 𝑆𝑇 𝐵 = Ψ 1I 𝐵2 𝐵. 𝑆𝑇 Therefore, in the range 0 < 𝑆𝑇 < 𝐵 for an up-and-out option

where for the reflection part, the payoff is only valid for

𝑉̃𝑢∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) =

(

𝑆𝑇 𝐵

)2𝛼

𝑉̂𝑢∕𝑜

(

𝐵2 , 𝑇 ; 𝐾, 𝐵, 𝑇 𝑆𝑇

) = 0.

Hence, the general solution of an up-and-out barrier option is 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉̂𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) −

(

𝑆𝑡 𝐵

)2𝛼

𝑉̂𝑢∕𝑜

(

𝐵2 , 𝑡; 𝐾, 𝐵, 𝑇 𝑆𝑡

) .

By writing the up-and-in barrier option as 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ), from the in–out parity 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) where 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is the Black–Scholes formula for a European option written on asset price 𝑆𝑡 with strike 𝐾 and expiry time 𝑇 . Thus, the up-and-in barrier option formula is 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑉̂𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) ) ( 2 ( )2𝛼 𝑆𝑡 𝐵 , 𝑡; 𝐾, 𝐵, 𝑇 . 𝑉̂𝑢∕𝑜 + 𝐵 𝑆𝑡

394

4.2.2 Reflection Principle Approach

4. Down-and-Out/In Barrier Options. Assume that at time 𝑡, the asset price 𝑆𝑡 follows the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 where 𝜇, 𝐷 and 𝜎 are the drift, continuous dividend yield and volatility, respectively, 𝑊𝑡 is the standard Wiener process and there is a risk-free asset which earns interest at a constant rate 𝑟. Let 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of a European option satisfying the Black–Scholes equation 𝜕𝑉 𝜕𝑉𝑏𝑠 1 2 2 𝜕 2 𝑉𝑏𝑠 + (𝑟 − 𝐷)𝑆𝑡 𝑏𝑠 − 𝑟𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 with boundary condition 𝑉𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = Ψ(𝑆𝑇 ) where Ψ(𝑆𝑇 ) is the option payoff for a strike price 𝐾 at expiry time 𝑇 > 𝑡. Let 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) and 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) be the European down-and-out/in options, respectively with common barrier 𝐵, strike price 𝐾 and expiry time 𝑇 . Using the reflection principle, show that the formula for a down-and-out barrier option is 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉̂𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) − ( 1 where 𝛼 = 2 equation 𝜕 𝑉̂𝑑∕𝑜 𝜕𝑡

1−

𝑟−𝐷 1 2 𝜎 2

(

𝑆𝑡 𝐵

)2𝛼

𝑉̂𝑑∕𝑜

(

𝐵2 , 𝑡; 𝐾, 𝐵, 𝑇 𝑆𝑡

)

) and 𝑉̂𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) satisfies the following Black–Scholes

𝜕 2 𝑉̂𝑑∕𝑜 𝜕 𝑉̂𝑑∕𝑜 1 + (𝑟 − 𝐷)𝑆 − 𝑟𝑉̂𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 0 + 𝜎 2 𝑆𝑡2 𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

with truncated payoff 𝑆𝑡 > 𝐵,

𝑡≤𝑇

𝑉̂𝑑∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = Ψ(𝑆𝑇 ). Deduce the general solution for the down-and-in barrier option price.

4.2.2 Reflection Principle Approach

395

Solution: By definition, the down-and-out barrier option 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) with strike price 𝐾 and barrier 𝐵 satisfies the PDE 𝜕𝑉𝑑∕𝑜 𝜕𝑡

𝜕 2 𝑉𝑑∕𝑜 𝜕𝑉𝑑∕𝑜 1 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 0 + 𝜎 2 𝑆𝑡2 2 2 𝜕𝑆𝑡 𝜕𝑆𝑡

with the following boundary conditions 𝑆𝑡 > 𝐵,

𝑡≤𝑇

𝑉𝑑∕𝑜 (𝐵, 𝑡; 𝐾, 𝐵, 𝑇 ) = 0 𝑉𝑑∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = Ψ(𝑆𝑇 ) where Ψ(𝑆𝑇 ) is the European option (Black–Scholes) payoff at expiry 𝑇 . As the downand-out barrier option has a European payoff at expiry time 𝑇 but becomes worthless on the barrier 𝐵, we can therefore write its solution as 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉̂𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) − 𝑉̃𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) so that 𝑉̂𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) satisfies the Black–Scholes equation 𝜕 𝑉̂𝑑∕𝑜 𝜕𝑡

𝜕 2 𝑉̂𝑑∕𝑜 𝜕 𝑉̂𝑑∕𝑜 1 + (𝑟 − 𝐷)𝑆 − 𝑟𝑉̂𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 0 + 𝜎 2 𝑆𝑡2 𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2

with discontinuous payoff 𝑆𝑡 > 𝐵,

𝑡≤𝑇

𝑉̂𝑑∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = Ψ(𝑆𝑇 ) whilst for 𝑉̃𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ), we require on the barrier 𝐵 𝑉̃𝑑∕𝑜 (𝐵, 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉̂𝑑∕𝑜 (𝐵, 𝑡; 𝐾, 𝐵, 𝑇 ) and at expiry 𝑇 𝑉̃𝑑∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = 0.

396

4.2.2 Reflection Principle Approach

From the reflection principle (see Problem 4.2.2.1, page 386), by writing 𝜉𝑡 =

𝐵2 𝑆𝑡

) ) ( 2 𝑆𝑡 2𝛼 𝐵 ̂ , 𝑡; 𝐾, 𝐵, 𝑇 𝑉𝑑∕𝑜 𝐵 𝑆𝑡 ( )2𝛼 𝐵 = 𝑉̂𝑑∕𝑜 (𝜉𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) 𝜉𝑡

𝑉̃𝑑∕𝑜 (𝑆, 𝑡; 𝐾, 𝐵, 𝑇 ) =

( 1 with 𝛼 = 2

1−

𝑟−𝐷

(

)

1 2 𝜎 2

also satisfies the Black–Scholes equation

] ( )2𝛼 [ ̂ 𝜕 𝑉̂𝑑∕𝑜 𝜕 𝑉𝑑∕𝑜 1 2 2 𝜕 2 𝑉̂𝑑∕𝑜 𝐵 + (𝑟 − 𝐷)𝜉𝑡 − 𝑟𝑉̂𝑑∕𝑜 (𝜉𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 0 + 𝜎 𝜉𝑡 𝜉𝑡 𝜕𝑡 2 𝜕𝜉𝑡 𝜕𝜉𝑡2 with the following boundary conditions 𝜉𝑡 > 𝐵,

𝑡≤𝑇

( )2𝛼 ( )2𝛼 𝐵 𝐵 Ψ(𝜉𝑇 ) 𝑉̂𝑑∕𝑜 (𝜉𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = 𝜉𝑡 𝜉𝑡 or 𝜕 𝑉̂𝑑∕𝑜 𝜕𝑡

2̂ 𝜕 𝑉̂𝑑∕𝑜 1 2 2 𝜕 𝑉𝑑∕𝑜 + (𝑟 − 𝐷)𝜉 − 𝑟𝑉̂𝑑∕𝑜 (𝜉𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 0 + 𝜎 𝜉𝑡 𝑡 2 𝜕𝜉𝑡 𝜕𝜉𝑡2

with boundary conditions 𝜉𝑡 > 𝐵,

𝑡≤𝑇

𝑉̂𝑑∕𝑜 (𝜉𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = Ψ(𝜉𝑇 ). Thus, we can set 𝑉̃𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) =

(

𝑆𝑡 𝐵

)2𝛼

𝑉̂𝑑∕𝑜

(

𝐵2 , 𝑡; 𝐾, 𝐵, 𝑇 𝑆𝑡

) .

To test the boundary conditions, we note that at the barrier 𝑆𝑡 = 𝐵 we have ) ( 2 ( )2𝛼 𝐵 𝐵 ̃ ̂ 𝑉𝑑∕𝑜 (𝐵, 𝑡; 𝐾, 𝐵, 𝑇 ) = , 𝑡; 𝐾, 𝐵, 𝑇 = 𝑉̂𝑑∕𝑜 (𝐵, 𝑡; 𝐾, 𝐵, 𝑇 ) 𝑉𝑑∕𝑜 𝐵 𝐵

4.2.2 Reflection Principle Approach

397

whilst at expiry time 𝑇 , we have (

𝑆𝑇 𝐵

)2𝛼

(

𝐵2 , 𝑇 ; 𝐾, 𝐵, 𝑇 𝑆𝑇 ( )2𝛼 ( 2 ) 𝑆𝑇 𝐵 = Ψ 1I 𝐵2 >𝐵 𝐵 𝑆𝑇 𝑆𝑇

𝑉̃𝑑∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) =

𝑉̂𝑑∕𝑜

)

𝐵2 > 𝐵 or 𝑆𝑇 < 𝐵. 𝑆𝑇 Therefore, in the range 𝑆𝑇 > 𝐵 for a down-and-out option

where for the reflection part, the payoff is only valid for

𝑉̃𝑑∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) =

(

𝑆𝑇 𝐵

)2𝛼

𝑉̂𝑑∕𝑜

(

𝐵2 , 𝑇 ; 𝐾, 𝐵, 𝑇 𝑆𝑇

) = 0.

Hence, the general solution of a down-and-out barrier option is 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉̂𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) −

(

𝑆𝑡 𝐵

)2𝛼

𝑉̂𝑑∕𝑜

(

𝐵2 , 𝑡; 𝐾, 𝐵, 𝑇 𝑆𝑡

) .

By writing the down-and-in barrier option as 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ), from the in–out parity 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) where 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is the Black–Scholes formula for a European option written on asset price 𝑆𝑡 with strike 𝐾 and expiry time 𝑇 . Thus, the down-and-in barrier option formula is 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑉̂𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) ) ( 2 ( )2𝛼 𝑆𝑡 𝐵 , 𝑡; 𝐾, 𝐵, 𝑇 . 𝑉̂𝑑∕𝑜 + 𝐵 𝑆𝑡 { } 5. Up-and-Out/Up-and-In Call Options. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. We consider a European up-and-out call option at time 𝑡, with expiry at time 𝑇 > 𝑡, strike 𝐾 and constant barrier 𝐵 > 𝐾. By constructing a payoff diagram and using the reflection principle, find the up-and-out call option price.

398

4.2.2 Reflection Principle Approach

Furthermore, deduce the up-and-in call option price using the knock-in and knock-out parity relationship. Solution: For the up-and-out barrier call option we know that the option expires worthless if the barrier 𝐵 is reached, and since we are dealing with an up-and-out call option, the barrier 𝐵 must be set above the strike 𝐾, i.e. 𝐵 > 𝐾, otherwise the payoff will be knock-out (see Figure 4.1). ∕

(

;

)

−

0 Figure 4.1

Up-and-out call payoff diagram.

The payoff diagram for the up-and-out call option 𝐶𝑢∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) can be constructed with the following portfolio ( Portfolio =

) Long one call with strike 𝐾 [( ) ( )] Long one call Long one digital call with − + with strike 𝐵 strike 𝐵 and payoff (𝐵 − 𝐾)

which is illustrated in Figure 4.2. At expiry time 𝑇 we can therefore write the up-and-out call option price as ( ( ) ) 𝐶𝑢∕𝑜 𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 = 𝐶𝑏𝑠 𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ( ( ) )] [ − 𝐶𝑏𝑠 𝑆𝑇 , 𝑇 ; 𝐵, 𝑇 + (𝐵 − 𝐾) 𝐶𝑑 𝑆𝑇 , 𝑇 ; 𝐵, 𝑇 . By discounting the entire payoff under the risk-neutral measure ℚ and using the reflection principle, we can write the solution for the up-and-out call option at time 𝑡 as ( ( ) ) 𝐶𝑢∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 ( ( ) )] [ − 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐵, 𝑇 + (𝐵 − 𝐾) 𝐶𝑑 𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) ( 2 ( )2𝛼 { 𝑆𝑡 𝐵 , 𝑡; 𝐾, 𝑇 𝐶𝑏𝑠 − 𝐵 𝑆𝑡 [ ) )]} ( 2 ( 2 𝐵 𝐵 − 𝐶𝑏𝑠 , 𝑡; 𝐵, 𝑇 + (𝐵 − 𝐾) 𝐶𝑑 , 𝑡; 𝐵, 𝑇 𝑆𝑡 𝑆𝑡 ( 1 where 𝛼 = 2

1−

𝑟−𝐷 1 2 𝜎 2

) .

4.2.2 Reflection Principle Approach

399 −

−

−

−

−

−

−

−

Figure 4.2

Construction of an up-and-out call payoff.

400

4.2.2 Reflection Principle Approach

From the knock-in and knock-out parity relationship ( ( ( ) ) ) 𝐶𝑢∕𝑖 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 + 𝐶𝑢∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 therefore the up-and-in call option price at time 𝑡 is ( ( ( ) ) ) 𝐶𝑢∕𝑖 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 − 𝐶𝑢∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ( ( ) ) = 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐵, 𝑇 + (𝐵 − 𝐾)𝐶𝑑 𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) ( 2 ( )2𝛼 { 𝑆𝑡 𝐵 , 𝑡; 𝐾, 𝑇 𝐶𝑏𝑠 + 𝐵 𝑆𝑡 [ ) )]} ( 2 ( 2 𝐵 𝐵 − 𝐶𝑏𝑠 , 𝑡; 𝐵, 𝑇 + (𝐵 − 𝐾)𝐶𝑑 , 𝑡; 𝐵, 𝑇 . 𝑆𝑡 𝑆𝑡

{ } 6. Down-and-Out/Down-and-In Call Options. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ , ℙ) and let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. We consider a European down-and-out call option at time 𝑡, with expiry at time 𝑇 > 𝑡, strike 𝐾 and constant barrier 𝐵. By constructing payoff diagrams for 𝐵 ≤ 𝐾 and 𝐵 > 𝐾, and using the reflection principle, find the down-and-out call option prices. Furthermore, using the knock-in and knock-out parity relationship, deduce the downand-in call options. Solution: From the definition of a down-and-out call option price, the payoff diagrams 𝐵 < 𝐾 and 𝐵 > 𝐾 are given in Figure 4.3.

(a) Figure 4.3

(b)

Down-and-out call payoff diagram for (a) 𝐵 ≤ 𝐾 and (b) 𝐵 > 𝐾.

4.2.2 Reflection Principle Approach

401

For the case when 𝐵 ≤ 𝐾, at expiry time 𝑇 the down-and-out call option price is the same as a European call option price ( ) 𝐶𝑑∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = 𝐶𝑏𝑠 𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 . By discounting the entire payoff under the risk-neutral measure ℚ and using the reflection principle, we can write the solution for the down-and-out call option at time 𝑡 as ( ) 𝐶𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 − ( 1 where 𝛼 = 2

1−

𝑟−𝐷 1 2 𝜎 2

(

𝑆𝑡 𝐵

)2𝛼

( 𝐶𝑏𝑠

𝐵2 , 𝑡; 𝐾, 𝑇 𝑆𝑡

)

) .

For the case when 𝐵 > 𝐾, the payoff diagram for the down-and-out call option price 𝐶𝑑∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) can be constructed with the following portfolio ( Portfolio =

Long one call with strike 𝐵

)

( +

Long one digital call with strike 𝐵 and payoff (𝐵 − 𝐾)

)

where Figure 4.4 graphically illustrates the construction of the payoff. Hence, at expiry time 𝑇 we can write ( ( ( ) ) ) 𝐶𝑑∕𝑜 𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 = 𝐶𝑏𝑠 𝑆𝑇 , 𝑇 ; 𝐵, 𝑇 + (𝐵 − 𝐾) 𝐶𝑑 𝑆𝑇 , 𝑇 ; 𝐵, 𝑇 . −

−

−

−

Figure 4.4

Down-and-out call payoff diagram for 𝐵 > 𝐾.

402

4.2.2 Reflection Principle Approach

By discounting the entire payoff under the risk-neutral measure ℚ and using the reflection principle, we can write the solution for the down-and-out call option for 𝐵 > 𝐾 as 𝐶𝑑∕𝑜

(

( )2𝛼 ( ( ) ) 𝑆𝑡 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐵, 𝑇 + (𝐵 − 𝐾) 𝐶𝑑 𝑆𝑡 , 𝑡; 𝐵, 𝑇 − 𝐵 [ ) )] ( 2 ( 2 𝐵 𝐵 × 𝐶𝑏𝑠 , 𝑡; 𝐵, 𝑇 + (𝐵 − 𝐾) 𝐶𝑑 , 𝑡; 𝐵, 𝑇 . 𝑆𝑡 𝑆𝑡 )

Hence, ( ) ⎧ 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ( 2 ⎪ ( 𝑆𝑡 )2𝛼 𝐵 ⎪− 𝐶𝑏𝑠 , 𝑡; 𝐾, 𝑇 𝐵 𝑆𝑡 ⎪ ⎪ ( ( ) ) ( ) ⎪ 𝑆𝑡 , 𝑡; 𝐵, 𝑇 + (𝐵 − 𝐾) 𝐶𝑑 𝑆𝑡 , 𝑡; 𝐵, 𝑇 𝐶𝑑∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = ⎨ 𝐶𝑏𝑠 ) ( 2 ( )2𝛼 [ 𝑆𝑡 ⎪ 𝐵 , 𝑡; 𝐵, 𝑇 𝐶 − 𝑏𝑠 ⎪ 𝐵 ( 2 𝑆𝑡 )] ⎪ ⎪ + (𝐵 − 𝐾) 𝐶𝑑 𝐵 , 𝑡; 𝐵, 𝑇 𝑆𝑡 ⎪ ⎩

if 𝐵 ≤ 𝐾

if 𝐵 > 𝐾.

Since ( ( ( ) ) ) 𝐶𝑑∕𝑖 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 + 𝐶𝑑∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 therefore ) ( 2 ⎧ ( )2𝛼 𝐵 ⎪ 𝑆𝑡 𝐶𝑏𝑠 , 𝑡; 𝐾, 𝑇 ⎪ 𝐵 𝑆𝑡 ⎪ ( ( ) ) ⎪ 𝐵, 𝑇 ⎪ 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 − 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; ( ) ( ) ⎪ ( ) 𝑆𝑡 2𝛼 𝐶𝑑∕𝑖 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = ⎨ − (𝐵 − 𝐾) 𝐶𝑑 𝑆𝑡 , 𝑡; 𝐵, 𝑇 + 𝐵 ⎪ [ ) ( 2 ⎪ 𝐵 ⎪ × 𝐶𝑏𝑠 𝑆 , 𝑡; 𝐵, 𝑇 𝑡 ( )] ⎪ 𝐵2 ⎪ + (𝐵 − 𝐾) 𝐶 , 𝑡; 𝐵, 𝑇 𝑑 ⎪ 𝑆𝑡 ⎩

if 𝐵 ≤ 𝐾

if 𝐵 > 𝐾.

{ } 7. Up-and-Out/Up-and-In Put Options. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡

4.2.2 Reflection Principle Approach

403

where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. We consider a European up-and-out put option at time 𝑡, with expiry at time 𝑇 > 𝑡, strike 𝐾 and constant barrier 𝐵. By constructing payoff diagrams for 𝐵 < 𝐾 and 𝐵 ≥ 𝐾, and using the reflection principle, find the up-and-out put option prices. Furthermore, using the knock-in and knock-out parity relationship, deduce the up-andin put option price. Solution: From the definition of an up-and-out put option price, the payoff diagrams for 𝐵 ≥ 𝐾 and 𝐵 < 𝐾 are given in Figure 4.5.

(a) Figure 4.5

(b)

Up-and-out put payoff diagrams for (a) 𝐵 ≥ 𝐾 and (b) 𝐵 < 𝐾.

For the case when 𝐵 ≥ 𝐾, at expiry time 𝑇 the up-and-out put price is the same as a European put option price with strike 𝐾 and therefore we can simply write ( ) 𝑃𝑢∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = 𝑃𝑏𝑠 𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 . By discounting the entire payoff under the risk-neutral measure ℚ and using the reflection principle, we can write the solution for the up-and-out put price as ( ) ( 𝑆 )2𝛼 𝑃𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑃𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 − 𝑃𝑏𝑠 𝐵 ( 1 where 𝛼 = 2

1−

𝑟−𝐷 1 2 𝜎 2

(

𝐵2 , 𝑡; 𝐾, 𝑇 𝑆𝑡

)

) .

For the case when 𝐵 < 𝐾, the payoff diagram for the up-and-out put option price 𝑃𝑢∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) can be constructed with the following portfolio ( Portfolio =

Long one put with strike 𝐵

)

( +

Long one digital put with strike 𝐵 and payoff (𝐾 − 𝐵)

)

where Figure 4.6 graphically illustrates the construction of the payoff. Hence, at expiry time 𝑇 we can write ( ( ( ) ) ) 𝑃𝑢∕𝑜 𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 = 𝑃𝑏𝑠 𝑆𝑇 , 𝑇 ; 𝐵, 𝑇 + (𝐾 − 𝐵) 𝑃𝑑 𝑆𝑇 , 𝑇 ; 𝐵, 𝑇 .

404

4.2.2 Reflection Principle Approach −

−

−

−

Figure 4.6

Up-and-out put payoff diagram for 𝐵 < 𝐾.

By discounting the entire payoff under the risk-neutral measure ℚ and using the reflection principle, we can write the solution for the up-and-out put option for 𝐵 < 𝐾 as 𝑃𝑢∕𝑜

(

( )2𝛼 ( ( ) ) 𝑆𝑡 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝑃𝑏𝑠 𝑆𝑡 , 𝑡; 𝐵, 𝑇 + (𝐾 − 𝐵) 𝑃𝑑 𝑆𝑡 , 𝑡; 𝐵, 𝑇 − 𝐵 [ ( 2 ) )] ( 2 𝐵 𝐵 × 𝑃𝑏𝑠 , 𝑡; 𝐵, 𝑇 + (𝐾 − 𝐵) 𝑃𝑑 , 𝑡; 𝐵, 𝑇 . 𝑆𝑡 𝑆𝑡 )

Therefore, ) ( 2 ( ) ( 𝑆 )2𝛼 ⎧ 𝐵 𝑆 , 𝑡; 𝐾, 𝑇 − 𝑃 , 𝑡; 𝐾, 𝑇 if 𝐵 ≥ 𝐾 𝑃 𝑏𝑠 ⎪ 𝑏𝑠 𝑡 𝐵 𝑆𝑡 ⎪ ⎪ ( ( ) ) ( ) ⎪ 𝑃𝑏𝑠 𝑆𝑡 , 𝑡; 𝐵, 𝑇 + (𝐾 − 𝐵) 𝑃𝑑 𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) ) [ ( ( 𝑃𝑢∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = ⎨ 𝑆𝑡 2𝛼 𝐵2 ⎪− , 𝑡; 𝐵, 𝑇 if 𝐵 < 𝐾. 𝑃𝑏𝑠 ⎪ 𝐵 )] ( 2𝑆𝑡 ⎪ 𝐵 , 𝑡; 𝐵, 𝑇 ⎪ + (𝐾 − 𝐵) 𝑃𝑑 𝑆𝑡 ⎩ From the knock-in and knock-out parity relationship ( ( ( ) ) ) 𝑃𝑢∕𝑖 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 + 𝑃𝑢∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝑃𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇

4.2.2 Reflection Principle Approach

405

the corresponding up-and-in put option price at time 𝑡 becomes ) ( 2 ⎧ ( 𝑆 )2𝛼 𝐵 𝑡 ⎪ 𝑃𝑏𝑠 , 𝑡; 𝐾, 𝑇 𝑆𝑡 ⎪ 𝐵 ⎪ ( ( ) ) ⎪ ( ) ⎪ 𝑃𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇( − 𝑃𝑏𝑠 𝑆)𝑡 , 𝑡; 𝐵, 𝑇 𝑃𝑢∕𝑖 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = ⎨ − (𝐾 − 𝐵) 𝑃𝑑 𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) ⎪ ( 𝑆𝑡 )2𝛼 [ ( 𝐵 2 ⎪+ , 𝑡; 𝐵, 𝑇 𝑃𝑏𝑠 𝐵 ⎪ )] ( 2𝑆𝑡 ⎪ 𝐵 (𝐾 , 𝑡; 𝐵, 𝑇 + − 𝐵) 𝑃 ⎪ 𝑑 𝑆𝑡 ⎩

if 𝐵 ≥ 𝐾

if 𝐵 < 𝐾.

{ } 8. Down-and-Out/Down-and-In Put Options. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. We consider a European down-and-out put option at time 𝑡, with expiry at time 𝑇 > 𝑡, strike 𝐾 and constant barrier 𝐵 < 𝐾. By constructing a payoff diagram and using the reflection principle, find the down-andout put option price. Furthermore, deduce the down-and-in put option price using the knock-in and knock-out parity relationship. Solution: For the down-and-out barrier put option we know that the option expires worthless if the barrier 𝐵 is triggered, and since we are dealing with a down-and-out put option, the barrier 𝐵 must be set below the strike 𝐾, i.e. 𝐵 < 𝐾; otherwise the payoff will be knock-out (see Figure 4.7).

Figure 4.7

Down-and-out put payoff diagram for 𝐵 < 𝐾.

406

4.2.2 Reflection Principle Approach

The payoff for the down-and-out put option 𝑃𝑑∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) can be constructed with the following portfolio (

) Long one put Portfolio = with strike 𝐾 [( ) ( )] Long one put Long one digital put with − + with strike 𝐵 strike 𝐵 and payoff (𝐾 − 𝐵) where Figure 4.8 graphically illustrates the construction of the payoff. At expiry time 𝑇 we can therefore write ( ( ) ) 𝑃𝑑∕𝑜 𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 = 𝑃𝑏𝑠 𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ( ) )] [ ( − 𝑃𝑏𝑠 𝑆𝑇 , 𝑇 ; 𝐵, 𝑇 + (𝐾 − 𝐵) 𝑃𝑑 𝑆𝑇 , 𝑇 ; 𝐵, 𝑇 . By discounting the entire payoff under the risk-neutral measure ℚ and using the reflection principle, we can write the solution for the down-and-out put option as ( ( ) ) 𝑃𝑑∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝑃𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 ( ) )] [ ( − 𝑃𝑏𝑠 𝑆𝑡 , 𝑡; 𝐵, 𝑇 + (𝐾 − 𝐵) 𝑃𝑑 𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) ( )2𝛼 { ( 2 𝑆𝑡 𝐵 , 𝑡; 𝐾, 𝑇 𝑃𝑏𝑠 − 𝐵 𝑆𝑡 [ ( 2 ) )]} ( 2 𝐵 𝐵 − 𝑃𝑏𝑠 , 𝑡; 𝐵, 𝑇 + (𝐾 − 𝐵) 𝑃𝑑 , 𝑡; 𝐵, 𝑇 𝑆𝑡 𝑆𝑡 ( 1 where 𝛼 = 2

1−

𝑟−𝐷 1 2 𝜎 2

) .

From the knock-in and knock-out parity relationship ( ( ( ) ) ) 𝑃𝑑∕𝑖 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 + 𝑃𝑑∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝑃𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 the corresponding down-and-in put option price at time 𝑡 is ( ( ( ) ) ) 𝑃𝑑∕𝑖 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 = 𝑃𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 − 𝑃𝑑∕𝑜 𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ( ( ) ) = 𝑃𝑏𝑠 𝑆𝑡 , 𝑡; 𝐵, 𝑇 + (𝐾 − 𝐵)𝑃𝑑 𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) ( )2𝛼 { ( 2 𝑆𝑡 𝐵 , 𝑡; 𝐾, 𝑇 + 𝑃𝑏𝑠 𝐵 𝑆𝑡 [ ( 2 ) )]} ( 2 𝐵 𝐵 − 𝑃𝑏𝑠 , 𝑡; 𝐵, 𝑇 + (𝐾 − 𝐵)𝑃𝑑 , 𝑡; 𝐵, 𝑇 . 𝑆𝑡 𝑆𝑡

4.2.2 Reflection Principle Approach

407 −

−

−

−

−

−

−

Figure 4.8

Construction of a down-and-out put payoff diagram for 𝐵 < 𝐾.

408

4.2.3 Further Barrier-Style Options

4.2.3

Further Barrier-Style Options

1. In–Out Parity with Rebate at Expiry. Assume we are in the Black–Scholes world where at time 𝑡 < 𝑇 , the asset price 𝑆𝑡 follows the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 where 𝜇, 𝐷 and 𝜎 > 0 are the drift, continuous dividend yield and volatility, respectively, 𝑊𝑡 is the standard Wiener process and there is a risk-free asset which earns interest at a constant rate 𝑟. Let 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of a European option satisfying the Black– Scholes equation 𝜕𝑉 𝜕𝑉𝑏𝑠 1 2 2 𝜕 2 𝑉𝑏𝑠 + (𝑟 − 𝐷)𝑆𝑡 𝑏𝑠 − 𝑟𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 with boundary condition 𝑉𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = Ψ(𝑆𝑇 ) where Ψ(𝑆𝑇 ) is the option payoff for a strike price 𝐾 at expiry time 𝑇 . 𝑅 (𝑆 , 𝑡; 𝐾, 𝐵, 𝑇 ), By incorporating a rebate 𝑅 payable at expiry time 𝑇 , we define 𝑉𝑢∕𝑜 𝑡

𝑅 (𝑆 , 𝑡; 𝐾, 𝐵, 𝑇 ), 𝑉 𝑅 (𝑆 , 𝑡; 𝐾, 𝐵, 𝑇 ) and 𝑉 𝑅 (𝑆 , 𝑡; 𝐾, 𝐵, 𝑇 ) as the European up-and𝑉𝑑∕𝑜 𝑡 𝑢∕𝑖 𝑡 𝑑∕𝑖 𝑡 out, down-and-out, up-and-in and down-and-in options, respectively with common barrier 𝐵, strike price 𝐾, rebate 𝑅 and expiry time 𝑇 . Show by setting up independent portfolios that 𝑅 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑅 𝑒−𝑟(𝑇 −𝑡) 𝑉𝑢∕𝑜

and 𝑅 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑅 𝑒−𝑟(𝑇 −𝑡) . 𝑉𝑑∕𝑜 𝑅 Solution: At time 𝑡 < 𝑇 , we first set up the portfolios Π𝑅 𝑢 (𝑆𝑡 , 𝑡) and Π𝑑 (𝑆𝑡 , 𝑡) with each having the following options 𝑅 𝑅 Π𝑅 𝑢 (𝑆𝑡 , 𝑡) = 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) 𝑅 𝑅 Π𝑅 𝑑 (𝑆𝑡 , 𝑡) = 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ).

First consider Π𝑅 𝑢 (𝑆𝑡 , 𝑡) where at expiry time 𝑇 , only one of the two barrier options can be active. If barrier 𝐵 is triggered (𝑆𝑇 ≥ 𝐵), then 𝑅 𝑅 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = 𝑅 and 𝑉𝑢∕𝑖 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑢∕𝑖 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) 𝑉𝑢∕𝑜

where 𝑉𝑢∕𝑖 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) is the European up-and-in barrier option payoff without rebate.

4.2.3 Further Barrier-Style Options

409

However, if the barrier 𝐵 is not triggered (𝑆𝑇 < 𝐵), then 𝑅 𝑅 𝑉𝑢∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑢∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) and 𝑉𝑢∕𝑖 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = 𝑅

where 𝑉𝑢∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) is the European up-and-out barrier option payoff without rebate. (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ), at expiry time 𝑇 if the barrier is triggered (𝑆𝑇 ≤ In the same vein for Π𝑅 𝑑 𝐵) then 𝑅 𝑅 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = 𝑅 and 𝑉𝑑∕𝑖 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑑∕𝑖 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) 𝑉𝑑∕𝑜

where 𝑉𝑑∕𝑖 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) is the European down-and-in barrier option payoff without rebate. Finally, if 𝐵 is never triggered (𝑆𝑇 > 𝐵) then 𝑅 𝑅 𝑉𝑑∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑑∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) and 𝑉𝑑∕𝑖 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = 𝑅

where 𝑉𝑑∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) is the European down-and-out barrier option payoff without rebate. Hence, at expiry time 𝑇 we have 𝑅 𝑅 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑢∕𝑖 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑢∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) 𝑉𝑢∕𝑜

+𝑉𝑢∕𝑖 (𝑆𝑇 , 𝑇 ; 𝐾, 𝐵, 𝑇 ) ] [ +𝑅 1I𝑆𝑇 ≥𝐵 + 1I𝑆𝑇 𝐵 = 𝑉𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) + 𝑅. By discounting the terminal payoffs back to time 𝑡, we can conclude that 𝑅 𝑅 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑅 𝑒−𝑟(𝑇 −𝑡)

and 𝑅 𝑅 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑅 𝑒−𝑟(𝑇 −𝑡) .

410

4.2.3 Further Barrier-Style Options

2. Up-and-Out/In with Rebates at Expiry. Assume we are in the Black–Scholes world where at time 𝑡 < 𝑇 , the asset price 𝑆𝑡 follows the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 where 𝜇, 𝐷 and 𝜎 > 0 are the drift, continuous dividend yield and volatility, respectively, 𝑊𝑡 is the standard Wiener process and there is a risk-free asset which earns interest at a constant rate 𝑟. Let 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of a European option satisfying the Black–Scholes equation 𝜕𝑉 𝜕𝑉𝑏𝑠 1 2 2 𝜕 2 𝑉𝑏𝑠 + (𝑟 − 𝐷)𝑆𝑡 𝑏𝑠 − 𝑟𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 with boundary condition 𝑉𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = Ψ(𝑆𝑇 ) where Ψ(𝑆𝑇 ) is the option payoff for a strike price 𝐾 at expiry time 𝑇 . 𝑅 (𝑆 , 𝑡; 𝐾, 𝐵, 𝑇 ) and By incorporating a rebate 𝑅 payable at expiry 𝑇 , we define 𝑉𝑢∕𝑜 𝑡

𝑅 (𝑆 , 𝑡; 𝐾, 𝐵, 𝑇 ) as the European up-and-out and European out-and-in options, respec𝑉𝑢∕𝑖 𝑡 tively with common barrier 𝐵, strike price 𝐾, rebate 𝑅 and expiry time 𝑇 . Using the put–call parity for European digital options and the reflection principle, show that the formula for an up-and-out barrier option incorporating a rebate 𝑅 payable at expiry 𝑇 is 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) 𝑉𝑢∕𝑜 [

+𝑅 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) +

(

𝑆𝑡 𝐵

)2𝛼

( 𝑃𝑑

𝐵2 , 𝑡; 𝐵, 𝑇 𝑆𝑡

)]

where 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) is the European up-and-out barrier option without rebate, 𝛼 = ( ) 𝑟−𝐷 1 1− 1 , 𝐶𝑑 (𝑋, 𝑡; 𝑌 , 𝑇 ) and 𝑃𝑑 (𝑋, 𝑡; 𝑌 , 𝑇 ) are the European digital call and put 2 𝜎2 2

options defined as ( 𝐶𝑑 (𝑋, 𝑡; 𝑌 , 𝑇 ) = 𝑒

−𝑟(𝑇 −𝑡)

Φ

log(𝑋∕𝑌 ) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

)

and ( 𝑃𝑑 (𝑋, 𝑡; 𝑌 , 𝑇 ) = 𝑒 respectively.

−𝑟(𝑇 −𝑡)

Φ

−log(𝑋∕𝑌 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

)

4.2.3 Further Barrier-Style Options

411

Finally, deduce the general solution for the up-and-in barrier option price with a rebate 𝑅 payable at expiry 𝑇 . Solution: For the up-and-out case, we can split the option price into two parts 𝑅

𝑅 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉 𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 )

where 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) is the usual European up-and-out barrier option price which only exists for 𝑆𝑡 < 𝐵, whilst when the asset reaches the barrier (i.e. 𝑆𝑡 = 𝐵) the option 𝑅

𝑉 𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) takes the form 𝑅

𝑉 𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) = 𝑅 𝑒−𝑟(𝑇 −𝑡) . From the put–call parity for European digital options we have 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) + 𝑃𝑑 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) such that at expiry 𝑇 { 𝐶𝑑 (𝑆𝑇 , 𝑇 ; 𝐵, 𝑇 ) =

0 if 𝑆𝑇 < 𝐵 1 if 𝑆𝑇 ≥ 𝐵

and { 𝑃𝑑 (𝑆𝑇 , 𝑇 ; 𝐵, 𝑇 ) =

1 if 𝑆𝑇 ≤ 𝐵 0 if 𝑆𝑇 > 𝐵.

Note that 𝑃𝑑 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) ≠ 0 for 𝑆𝑡 ≤ 𝐵, 𝑡 ≤ 𝑇 but its reflection does vanish such that at expiry time 𝑇 (

𝑆𝑇 𝐵

)2𝛼

( 𝑃𝑑

𝐵2 , 𝑇 ; 𝐵, 𝑇 𝑆𝑇

)

( = ( =

𝑆𝑇 𝐵 𝑆𝑇 𝐵

)2𝛼 1I{ 𝐵2 𝑆𝑇

)2𝛼

≤𝐵

}

1I{𝑆𝑇 ≥𝐵 }

where for the reflection part the payoff is only valid for 𝑆𝑇 ≥ 𝐵. Since we only consider the range 0 < 𝑆𝑇 < 𝐵 for an up-and-out barrier option, we can set 𝑉

𝑅 𝑢∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐵, 𝑇 )

[ = 𝑅 𝐶𝑑 (𝑆𝑇 , 𝑇 ; 𝐵, 𝑇 ) +

(

𝑆𝑇 𝐵

)2𝛼

( 𝑃𝑑

𝐵2 , 𝑇 ; 𝐵, 𝑇 𝑆𝑇

)] .

412

4.2.3 Further Barrier-Style Options

By discounting the entire payoff back to time 𝑡 using the risk-free interest rate 𝑟, we have 𝑅 𝑉 𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 )

[

(

= 𝑅 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) +

𝑆𝑡 𝐵

)2𝛼

( 𝑃𝑑

𝐵2 , 𝑡; 𝐵, 𝑇 𝑆𝑡

)]

where for 𝑆𝑡 = 𝐵 ] [ 𝑅 𝑉 𝑢∕𝑜 (𝐵, 𝑡; 𝐵, 𝑇 ) = 𝑅 𝐶𝑑 (𝐵, 𝑡; 𝐵, 𝑇 ) + 𝑃𝑑 (𝐵, 𝑡; 𝐵, 𝑇 ) = 𝑅 𝑒−𝑟(𝑇 −𝑡) . Hence, by adding back the up-and-out barrier option without rebate, the general solution of an up-and-out barrier option with rebate payable at expiry 𝑇 becomes 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) 𝑉𝑢∕𝑜 [

+𝑅 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) +

(

𝑆𝑡 𝐵

)2𝛼

( 𝑃𝑑

𝐵2 , 𝑡; 𝐵, 𝑇 𝑆𝑡

)] .

From the in–out parity for barriers with rebates at expiry 𝑅 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑅 𝑒−𝑟(𝑇 −𝑡) 𝑉𝑢∕𝑜

where 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is the Black–Scholes formula for a European option written on asset 𝑆𝑡 with strike 𝐾 and expiry time 𝑇 . Therefore, the up-and-in barrier option price with a rebate 𝑅 payable at expiry 𝑇 is 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) 𝑉𝑢∕𝑖 [ +𝑅 𝑒−𝑟(𝑇 −𝑡) − 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) )] ( )2𝛼 ( 2 𝑆𝑡 𝐵 𝑃𝑑 , 𝑡; 𝐵, 𝑇 . − 𝐵 𝑆𝑡

3. Down-and-Out/In with Rebates at Expiry. Assume we are in the Black–Scholes world where at time 𝑡 < 𝑇 , the asset price 𝑆𝑡 follows the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 where 𝜇, 𝐷 and 𝜎 > 0 are the drift, continuous dividend yield and volatility, respectively, 𝑊𝑡 is the standard Wiener process and there is a risk-free asset which earns interest at a constant rate 𝑟.

4.2.3 Further Barrier-Style Options

413

Let 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of a European option satisfying the Black–Scholes equation 𝜕𝑉𝑏𝑠 𝜕𝑉𝑏𝑠 1 2 2 𝜕 2 𝑉𝑏𝑠 + (𝑟 − 𝐷)𝑆 − 𝑟𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 𝑆𝑡 𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 with 𝑉𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = Ψ(𝑆𝑇 ) where Ψ(𝑆𝑇 ) is the option payoff for a strike price 𝐾 at expiry time 𝑇 . By incorporating a 𝑅 (𝑆 , 𝑡; 𝐾, 𝐵, 𝑇 ) and 𝑉 𝑅 (𝑆 , 𝑡; 𝐾, 𝐵, 𝑇 ) be the Eurorebate 𝑅 payable at expiry 𝑇 , let 𝑉𝑑∕𝑜 𝑡 𝑑∕𝑖 𝑡 pean up-and-out/in options, respectively with common barrier 𝐵, strike price 𝐾, rebate 𝑅 and expiry time 𝑇 . Using the put–call parity for European digital options and the reflection principle, show that the formula for an up-and-out barrier option incorporating a rebate 𝑅 payable at expiry 𝑇 is 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) 𝑉𝑑∕𝑜 [( ) ] ) ( 2 𝑆𝑡 2𝛼 𝐵 +𝑅 𝐶𝑑 , 𝑡; 𝐵, 𝑇 + 𝑃𝑑 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) 𝐵 𝑆𝑡

where 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) is the European down-and-out barrier option without rebate, ( ) 𝑟−𝐷 1 1− 1 , 𝐶𝑑 (𝑋, 𝑡; 𝑌 , 𝑇 ) and 𝑃𝑑 (𝑋, 𝑡; 𝑌 , 𝑇 ) are the European digital call and 𝛼= 2 𝜎2 2

put options defined as ( 𝐶𝑑 (𝑋, 𝑡; 𝑌 , 𝑇 ) = 𝑒

−𝑟(𝑇 −𝑡)

Φ

log(𝑋∕𝑌 ) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

)

and ( 𝑃𝑑 (𝑋, 𝑡; 𝑌 , 𝑇 ) = 𝑒

−𝑟(𝑇 −𝑡)

Φ

−log(𝑋∕𝑌 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

)

respectively. Finally, deduce the general solution for the down-and-in barrier option price with a rebate 𝑅 payable at expiry 𝑇 . Solution: For the down-and-out case, we can split the option price into two parts 𝑅

𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉 𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) 𝑉𝑑∕𝑜

414

4.2.3 Further Barrier-Style Options

where 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) is the usual European up-and-out barrier option price which 𝑅

only exists for 𝑆𝑡 > 𝐵, whilst when 𝑆𝑡 = 𝐵 the option 𝑉 𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) takes the form 𝑅

𝑉 𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) = 𝑅 𝑒−𝑟(𝑇 −𝑡) . From the put–call parity for European digital options we have 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) + 𝑃𝑑 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) such that at expiry 𝑇 { 𝐶𝑑 (𝑆𝑇 , 𝑇 ; 𝐵, 𝑇 ) =

0 if 𝑆𝑇 < 𝐵 1 if 𝑆𝑇 ≥ 𝐵

and { 𝑃𝑑 (𝑆𝑇 , 𝑇 ; 𝐵, 𝑇 ) =

1 if 𝑆𝑇 ≤ 𝐵 0 if 𝑆𝑇 > 𝐵.

Note that 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) ≠ 0 for 𝑆𝑡 ≥ 𝐵, 𝑡 ≤ 𝑇 but its reflection is where at expiry time 𝑇 (

𝑆𝑇 𝐵

)2𝛼

( 𝐶𝑑

𝐵2 , 𝑇 ; 𝐵, 𝑇 𝑆𝑇

)

( = ( =

𝑆𝑇 𝐵 𝑆𝑇 𝐵

)2𝛼 1I{ 𝐵2 )2𝛼

𝑆𝑇

≥𝐵

}

1I{𝑆𝑇 ≤𝐵 }

where for the reflection part the payoff is only valid for 𝑆𝑇 ≤ 𝐵. Since we only consider the range 𝑆𝑇 > 𝐵, for a down-and-out barrier option we can set [( ) ] ) ( 2 𝑆𝑇 2𝛼 𝑅 𝐵 𝑉 𝑑∕𝑜 (𝑆𝑇 , 𝑇 ; 𝐵, 𝑇 ) = 𝑅 𝐶𝑑 , 𝑇 ; 𝐵, 𝑇 + 𝑃𝑑 (𝑆𝑇 , 𝑇 ; 𝐵, 𝑇 ) . 𝐵 𝑆𝑇 By discounting the entire payoff back to time 𝑡 using the risk-free interest rate 𝑟, we have 𝑅 𝑉 𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 )

[( =𝑅

𝑆𝑡 𝐵

)2𝛼

( 𝐶𝑑

𝐵2 , 𝑡; 𝐵, 𝑇 𝑆𝑡

]

) + 𝑃𝑑 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 )

where for 𝑆𝑡 = 𝐵 [ ] 𝑅 𝑉 𝑑∕𝑜 (𝐵, 𝑡; 𝐵, 𝑇 ) = 𝑅 𝐶𝑑 (𝐵, 𝑡; 𝐵, 𝑇 ) + 𝑃𝑑 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) = 𝑅 𝑒−𝑟(𝑇 −𝑡) .

4.2.3 Further Barrier-Style Options

415

Hence, by adding back the down-and-out barrier option without rebate, the general solution of a down-and-out barrier option with rebate payable at expiry 𝑇 becomes 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) 𝑉𝑑∕𝑜 [( ) ] ) ( 2 𝑆𝑡 2𝛼 𝐵 +𝑅 𝐶𝑑 , 𝑡; 𝐵, 𝑇 + 𝑃𝑑 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) . 𝐵 𝑆𝑡

From the in–out parity for barriers with rebates at expiry 𝑅 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑅 𝑒−𝑟(𝑇 −𝑡) 𝑉𝑑∕𝑜

where 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is the Black–Scholes formula for a European option written on asset 𝑆𝑡 with strike 𝐾 and expiry time 𝑇 . Therefore, the down-and-in barrier option price with a rebate 𝑅 payable at expiry 𝑇 is 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) 𝑉𝑑∕𝑖 [ ( )2𝛼 ) ( 2 𝑆𝑡 𝐵 −𝑟(𝑇 −𝑡) − 𝐶𝑑 , 𝑡; 𝐵, 𝑇 +𝑅 𝑒 𝐵 𝑆𝑡 ] −𝑃𝑑 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) .

{ } 4. Up-and-Out/In Barrier Options with Immediate Rebates. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ , ℙ) and assume that we are in the Black– Scholes world where at time 𝑡 < 𝑇 , the asset price 𝑆𝑡 follows the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 where 𝜇, 𝐷 and 𝜎 > 0 are the drift, dividend yield and volatility, respectively, and there is a risk-free asset which earns interest at a constant rate 𝑟. Let 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of a European option satisfying the Black–Scholes equation 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 with 𝑉𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = Ψ(𝑆𝑇 ) where Ψ(𝑆𝑇 ) is the option payoff for a strike price 𝐾 at expiry time 𝑇 . By incorporating a rebate 𝑅 payable at knock-out/knock-in time 𝜏, 𝑡 ≤ 𝜏 ≤ 𝑇 , let 𝑅 (𝑆 , 𝑡; 𝐾, 𝐵, 𝑇 ) and 𝑉 𝑅 (𝑆 , 𝑡; 𝐾, 𝐵, 𝑇 ) be the European up-and-out and up-and-in bar𝑉𝑢∕𝑜 𝑡 𝑢∕𝑖 𝑡 rier options, respectively with common barrier 𝐵, strike price 𝐾, rebate 𝑅 and expiry time 𝑇 .

416

4.2.3 Further Barrier-Style Options

By writing 𝑅 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉̃𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) 𝑉𝑢∕𝑜 𝑅 𝑅 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉̃𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 )

where 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) and 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) are the European up-and-out and up𝑅 (𝑆 , 𝑡; 𝐵, 𝑇 ) and and-in barrier option prices without rebates, respectively, whilst 𝑉̃𝑢∕𝑜 𝑡 𝑉̃ 𝑅 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) are the corresponding option prices associated with immediate rebate at 𝑢∕𝑖

knock-out/knock-in time, show that 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑅 𝐶𝑑𝐴𝑚 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) 𝑉̃𝑢∕𝑜 𝑅 𝑉̃𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑅 𝑃𝑑𝐴𝑚 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 )

where 𝐶𝑑𝐴𝑚 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) and 𝑃𝑑𝐴𝑚 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) are the immediate-touch call and put options, respectively defined as 𝐶𝑑𝐴𝑚 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) =

(

𝑆𝑡 𝐵

(

)𝜆+ Φ(𝑑+ ) +

𝑆𝑡 𝐵

)𝜆− Φ(𝑑− )

and 𝑃𝑑𝐴𝑚 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 )

( =

𝑆𝑡 𝐵

(

)𝜆+ Φ(−𝑑+ ) +

𝑆𝑡 𝐵

)𝜆− Φ(−𝑑− )

such that 𝜆± =

−(𝑟 − 𝐷 − 12 𝜎 2 ) −

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟

𝜎2 √ log(𝑆𝑡 ∕𝐵) ± (𝑇 − 𝑡) (𝑟 − 𝐷 − 12 𝜎 2 ) + 2𝜎 2 𝑟 𝑑± = √ 𝜎 𝑇 −𝑡 and 𝑥

Φ(𝑥) =

∫−∞

1 − 1 𝑢2 √ 𝑒 2 𝑑𝑢 2𝜋

is the cdf of a standard normal. Finally, verify that 𝑅 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) 𝑉𝑢∕𝑜 [( ) ( )𝜆− ] 𝑆𝑡 𝑆𝑡 𝜆+ + . +𝑅 𝐵 𝐵

4.2.3 Further Barrier-Style Options

417

Solution: At time 𝑡 we first set up the portfolio Π𝑅 𝑢 (𝑆𝑡 , 𝑡) having the following options: 𝑅 𝑅 Π𝑅 𝑢 (𝑆𝑡 , 𝑡) = 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 )

where 𝑅 𝑅 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉̃𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) 𝑅 𝑅 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉̃𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ).

At any time 𝜏, 𝑡 ≤ 𝜏 ≤ 𝑇 , only one of the two barrier options can be active. If barrier 𝐵 is triggered, that is max 𝑆𝜏 ≥ 𝐵 such that 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) vanishes, then under the 𝑡≤𝜏≤𝑇

risk-neutral measure ℚ the rebate price is 𝑅 𝑅 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉̃𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) [ ] | = 𝑅 𝔼ℚ max 𝑒−𝑟(𝑇 −𝜏) 1I𝑆𝜏 ≥𝐵 || ℱ𝑡 𝑡≤𝜏≤𝑇 | 𝐴𝑚 = 𝑅 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ). 𝑅 (𝑆 , 𝑡; 𝐵, 𝑇 ) vanishes so that Correspondingly, if max 𝑆𝜏 ≥ 𝐵 then the rebate price 𝑉̃𝑢∕𝑖 𝑡 𝑡≤𝜏≤𝑇

𝑅 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ).

In contrast, if the barrier is not triggered, max 𝑆𝜏 < 𝐵, then under the risk-neutral measure ℚ

𝑡≤𝜏≤𝑇

𝑅 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 )

and 𝑅 𝑅 𝑉𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉̃𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) [ ] | ℚ −𝑟(𝑇 −𝜏) | = 𝑅𝔼 1I𝑆𝜏 0 are the drift, dividend yield and volatility, respectively, and there is a risk-free asset which earns interest at a constant rate 𝑟. Let 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of a European option satisfying the Black–Scholes equation 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 with 𝑉 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = Ψ(𝑆𝑇 ) where Ψ(𝑆𝑇 ) is the option payoff for a strike price 𝐾 at expiry time 𝑇 . When incorporating 𝑅 (𝑆 , 𝑡; 𝐾, 𝐵, 𝑇 ) and a rebate 𝑅 payable at knock-out/knock-in time 𝜏, 𝑡 ≤ 𝜏 ≤ 𝑇 , let 𝑉𝑑∕𝑜 𝑡

𝑅 (𝑆 , 𝑡; 𝐾, 𝐵, 𝑇 ) be the European down-and-out and down-and-in options, respectively 𝑉𝑑∕𝑖 𝑡 with common barrier 𝐵, strike price 𝐾, rebate 𝑅 and expiry time 𝑇 . By writing 𝑅 𝑅 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉̃𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) 𝑅 𝑅 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉̃𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 )

where 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) and 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) are the European down-and-out and 𝑅 (𝑆 , 𝑡; 𝐵, 𝑇 ) down-and-in barrier option prices without rebates, respectively, whilst 𝑉̃𝑑∕𝑜 𝑡 𝑅 ̃ and 𝑉 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) are the corresponding option prices associated with immediate rebate 𝑑∕𝑖

at knock-out/knock-in time, show that 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑅 𝐶𝑑𝐴𝑚 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) 𝑉̃𝑑∕𝑜 𝑅 𝑉̃𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑅 𝑃𝑑𝐴𝑚 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 )

4.2.3 Further Barrier-Style Options

419

where 𝐶𝑑𝐴𝑚 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) and 𝑃𝑑𝐴𝑚 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) are the immediate-touch call and put options, respectively defined as 𝐶𝑑𝐴𝑚 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) =

(

𝑆𝑡 𝐵

(

)𝜆+ Φ(𝑑+ ) +

𝑆𝑡 𝐵

)𝜆− Φ(𝑑− )

and 𝑃𝑑𝐴𝑚 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 )

( =

𝑆𝑡 𝐵

(

)𝜆+ Φ(−𝑑+ ) +

𝑆𝑡 𝐵

)𝜆− Φ(−𝑑− )

such that

𝜆± =

−(𝑟 − 𝐷 − 12 𝜎 2 ) −

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟 𝜎2

√ log(𝑆𝑡 ∕𝐵) ± (𝑇 − 𝑡) (𝑟 − 𝐷 − 12 𝜎 2 ) + 2𝜎 2 𝑟 𝑑± = √ 𝜎 𝑇 −𝑡 and 𝑥

Φ(𝑥) =

1 − 21 𝑢2 𝑒 𝑑𝑢 ∫−∞ √2𝜋

is the cdf of a standard normal. Finally, verify that 𝑅 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) 𝑉𝑑∕𝑜 [( ) ( )𝜆− ] 𝑆𝑡 𝜆+ 𝑆𝑡 +𝑅 + . 𝐵 𝐵

(𝑆𝑡 , 𝑡) having the following options Solution: At time 𝑡 we first set up the portfolio Π𝑅 𝑑 𝑅 𝑅 Π𝑅 𝑑 (𝑆𝑡 , 𝑡) = 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 )

where 𝑅 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉̃𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ), 𝑉𝑑∕𝑜 𝑅 𝑅 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉̃𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ).

420

4.2.3 Further Barrier-Style Options

At any time 𝜏, 𝑡 ≤ 𝜏 ≤ 𝑇 , only one of the two barrier options can be active. If barrier 𝐵 is triggered that is max 𝑆𝜏 ≤ 𝐵 such that 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) vanishes then under the 𝑡≤𝜏≤𝑇

risk-neutral measure ℚ the rebate price is 𝑅 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉̃𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) 𝑉𝑑∕𝑜 [ ] | ℚ −𝑟(𝑇 −𝜏) | = 𝑅𝔼 1I𝑆𝜏 ≤𝐵 | ℱ𝑡 max 𝑒 𝑡≤𝜏≤𝑇 | = 𝑅 𝑃𝑑𝐴𝑚 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ). 𝑅 (𝑆 , 𝑡; 𝐵, 𝑇 ) vanishes so that Correspondingly, if max 𝑆𝜏 ≤ 𝐵 then the rebate price 𝑉̃𝑑∕𝑖 𝑡 𝑡≤𝜏≤𝑇

𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ). 𝑉𝑑∕𝑖

Conversely if 𝐵 is never triggered, max 𝑆𝜏 > 𝐵 then under the risk-neutral measure ℚ 𝑡≤𝜏≤𝑇

𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) 𝑉𝑑∕𝑜

and 𝑅 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉̃𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) 𝑉𝑑∕𝑖 [ ] | ℚ −𝑟(𝑇 −𝜏) | = 𝑅𝔼 1I𝑆𝜏 >𝐵 | ℱ𝑡 max 𝑒 𝑡≤𝜏≤𝑇 | = 𝑅 𝐶𝑑𝐴𝑚 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ).

Thus, we can conclude that the European down-and-out/in barrier option prices with immediate rebates are 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑅 𝑃𝑑𝐴𝑚 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) 𝑉𝑑∕𝑜 𝑅 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑅 𝐶𝑑𝐴𝑚 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ).

Finally, by subsituting the values of 𝐶𝑑𝐴𝑚 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) and 𝑃𝑑𝐴𝑚 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) (see Problems 3.2.3.15 and 3.2.3.16, pages 339–345) and using the identity of standard normal density, we can write 𝑅 𝑅 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = 𝑉𝑑∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) + 𝑉𝑑∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) 𝑉𝑑∕𝑜 [ ] +𝑅 𝑃𝑑𝐴𝑚 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) + 𝐶𝑑𝐴𝑚 (𝑆𝑡 , 𝑡; 𝐵, 𝑇 ) = 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) [( ) ( )𝜆− ] 𝑆𝑡 𝜆+ 𝑆𝑡 +𝑅 + . 𝐵 𝐵

4.2.3 Further Barrier-Style Options

421

6. Reflection Principle Properties for Black–Scholes Equation. Assume we are in the Black– Scholes world where at time 𝑡, the asset price 𝑆𝑡 follows the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 where 𝜇, 𝐷 and 𝜎 > 0 are the drift, dividend yield and volatility, respectively, 𝑊𝑡 is the standard Wiener process and there is a risk-free asset which earns interest at a constant rate 𝑟. Let 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be the price of a European option satisfying the Black–Scholes equation 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 with 𝑉 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = Ψ(𝑆𝑇 ) where Ψ(𝑆𝑇 ) is the option payoff for a strike price 𝐾 at expiry time 𝑇 . Show that 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) and 𝐾𝑒−𝑟(𝑇 −𝑡) satisfy the Black–Scholes equation. Hence, deduce from the reflection principle that for a constant 𝐵 ( 𝑆𝑡

𝑆𝑡 𝐵

(

)−1− 𝑟−𝐷

1 𝜎2 2

𝑒

−𝐷(𝑇 −𝑡)

and 𝐾

𝑆𝑡 𝐵

)1− 𝑟−𝐷

1 𝜎2 2

𝑒−𝑟(𝑇 −𝑡)

also satisfy the Black–Scholes equation. Solution: By setting 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) we have 𝜕𝑉 𝜕2𝑉 𝜕𝑉 = 𝑒−𝐷(𝑇 −𝑡) and =0 = 𝐷𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) , 𝜕𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡2 and substituting the partial derivatives into the Black–Scholes equation we have 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 = 𝐷𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) + (𝑟 − 𝐷)𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝑟𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) = 0. In contrast, by setting 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑒−𝑟(𝑇 −𝑡) and taking partial derivatives 𝜕𝑉 𝜕2𝑉 𝜕𝑉 = 0 and =0 = 𝑟𝐾𝑒−𝑟(𝑇 −𝑡) , 𝜕𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡2 and substituting them into the Black–Scholes equation we have 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0. + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 Therefore, we can conclude that both 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) and 𝐾𝑒−𝑟(𝑇 −𝑡) satisfy the Black–Scholes equation.

422

4.2.3 Further Barrier-Style Options

If 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) satisfies the Black–Scholes equation then from the reflection principle ( )2𝛼 ( 2 ) for a constant 𝐵, 𝑆𝑡 ∕𝐵 𝑉 𝐵 ∕𝑆𝑡 , 𝑡; 𝐾, 𝑇 also satisfies the Black–Scholes equation ( ) 1 𝑟−𝐷 provided 𝛼 = 1− 1 . 2 𝜎2 2

Hence, using the reflection principle (

𝑆𝑡 𝐵

)2𝛼 (

𝐵2 𝑆𝑡

)

( 𝑒

−𝐷(𝑇 −𝑡)

= 𝑆𝑡

𝑆𝑡 𝐵

)2(𝛼−1)

( 𝑒

−𝐷(𝑇 −𝑡)

= 𝑆𝑡

𝑆𝑡 𝐵

)−1− 𝑟−𝐷

1 𝜎2 2

𝑒−𝐷(𝑇 −𝑡)

and (

𝑆𝑡 𝐵

(

)2𝛼 𝐾𝑒

−𝑟(𝑇 −𝑡)

=𝐾

𝑆𝑡 𝐵

)1− 𝑟−𝐷

1 𝜎2 2

𝑒−𝑟(𝑇 −𝑡)

would yield an additional two solutions to the Black–Scholes equation. 7. Reflection Principle for Black Equation. Assume we are in the Black–Scholes world where at time 𝑡, the asset price 𝑆𝑡 follows the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 where 𝜇, 𝐷 and 𝜎 > 0 are the drift, dividend yield and volatility, respectively, 𝑊𝑡 is the standard Wiener process and there is a risk-free asset which earns interest at a constant rate 𝑟. Recall that the price of a futures contract at delivery time 𝑇 on an asset 𝑆𝑡 at time 𝑡 is 𝐹 (𝑡, 𝑇 ) = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) and for an option with value 𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) on a futures 𝐹 (𝑡, 𝑇 ) with strike price 𝐾, expiry time 𝜏 < 𝑇 the corresponding Black formula is 𝜕𝑉 𝜕2𝑉 1 + 𝜎 2 𝐹 (𝑡, 𝑇 )2 2 − 𝑟𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) = 0. 𝜕𝑡 2 𝜕𝐹 Show that if 𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) is a solution to the above equation, then so is ) ( (𝐹 (𝑡, 𝑇 )∕𝐵)𝑉 𝐵 2 ∕𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏 for any constant 𝐵. Hence, for a constant 𝐵, show that the prices of European call 𝐶𝑏𝑠 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) and put 𝑃𝑏𝑠 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) options on 𝐹 (𝑡, 𝑇 ) with the same strike 𝐾 and expiry 𝜏 < 𝑇 have the following relationships ) ) ( ( 𝐹 (𝑡, 𝑇 ) 𝐵2 𝐵2 𝐾 𝐶𝑏𝑠 𝐹 (𝑡, 𝑇 ), 𝑡; ,𝜏 = 𝑃𝑏𝑠 , 𝑡; 𝐾, 𝜏 𝐵 𝐾 𝐵 𝐹 (𝑡, 𝑇 ) and ) ) ( ( 𝐹 (𝑡, 𝑇 ) 𝐾 𝐵2 𝐵2 𝑃 ,𝜏 = 𝐶𝑏𝑠 , 𝑡; 𝐾, 𝜏 . 𝐹 (𝑡, 𝑇 ), 𝑡; 𝐵 𝑏𝑠 𝐾 𝐵 𝐹 (𝑡, 𝑇 )

4.2.3 Further Barrier-Style Options

Solution: Let 𝐹̂(𝑡, 𝑇 ) =

423

𝐵2 and define 𝐹 (𝑡, 𝑇 )

) 𝐹 (𝑡, 𝑇 ) ( ̂ 𝑉 𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏 . 𝑉̂ (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) = 𝐵 By differentiating 𝑉̂ (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) with respect to 𝑡 and 𝐹 (𝑡, 𝑇 ) we have 𝐹 (𝑡, 𝑇 ) 𝜕𝑉 𝜕 𝑉̂ = 𝜕𝑡 𝐵 𝜕𝑡 𝑉 (𝐹̂(𝑡, 𝑇 ), 𝑡) 𝐹 (𝑡, 𝑇 ) 𝜕𝑉 𝜕 𝐹̂ 𝜕 𝑉̂ = + 𝜕𝐹 𝐵 𝐵 𝜕 𝐹̂ 𝜕𝐹 ) ( ̂ 𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡) 𝐹 (𝑡, 𝑇 ) 𝜕𝑉 𝐵2 = + − 𝐵 𝐵 𝜕 𝐹̂ 𝐹 (𝑡, 𝑇 )2 𝑉 (𝐹̂(𝑡, 𝑇 ), 𝑡) 𝐵 𝜕𝑉 = − 𝐵 𝐹 (𝑡, 𝑇 ) 𝜕 𝐹̂ and 1 𝜕𝑉 𝜕 𝐹̂ 𝜕𝑉 𝐵 𝐵 𝜕 2 𝑉 𝜕 𝐹̂ 𝜕 2 𝑉̂ = + − 𝐵 𝜕 𝐹̂ 𝜕𝐹 𝜕𝐹 2 𝐹 (𝑡, 𝑇 )2 𝜕 𝐹̂ 𝐹 (𝑡, 𝑇 ) 𝜕 𝐹̂2 𝜕𝐹 )3 2 ( 𝐵 𝜕𝑉 𝜕𝑉 𝜕 𝑉 𝐵 𝐵 =− + + 𝐹 (𝑡, 𝑇 ) 𝐹 (𝑡, 𝑇 )2 𝜕 𝐹̂ 𝐹 (𝑡, 𝑇 )2 𝜕 𝐹̂ 𝜕 𝐹̂2 ( )3 2 𝐵 𝜕 𝑉 = . 𝐹 (𝑡, 𝑇 ) 𝜕 𝐹̂2 By substituting the above expressions into the Black formula we have 𝜕 2 𝑉̂ 𝜕 𝑉̂ 1 2 + 𝜎 𝐹 (𝑡, 𝑇 )2 2 − 𝑟𝑉̂ (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) 𝜕𝑡 2 𝜕𝐹 )3 2 ( 𝐹 (𝑡, 𝑇 ) 𝜕𝑉 𝐹 (𝑡, 𝑇 ) ̂ 𝜕 𝑉 1 𝐵 = −𝑟 + 𝜎 2 𝐹 (𝑡, 𝑇 )2 𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) 2 𝐵 𝜕𝑡 2 𝐹 (𝑡, 𝑇 ) 𝐵 ̂ 𝜕𝐹 ] [ )2 2 ( 𝐹 (𝑡, 𝑇 ) 𝜕𝑉 𝜕 𝑉 1 2 𝐵 − 𝑟𝑉 (𝐹̂(𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) = + 𝜎 𝐹 (𝑡, 𝑇 )2 𝐵 𝜕𝑡 2 𝐹 (𝑡, 𝑇 ) 𝜕 𝐹̂2 =

] [ 𝐹 (𝑡, 𝑇 ) 𝜕𝑉 𝜕2𝑉 1 − 𝑟𝑉 (𝐹̂(𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) + 𝜎 2 𝐹̂(𝑡, 𝑇 )2 𝐵 𝜕𝑡 2 𝜕 𝐹̂2

=0 since 𝜕𝑉 𝜕2𝑉 1 − 𝑟𝑉 (𝐹̂(𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) = 0. + 𝜎 2 𝐹̂(𝑡, 𝑇 )2 𝜕𝑡 2 𝜕 𝐹̂2 ) ( Hence, for a constant 𝐵, (𝐹 (𝑡, 𝑇 )∕𝐵)𝑉 𝐵 2 ∕𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏 also satisfies the Black formula.

424

4.2.3 Further Barrier-Style Options

Finally, to show the relationship between the call and put options on 𝐹 (𝑡, 𝑇 ) we note that at expiry 𝜏 < 𝑇 𝐶𝑏𝑠 (𝐹 (𝜏, 𝑇 ), 𝜏; 𝐾, 𝜏) = max {𝐹 (𝜏, 𝑇 ) − 𝐾, 0} 𝑃𝑏𝑠 (𝐹 (𝜏, 𝑇 ), 𝜏; 𝐾, 𝜏) = max {𝐾 − 𝐹 (𝜏, 𝑇 ), 0} . Thus, 𝐹 (𝜏, 𝑇 ) 𝑃𝑏𝑠 𝐵

(

𝐵2 , 𝜏; 𝐾, 𝜏 𝐹 (𝜏, 𝑇 )

)

{ } 𝐹 (𝜏, 𝑇 ) 𝐵2 max 𝐾 − ,0 𝐵 𝐹 (𝜏, 𝑇 ) { } 𝐹 (𝜏, 𝑇 ) 𝐾 𝐵2 = max 𝐹 (𝜏, 𝑇 ) − ,0 𝐵 𝐹 (𝜏, 𝑇 ) 𝐾 { } 2 𝐾 𝐵 = max 𝐹 (𝜏, 𝑇 ) − ,0 𝐵 𝐾 ) ( 𝐾 𝐵2 = 𝐶𝑏𝑠 𝐹 (𝜏, 𝑇 ), 𝜏; ,𝜏 𝐵 𝐾

=

and similarly 𝐹 (𝜏, 𝑇 ) 𝐶𝑏𝑠 𝐵

(

𝐵2 , 𝜏; 𝐾, 𝜏 𝐹 (𝜏, 𝑇 )

) = = = =

{ } 𝐹 (𝜏, 𝑇 ) 𝐵2 max − 𝐾, 0 𝐵 𝐹 (𝜏, 𝑇 ) { 2 } 𝐹 (𝜏, 𝑇 ) 𝐾 𝐵 max − 𝐹 (𝜏, 𝑇 ), 0 𝐵 𝐹 (𝜏, 𝑇 ) 𝐾 { 2 } 𝐾 𝐵 max − 𝐹 (𝜏, 𝑇 ), 0 𝐵 𝐾 ) ( 𝐾 𝐵2 𝑃𝑏𝑠 𝐹 (𝜏, 𝑇 ), 𝜏; ,𝜏 . 𝐵 𝐾

Hence, for all 𝑡 < 𝜏 we can deduce that ) ) ( ( 𝐹 (𝑡, 𝑇 ) 𝐾 𝐵2 𝐵2 𝐶 ,𝜏 = 𝑃𝑏𝑠 , 𝑡; 𝐾, 𝜏 𝐹 (𝑡, 𝑇 ), 𝑡; 𝐵 𝑏𝑠 𝐾 𝐵 𝐹 (𝑡, 𝑇 ) and ) ) ( ( 𝐹 (𝑡, 𝑇 ) 𝐾 𝐵2 𝐵2 𝑃𝑏𝑠 𝐹 (𝑡, 𝑇 ), 𝑡; ,𝜏 = 𝐶𝑏𝑠 , 𝑡; 𝐾, 𝜏 . 𝐵 𝐾 𝐵 𝐹 (𝑡, 𝑇 )

8. Knock-Out and Knock-In Options on Futures. Let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and assume we are in the Black–Scholes world where at time 𝑡, the asset price 𝑆𝑡 follows the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 where 𝜇, 𝐷 and 𝜎 > 0 are the drift, dividend yield and volatility, respectively, and there is a risk-free asset which earns interest at a constant rate 𝑟.

4.2.3 Further Barrier-Style Options

425

Given that the price of a futures contract at delivery time 𝑇 on an asset 𝑆𝑡 at time 𝑡 is 𝐹 (𝑡, 𝑇 ) = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) and under the risk-neutral measure ℚ, find the European up-andout/in and down-and-out/in call and put options on futures at time 𝑡, expiring at time 𝜏 ≤ 𝑇 with strike price 𝐾 and barrier 𝐵. Solution: From Girsanov’s theorem, under the risk-neutral measure ℚ, 𝑆𝑡 follows

where 𝑊𝑡ℚ =

(𝜇 − 𝑟) 𝜎

𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡 𝑡 + 𝑊𝑡 is a ℚ-standard Wiener process. For 𝑇 > 𝑡 and given

𝐹 (𝑡, 𝑇 ) = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) , from Taylor’s theorem 𝑑𝐹 (𝑡, 𝑇 ) =

𝜕𝐹 (𝑡, 𝑇 ) 𝜕𝐹 (𝑡, 𝑇 ) 1 𝜕 2 𝐹 (𝑡, 𝑇 ) 𝑑𝑆𝑡 + (𝑑𝑡)2 + ⋯ 𝑑𝑡 + 𝜕𝑡 𝜕𝑆𝑡 2 𝜕𝑡2

and by applying It¯o’s lemma we have 𝑑𝐹 (𝑡, 𝑇 ) = 𝜎𝑑𝑊𝑡ℚ . 𝐹 (𝑡, 𝑇 ) Hence, for 𝜏 ≤ 𝑇 the solution to the above SDE is ̂

𝐹 (𝜏, 𝑇 ) = 𝐹 (𝑡, 𝑇 )𝑒𝜎 𝑊𝜏−𝑡 ̂𝜏−𝑡 = − 1 𝜎(𝑇 − 𝑡) + 𝑊 ℚ . where 𝑊 𝜏−𝑡 2 By comparing both the SDEs for an asset price 𝑆𝑡 and the futures price 𝐹 (𝑡, 𝑇 ), the equivalent up-and-out/in and down-and-out/in call and put option prices for futures expiring at 𝜏 ≤ 𝑇 can by setting 𝑟 − 𝐷 = 0 and replacing 𝑆𝑡 with 𝐹 (𝑡, 𝑇 ) so that 𝐶𝑢∕𝑜 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) = 𝐶𝑏𝑠 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) [ ] − 𝐶𝑏𝑠 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐵, 𝜏) + (𝐵 − 𝐾)𝐶𝑑 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐵, 𝜏) { ) ( 𝐹 (𝑡, 𝑇 ) 𝐵2 − 𝐶𝑏𝑠 , 𝑡; 𝐾, 𝜏 𝐵 𝐹 (𝑡, 𝑇 ) [ ) ( 𝐵2 − 𝐶𝑏𝑠 , 𝑡; 𝐵, 𝜏 𝐹 (𝑡, 𝑇 ) )]} ( 𝐵2 +(𝐵 − 𝐾)𝐶𝑑 , 𝑡; 𝐵, 𝜏, 𝐵 𝐹 (𝑡, 𝑇 ) 𝐶𝑢∕𝑖 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) = 𝐶𝑏𝑠 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐵, 𝜏) + (𝐵 − 𝐾)𝐶𝑑 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐵, 𝜏) { ) ( 𝐹 (𝑡, 𝑇 ) 𝐵2 𝐶𝑏𝑠 , 𝑡; 𝐾, 𝜏 + 𝐵 𝐹 (𝑡, 𝑇 ) [ ) ( 𝐵2 − 𝐶𝑏𝑠 , 𝑡; 𝐵, 𝜏 𝐹 (𝑡, 𝑇 ) )]} ( 𝐵2 +(𝐵 − 𝐾)𝐶𝑑 , 𝑡; 𝐵, 𝜏 𝐹 (𝑡, 𝑇 )

426

4.2.3 Further Barrier-Style Options

⎧ 𝐶𝑏𝑠 (𝐹 (𝑡, 𝑇 ), 𝑡;(𝐾, 𝜏) ) ⎪ 𝐹 (𝑡, 𝑇 ) 𝐵2 ⎪ − 𝐵 𝐶𝑏𝑠 𝐹 (𝑡, 𝑇 ) , 𝑡; 𝐾, 𝜏 ⎪ ⎪ ⎪ 𝐶 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝜏, 𝐵) 𝐶𝑑∕𝑜 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) = ⎨ 𝑏𝑠 +(𝐵 − 𝐾)𝐶 [ 𝑑 (𝐹((𝑡, 𝑇 ),2 𝑡; 𝐵, 𝜏) ) ⎪ 𝐹 (𝑡, 𝑇 ) 𝐵 ⎪− 𝐶𝑏𝑠 , 𝑡; 𝐵, 𝜏 ⎪ 𝐵 𝐹 (𝑡, 𝑇) )] ( ⎪ 2 𝐵 ⎪ +(𝐵 − 𝐾)𝐶𝑑 , 𝑡; 𝐵, 𝜏 ⎩ 𝐹 (𝑡, 𝑇 ) ) ( 𝐵2 ⎧ 𝐹 (𝑡, 𝑇 ) 𝐶 , 𝑡; 𝐾, 𝜏 𝑏𝑠 ⎪ 𝐵 𝐹 (𝑡, 𝑇 ) ⎪ ⎪ ⎪ 𝐶𝑏𝑠 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) ⎪ −𝐶𝑏𝑠 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐵, 𝜏) 𝐶𝑑∕𝑖 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) = ⎨ −(𝐵 − 𝐾)𝐶 [ 𝑑 (𝐹((𝑡, 𝑇 ),2 𝑡; 𝐵, 𝜏) ) ⎪ 𝐹 (𝑡, 𝑇 ) 𝐵 ⎪+ 𝐶𝑏𝑠 , 𝑡; 𝐵, 𝜏 ⎪ 𝐵 𝐹 (𝑡, 𝑇) )] ( ⎪ 2 𝐵 ⎪ +(𝐵 − 𝐾)𝐶𝑑 , 𝑡; 𝐵, 𝜏 ⎩ 𝐹 (𝑡, 𝑇 ) ⎧ 𝑃𝑏𝑠 (𝐹 (𝑡, 𝑇 ), 𝑡;(𝐾, 𝜏) ) ⎪ 𝐹 (𝑡, 𝑇 ) 𝐵2 − 𝑃 , 𝑡; 𝐾, 𝜏 ⎪ 𝑏𝑠 𝐵 𝐹 (𝑡, 𝑇 ) ⎪ ⎪ ⎪ 𝑃 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐵, 𝜏) 𝑃𝑢∕𝑜 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) = ⎨ 𝑏𝑠 +(𝐾 − 𝐵)𝑃 [ 𝑑 (𝐹((𝑡, 𝑇 ),2 𝑡; 𝐵, 𝜏) ) ⎪ 𝐹 (𝑡, 𝑇 ) 𝐵 ⎪− 𝑃𝑏𝑠 , 𝑡; 𝐵, 𝜏 ⎪ 𝐵 𝐹 (𝑡, 𝑇) )] ( ⎪ 2 𝐵 ⎪ +(𝐾 − 𝐵)𝑃𝑑 , 𝑡; 𝐵, 𝜏 ⎩ 𝐹 (𝑡, 𝑇 ) ) ( ⎧ 𝐹 (𝑡, 𝑇 ) 𝐵2 ⎪ 𝐵 𝑃𝑏𝑠 𝐹 (𝑡, 𝑇 ) , 𝑡; 𝐾, 𝜏 ⎪ ⎪ [ ( ) ⎪ 𝐹 (𝑡, 𝑇 ) 𝐵2 𝑃𝑏𝑠 , 𝑡; 𝐵, 𝜏 𝑃𝑢∕𝑖 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) = ⎨ )] (𝐹 (𝑡, 2𝑇 ) ⎪ 𝐵 𝐵 ⎪ +(𝐾 − 𝐵)𝑃 , 𝑡; 𝐵, 𝜏 𝑑 ⎪ 𝐹 (𝑡, 𝑇 ) ⎪ −(𝐾 − 𝐵)𝑃 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐵, 𝜏) 𝑑 ⎩

if 𝐵 ≤ 𝐾

if 𝐵 > 𝐾

if 𝐵 ≤ 𝐾

if 𝐵 > 𝐾

if 𝐵 ≥ 𝐾

if 𝐵 < 𝐾

if 𝐵 ≥ 𝐾

if 𝐵 < 𝐾

4.2.3 Further Barrier-Style Options

427

𝑃𝑑∕𝑜 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) = 𝑃𝑏𝑠 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) [ ] − 𝑃𝑏𝑠 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐵, 𝜏) + (𝐾 − 𝐵)𝑃𝑑 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐵, 𝜏) { ( ) 𝐹 (𝑡, 𝑇 ) 𝐵2 𝑃𝑏𝑠 , 𝑡; 𝐾, 𝜏 − 𝐵 𝐹 (𝑡, 𝑇 ) [ ( ) 2 𝐵 − 𝑃𝑏𝑠 , 𝑡; 𝐵, 𝜏 𝐹 (𝑡, 𝑇 ) )]} ( 𝐵2 +(𝐾 − 𝐵)𝑃𝑑 , 𝑡; 𝐵, 𝜏 𝐹 (𝑡, 𝑇 )

𝑃𝑑∕𝑖 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝜏) = 𝑃𝑏𝑠 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐵, 𝜏) + (𝐾 − 𝐵)𝑃𝑑 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐵, 𝜏) { ( ) 𝐹 (𝑡, 𝑇 ) 𝐵2 𝑃𝑏𝑠 , 𝑡; 𝐾, 𝜏 + 𝐵 𝐹 (𝑡, 𝑇 ) [ ( ) 2 𝐵 − 𝑃𝑏𝑠 , 𝑡; 𝐵, 𝜏 𝐹 (𝑡, 𝑇 ) )]} ( 𝐵2 +(𝐾 − 𝐵)𝐶𝑑 , 𝑡; 𝐵, 𝜏 𝐹 (𝑡, 𝑇 ) where 𝐶𝑏𝑠 (𝑋, 𝑡; 𝑌 , 𝜏) and 𝐶𝑑 (𝑋, 𝑡; 𝑌 , 𝜏) are the vanilla and digital call options on futures defined as [

(

𝐶𝑏𝑠 (𝑋, 𝑡; 𝑌 , 𝜏) = 𝑒−𝑟(𝑇 −𝑡) 𝑋Φ −𝑌 Φ

log(𝑋∕𝑌 ) + 12 𝜎 2 (𝜏 − 𝑡)

)

𝜎(𝜏 − 𝑡) ( )] log(𝑋∕𝑌 ) − 12 𝜎 2 (𝜏 − 𝑡) 𝜎(𝜏 − 𝑡)

and ( 𝐶𝑑 (𝑋, 𝑡; 𝑌 , 𝜏) = 𝑒

−𝑟(𝑇 −𝑡)

Φ

log(𝑋∕𝑌 ) − 12 𝜎 2 (𝜏 − 𝑡)

)

𝜎(𝜏 − 𝑡)

and 𝑃𝑏𝑠 (𝑋, 𝑡; 𝑌 , 𝜏) and 𝑃𝑑 (𝑋, 𝑡; 𝑌 , 𝜏) are the vanilla and digital put options on futures defined as [ 𝑃𝑏𝑠 (𝑋, 𝑡; 𝑌 , 𝜏) = 𝑒

−𝑟(𝑇 −𝑡)

( −𝑋Φ

𝑌Φ

(

− log(𝑋∕𝑌 ) + 12 𝜎 2 (𝜏 − 𝑡) 𝜎(𝜏 − 𝑡)

− log(𝑋∕𝑌 ) −

1 2 𝜎 (𝜏 2

𝜎(𝜏 − 𝑡)

− 𝑡)

)]

)

428

4.2.3 Further Barrier-Style Options

and ( 𝑃𝑑 (𝑋, 𝑡; 𝑌 , 𝜏) = 𝑒

−𝑟(𝑇 −𝑡)

𝑥

respectively. Note that Φ(𝑥) =

Φ

− log(𝑋∕𝑌 ) + 12 𝜎 2 (𝜏 − 𝑡)

)

𝜎(𝜏 − 𝑡)

1 − 1 𝑢2 √ 𝑒 2 𝑑𝑢 is the cdf of a standard normal. 2𝜋

∫−∞

9. Knock-Out Equity Accumulator. Assume we are in the Black–Scholes world where at time 𝑡, the asset price 𝑆𝑡 follows the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 where 𝜇, 𝐷 and 𝜎 > 0 are the drift, dividend yield and volatility, respectively, 𝑊𝑡 is the ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and there is a risk-free asset which earns interest at a constant rate 𝑟. An accumulator is a product that accumulates a specific asset at a fixed strike price 𝐾 every day until the contract expires at time 𝑇 or the asset price rises above a preset barrier level 𝐵. Here, 𝐾 is set below the asset price 𝑆𝑡 at initial time 𝑡 < 𝑇 and 𝐾 < 𝑆𝑡 < 𝐵. By denoting 𝑁 as the number of days between 𝑡 and 𝑇 , show that under the risk-neutral measure ℚ the present value of this product at time 𝑡 is 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) = (

𝑁 ∑ [ 𝑖=1

] 𝐶𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑡𝑖 ) − 𝑃𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑡𝑖 )

)

𝑖−1 (𝑇 − 𝑡), 𝑖 = 1, 2, … , 𝑁, 𝐶𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑡𝑖 ) and 𝑃𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑡𝑖 ) where 𝑡𝑖 = 𝑡 + 𝑁−1 are the up-and-out European call and put options evaluated at time 𝑡 with common strike 𝐾, barrier 𝐵 and expiry time 𝑡𝑖 , respectively.

Solution: From the definition of the equity accumulator, the payoff on date 𝑡𝑖 is a futures contract with a knock-out barrier 𝐵 where it can be decomposed into a long up-and-out European call option and a short up-and-out European put option (see Figure 4.9) for the construction of the accumulator payoff. By denoting Ψ(𝑆𝑡𝑖 ) as the payoff on date 𝑡𝑖 , 𝑖 = 1, 2, … , 𝑁 we can write { } Ψ(𝑆𝑡𝑖 ) = max 𝑆𝑡𝑖 − 𝐾, 0 1I{ max 𝑆 < 𝐵} 𝑡≤𝑢≤𝑡𝑖 𝑢 { } − max 𝐾 − 𝑆𝑡𝑖 , 0 1I{ max 𝑆 < 𝐵} 𝑢 𝑡≤𝑢≤𝑡𝑖

and the total accumulated payoff is therefore Ψ(𝑆𝑡1 , 𝑆𝑡2 , … , 𝑆𝑡𝑛 ) =

𝑁 ∑ 𝑖=1

Ψ(𝑆𝑡𝑖 ).

4.2.3 Further Barrier-Style Options

429 −

−

−

−

−

Figure 4.9

Accumulator payoff diagram.

From the risk-neutral measure ℚ, the value of the accumulator contract at time 𝑡 is [ ] | 𝑒−𝑟(𝑡𝑖 −𝑡) 𝔼ℚ Ψ(𝑆𝑡𝑖 )| ℱ𝑡 | 𝑖=1 [ ] 𝑁 } { | ∑ | −𝑟(𝑡𝑖 −𝑡) ℚ 𝑒 𝔼 max 𝑆𝑡𝑖 − 𝐾, 0 1I{ max 𝑆 < 𝐵} | ℱ𝑡 = | 𝑡≤𝑢≤𝑡𝑖 𝑢 | 𝑖=1 [ ] 𝑁 } { | ∑ | −𝑟(𝑡𝑖 −𝑡) ℚ − 𝑒 𝔼 max 𝐾 − 𝑆𝑡𝑖 , 0 1I{ max 𝑆 < 𝐵} | ℱ𝑡 | 𝑡≤𝑢≤𝑡𝑖 𝑢 | 𝑖=1

𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) =

=

𝑁 ∑

𝑁 ∑ [ 𝑖=1

] 𝐶𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑡𝑖 ) − 𝑃𝑢∕𝑜 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑡𝑖 ) .

10. Merton Model for Default of a Company II. At time 𝑡, we assume that the asset 𝐴𝑡 of a company satisfies the SDE 𝑑𝐴𝑡 = 𝜇𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝐴𝑡

430

4.2.3 Further Barrier-Style Options

where 𝜇 is the drift parameter, 𝜎 > 0 is the volatility and {𝑊𝑡 : 𝑡 ≥ 0} is a standard Wiener process on the probability space (Ω, ℱ , ℙ). The risk-free interest rate is denoted by 𝑟. In financial accounting, the asset 𝐴𝑡 is a combination of equity 𝐸𝑡 and debt 𝐷𝑡 so that 𝐴𝑡 = 𝐸𝑡 + 𝐷𝑡 where, if at any time 𝑡 ≤ 𝑇 , the company will be in default if 𝐴𝑡 < 𝐹 where 𝐹 > 0 is the face value and the debt holders will receive 𝐴𝑡 whilst the equity holders will receive nothing. In contrast, if the company is not in default by time 𝑡 ≤ 𝑇 , the debt holders will receive 𝐹 and the equity holders will receive the rest of the value of the company. By constructing the payoff diagrams for 𝐸𝑇 and 𝐷𝑇 , and using the reflection principle, find the values of 𝐸𝑡 and 𝐷𝑡 for all 𝑡 ≤ 𝑇 under the Black–Scholes framework. Solution: Following Problem 2.2.2.30 (page 158) at terminal time 𝑇 , the payoff diagram for equity shareholders is given in Figure 4.10. Because the company can default at any time 𝑡 ≤ 𝑇 , we can write 𝐸𝑇 as 𝐸𝑇 = max{𝐴𝑇 − 𝐹 , 0}1I{ min 𝐴 > 𝐹 } 𝑢 𝑡≤𝑢≤𝑇

which is a down-and-out European call option on the assets with strike and barrier equal to the face value. By discounting the entire payoff under the risk-neutral measure ℚ and using the reflection principle, we can write 𝐸𝑡 as ( 𝐸𝑡 = 𝐶(𝐴𝑡 , 𝑡; 𝐹 , 𝑇 ) −

Figure 4.10

𝐴𝑡 𝐹

(

)2𝛼 𝐶

𝐹2 , 𝑡; 𝐹 , 𝑇 𝐴𝑡

Payment to equity holders at time 𝑇 .

)

4.2.3 Further Barrier-Style Options

Figure 4.11

( 1 where 𝛼 = 2

1−

𝑟 1 2 𝜎 2

431

Payment to debt holders at time 𝑇 .

) , 𝑟 is the risk-free interest rate,

(

) log(𝑋∕𝑌 ) + (𝑟 + 12 𝜎 2 )(𝑇 − 𝑡) 𝐶 (𝑋, 𝑡; 𝑌 , 𝑇 ) = 𝑋Φ √ 𝜎 𝑇 −𝑡 ) ( log(𝑋∕𝑌 ) + (𝑟 − 12 𝜎 2 )(𝑇 − 𝑡) −𝑟(𝑇 −𝑡) −𝑌 𝑒 Φ √ 𝜎 𝑇 −𝑡 𝑥

1 − 1 𝑢2 √ 𝑒 2 𝑑𝑢 is the cdf of a standard normal. ∫−∞ 2𝜋 As for payment to the debt holders, the payoff diagram at time 𝑇 is given in Figure 4.11. Given that the debt at terminal time 𝑇 is

and Φ(𝑥) =

𝐷𝑇 = 𝐴𝑇 − 𝐸𝑇 by discounting the payoff under the risk-neutral measure ℚ and from the results of the equity holders at time 𝑡 𝐷𝑡 = 𝐴𝑡 − 𝐸𝑡 = 𝐴𝑡 − 𝐶(𝐴𝑡 , 𝑡; 𝐹 , 𝑇 ) +

(

𝐴𝑡 𝐹

(

)2𝛼 𝐶

𝐹2 , 𝑡; 𝐹 , 𝑇 𝐴𝑡

) .

432

4.2.3 Further Barrier-Style Options

11. Consider a binomial tree model for an underlying asset process {𝑆𝑛 : 0 ≤ 𝑛 ≤ 3} following the Cox–Ross–Rubinstein model. With the initial asset price 𝑆0 = 100, let

𝑆𝑛+1

⎧ 𝑢𝑆 ⎪ 𝑛 =⎨ ⎪ 𝑑𝑆𝑛 ⎩

with probability 𝜋 with probability 1 − 𝜋

√

where 𝑢 = 𝑒𝜎 Δ𝑡 and 𝑑 = 1∕𝑢, with 𝜎 the volatility and Δ𝑡 the binomial time step. Assuming the risk-free interest rate 𝑟 = 3%, continuous dividend yield 𝐷 = 1% and volatility 𝜎 = 4%, find the price of a European-style up-and-out barrier call option at time 𝑡 = 0 with strike 𝐾 = 95, barrier 𝐵 = 102 with expiry time 𝑇 = 1 year in a 3-period binomial tree model. Solution: Given 𝑆0 = 100, 𝐾 = 95, 𝑟 = 0.03, 𝐷 = 0.01, 𝜎 = 0.04 and time step Δ𝑡 = 13 , we have 𝑢 = 𝑒𝜎

√ Δ𝑡

=𝑒

0.04 √ 3

= 1.0233

and 𝑑 = 𝑒−𝜎

√ Δ𝑡

=

1 1 = = 0.9772. 𝑢 1.0233

Therefore, the risk-neutral probabilities are 𝜋=

𝑒(𝑟−𝐷)Δ𝑡 − 𝑑 = 0.6390 𝑢−𝑑

and 1 − 𝜋 = 1 − 0.6390 = 0.3610. The binomial tree in Figure 4.12 shows the price movement of 𝑆0 in a 3-period binomial model. By setting 𝑆𝑖(𝑗) = 𝑢𝑗 𝑑 𝑖−𝑗 𝑆0 ,

𝑖 = 0, 1, … , 𝑛,

𝑗 = 0, 1, … , 𝑖

the up-and-out call option price at each of the lattice points is

𝑉𝑖(𝑗)

[ ] ⎧ 𝑒−𝑟Δ𝑡 𝜋𝑉 (𝑗+1) + (1 − 𝜋)𝑉 (𝑗) 𝑖+1 𝑖+1 ⎪ =⎨ ⎪0 ⎩

where 𝑖 = 0, 1, … , 𝑛 and 𝑗 = 0, 1, … , 𝑖.

if 𝑆𝑖(𝑗) < 𝐵 if 𝑆𝑖(𝑗) ≥ 𝐵

4.2.3 Further Barrier-Style Options

Figure 4.12

433

A 3-period binomial tree model.

At terminal node 𝑖 = 𝑛

𝑉𝑛(𝑗)

{ } ⎧ max 𝑆 (𝑗) − 𝐾, 0 if 𝑆𝑛(𝑗) < 𝐵 𝑛 ⎪ (𝑗) = Ψ(𝑆𝑛 ) = ⎨ ⎪0 if 𝑆𝑛(𝑗) ≥ 𝐵. ⎩

At time period 𝑛 = 3 (i.e., option expiry time 𝑇 = 1 year) 𝑉3(0) = Ψ(𝑆3(0) ) ⎧ max {𝑑 3 𝑆 − 𝐾, 0} if 𝑑 3 𝑆 < 𝐵 0 0 ⎪ =⎨ ⎪0 if 𝑑 3 𝑆0 ≥ 𝐵 ⎩ = max{0.97723 × 100 − 95, 0} =0 𝑉3(1) = Ψ(𝑆3(1) ) ⎧ max {𝑢𝑑 2 𝑆 − 𝐾, 0} if 𝑢𝑑 2 𝑆 < 𝐵 0 0 ⎪ =⎨ ⎪0 if 𝑢𝑑 2 𝑆0 ≥ 𝐵 ⎩ = max{1.0233 × 0.97722 × 100 − 95, 0} = 0.27169

434

4.2.3 Further Barrier-Style Options

𝑉3(2) = Ψ(𝑆3(2) ) ⎧ max {𝑢2 𝑑𝑆 − 𝐾, 0} if 𝑢2 𝑑𝑆 < 𝐵 0 0 ⎪ =⎨ ⎪0 if 𝑢2 𝑑𝑆0 ≥ 𝐵 ⎩ =0 𝑉3(3) = Ψ(𝑆3(3) ) ⎧ max {𝑢3 𝑆 − 𝐾, 0} if 𝑢3 𝑆 < 𝐵 0 0 ⎪ =⎨ ⎪0 if 𝑢3 𝑆0 ≥ 𝐵 ⎩ = 0. At time period 𝑛 = 2

𝑉2(0)

[ ] ⎧ 𝑒−𝑟Δ𝑡 𝜋𝑉 (1) + (1 − 𝜋)𝑉 (0) if 𝑑 2 𝑆0 < 𝐵 3 3 ⎪ =⎨ ⎪0 if 𝑑 2 𝑆0 ≥ 𝐵 ⎩ = 𝑒−

0.03 3

[0.6390 × 2.7169 + 0.3610 × 0] = 1.7188

𝑉2(1)

[ ] ⎧ 𝑒−𝑟Δ𝑡 𝜋𝑉 (2) + (1 − 𝜋)𝑉 (1) 3 3 ⎪ =⎨ ⎪0 ⎩

if 𝑢𝑑𝑆0 < 𝐵 if 𝑢𝑑𝑆0 ≥ 𝐵

− 0.03 3

=𝑒

[0.6390 × 0 + 0.3610 × 2.7169] = 0.9710

𝑉2(2)

[ ] ⎧ 𝑒−𝑟Δ𝑡 𝜋𝑉 (1) + (1 − 𝜋)𝑉 (0) 3 3 ⎪ =⎨ ⎪0 ⎩

if 𝑢2 𝑆0 < 𝐵 if 𝑢2 𝑆0 ≥ 𝐵

= 0. At time period 𝑛 = 1

𝑉1(0)

[ ] ⎧ 𝑒−𝑟Δ𝑡 𝜋𝑉 (1) + (1 − 𝜋)𝑉 (0) if 𝑑𝑆0 < 𝐵 2 2 ⎪ =⎨ ⎪0 if 𝑑𝑆0 ≥ 𝐵 ⎩ = 𝑒−

0.03 3

[0.6390 × 0.9710 + 0.3610 × 1.7188] = 1.2286

4.2.3 Further Barrier-Style Options

𝑉1(1)

435

[ ] ⎧ 𝑒−𝑟Δ𝑡 𝜋𝑉 (2) + (1 − 𝜋)𝑉 (1) if 𝑢𝑆0 < 𝐵 2 2 ⎪ =⎨ ⎪0 if 𝑢𝑆0 ≥ 𝐵 ⎩ = 0.

Finally, at time period 𝑛 = 0

𝑉0(0)

[ ] ⎧ 𝑒−𝑟Δ𝑡 𝜋𝑉 (1) + (1 − 𝜋)𝑉 (0) 1 1 ⎪ =⎨ ⎪0 ⎩

if 𝑆0 < 𝐵 if 𝑆0 ≥ 𝐵

− 0.03 3

=𝑒

[0.6390 × 0 + 0.3610 × 1.2286] = 0.4391. Therefore, the price of the up-and-out call option based on a 3-period binomial model is 𝑉0(0) = 0.4391. 12. Table 4.1 shows the up-and-in call option 𝐶𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 ) prices on an asset 𝑆𝑡 = $30, strike 𝐾 = $35, time to expiry 𝑇 − 𝑡 = 9 months for various barrier values 𝐵. Table 4.1 European call up-and-in prices for different barriers. 𝐵

𝐶𝑢∕𝑖 (𝑆𝑡 , 𝑡; 𝐾, 𝐵, 𝑇 )

0 10 20 30 40 50 60

1.76119 1.76100 1.76076 1.76061 1.66433 0.75711 0.19322

By considering the following payoff ⎧ 3 max{𝑆𝑇 − 𝐾, 0} if 0 ≤ 𝑆𝑇 < 40 ⎪ ⎪ Ψ(𝑆𝑇 ) = ⎨ 2 max{𝑆𝑇 − 𝐾, 0} if 40 ≤ 𝑆𝑇 < 60 ⎪ ⎪0 if 𝑆𝑇 ≥ 60 ⎩ calculate the price of this 9-month 35-strike European “partial barrier” option. Solution: By setting 𝐵1 = 40 and 𝐵2 = 60, the payoff Ψ(𝑆𝑇 ) can be constructed by adding two payoffs Ψ1 (𝑆𝑇 ) = 3 max{𝑆𝑇 − 𝐾, 0}1I0≤𝑆𝑇 𝑡, let the arithmetic average rate (fixed strike) Asian call and put options be [ ] (𝑎) (𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐴𝑛 (𝜏, 𝑇 ) − 𝐾, 0}|| ℱ𝑡 𝐶𝑎𝑟 [ ] (𝑎) (𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐾 − 𝐴𝑛 (𝜏, 𝑇 ), 0}|| ℱ𝑡 𝑃𝑎𝑟 where the expectation is defined under the risk-neutral measure ℚ. In contrast, let the corresponding geometric average rate (fixed strike) Asian call and put options be [ ] (𝑔) (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐺𝑛 (𝜏, 𝑇 ) − 𝐾, 0}|| ℱ𝑡 𝐶𝑎𝑟 [ ] (𝑔) (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐾 − 𝐺𝑛 (𝜏, 𝑇 ), 0}|| ℱ𝑡 . 𝑃𝑎𝑟 Show that (𝑔) (𝑎) 𝐶𝑎𝑟 (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) ≤ 𝐶𝑎𝑟 (𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 )

{ [ ] [ ]} (𝑔) (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) + 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 )|| ℱ𝑡 − 𝔼ℚ 𝐺𝑛 (𝜏, 𝑇 )|| ℱ𝑡 ≤ 𝐶𝑎𝑟 and { [ ] [ ]} (𝑔) 𝑃𝑎𝑟 (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) − 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 )|| ℱ𝑡 − 𝔼ℚ 𝐺𝑛 (𝜏, 𝑇 )|| ℱ𝑡 (𝑔) (𝑎) (𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) ≤ 𝑃𝑎𝑟 (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ). ≤ 𝑃𝑎𝑟

Solution: From Problem 5.2.1.3 (see page 444) we have 𝐺𝑛 (𝜏, 𝑇 ) ≤ 𝐴𝑛 (𝜏, 𝑇 ), therefore [ ] [ ] 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐺𝑛 (𝜏, 𝑇 ) − 𝐾, 0}|| ℱ𝑡 ≤ 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐴𝑛 (𝜏, 𝑇 ) − 𝐾, 0}|| ℱ𝑡

5.2.1 Discrete Sampling

447

or (𝑔) (𝑎) (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) ≤ 𝐶𝑎𝑟 (𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ). 𝐶𝑎𝑟

In addition, we note that max{𝐴𝑛 (𝜏, 𝑇 ) − 𝐾, 0} = max{𝐴𝑛 (𝜏, 𝑇 ) − 𝐺𝑛 (𝜏, 𝑇 ) + 𝐺𝑛 (𝜏, 𝑇 ) − 𝐾, 0} ≤ 𝐴𝑛 (𝜏, 𝑇 ) − 𝐺𝑛 (𝜏, 𝑇 ) + max{𝐺𝑛 (𝜏, 𝑇 ) − 𝐾, 0}. Thus, [ ] (𝑎) (𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐴𝑛 (𝜏, 𝑇 ) − 𝐾, 0}|| ℱ𝑡 𝐶𝑎𝑟 [ ] ≤ 𝑒−𝑟(𝑇 −𝑇 ) 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 ) − 𝐺𝑛 (𝜏, 𝑇 )|| ℱ𝑡 [ ] +𝑒−𝑟(𝑇 −𝑇 ) 𝔼ℚ max{𝐺𝑛 (𝜏, 𝑇 ) − 𝐾, 0}|| ℱ𝑡 { [ ] [ ]} = 𝑒−𝑟(𝑇 −𝑇 ) 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 )|| ℱ𝑡 − 𝔼ℚ 𝐺𝑛 (𝜏, 𝑇 )|| ℱ𝑡 (𝑔) (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ). +𝐶𝑎𝑟 In contrast, for the average rate put options [ ] [ ] 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐾 − 𝐴𝑛 (𝜏, 𝑇 ), 0}|| ℱ𝑡 ≤ 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐾 − 𝐺𝑛 (𝜏, 𝑇 ), 0}|| ℱ𝑡 or (𝑎) (𝑔) (𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) ≤ 𝑃𝑎𝑟 (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ). 𝑃𝑎𝑟

Further, we note that max{𝐾 − 𝐴𝑛 (𝜏, 𝑇 ), 0} = max{𝐾 − 𝐺𝑛 (𝜏, 𝑇 ) + 𝐺𝑛 (𝜏, 𝑇 ) − 𝐴𝑛 (𝜏, 𝑇 ), 0} = max{𝐾 − 𝐺𝑛 (𝜏, 𝑇 ) − (𝐴𝑛 (𝜏, 𝑇 ) − 𝐺𝑛 (𝜏, 𝑇 )), 0} ≥ max{𝐾 − 𝐺𝑛 (𝜏, 𝑇 ), 0} − (𝐴𝑛 (𝜏, 𝑇 ) − 𝐺𝑛 (𝜏, 𝑇 )). Therefore, [ ] (𝑎) (𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐾 − 𝐴𝑛 (𝜏, 𝑇 ), 0}|| ℱ𝑡 𝑃𝑎𝑟 [ ] ≥ 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐾 − 𝐺𝑛 (𝜏, 𝑇 ), 0}|| ℱ𝑡 [ ] −𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 ) − 𝐺𝑛 (𝜏, 𝑇 )|| ℱ𝑡 (𝑔) (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝑃𝑎𝑟 { [ ] [ ]} −𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 )|| ℱ𝑡 − 𝔼ℚ 𝐺𝑛 (𝜏, 𝑇 )|| ℱ𝑡 .

448

5.2.1 Discrete Sampling

5. Arithmetic–Geometric Average Strike (Floating Strike) Identity. Let the asset price 𝑆𝑡 follow the GBM 𝑑𝑆𝑡 = (𝜇 − 𝐷) + 𝜎 𝑑𝑊𝑡 𝑆𝑡 where 𝑊𝑡 is the ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ), 𝜇, 𝐷 and 𝜎 are the drift parameter, continuous dividend yield and volatility, respectively. In addition, let 𝑟 be the risk-free interest rate. Let the time interval [𝜏, 𝑇 ] be partitioned into 𝑛 equal subintervals each of length Δ𝑡 = (𝑇 − 𝜏)∕𝑛 and let 𝑆𝑡𝑖 , 𝑡𝑖 = 𝜏 + 𝑖Δ𝑡 be the stock price at the end of the 𝑖th interval, 𝑖 = 1, 2, … , 𝑛. Define the discretely measured arithmetic average as 𝑛

𝐴𝑛 (𝜏, 𝑇 ) =

1∑ 𝑆 𝑛 𝑖=1 𝑡𝑖

and the discretely measured geometric average as

𝐺𝑛 (𝜏, 𝑇 ) =

( 𝑛 ∏ 𝑖=1

)1 𝑆 𝑡𝑖

𝑛

.

At time 𝑡, for a common strike value 𝐾 and expiry time 𝑇 > 𝑡, let the arithmetic average strike (floating strike) Asian call and put options be [ ] (𝑎) 𝐶𝑎𝑠 (𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑆𝑇 − 𝐴𝑛 (𝜏, 𝑇 ), 0}|| ℱ𝑡 [ ] (𝑎) (𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐴𝑛 (𝜏, 𝑇 ) − 𝑆𝑇 , 0}|| ℱ𝑡 𝑃𝑎𝑠 where the expectation is under the risk-neutral measure ℚ. In contrast, let the corresponding geometric average strike (floating strike) Asian call and put options be [ ] (𝑔) (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑆𝑇 − 𝐺𝑛 (𝜏, 𝑇 ), 0}|| ℱ𝑡 𝐶𝑎𝑠 [ ] (𝑔) (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐺𝑛 (𝜏, 𝑇 ) − 𝑆𝑇 , 0}|| ℱ𝑡 . 𝑃𝑎𝑠 Show that (𝑔) (𝑎) 𝐶𝑎𝑠 (𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝑇 ) ≤ 𝐶𝑎𝑠 (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝑇 )

{ [ ] [ ]} (𝑎) ≤ 𝐶𝑎𝑠 (𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝑇 ) + 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 )|| ℱ𝑡 − 𝔼ℚ 𝐺𝑛 (𝜏, 𝑇 )|| ℱ𝑡 and { [ ] [ ]} (𝑎) (𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝑇 ) − 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 )|| ℱ𝑡 − 𝔼ℚ 𝐺𝑛 (𝜏, 𝑇 )|| ℱ𝑡 𝑃𝑎𝑠 (𝑔) (𝑎) (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝑇 ) ≤ 𝑃𝑎𝑠 (𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝑇 ). ≤ 𝑃𝑎𝑠

5.2.1 Discrete Sampling

449

Solution: Following Problem 5.2.1.3 (see page 444) we have 𝐺𝑛 (𝜏, 𝑇 ) ≤ 𝐴𝑛 (𝜏, 𝑇 ), therefore [ ] [ ] 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑆𝑇 − 𝐴𝑛 (𝜏, 𝑇 ), 0}|| ℱ𝑡 ≤ 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑆𝑇 − 𝐺𝑛 (𝜏, 𝑇 ), 0}|| ℱ𝑡 or (𝑎) (𝑔) (𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝑇 ) ≤ 𝐶𝑎𝑠 (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝑇 ). 𝐶𝑎𝑠

In addition, we note that max{𝑆𝑇 − 𝐺𝑛 (𝜏, 𝑇 ), 0} = max{𝑆𝑇 − 𝐴𝑛 (𝜏, 𝑇 ) + 𝐴𝑛 (𝜏, 𝑇 ) − 𝐺𝑛 (𝜏, 𝑇 ), 0} ≤ max{𝑆𝑇 − 𝐴𝑛 (𝜏, 𝑇 ), 0} + 𝐴𝑛 (𝜏, 𝑇 ) − 𝐺𝑛 (𝜏, 𝑇 ). Thus, [ ] (𝑔) (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑆𝑇 − 𝐺𝑛 (𝜏, 𝑇 ), 0}|| ℱ𝑡 𝐶𝑎𝑠 [ ] ≤ 𝑒−𝑟(𝑇 −𝑇 ) 𝔼ℚ max{𝑆𝑇 − 𝐴𝑛 (𝜏, 𝑇 ), 0}|| ℱ𝑡 [ ] +𝑒−𝑟(𝑇 −𝑇 ) 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 ) − 𝐺𝑛 (𝜏, 𝑇 )|| ℱ𝑡 (𝑎) (𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝑇 ) = 𝐶𝑎𝑠 { ℚ[ ] [ ]} −𝑟(𝑇 −𝑇 ) 𝔼 𝐴𝑛 (𝜏, 𝑇 )|| ℱ𝑡 − 𝔼ℚ 𝐺𝑛 (𝜏, 𝑇 )|| ℱ𝑡 . +𝑒

In contrast, for the average strike put options [ ] [ ] 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐺𝑛 (𝜏, 𝑇 ) − 𝑆𝑇 , 0}|| ℱ𝑡 ≤ 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐴𝑛 (𝜏, 𝑇 ) − 𝑆𝑇 , 0}|| ℱ𝑡 or (𝑔) (𝑎) (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝑇 ) ≤ 𝑃𝑎𝑠 (𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝑇 ). 𝑃𝑎𝑠

In addition, we note that max{𝐺𝑛 (𝜏, 𝑇 ) − 𝑆𝑇 , 0} = max{𝐺𝑛 (𝜏, 𝑇 ) − 𝐴𝑛 (𝜏, 𝑇 ) + 𝐴𝑛 (𝜏, 𝑇 ) − 𝑆𝑇 , 0} = max{𝐴𝑛 (𝜏, 𝑇 ) − 𝑆𝑇 − (𝐴𝑛 (𝜏, 𝑇 ) − 𝐺𝑛 (𝜏, 𝑇 )), 0} ≥ max{𝐴𝑛 (𝜏, 𝑇 ) − 𝑆𝑇 , 0} − (𝐴𝑛 (𝜏, 𝑇 ) − 𝐺𝑛 (𝜏, 𝑇 )). Therefore, [ ] (𝑔) (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐺𝑛 (𝜏, 𝑇 ) − 𝑆𝑇 , 0}|| ℱ𝑡 𝑃𝑎𝑠 [ ] ≥ 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐴𝑛 (𝜏, 𝑇 ) − 𝑆𝑇 , 0}|| ℱ𝑡 [ ] −𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 ) − 𝐺𝑛 (𝜏, 𝑇 )|| ℱ𝑡 (𝑎) (𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝑇 ) = 𝑃𝑎𝑠 { [ ] [ ]} −𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 )|| ℱ𝑡 − 𝔼ℚ 𝐺𝑛 (𝜏, 𝑇 )|| ℱ𝑡 .

450

5.2.1 Discrete Sampling

6. Discrete Geometric Average Rate (Fixed Strike) Asian Option. Let the asset price 𝑆𝑡 follow the GBM 𝑑𝑆𝑡 = (𝜇 − 𝐷) + 𝜎 𝑑𝑊𝑡 𝑆𝑡 where 𝑊𝑡 is the ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ), 𝜇, 𝐷 and 𝜎 are the drift parameter, continuous dividend yield and volatility, respectively. In addition, let 𝑟 be the risk-free interest rate. Let the time interval [𝜏, 𝑇 ] be partitioned into 𝑛 equal subintervals each of length Δ𝑡 = (𝑇 − 𝜏)∕𝑛 and let 𝑆𝑡𝑖 , 𝑡𝑖 = 𝜏 + 𝑖Δ𝑡 be the asset price at the end of the 𝑖th interval, 𝑖 = 1, 2, … , 𝑛. We define the discretely measured geometric average of asset prices as ( 𝐺𝑛 (𝜏, 𝑇 ) =

𝑛 ∏ 𝑖=1

)1 𝑆 𝑡𝑖

𝑛

and we consider a fixed strike geometric Asian call option at expiry 𝑇 > 𝑡 > 𝜏 with payoff Ψ(𝑆𝑇 , 𝐺𝑛 (𝜏, 𝑇 )) = max{𝐺𝑛 (𝜏, 𝑇 ) − 𝐾, 0} where 𝐾 > 0 is the strike price. Show that if 𝑋 ∼  (𝜇𝑥 , 𝜎𝑥2 ) then for a constant 𝐾 > 0 [

1 2 ] 𝔼 max{𝛿(𝑒 − 𝐾), 0} = 𝛿𝑒𝜇𝑥 + 2 𝜎𝑥 Φ

𝑋

(

𝛿(𝜇𝑥 + 𝜎𝑥2 − log 𝐾) 𝜎𝑥

)

( − 𝛿𝐾Φ

𝛿(𝜇𝑥 − log 𝐾) 𝜎𝑥

)

where 𝛿 ∈ {−1, 1} and Φ(⋅) denotes the cdf of a standard normal. Suppose 𝑆𝑡1 , 𝑆𝑡2 , …, 𝑆𝑡𝑘 , 1 ≤ 𝑘 ≤ 𝑛 − 1 have been observed and let 𝑡𝑘 ≤ 𝑡 < 𝑡𝑘+1 where 𝑡 = 𝑡𝑘 + 𝜖Δ𝑡, 𝜖 ∈ [0, 1). Show that under the risk-neutral measure ℚ ( 𝐺𝑛 (𝜏, 𝑇 ) =

𝑘 ∏ 𝑖=1

) 1𝑛 𝑆 𝑡𝑖

𝑛−𝑘 𝑛

𝑆𝑡

1 )( )2 ( )3 ( ( )𝑛−𝑘 ⎤ 𝑛 ⎡ 𝑆 𝑆 𝑆 𝑆 𝑡𝑘+1 𝑡𝑛 𝑡𝑛−1 𝑡𝑛−2 ⎢ ⎥ ⋯ ⎢ 𝑆𝑡𝑛−1 ⎥ 𝑆𝑡𝑛−2 𝑆𝑡𝑛−3 𝑆𝑡 ⎣ ⎦

and deduce that log 𝐺𝑛 (𝜏, 𝑇 ) ∼  (𝑚, 𝑠2 )

5.2.1 Discrete Sampling

451

where ) 1𝑛 ⎡( 𝑘 ⎤ 𝑛−𝑘 ⎢ ∏ ⎥ 𝑛 𝑚 = log ⎢ 𝑆 𝑡𝑖 𝑆𝑡 ⎥ ⎢ 𝑖=1 ⎥ ⎣ ⎦ ) [ (𝑛 − 𝑘 − 1)(𝑛 − 𝑘) (𝑛 − 𝑘)(1 − 𝜖) ] ( 1 + 𝑟 − 𝐷 − 𝜎2 + Δ𝑡 2 2𝑛 𝑛 and [ 𝑠2 = 𝜎 2

] (𝑛 − 𝑘 − 1)(𝑛 − 𝑘)(2𝑛 − 2𝑘 − 1) (𝑛 − 𝑘)2 (1 − 𝜖) + Δ𝑡. 6𝑛2 𝑛2

Using the above information show that under the risk-neutral measure ℚ, the discrete geometric average rate Asian call option price at time 𝑡, 𝜏 < 𝑡 < 𝑇 is

(𝑔) (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝐶𝑎𝑟

( 𝑘 ∏ 𝑖=1

) 1𝑛 𝑆 𝑡𝑖

𝑛−𝑘 𝑛

𝑆𝑡

(𝑛)

where [ log (𝑛)

(∏

𝑘 𝑖=1 𝑆𝑡𝑖

)1 𝑛

𝑛−𝑘 𝑛

]

𝑆𝑡

𝑑± =

) ( − log 𝐾 + 𝑟 − 𝐷𝑛 ± 12 𝜎𝑛2 (𝑇 − 𝑡)

√ 𝜎𝑛 𝑇 − 𝑡 [

] 𝑛(𝑇 − 𝑡) (𝑛 − 𝑘 − 1)(𝑛 − 𝑘) (𝑛 − 𝑘)(1 − 𝜖) Δ𝑡 − − 𝑇 −𝜏 2𝑛 𝑛 𝑇 −𝑡 [ ] (𝑛 − 𝑘 − 1)(𝑛 − 𝑘) (𝑛 − 𝑘)(1 − 𝜖) Δ𝑡 +𝐷 + 2𝑛 𝑛 𝑇 −𝑡 [ 2 𝜎 (𝑛 − 𝑘 − 1)(𝑛 − 𝑘) (𝑛 − 𝑘 − 1)(𝑛 − 𝑘)(2𝑛 − 2𝑘 − 1) + − 2 2𝑛 6𝑛2 ] 2 (𝑛 − 𝑘)(1 − 𝜖) (𝑛 − 𝑘) (1 − 𝜖) Δ𝑡 + − 𝑛 𝑛 𝑇 −𝑡

𝐷𝑛 = 𝑟

and [ 𝜎 2𝑛

=𝜎

2

] (𝑛 − 𝑘 − 1)(𝑛 − 𝑘)(2𝑛 − 2𝑘 − 1) (𝑛 − 𝑘)2 (1 − 𝜖) Δ𝑡 + . 𝑇 −𝑡 6𝑛2 𝑛2

Finally, show that in the limit 𝑛 → ∞, 𝐼𝑡

𝑇 −𝑡

(𝑔) lim 𝐶𝑎𝑟 (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝑒 𝑇 𝑆𝑡 𝑇 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑 + ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑 − )

𝑛→∞

(𝑛)

𝑒−𝐷𝑛 (𝑇 −𝑡) Φ(𝑑 + ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑 − )

452

5.2.1 Discrete Sampling

where 𝐼𝑡 =

𝑡

∫𝜏

log 𝑆𝑢 𝑑 𝑢

) ) ( ( 𝐼𝑡 1 𝑇 −𝑡 log 𝑆𝑡 − log 𝐾 + 𝑟 − 𝐷 ± 𝜎 2 (𝑇 − 𝑡) + 𝑇 −𝜏 2 𝑑± = 𝑇 − 𝜏 √ 𝜎 𝑇 −𝑡 ( ) ( ) ( )( ) 𝐷 𝑇 −𝑡 𝜎2 𝑇 − 𝑡 𝑇 + 2𝑡 − 3𝜏 𝑟 𝑇 + 𝑡 − 2𝜏 + + 𝐷= 2 𝑇 −𝜏 2 𝑇 −𝜏 12 𝑇 − 𝜏 𝑇 −𝜏 2 ( 𝑇 − 𝑡 )2 𝜎 𝜎2 = . 3 𝑇 −𝜏 Solution: For the first part of the results refer to Problem 1.2.2.7 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus. From Girsanov’s theorem, under the risk-neutral measure ℚ, the asset price 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡ℚ 𝑆𝑡 where 𝑊𝑡ℚ is a ℚ-standard Wiener process. Let the time interval [𝜏, 𝑇 ] be partitioned into 𝑛 equal subintervals each of length Δ𝑡 = (𝑇 − 𝜏)∕𝑛 and let 𝑆𝑡𝑖 , 𝑡𝑖 = 𝜏 + 𝑖Δ𝑡 be the asset price at the end of the 𝑖th interval. Suppose 𝑆𝑡1 , 𝑆𝑡2 , …, 𝑆𝑡𝑘 , 1 ≤ 𝑘 ≤ 𝑛 − 1 have been observed and let 𝑡𝑘 ≤ 𝑡 < 𝑡𝑘+1 where 𝑡 = 𝑡𝑘 + 𝜖Δ𝑡, 𝜖 ∈ [0, 1). By definition ( 𝐺𝑛 (𝜏, 𝑇 ) = ( = ( =

𝑛 ∏ 𝑖=1 𝑘 ∏ 𝑖=1 𝑘 ∏ 𝑖=1

( =

𝑘 ∏ 𝑖=1

)1 𝑆 𝑡𝑖

𝑛

) 1𝑛 ( 𝑆 𝑡𝑖 ) 1𝑛

𝑛 ∏

𝑖=𝑘+1

(

)1 𝑆 𝑡𝑖

𝑛

𝑆𝑡𝑛 𝑆𝑡𝑛−1 𝑆𝑡𝑛−3 ⋯ 𝑆𝑡𝑘+2 𝑆𝑡𝑘+1

𝑆 𝑡𝑖 ) 1𝑛 𝑆 𝑡𝑖

𝑛−𝑘 𝑛

𝑆𝑡

)1 𝑛

1 ( )( )2 ( )3 ( )𝑛−𝑘 ⎤ 𝑛 ⎡ 𝑆 𝑆 𝑆 𝑆 𝑡𝑘+1 𝑡𝑛 𝑡𝑛−1 𝑡𝑛−2 ⎢ ⎥ . ⋯ ⎢ 𝑆𝑡𝑛−1 ⎥ 𝑆𝑡𝑛−2 𝑆𝑡𝑛−3 𝑆𝑡 ⎣ ⎦

From It¯o’s lemma we can easily show for 𝑖 = 𝑘 + 2, 𝑘 + 3, … , 𝑛 − 1, 𝑛 ( log

𝑆 𝑡𝑖 𝑆𝑡𝑖−1

)

) ( 1 = 𝑟 − 𝐷 − 𝜎 2 (𝑡𝑖 − 𝑡𝑖−1 ) + 𝜎𝑊𝑡ℚ−𝑡 𝑖 𝑖−1 2 ) ( 1 = 𝑟 − 𝐷 − 𝜎 2 Δ𝑡 + 𝜎𝑊𝑡ℚ−𝑡 𝑖 𝑖−1 2

5.2.1 Discrete Sampling

453

and log

(𝑆

𝑡𝑘+1

𝑆𝑡

)

) ( 1 = 𝑟 − 𝐷 − 𝜎 2 (𝑡𝑘+1 − 𝑡) + 𝜎𝑊𝑡ℚ −𝑡 𝑘+1 2 ) ( 1 = 𝑟 − 𝐷 − 𝜎 2 (1 − 𝜖)Δ𝑡 + 𝜎𝑊𝑡ℚ −𝑡 . 𝑘+1 2

From the independent increment property of the standard Wiener process we can deduce that ) ( ) ( ) ( ) ( (𝑆 ) 𝑆𝑡𝑘+2 𝑆𝑡𝑛−1 𝑆𝑡𝑛−2 𝑆 𝑡𝑛 𝑡𝑘+1 , log , log , … , log , log log 𝑆𝑡𝑛−1 𝑆𝑡𝑛−2 𝑆𝑡𝑛−3 𝑆𝑡𝑘+1 𝑆𝑡 are mutually independent. Thus, ) 1𝑛 ⎤ ⎡( 𝑘 𝑛−𝑘 ⎥ ⎢ ∏ 𝑛 log 𝐺𝑛 (𝜏, 𝑇 ) = log ⎢ 𝑆 𝑡𝑖 𝑆𝑡 ⎥ ⎥ ⎢ 𝑖=1 ⎦ ⎣ [ ( ) ( ) 𝑆 𝑡𝑛 𝑆𝑡𝑛−1 1 + log + 2 log 𝑛 𝑆𝑡𝑛−1 𝑆𝑡𝑛−2 ( ) ( 𝑆 )] 𝑆𝑡𝑛−2 𝑡𝑘+1 +3 log + … + (𝑛 − 𝑘) log 𝑆𝑡𝑛−3 𝑆𝑡 and because ) [ ( ] 1 log 𝑆𝑡𝑖 ∼  log 𝑆𝑡𝑖−1 + 𝑟 − 𝐷 − 𝜎 2 Δ𝑡, 𝜎 2 Δ𝑡 , 𝑖 = 𝑘 + 2, 𝑘 + 3, … , 𝑛 2 ) [ ( ] 1 2 log 𝑆𝑡𝑘+1 ∼  log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 (1 − 𝜖)Δ𝑡, 𝜎 2 (1 − 𝜖)Δ𝑡 2 we have

𝔼

[ ℚ

) 1𝑛 ⎤ ⎡( 𝑘 𝑛−𝑘 ∏ ⎥ ⎢ 𝑛 log 𝐺𝑛 (𝜏, 𝑇 ) = log ⎢ 𝑆 𝑡𝑖 𝑆𝑡 ⎥ ⎥ ⎢ 𝑖=1 ⎦ ⎣ ( ) 1 1 2 + 𝑟−𝐷− 𝜎 𝑛 2 × (1 + 2 + 3 + … + (𝑛 − 𝑘 − 1) + (𝑛 − 𝑘)(1 − 𝜖)) Δ𝑡 ) 1𝑛 ⎤ ⎡( 𝑘 𝑛−𝑘 ⎥ ⎢ ∏ 𝑛 𝑆 𝑡𝑖 𝑆𝑡 ⎥ = log ⎢ ⎥ ⎢ 𝑖=1 ⎦ ⎣ [ ) (𝑛 − 𝑘 − 1)(𝑛 − 𝑘) (𝑛 − 𝑘)(1 − 𝜖) ] ( 1 + 𝑟 − 𝐷 − 𝜎2 + Δ𝑡 2 2𝑛 𝑛 ]

454

5.2.1 Discrete Sampling

and [ ] 𝜎2 ( ) Varℚ log 𝐺𝑛 (𝜏, 𝑇 ) = 2 12 + 22 + 32 + … + (𝑛 − 𝑘 − 1)2 + (𝑛 − 𝑘)2 (1 − 𝜖) Δ𝑡 𝑛 [ ] (𝑛 − 𝑘)2 (1 − 𝜖) 2 (𝑛 − 𝑘 − 1)(𝑛 − 𝑘)(2𝑛 − 2𝑘 − 1) =𝜎 + Δ𝑡. 6𝑛2 𝑛2 Thus, using the above information we can deduce that the discrete geometric average rate Asian call option price at time 𝜏 < 𝑡 < 𝑇 is [ ] (𝑔) (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐺𝑛 (𝜏, 𝑇 ) − 𝐾, 0}|| ℱ𝑡 𝐶𝑎𝑟 1 2 −𝑟(𝑇 −𝑡)

= 𝑒𝑚+ 2 𝑠

(𝑛)

(𝑛)

Φ(𝑑 + ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑 − )

where ) 1𝑛 ⎡( 𝑘 ⎤ 𝑛−𝑘 ⎢ ∏ ⎥ 𝑛 𝑚 = log ⎢ 𝑆 𝑡𝑖 𝑆𝑡 ⎥ ⎢ 𝑖=1 ⎥ ⎣ ⎦ [ ( ) (𝑛 − 𝑘 − 1)(𝑛 − 𝑘) (𝑛 − 𝑘)(1 − 𝜖) ] 1 + 𝑟 − 𝐷 − 𝜎2 + Δ𝑡, 2 2𝑛 𝑛 [ 𝑠 =𝜎 2

2

] (𝑛 − 𝑘 − 1)(𝑛 − 𝑘)(2𝑛 − 2𝑘 − 1) (𝑛 − 𝑘)2 (1 − 𝜖) + Δ𝑡, 6𝑛2 𝑛2

𝑚 + 𝑠2 − log 𝐾 𝑠

(𝑛)

𝑑+ = and

(𝑛)

𝑑 − = 𝑑+(𝑛) − 𝑠. Therefore,

𝑒

𝑚+ 12 𝑠2 −𝑟(𝑇 −𝑡)

=

( 𝑘 ∏ 𝑖=1

log (𝑛)

𝑑± =

) 1𝑛 𝑆 𝑡𝑖

[ (∏

𝑛−𝑘 𝑛

𝑆𝑡

𝑘 𝑖=1 𝑆𝑡𝑖

𝑒−𝐷𝑛 (𝑇 −𝑡)

)1 𝑛

𝑛−𝑘 𝑛

𝑆𝑡

]

) ( − log 𝐾 + 𝑟 − 𝐷𝑛 ± 12 𝜎 2𝑛 (𝑇 − 𝑡)

√ 𝜎𝑛 𝑇 − 𝑡

5.2.1 Discrete Sampling

455

where ) [ (𝑛 − 𝑘 − 1)(𝑛 − 𝑘) (𝑛 − 𝑘)(1 − 𝜖) ] ( 1 Δ𝑡 𝐷𝑛 = 𝑟 − 𝑟 − 𝐷 − 𝜎 2 + 2 2𝑛 𝑛 𝑇 −𝑡 [ ] 𝜎 2 (𝑛 − 𝑘 − 1)(𝑛 − 𝑘)(2𝑛 − 2𝑘 − 1) (𝑛 − 𝑘)2 (1 − 𝜖) Δ𝑡 − + 2 𝑇 −𝑡 6𝑛2 𝑛2 [ ] 𝑛(𝑇 − 𝑡) (𝑛 − 𝑘 − 1)(𝑛 − 𝑘) (𝑛 − 𝑘)(1 − 𝜖) Δ𝑡 =𝑟 − − 𝑇 −𝜏 2𝑛 𝑛 𝑇 −𝑡 [ ] (𝑛 − 𝑘 − 1)(𝑛 − 𝑘) (𝑛 − 𝑘)(1 − 𝜖) Δ𝑡 +𝐷 + 2𝑛 𝑛 𝑇 −𝑡 [ 𝜎 2 (𝑛 − 𝑘 − 1)(𝑛 − 𝑘) (𝑛 − 𝑘 − 1)(𝑛 − 𝑘)(2𝑛 − 2𝑘 − 1) + − 2 2𝑛 6𝑛2 ] (𝑛 − 𝑘)(1 − 𝜖) (𝑛 − 𝑘)2 (1 − 𝜖) Δ𝑡 + − 𝑛 𝑛 𝑇 −𝑡 and [ 𝜎 2𝑛 = 𝜎 2

] (𝑛 − 𝑘 − 1)(𝑛 − 𝑘)(2𝑛 − 2𝑘 − 1) (𝑛 − 𝑘)2 (1 − 𝜖) Δ𝑡 + . 𝑇 −𝑡 6𝑛2 𝑛2

Thus, we can write the discrete geometric average rate Asian call option price at time 𝑡, 𝜏 < 𝑡 < 𝑇 as ( (𝑔) (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝐶𝑎𝑟

𝑘 ∏ 𝑖=1

) 1𝑛 𝑆 𝑡𝑖

𝑛−𝑘 𝑛

𝑆𝑡

(𝑛)

(𝑛)

𝑒−𝐷𝑛 (𝑇 −𝑡) Φ(𝑑 + ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑 − ).

For the limiting case, we note that since 𝑇 = 𝜏 + 𝑛Δ𝑡 and 𝑡 = 𝑡𝑘 + 𝜖Δ𝑡 = 𝜏 + (𝑘 + 𝜖)Δ𝑡, 𝜖 ∈ [0, 1) therefore lim (𝑛 − 𝑘)Δ𝑡 = lim (𝑇 − 𝑡 + 𝜖Δ𝑡) = 𝑇 − 𝑡

𝑛→∞

𝑛→∞

lim (𝑛 − 𝑘 − 1)Δ𝑡 = lim (𝑇 − 𝑡 + (𝜖 − 1)Δ𝑡) = 𝑇 − 𝑡

𝑛→∞

𝑛→∞

and lim (2𝑛 − 2𝑘 − 1)Δ𝑡 = lim (2(𝑇 − 𝑡 + 𝜖Δ𝑡) + Δ𝑡) = 2(𝑇 − 𝑡).

𝑛→∞

𝑛→∞

456

5.2.1 Discrete Sampling

By setting 𝐼𝑡 =

𝑡

∫𝜏

lim

log 𝑆𝑢 𝑑 𝑢,

( 𝑛 ∏

𝑛→∞

𝑖=1

)1 𝑆 𝑡𝑖

𝑛

𝑛−𝑘 𝑛

𝑆𝑡

( = lim

𝑛→∞

𝑛 ∏

𝑆 𝑡𝑖

𝑛

𝑛→∞

𝑛−𝑘 𝑛

lim 𝑆𝑡

𝑛→∞

𝑖=1 1 ∑𝑘

= lim 𝑒 𝑇 −𝜏 =𝑒

)1

𝑖=1 log 𝑆𝑡𝑖 Δ𝑡

1 𝑇 −𝜏

𝑡 ∫𝜏 log 𝑆𝑢 𝑑𝑢

𝐼𝑡

𝑇 −𝑡 𝑇 −𝜏

(𝑛−𝑘)Δ𝑡 𝑛Δ𝑡

⋅ lim 𝑆𝑡 𝑛→∞

𝑇 −𝑡 𝑇 −𝜏

𝑆𝑡

= 𝑒 𝑇 𝑆𝑡 [

lim 𝐷𝑛 =

𝑛→∞

=

= =

] 𝑛(𝑇 − 𝑡) (𝑛 − 𝑘 − 1)(𝑛 − 𝑘) (𝑛 − 𝑘)(1 − 𝜖) Δ𝑡 lim 𝑟 − − 𝑛→∞ 𝑇 −𝜏 2𝑛 𝑛 𝑇 −𝑡 [ ] (𝑛 − 𝑘 − 1)(𝑛 − 𝑘) (𝑛 − 𝑘)(1 − 𝜖) Δ𝑡 + lim 𝐷 + 𝑛→∞ 2𝑛 𝑛 𝑇 −𝑡 [ 𝜎 2 (𝑛 − 𝑘 − 1)(𝑛 − 𝑘) (𝑛 − 𝑘 − 1)(𝑛 − 𝑘)(2𝑛 − 2𝑘 − 1) + lim − 𝑛→∞ 2 2𝑛 6𝑛2 ] (𝑛 − 𝑘)(1 − 𝜖) (𝑛 − 𝑘)2 (1 − 𝜖) Δ𝑡 + − 𝑛 𝑛 𝑇 −𝑡 [ ] 2 (𝑛 − 𝑘 − 1)(𝑛 − 𝑘)Δ𝑡 (𝑛 − 𝑘)(1 − 𝜖)Δ𝑡2 lim 𝑟 1 − − 𝑛→∞ 2𝑛Δ𝑡(𝑇 − 𝑡) 𝑛Δ𝑡(𝑇 − 𝑡) [ ] 2 (𝑛 − 𝑘 − 1)(𝑛 − 𝑘)Δ𝑡 (𝑛 − 𝑘)(1 − 𝜖)Δ𝑡2 + lim 𝐷 + 𝑛→∞ 2𝑛Δ𝑡(𝑇 − 𝑡) 𝑛Δ𝑡(𝑇 − 𝑡) [ 2 2 (𝑛 − 𝑘 − 1)(𝑛 − 𝑘)(2𝑛 − 2𝑘 − 1)Δ𝑡3 𝜎 (𝑛 − 𝑘 − 1)(𝑛 − 𝑘)Δ𝑡 + lim − 𝑛→∞ 2 2𝑛Δ𝑡(𝑇 − 𝑡) 6𝑛2 Δ𝑡2 (𝑇 − 𝑡) ] 2 2 2 (𝑛 − 𝑘) (1 − 𝜖)Δ𝑡 (𝑛 − 𝑘)(1 − 𝜖)Δ𝑡 − + 𝑛Δ𝑡(𝑇 − 𝑡) 𝑛Δ𝑡(𝑇 − 𝑡) ( ) ( ) [ ] (𝑇 − 𝑡)2 𝑇 −𝑡 𝑇 −𝑡 𝜎2 𝑇 −𝑡 𝑟 1− +𝐷 + + 2(𝑇 − 𝜏) 2(𝑇 − 𝜏) 2 2(𝑇 − 𝜏) 3(𝑇 − 𝜏)2 ( ) ( ) ( )( ) 𝐷 𝑇 −𝑡 𝜎2 𝑇 − 𝑡 𝑇 + 2𝑡 − 3𝜏 𝑟 𝑇 + 𝑡 − 2𝜏 + + 2 𝑇 −𝜏 2 𝑇 −𝜏 12 𝑇 − 𝜏 𝑇 −𝜏

and [

] (𝑛 − 𝑘 − 1)(𝑛 − 𝑘)(2𝑛 − 2𝑘 − 1) (𝑛 − 𝑘)2 (1 − 𝜖) Δ𝑡 + 𝑛→∞ 𝑇 −𝑡 6𝑛2 𝑛2 [ ] 3 2 (𝑛 − 𝑘 − 1)(𝑛 − 𝑘)(2𝑛 − 2𝑘 − 1)Δ𝑡 (𝑛 − 𝑘) (1 − 𝜖)Δ𝑡3 = lim 𝜎 2 + 𝑛→∞ 6𝑛2 Δ𝑡2 (𝑇 − 𝑡) 𝑛2 Δ𝑡2 (𝑇 − 𝑡) ( )2 2 𝑇 −𝑡 𝜎 . = 3 𝑇 −𝜏

lim 𝜎 2𝑛 = lim 𝜎 2

𝑛→∞

5.2.1 Discrete Sampling

457

Thus, in the limit 𝑛 → ∞, 𝑇 −𝑡

𝐼𝑡

(𝑔) lim 𝐶𝑎𝑟 (𝑆𝑡 , 𝐺𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝑒 𝑇 −𝜏 𝑆𝑡𝑇 −𝜏 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑 + ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑 − )

𝑛→∞

where ) ) ( ( 𝐼𝑡 1 𝑇 −𝑡 log 𝑆𝑡 − log 𝐾 + 𝑟 − 𝐷 ± 𝜎 2 (𝑇 − 𝑡) + 𝑇 −𝜏 2 𝑑± = 𝑇 − 𝜏 √ 𝜎 𝑇 −𝑡

𝐷=

( ) ( ) ( )( ) 𝐷 𝑇 −𝑡 𝜎2 𝑇 − 𝑡 𝑇 + 2𝑡 − 3𝜏 𝑟 𝑇 + 𝑡 − 2𝜏 + + 2 𝑇 −𝜏 2 𝑇 −𝜏 12 𝑇 − 𝜏 𝑇 −𝜏

and 𝜎2 =

𝜎2 3

(

𝑇 −𝑡 𝑇 −𝜏

)2

.

7. Levy Approximation – Discrete Arithmetic Average Rate (Fixed Strike) Asian Option. Let the asset price 𝑆𝑡 follow the GBM 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡 𝑆𝑡 where 𝑊𝑡 is the ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ), 𝜇, 𝐷 and 𝜎 are the drift parameter, continuous dividend yield and volatility, respectively. In addition, let 𝑟 be the risk-free interest rate. Let the time interval [𝜏, 𝑇 ] be partitioned into 𝑛 equal subintervals each of length Δ𝑡 = (𝑇 − 𝜏)∕𝑛 and let 𝑆𝑡𝑖 , 𝑡𝑖 = 𝜏 + 𝑖Δ𝑡 be the asset price at the end of the 𝑖th interval, 𝑖 = 1, 2, … , 𝑛. We define the discretely measured arithmetic average of asset prices as 𝑛

1∑ 𝑆 𝐴𝑛 (𝜏, 𝑇 ) = 𝑛 𝑖=1 𝑡𝑖 and we consider an arithmetic average rate Asian put option with expiry 𝑇 > 𝑡 > 𝜏 and payoff Ψ(𝑆𝑇 , 𝐴𝑛 (𝜏, 𝑇 )) = max{𝐾 − 𝐴𝑛 (𝜏, 𝑇 ), 0} where 𝐾 > 0 is the strike price. Show that if 𝑋 ∼  (𝜇𝑥 , 𝜎𝑥2 ) then for a constant 𝐾 > 0 and 𝛼 ∈ ℕ, 1 2 2 𝜎𝑥

𝔼(𝑋 𝛼 ) = 𝑒𝛼𝜇𝑥 + 2 𝛼

458

5.2.1 Discrete Sampling

and [

1 2 ] 𝔼 max{𝛿(𝑒𝑋 − 𝐾), 0} = 𝛿𝑒𝜇𝑥 + 2 𝜎𝑥 Φ

(

𝛿(𝜇𝑥 + 𝜎𝑥2 − log 𝐾)

)

𝜎𝑥

( − 𝛿𝐾Φ

𝛿(𝜇𝑥 − log 𝐾) 𝜎𝑥

)

where 𝛿 ∈ {−1, 1} and Φ(⋅) denotes the cdf of a standard normal. Suppose 𝑆𝑡1 , 𝑆𝑡2 , …, 𝑆𝑡𝑘 , 1 ≤ 𝑘 ≤ 𝑛 − 1 have been observed and let 𝑡𝑘 ≤ 𝑡 < 𝑡𝑘+1 where 𝑡 = 𝑡𝑘 + 𝜖Δ𝑡, 𝜖 ∈ [0, 1) and 𝐴𝑛 (𝜏, 𝑇 ) = 𝐴𝑛 (𝑡1 , 𝑡𝑘 ) + 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 ) where 𝐴𝑛 (𝑡1 , 𝑡𝑘 ) = neutral measure ℚ

𝑘 𝑛 1∑ 1 ∑ 𝑆𝑡𝑖 and 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 ) = 𝑆 . Show that under the risk𝑛 𝑖=1 𝑛 𝑖=𝑘+1 𝑡𝑖

[ ] | 𝔼ℚ 𝑆𝑡𝑖 | 𝑆𝑡 = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑖−𝑘−𝜖)Δ𝑡 | [ ] 2 | ℚ 𝔼 𝑆𝑡𝑖 𝑆𝑡𝑗 | ℱ𝑡 = 𝑆𝑡2 𝑒(𝑟−𝐷)(𝑖+𝑗−2𝑘−2𝜖)Δ𝑡+𝜎 (min{𝑖,𝑗}−𝑘−𝜖)Δ𝑡 | where 𝑡𝑖 , 𝑡𝑗 > 𝑡𝑘 . By assuming 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 ) ∻ log- (𝑚, 𝑠2 ) and using the moment-matching technique up to second order show that [ ] [ ] 1 | 𝑚 ≈ 2 log 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )|| 𝑆𝑡 − log 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )2 | 𝑆𝑡 | 2 and [ ] [ ] | 𝑠2 ≈ log 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )2 | 𝑆𝑡 − 2 log 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )|| 𝑆𝑡 . | [ ] [ ] Find 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )|| 𝑆𝑡 and 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )2 || 𝑆𝑡 , and hence show that the discrete arithmetic average rate Asian put option at time 𝜏 < 𝑡 < 𝑇 can be approximated by (𝑛)

1 2 −𝑟(𝑇 −𝑡)

(𝑎) 𝑃𝑎𝑟 (𝑆𝑡 , 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 ), 𝑡; 𝐾, 𝑇 ) ≈ 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑 − ) − 𝑒𝑚+ 2 𝑠

(𝑛) (𝑛) 𝑚 + 𝑠2 − log 𝐾 where 𝐾 = 𝐾 − 𝐴𝑛 (𝑡1 , 𝑡𝑘 ), 𝑑 + = and 𝑑 − = 𝑑+ − 𝑠. 𝑠 Finally, show that in the limit 𝑛 → ∞,

[ ] 𝑆 (𝑒(𝑟−𝐷)(𝑇 −𝑡) − 1) lim 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )|| 𝑆𝑡 = 𝑡 𝑛→∞ (𝑟 − 𝐷)(𝑇 − 𝜏)

(𝑛)

Φ(−𝑑 + )

5.2.1 Discrete Sampling

459

and [

| lim 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )2 | 𝑆𝑡 |

𝑛→∞

]

) ( ⎡ (𝑟−𝐷+𝜎 2 )(𝑇 −𝑡) − 1 𝑒(𝑟−𝐷)(𝑇 −𝑡) 2 𝑒 ⎢ = ⎢ (𝑟 − 𝐷)(𝑇 − 𝜏)2 ⎢ 𝑟 − 𝐷 + 𝜎2 ⎣ ( ) 1 2 ⎤ 𝑒2(𝑟−𝐷+ 2 𝜎 )(𝑇 −𝑡) − 1 ⎥ − ⎥. 𝑟 − 𝐷 + 12 𝜎 2 ⎥ ⎦ 𝑆𝑡2

Solution: For the first two parts of the results see Problems 1.2.2.7 and 1.2.2.10 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus. Under the risk-neutral measure ℚ, 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡ℚ 𝑆𝑡 (𝜇 − 𝑟) 𝑡 + 𝑊𝑡 is a ℚ-standard Wiener process. Thus, conditional on 𝑆𝑡 for where 𝑊𝑡ℚ = 𝜎 𝑡𝑖 ≠ 𝑡𝑗 , 𝑡𝑖 , 𝑡𝑗 > 𝑡 1 2 )(𝑡

𝑆𝑡𝑖 = 𝑆𝑡 𝑒(𝑟−𝐷− 2 𝜎 𝑆𝑡𝑖 𝑆𝑡𝑗 = 𝑆𝑡2 𝑒

𝑖 −𝑡)+𝜎𝑊𝑡𝑖 −𝑡

(𝑟−𝐷− 21 𝜎 2 )(𝑡𝑖 −𝑡+𝑡𝑗 −𝑡)+𝜎(𝑊𝑡𝑖 −𝑡 +𝑊𝑡𝑗 −𝑡 )

.

Since 𝑡 = 𝑡𝑘 + 𝜖Δ𝑡 and 𝑡𝑗 = 𝜏 + 𝑗Δ𝑡, 𝑗 = 𝑖, 𝑘 we have [ ] | 𝔼ℚ 𝑆𝑡𝑖 | 𝑆𝑡 = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑡𝑖 −𝑡) = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑡𝑖 −𝑡𝑘 −𝜖Δ𝑡) = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑖−𝑘−𝜖)Δ𝑡 . | From Problem 2.2.1.4 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus we can deduce Cov(𝑊𝑡𝑖 −𝑡 , 𝑊𝑡𝑗 −𝑡 ) = min{𝑡𝑖 − 𝑡, 𝑡𝑗 − 𝑡} = min{𝑡𝑖 , 𝑡𝑗 } − 𝑡 and hence 𝑊𝑡𝑖 −𝑡 + 𝑊𝑡𝑗 −𝑡 ∼  (0, 𝑡𝑖 + 𝑡𝑗 + 2 min{𝑡𝑖 , 𝑡𝑗 } − 4𝑡).

460

5.2.1 Discrete Sampling

Therefore, conditional on 𝑆𝑡 [ ] 1 2 1 2 | 𝔼ℚ 𝑆𝑡𝑖 𝑆𝑡𝑗 | 𝑆𝑡 = 𝑆𝑡2 𝑒(𝑟−𝐷− 2 𝜎 )(𝑡𝑖 +𝑡𝑗 −2𝑡)+ 2 𝜎 (𝑡𝑖 +𝑡𝑗 +2 min{𝑡𝑖 ,𝑡𝑗 }−4𝑡) | 2 (min{𝑡

𝑖 ,𝑡𝑗 }−𝑡)

= 𝑆𝑡2 𝑒(𝑟−𝐷)(𝑡𝑖 +𝑡𝑗 −2𝑡𝑘 −2𝜖Δ𝑡)+𝜎

2 (min{𝑡

= 𝑆𝑡2 𝑒(𝑟−𝐷)(𝑡𝑖 +𝑡𝑗 −2𝑡)+𝜎

= 𝑆𝑡2 𝑒(𝑟−𝐷)(𝑖+𝑗−2𝑘−2𝜖)Δ𝑡+𝜎 Let 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑘 ) = order we have

𝑖 ,𝑡𝑗 }−𝑡𝑘 −𝜖Δ)

2 (min{𝑖,𝑗}−𝑘−𝜖)Δ𝑡

.

𝑛 1 ∑ 𝑆 ∻ log- (𝑚, 𝑠2 ) and by matching the moments up to second 𝑛 𝑖=𝑘+1 𝑡𝑖

1 2 [ ] 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑘 )|| 𝑆𝑡 ≈ 𝑒𝑚+ 2 𝑠

and

[ ] 2 | 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )2 | 𝑆𝑡 ≈ 𝑒2𝑚+2𝑠 |

and

𝑚 + 𝑠2 ≈

or [ ] 1 𝑚 + 𝑠2 ≈ log 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )|| 𝑆𝑡 2

[ ] 1 | log 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )2 | 𝑆𝑡 . | 2

Solving the approximate equations simultaneously we have [ ] [ ] 1 | 𝑚 ≈ 2 log 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )|| 𝑆𝑡 − log 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )2 | 𝑆𝑡 | 2 and [ ] [ ] | 𝑠2 ≈ log 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )2 | 𝑆𝑡 − 2 log 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )|| 𝑆𝑡 . | By definition 𝑛 [ ] [ ] 1 ∑ | 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )|| 𝑆𝑡 = 𝔼 ℚ 𝑆 𝑡𝑖 | 𝑆 𝑡 | 𝑛 𝑖=𝑘+1

𝑛 1 ∑ = 𝑆 𝑒(𝑟−𝐷)(𝑖−𝑘−𝜖)Δ𝑡 𝑛 𝑖=𝑘+1 𝑡

𝑛 𝑆𝑡 𝑒−(𝑟−𝐷)(𝑘+𝜖)Δ𝑡 ∑ (𝑟−𝐷)𝑖Δ𝑡 𝑒 𝑛 𝑖=𝑘+1 [ ( )] 𝑆𝑡 𝑒−(𝑟−𝐷)(𝑘+𝜖)Δ𝑡 𝑒(𝑟−𝐷)(𝑘+1)Δ𝑡 1 − 𝑒(𝑟−𝐷)(𝑛−𝑘)Δ𝑡 = 𝑛 1 − 𝑒(𝑟−𝐷)Δ𝑡 [ ] 𝑆𝑡 𝑒(𝑟−𝐷)(1−𝜖)Δ𝑡 1 − 𝑒(𝑟−𝐷)(𝑛−𝑘)Δ𝑡 = 𝑛 1 − 𝑒(𝑟−𝐷)Δ𝑡

=

5.2.1 Discrete Sampling

461

whilst for 𝑛 𝑛 [ ] [ ] 1 ∑ ∑ ℚ | | 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )2 | 𝑆𝑡 = 2 𝔼 𝑆 𝑡𝑖 𝑆 𝑡𝑗 | 𝑆 𝑡 | | 𝑛 𝑖=𝑘+1 𝑗=𝑘+1 1 2 )𝜖Δ𝑡

𝑆𝑡2 𝑒−2(𝑟−𝐷+ 2 𝜎

=

×

𝑛 ∑

𝑛2 𝑛 ∑

𝑒(𝑟−𝐷)(𝑖+𝑗−2𝑘)Δ𝑡+𝜎

2 (min{𝑖, 𝑗}−𝑘)Δ𝑡

𝑖=𝑘+1 𝑗=𝑘+1 1 2 )𝜖Δ𝑡

=

𝑆𝑡2 𝑒−2(𝑟−𝐷+ 2 𝜎 𝑛2

(𝐵1 + 𝐵2 + 𝐵3 )

where 𝑛 𝑛 ∑ ∑

𝐵1 =

𝑒(𝑟−𝐷)(𝑖+𝑗−2𝑘)Δ𝑡+𝜎

2 (min{𝑖,𝑗}−𝑘)Δ𝑡

𝑒(𝑟−𝐷)(𝑖+𝑗−2𝑘)Δ𝑡+𝜎

2 (min{𝑖,𝑗}−𝑘)Δ𝑡

𝑖=𝑘+1 𝑗=𝑘+1 𝑖=𝑗

𝑛 𝑛 ∑ ∑

𝐵2 =

𝑖=𝑘+1 𝑗=𝑘+1 𝑖𝑗

For the case 𝐵1 =

=

𝑛 𝑛 ∑ ∑

𝑒(𝑟−𝐷)(𝑖+𝑗−2𝑘)Δ𝑡+𝜎

2 (min{𝑖,𝑗}−𝑘)Δ𝑡

𝑖=𝑘+1 𝑗=𝑘+1 𝑖=𝑗 𝑛 ∑ 2(𝑟−𝐷)(𝑖−𝑘)Δ𝑡+𝜎 2 (𝑖−𝑘)Δ𝑡

𝑒

𝑖=𝑘+1

( ) ⎡ 2(𝑟−𝐷+ 21 𝜎 2 )(𝑘+1)Δ𝑡 2(𝑟−𝐷+ 21 𝜎 2 )(𝑛−𝑘)Δ𝑡 ⎤ 1 − 𝑒 𝑒 1 2 ⎢ ⎥ = 𝑒−2(𝑟−𝐷+ 2 𝜎 )𝑘Δ𝑡 ⎢ ⎥ 1 2 ⎢ ⎥ 1 − 𝑒2(𝑟−𝐷+ 2 𝜎 )Δ𝑡 ⎣ ⎦ ( ) 2(𝑟−𝐷+ 12 𝜎 2 )Δ𝑡 2(𝑟−𝐷+ 21 𝜎 2 )(𝑛−𝑘)Δ𝑡 1−𝑒 𝑒 = 1 2 1 − 𝑒2(𝑟−𝐷+ 2 𝜎 )Δ𝑡

462

5.2.1 Discrete Sampling

whilst for

𝐵2 =

𝑛 𝑛 ∑ ∑

𝑒(𝑟−𝐷)(𝑖+𝑗−2𝑘)Δ𝑡+𝜎

2 (min{𝑖, 𝑗}−𝑘)Δ𝑡

𝑖=𝑘+1 𝑗=𝑘+1 𝑖𝑗

=

𝑛 𝑖−1 ∑ ∑ 𝑖=𝑘+1 𝑗=𝑘+1

=

(

1 2

1 − 𝑒2(𝑟−𝐷+ 2 𝜎 )(𝑛−𝑘−1)Δ𝑡 ) 1 2 ( )( 1 − 𝑒(𝑟−𝐷)Δ𝑡 1 − 𝑒2(𝑟−𝐷+ 2 𝜎 )Δ𝑡

𝑒(3(𝑟−𝐷)+𝜎

2 )Δ𝑡

)

( ) 2 2 𝑒((𝑛−𝑘+2)(𝑟−𝐷)+𝜎 )Δ𝑡 1 − 𝑒(𝑟−𝐷+𝜎 )(𝑛−𝑘−1)Δ𝑡 − . ( )( ) 1 − 𝑒(𝑟−𝐷)Δ𝑡 1 − 𝑒(𝑟−𝐷+𝜎 2 )Δ𝑡

Therefore

[

| 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )2 | 𝑆𝑡 |

]

( ) ⎡ 2(𝑟−𝐷+ 12 𝜎 2 )Δ𝑡 2(𝑟−𝐷+ 21 𝜎 2 )(𝑛−𝑘)Δ𝑡 1 − 𝑒 𝑒 ⎢ = ⎢ 1 2 𝑛2 ⎢ 1 − 𝑒2(𝑟−𝐷+ 2 𝜎 )Δ𝑡 ⎣ ( ) 1 2 2 2𝑒(3(𝑟−𝐷)+𝜎 )Δ𝑡 1 − 𝑒2(𝑟−𝐷+ 2 𝜎 )(𝑛−𝑘−1)Δ𝑡 + ) 1 2 ( )( 1 − 𝑒(𝑟−𝐷)Δ𝑡 1 − 𝑒2(𝑟−𝐷+ 2 𝜎 )Δ𝑡 1 2 )𝜖Δ𝑡

𝑆𝑡2 𝑒−2(𝑟−𝐷+ 2 𝜎

( ) 2 2 ⎤ 2𝑒((𝑛−𝑘+2)(𝑟−𝐷)+𝜎 )Δ𝑡 1 − 𝑒(𝑟−𝐷+𝜎 )(𝑛−𝑘−1)Δ𝑡 ⎥ − ( )( ) ⎥. 1 − 𝑒(𝑟−𝐷)Δ𝑡 1 − 𝑒(𝑟−𝐷+𝜎 2 )Δ𝑡 ⎥ ⎦ Thus, using the above information we can deduce that the discrete arithmetic average rate Asian put option price at time 𝑡 < 𝑇 can be approximated by [ (𝑎) 𝑃𝑎𝑟 (𝑆𝑡 , 𝐴𝑛 , 𝑡; 𝐾, 𝑇 )

=𝑒

−𝑟(𝑇 −𝑡) ℚ

𝔼

}| ] 𝑛 | 1∑ 𝑆𝑡𝑖 , 0 || ℱ𝑡 𝐾− 𝑛 𝑖=1 | |

{ max (𝑛)

1 2 −𝑟(𝑇 −𝑡)

≈ 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑 − ) − 𝑒𝑚+ 2 𝑠

(𝑛)

Φ(−𝑑 + )

464

5.2.1 Discrete Sampling

where 𝐾 = 𝐾 − 𝐴𝑛 (𝑡1 , 𝑡𝑘 ) (𝑛)

𝑑+ =

𝑚 + 𝑠2 − log 𝐾 𝑠

(𝑛)

(𝑛)

𝑑− = 𝑑+ − 𝑠

[ ] [ ] 1 | 𝑚 ≈ 2 log 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )|| 𝑆𝑡 − log 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )2 | 𝑆𝑡 | [ ] 2 [ ] 2 ℚ 2| ℚ 𝑠 ≈ log 𝔼 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 ) | 𝑆𝑡 − 2 log 𝔼 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )|| 𝑆𝑡 . |

Since 𝑇 = 𝜏 + 𝑛Δ𝑡 and 𝑡 = 𝑡𝑘 + 𝜖Δ𝑡 = 𝜏 + (𝑘 + 𝜖)Δ𝑡, 𝜖 ∈ [0, 1) therefore lim (𝑛 − 𝑘)Δ𝑡 = lim (𝑇 − 𝑡 + 𝜖Δ𝑡) = 𝑇 − 𝑡

𝑛→∞

𝑛→∞

and lim (𝑛 − 𝑘 − 1)Δ𝑡 = lim (𝑇 − 𝑡 + (𝜖 − 1)Δ𝑡) = 𝑇 − 𝑡.

𝑛→∞

𝑛→∞

Taking limits 𝑛 → ∞, [ ] [ ] 𝑆 𝑒(𝑟−𝐷)(1−𝜖)Δ𝑡 1 − 𝑒(𝑟−𝐷)(𝑛−𝑘)Δ𝑡 lim 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )|| 𝑆𝑡 = lim 𝑡 𝑛→∞ 𝑛→∞ 𝑛 1 − 𝑒(𝑟−𝐷)Δ𝑡 (𝑟−𝐷)(1−𝜖)Δ𝑡 𝑆𝑒 (1 − 𝑒(𝑟−𝐷)(𝑛−𝑘)Δ𝑡 ) = lim 𝑡 𝑛→∞ 𝑛Δ𝑡 Δ𝑡 × lim Δ𝑡→0 1 − 𝑒(𝑟−𝐷)Δ𝑡 𝑆 (1 − 𝑒(𝑟−𝐷)(𝑇 −𝑡) ) Δ𝑡 = 𝑡 . lim Δ𝑡→0 1 − 𝑒(𝑟−𝐷)Δ𝑡 𝑇 −𝜏 From L’Ĥopital’s rule lim

Δ𝑡→0

1 Δ𝑡 1 = lim =− 𝑟−𝐷 1 − 𝑒(𝑟−𝐷)Δ𝑡 Δ𝑡→0 −(𝑟 − 𝐷)𝑒(𝑟−𝐷)Δ𝑡

and hence [ ] 𝑆 (𝑒(𝑟−𝐷)(𝑇 −𝑡) − 1) lim 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )|| 𝑆𝑡 = 𝑡 . 𝑛→∞ (𝑟 − 𝐷)(𝑇 − 𝜏)

5.2.1 Discrete Sampling

465

In contrast,

lim 𝔼

𝑛→∞

ℚ

[

1 2 )𝜖Δ𝑡

]

2|

𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 ) | 𝑆𝑡 = lim | 𝑛→∞

𝑆𝑡2 𝑒−2(𝑟−𝐷+ 2 𝜎 𝑛2

( ) ⎡ 2(𝑟−𝐷+ 12 𝜎 2 )Δ𝑡 2(𝑟−𝐷+ 21 𝜎 2 )(𝑛−𝑘)Δ𝑡 𝑒 1 − 𝑒 ⎢ ×⎢ 1 2 ⎢ 1 − 𝑒2(𝑟−𝐷+ 2 𝜎 )Δ𝑡 ⎣ ( ) 1 2 2 2𝑒(3(𝑟−𝐷)+𝜎 )Δ𝑡 1 − 𝑒2(𝑟−𝐷+ 2 𝜎 )(𝑛−𝑘−1)Δ𝑡 + ) 1 2 ( )( 1 − 𝑒(𝑟−𝐷)Δ𝑡 1 − 𝑒2(𝑟−𝐷+ 2 𝜎 )Δ𝑡 ( ) 2 2 ⎤ 2𝑒((𝑛−𝑘+2)(𝑟−𝐷)+𝜎 )Δ𝑡 1 − 𝑒(𝑟−𝐷+𝜎 )(𝑛−𝑘−1)Δ𝑡 ⎥ − ( )( ) ⎥ 1 − 𝑒(𝑟−𝐷)Δ𝑡 1 − 𝑒(𝑟−𝐷+𝜎 2 )Δ𝑡 ⎥ ⎦ 1 2 )𝜖Δ𝑡

= lim

𝑆𝑡2 𝑒−2(𝑟−𝐷+ 2 𝜎 𝑛2 Δ𝑡2

𝑛→∞

( ) ⎡ 2(𝑟−𝐷+ 12 𝜎 2 )Δ𝑡 2(𝑟−𝐷+ 21 𝜎 2 )(𝑛−𝑘)Δ𝑡 𝑒 1 − 𝑒 Δ𝑡2 ⎢ ×⎢ 1 2 ⎢ 1 − 𝑒2(𝑟−𝐷+ 2 𝜎 )Δ𝑡 ⎣ ( ) 1 2 2 2𝑒(3(𝑟−𝐷)+𝜎 )Δ𝑡 1 − 𝑒2(𝑟−𝐷+ 2 𝜎 )(𝑛−𝑘−1)Δ𝑡 Δ𝑡2 + ) 1 2 ( )( 1 − 𝑒(𝑟−𝐷)Δ𝑡 1 − 𝑒2(𝑟−𝐷+ 2 𝜎 )Δ𝑡 ( ) 2 2 ⎤ 2𝑒((𝑛−𝑘+2)(𝑟−𝐷)+𝜎 )Δ𝑡 1 − 𝑒(𝑟−𝐷+𝜎 )(𝑛−𝑘−1)Δ𝑡 Δ𝑡2 ⎥ − ( )( ) ⎥ 1 − 𝑒(𝑟−𝐷)Δ𝑡 1 − 𝑒(𝑟−𝐷+𝜎 2 )Δ𝑡 ⎥ ⎦ = 𝐶1 + 𝐶2 − 𝐶3 where 1 2 )(1−𝜖)Δ𝑡

𝐶1 = lim

𝑆𝑡2 𝑒2(𝑟−𝐷+ 2 𝜎

Δ𝑡→0

Δ𝑡2 1 2 )Δ𝑡

1 − 𝑒2(𝑟−𝐷+ 2 𝜎

1 2 )(𝑛−𝑘)Δ𝑡

1 − 𝑒2(𝑟−𝐷+ 2 𝜎

𝑛2 Δ𝑡2

𝑛→∞

× lim

(

)

466

5.2.1 Discrete Sampling 1 2 )(1−𝜖)Δ𝑡

𝐶2 = lim

2𝑆𝑡2 𝑒2(𝑟−𝐷+ 2 𝜎

(

1 2 )(𝑛−𝑘−1)Δ𝑡

1 − 𝑒2(𝑟−𝐷+ 2 𝜎

)

𝑒(𝑟−𝐷)Δ𝑡

𝑛2 Δ𝑡2

𝑛→∞

Δ𝑡2

× lim ( Δ𝑡→0

1 − 𝑒(𝑟−𝐷)Δ𝑡

)(

1 2 )Δ𝑡

1 − 𝑒2(𝑟−𝐷+ 2 𝜎

)

and 1 2 )(1−𝜖)Δ𝑡

𝐶3 = lim

2𝑆𝑡2 𝑒2(𝑟−𝐷+ 2 𝜎

(

1 − 𝑒(𝑟−𝐷+𝜎

2 )(𝑛−𝑘−1)Δ𝑡

)

𝑒(𝑟−𝐷)(𝑛−𝑘)Δ𝑡

𝑛2 Δ𝑡2

𝑛→∞

Δ𝑡2

× lim ( Δ𝑡→0

1 − 𝑒(𝑟−𝐷)Δ𝑡

)( ). 1 − 𝑒(𝑟−𝐷+𝜎 2 )Δ𝑡

By taking limits 𝑛 → ∞ and by applying L’Ĥopital’s rule

𝐶1 =

( ) 1 2 𝑆𝑡2 1 − 𝑒2(𝑟−𝐷+ 2 𝜎 )(𝑇 −𝑡) (𝑇 − 𝜏)2 (

𝐶2 =

1 2 )(𝑇 −𝑡)

2𝑆𝑡2 1 − 𝑒2(𝑟−𝐷+ 2 𝜎

(

lim

Δ𝑡→0

−2 𝑟 − 𝐷 +

𝑒

2(𝑟−𝐷+ 12 𝜎 2 )Δ𝑡

)

(𝑇 − 𝜏)2 × lim

2Δ𝑡 ( ) 1 2 1 − 𝑒2(𝑟−𝐷+ 2 𝜎 )Δ𝑡

⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ( ) ⎢ −2 𝑟 − 𝐷 + 1 𝜎 2 𝑒2(𝑟−𝐷+ 12 𝜎 2 )Δ𝑡 (1 − 𝑒(𝑟−𝐷)Δ𝑡 ) ⎥ ⎣ ⎦ 2 ( ) 1 2 2𝑆𝑡2 1 − 𝑒2(𝑟−𝐷+ 2 𝜎 )(𝑇 −𝑡) Δ𝑡→0

=

2Δ𝑡 )

1 2 𝜎 2

−(𝑟 − 𝐷)𝑒(𝑟−𝐷)Δ𝑡

(𝑇 − 𝜏)2

2 ( ) 1 2 ⎤ ⎡ 2 (𝑟−𝐷)Δ𝑡 1 − 𝑒2(𝑟−𝐷+ 2 𝜎 )Δ𝑡 −(𝑟 − 𝐷) 𝑒 ⎥ ⎢ ⎥ ⎢ ) ( ⎥ ⎢ 2 )Δ𝑡 1 2 (3(𝑟−𝐷)+𝜎 +4(𝑟 − 𝐷) 𝑟 − 𝐷 + 2 𝜎 𝑒 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ )2 ( 1 2 ( ) ⎢ −4 𝑟 − 𝐷 + 1 𝜎 2 𝑒2(𝑟−𝐷+ 2 𝜎 )Δ𝑡 1 − 𝑒(𝑟−𝐷)Δ𝑡 ⎥ ⎦ ⎣ 2 ( ) 1 2 𝑆𝑡2 1 − 𝑒2(𝑟−𝐷+ 2 𝜎 )(𝑇 −𝑡) = ) ( (𝑟 − 𝐷) 𝑟 − 𝐷 + 12 𝜎 2 (𝑇 − 𝜏)2 × lim

Δ𝑡→0

=0

5.2.1 Discrete Sampling

467

and

𝐶3 =

( ) 2 2𝑆𝑡2 1 − 𝑒(𝑟−𝐷+𝜎 )(𝑇 −𝑡) 𝑒(𝑟−𝐷)(𝑇 −𝑡) (𝑇 − 𝜏)2 2Δ𝑡 ( ) ⎡ −(𝑟 − 𝐷)𝑒(𝑟−𝐷)Δ𝑡 1 − 𝑒(𝑟−𝐷+𝜎 2 )Δ𝑡 ⎤ ⎢ ⎥ ⎢ ⎥ ( ) ⎢ (𝑟−𝐷+𝜎 2 ) 1−𝑒(𝑟−𝐷)Δ𝑡 ⎥ 2 ⎣ −(𝑟 − 𝐷 + 𝜎 )𝑒 ⎦ ( ) 2 2𝑆𝑡2 1 − 𝑒(𝑟−𝐷+𝜎 )(𝑇 −𝑡) 𝑒(𝑟−𝐷)(𝑇 −𝑡)

× lim

Δ𝑡→0

=

(𝑇 − 𝜏)2 2 (

) ⎡ −(𝑟 − 𝐷)2 𝑒(𝑟−𝐷)Δ𝑡 1 − 𝑒(𝑟−𝐷+𝜎 2 )Δ𝑡 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ 2(𝑟−𝐷+ 21 𝜎 2 )Δ𝑡 ⎥ 2 ⎢ +2(𝑟 − 𝐷)(𝑟 − 𝐷 + 𝜎 )𝑒 ⎥ ⎢ ⎥ )⎥ 2 )Δ𝑡 ( ⎢ 2 (𝑟−𝐷+𝜎 (𝑟−𝐷)Δ𝑡 1−𝑒 ⎣ −(𝑟 − 𝐷 + 𝜎 )𝑒 ⎦ ( ) 2 2𝑆𝑡2 1 − 𝑒(𝑟−𝐷+𝜎 )(𝑇 −𝑡) 𝑒(𝑟−𝐷)(𝑇 −𝑡) . = (𝑟 − 𝐷)(𝑟 − 𝐷 + 𝜎 2 )(𝑇 − 𝜏)2 × lim

Δ𝑡→0

Thus, [

| lim 𝔼ℚ 𝐴𝑛 (𝑡𝑘+1 , 𝑡𝑛 )2 | 𝑆𝑡 |

𝑛→∞

]

) ( ⎡ (𝑟−𝐷+𝜎 2 )(𝑇 −𝑡) − 1 𝑒(𝑟−𝐷)(𝑇 −𝑡) 2 𝑒 ⎢ = ⎢ (𝑟 − 𝐷)(𝑇 − 𝜏)2 ⎢ 𝑟 − 𝐷 + 𝜎2 ⎣ ( ) 1 2 ⎤ 𝑒2(𝑟−𝐷+ 2 𝜎 )(𝑇 −𝑡) − 1 ⎥ − ⎥. 𝑟 − 𝐷 + 12 𝜎 2 ⎥ ⎦ 𝑆𝑡2

8. Curran Approximation – Discrete Arithmetic Average Rate (Fixed Strike) Asian Option. Let the asset price 𝑆𝑡 follow the GBM 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡 𝑆𝑡

468

5.2.1 Discrete Sampling

where 𝑊𝑡 is the ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ), 𝜇, 𝐷 and 𝜎 are the drift parameter, continuous dividend yield and volatility, respectively. In addition, let 𝑟 be the risk-free interest rate. Let the time interval [𝜏, 𝑇 ] be partitioned into 𝑛 equal subintervals each of length Δ𝑡 = (𝑇 − 𝜏)∕𝑛 and let 𝑆𝑡𝑖 , 𝑡𝑖 = 𝜏 + 𝑖Δ𝑡 be the asset price at the end of the 𝑖th interval, 𝑖 = 1, 2, … , 𝑛. We define the discretely measured arithmetic average of asset prices as 𝑛

𝐴𝑛 (𝜏, 𝑇 ) =

1∑ 𝑆 𝑛 𝑖=1 𝑡𝑖

and the discretely measured geometric average of asset prices as

𝐺𝑛 (𝜏, 𝑇 ) =

( 𝑛 ∏ 𝑖=1

)1 𝑆 𝑡𝑖

𝑛

.

We consider an arithmetic average rate Asian call option with expiry 𝑇 > 𝑡 > 𝜏 and payoff Ψ(𝑆𝑇 , 𝐴𝑛 (𝜏, 𝑇 )) = max{𝐴𝑛 (𝜏, 𝑇 ) − 𝐾, 0} where 𝐾 > 0 is the strike price. Show that if the random variables 𝑋 and 𝑌 follow a joint normal distribution (𝑋, 𝑌 ) ∼ 2 (𝝁, 𝚺) ] [ ] [ ] 𝜎𝑥2 𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 𝜇𝑥 Var(𝑋) Cov(𝑋, 𝑌 ) and 𝚺 = = where 𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 𝜎𝑦2 𝜇𝑦 Cov(𝑋, 𝑌 ) Var(𝑌 ) 𝜇𝑥 , 𝜇𝑦 are the means, 𝜎𝑥2 , 𝜎𝑦2 are the variances and 𝜌𝑥𝑦 ∈ (−1, 1) is the correlation coefficient, then the conditional distribution [

where 𝝁 =

( 𝑋| 𝑌 = 𝑦 ∼ 

) 𝜎𝑥 2 2 𝜇𝑥 + 𝜌𝑥𝑦 (𝑦 − 𝜇𝑦 ), (1 − 𝜌𝑥𝑦 )𝜎𝑥 . 𝜎𝑦

Suppose 𝑆𝑡1 , 𝑆𝑡2 , …, 𝑆𝑡𝑘 , 1 ≤ 𝑘 ≤ 𝑛 − 1 have been observed and let 𝑡𝑘 ≤ 𝑡 < 𝑡𝑘+1 where 𝑡 = 𝑡𝑘 + 𝜖Δ𝑡, 𝜖 ∈ [0, 1). Show that under the risk-neutral measure ℚ, for 𝑡𝑖 > 𝑡𝑘 log 𝑆𝑡𝑖 ∼  (𝜇𝑖 , 𝜎𝑖2 ) 2 ) log 𝐺𝑛 (𝜏, 𝑇 ) ∼  (𝜇𝐺 , 𝜎𝐺

where ) ( 1 𝜇𝑖 = log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑖 − 𝑘 − 𝜖)Δ𝑡 2 𝜎𝑖2 = 𝜎 2 (𝑖 − 𝑘 − 𝜖)Δ𝑡

5.2.1 Discrete Sampling

469

and ) 1𝑛 ⎤ ⎡( 𝑘 𝑛−𝑘 ⎥ ⎢ ∏ 𝑛 𝑆 𝑡𝑗 𝑆𝑡 ⎥ 𝜇𝐺 = log ⎢ ⎥ ⎢ 𝑗=1 ⎦ ⎣ [ ) (𝑛 − 𝑘 − 1)(𝑛 − 𝑘) (1 − 𝜖)(𝑛 − 𝑘) ] ( 1 + 𝑟 − 𝐷 − 𝜎2 + Δ𝑡 2 2𝑛 𝑛 [ ] (𝑛 − 𝑘)(2(𝑛 − 𝑘) − 1)(𝑛 − 𝑘 − 1) (1 − 𝜖)(𝑛 − 𝑘)2 2 𝜎𝐺 = 𝜎2 + Δ𝑡. 6𝑛2 𝑛2 ) ( Given the pair of random variables log 𝑆𝑡𝑖 , log 𝐺𝑛 (𝜏, 𝑇 ) , 𝑡𝑖 > 𝑡𝑘 show that it has the covariance matrix ⎡ 𝜎𝑖2 𝚺=⎢ ⎢ ⎣ 𝜎𝑖𝐺

𝜎𝑖𝐺 ⎤ ⎥ 2 ⎥ 𝜎𝐺 ⎦

where 𝜎𝑖𝐺 =

𝜎2 [(𝑖 − 𝑘 − 𝜖)(2𝑛 − 𝑖 − 𝑘) + (1 − 𝜖)(𝑖 − 𝑘)] Δ𝑡 2𝑛

( ) and log 𝑆𝑡𝑖 , log 𝐺𝑛 (𝜏, 𝑇 ) follows a bivariate normal distribution. Hence, deduce that the conditional distribution of log 𝑆𝑡𝑖 , 𝑡𝑖 > 𝑡𝑘 given log 𝐺𝑛 (𝜏, 𝑇 ) is ( log 𝑆𝑡𝑖 | log 𝐺𝑛 (𝜏, 𝑇 ) = 𝑦 ∼ 

𝜎𝑖𝐺

𝜇𝑖 + (𝑦 − 𝜇𝐺 ) 2 , 𝜎𝑖2 𝜎𝐺

−

2 𝜎𝑖𝐺 2 𝜎𝐺

) .

Show that for any random variable 𝑍 under the filtration 𝒢 , 0 ≤ 𝔼 ( max{𝑍, 0}| 𝒢 ) − max {𝔼 ( 𝑍| 𝒢 ) , 0} ≤

1√ Var ( 𝑍| 𝒢 ). 2

By defining the arithmetic average rate (fixed strike) call option price at time 𝑡, 𝜏 < 𝑡𝑘 < 𝑡 < 𝑇 as [ ] (𝑎) (𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐴𝑛 (𝜏, 𝑇 ) − 𝐾, 0}|| ℱ𝑡 𝐶𝑎𝑟 and by setting 𝐴𝑛 (𝑡1 , 𝑡𝑘 ) =

1 𝑛

∑𝑘

𝑗=1 𝑆𝑡𝑗

and using the following tower property

[ [ ] [ ]| ] 𝔼ℚ max{𝐴𝑛 (𝜏, 𝑇 ) − 𝐾, 0}|| ℱ𝑡 = 𝔼ℚ 𝔼ℚ max{𝐴𝑛 (𝜏, 𝑇 ) − 𝐾, 0}|| 𝐺𝑛 (𝜏, 𝑇 ) | ℱ𝑡 |

470

5.2.1 Discrete Sampling

show that (𝑎) 𝐶𝑙𝑜𝑤𝑒𝑟 ≤ 𝐶𝑎𝑟 (𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) ≤ 𝐶𝑢𝑝𝑝𝑒𝑟

where

( ) 𝑛 ̃ + 𝜎𝑖𝐺 𝜇𝐺 − log 𝐾 1 ∑ 𝜇𝑖 + 12 𝜎𝑖2 =𝑒 𝑒 Φ 𝑛 𝑖=𝑘+1 𝜎𝐺 ( )] ̃ ( ) 𝜇𝐺 − log 𝐾 + 𝐴𝑛 (𝑡1 , 𝑡𝑘 ) − 𝐾 Φ 𝜎𝐺 [

𝐶𝑙𝑜𝑤𝑒𝑟

𝐶𝑢𝑝𝑝𝑒𝑟

−𝑟(𝑇 −𝑡)

√ [ ]| 𝑛 ⎡√ | | ⎤ 1 −𝑟(𝑇 −𝑡) ℚ ⎢ √ 1 ∑ | | √ = 𝐶𝑙𝑜𝑤𝑒𝑟 + 𝑒 𝔼 Var 𝑆𝑡𝑖 | 𝐺𝑛 (𝜏, 𝑇 ) | ℱ𝑡 ⎥ | ⎥ ⎢ 2 𝑛 𝑖=𝑘+1 || | ⎦ ⎣ |

{

𝑛 | 2 ∕𝜎 2 ) 1 ∑ 𝜇𝑖 +(log 𝑧−𝜇𝐺 )𝜎𝑖𝐺 ∕𝜎𝐺2 + 21 (𝜎𝑖2 −𝜎𝑖𝐺 | 𝐺 = 𝐾 𝑧 |𝐴𝑛 (𝜏, 𝑡𝑘 ) + 𝑒 | 𝑛 𝑖=𝑘+1 |

̃= 𝐾 𝑥

and Φ(𝑥) =

∫−∞

}

1 − 1 𝑢2 √ 𝑒 2 𝑑𝑢 is the cdf of a standard normal. 2𝜋

Solution: If (𝑋, 𝑌 ) follows a bivariate normal distribution then the joint pdf is given as [

𝑓𝑋𝑌 (𝑥, 𝑦) =

1 𝑒 √ 2𝜋𝜎𝑥 𝜎𝑦 1 − 𝜌2𝑥𝑦

−

1 2(1−𝜌2𝑥𝑦 )

(

𝑥−𝜇𝑥 𝜎𝑥

)2

( −2𝜌𝑥𝑦

𝑥−𝜇𝑥 𝜎𝑥

] )( 𝑦−𝜇 ) ( 𝑦−𝜇 )2 𝑦 𝑦 + 𝜎 𝜎 𝑦

𝑦

.

The pdf of the conditional distribution 𝑓𝑋|𝑌 (𝑥|𝑦) =

=

𝑓𝑋𝑌 (𝑥, 𝑦) 𝑓𝑌 (𝑦)

2𝜋𝜎𝑥 𝜎𝑦

1 √

1−𝜌2𝑥𝑦

𝑒

−

1 2(1−𝜌2𝑥𝑦 )

[ (

𝑥−𝜇𝑥 𝜎𝑥

)2

( 1

−2 1 √ 𝑒 𝜎𝑦 2𝜋 [ (

1 = √ 𝑒 √ 2𝜋𝜎𝑥 1 − 𝜌2𝑥𝑦

−

1 2(1−𝜌2𝑥𝑦 )

(

1 = √ 𝑒 √ 2𝜋𝜎𝑥 1 − 𝜌2𝑥𝑦

( −2𝜌𝑥𝑦

𝑦−𝜇𝑦 𝜎𝑦

𝑥−𝜇𝑥 𝜎𝑥

𝑦

𝑦

)2

)2

( −2𝜌𝑥𝑦

𝑥−𝜇𝑥 𝜎𝑥

) 2

⎡ 𝑥− 𝜇𝑥 +𝜌𝑥𝑦 𝜎𝜎𝑥 (𝑦−𝜇𝑦 ) ⎤ ⎥ √ 𝑦 ⎥ 𝜎 1−𝜌2 𝑥 𝑥𝑦 ⎣ ⎦

− 21 ⎢ ⎢

] )( 𝑦−𝜇 ) ( 𝑦−𝜇 )2 𝑦 𝑦 + 𝜎 𝜎

𝑥−𝜇𝑥 𝜎𝑥

.

( )2 ] )( 𝑦−𝜇 ) 𝑦−𝜇𝑦 𝑦 2 +𝜌 𝑥𝑦 𝜎 𝜎 𝑦

𝑦

5.2.1 Discrete Sampling

471

(

) 𝜎𝑥 2 2 Hence, 𝑋| 𝑌 = 𝑦 ∼  𝜇𝑥 + 𝜌𝑥𝑦 (𝑦 − 𝜇𝑦 ), (1 − 𝜌𝑥𝑦 )𝜎𝑥 . 𝜎𝑦 Under the risk-neutral measure ℚ, 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡ℚ 𝑆𝑡

(𝜇 − 𝑟) 𝑡 + 𝑊𝑡 is a ℚ-standard Wiener process. where 𝑊𝑡ℚ = 𝜎 Thus, conditional on 𝑆𝑡 for 𝑡𝑖 > 𝑡𝑘 1 2 )(𝑡

𝑆𝑡𝑖 = 𝑆𝑡 𝑒(𝑟−𝐷− 2 𝜎 or

𝑖 −𝑡)+𝜎𝑊𝑡𝑖 −𝑡

) ( 1 log 𝑆𝑡𝑖 = log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑡𝑖 − 𝑡) + 𝜎𝑊𝑡𝑖 −𝑡 2

where 𝑊𝑡𝑖 −𝑡 ∼  (0, 𝑡𝑖 − 𝑡). Hence, we can easily deduce that log 𝑆𝑡𝑖 ∼  (𝜇𝑖 , 𝜎𝑖2 ) [ ] [ ] | | where 𝜇𝑖 = 𝔼ℚ log 𝑆𝑡𝑖 | 𝑆𝑡 and 𝜎𝑖2 = Varℚ log 𝑆𝑡𝑖 | 𝑆𝑡 . | | Since 𝑡 = 𝑡𝑘 + 𝜖Δ𝑡, 𝑡𝑖 = 𝜏 + 𝑖Δ𝑡 and 𝑡𝑘 = 𝜏 + 𝑘Δ𝑡, [ ] | 𝜇𝑖 = 𝔼ℚ log 𝑆𝑡𝑖 | 𝑆𝑡 | ) ( 1 = log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑡𝑖 − 𝑡) 2 ) ( 1 = log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑖 − 𝑘 − 𝜖)Δ𝑡 2 and

[ ] | 𝜎𝑖2 = Varℚ log 𝑆𝑡𝑖 | 𝑆𝑡 | 2 = 𝜎 (𝑡𝑖 − 𝑡) = 𝜎 2 (𝑖 − 𝑘 − 𝜖)Δ𝑡.

For the geometric average, since 𝑆𝑡1 , 𝑆𝑡2 , …, 𝑆𝑡𝑘 , 1 ≤ 𝑘 ≤ 𝑛 − 1 have been observed, by taking logarithms ) ( 𝑛 ∏ 1 𝑆 𝑡𝑗 log 𝐺𝑛 (𝜏, 𝑇 ) = log 𝑛 𝑗=1 𝑛

=

1∑ log 𝑆𝑡𝑗 𝑛 𝑗=1

=

𝑘 𝑛 1∑ 1 ∑ log 𝑆𝑡𝑗 + log 𝑆𝑡𝑗 𝑛 𝑗=1 𝑛 𝑗=𝑘+1

472

5.2.1 Discrete Sampling 𝑘 𝑛 [ ) ] ( 1∑ 1 ∑ 1 log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑡𝑗 − 𝑡) + 𝜎𝑊𝑡𝑗 −𝑡 = log 𝑆𝑡𝑗 + 𝑛 𝑗=1 𝑛 𝑗=𝑘+1 2

) 1𝑛 ⎡( 𝑘 ⎤ 𝑛 ) ∑ 𝑛−𝑘 ⎢ ∏ ⎥ 1( 1 𝑛 𝑟 − 𝐷 − 𝜎2 = log ⎢ 𝑆 𝑡𝑗 𝑆𝑡 ⎥ + (𝑡𝑗 − 𝑡) 2 ⎢ 𝑗=1 ⎥ 𝑛 𝑗=𝑘+1 ⎣ ⎦ 𝑛 𝜎 ∑ + 𝑊 . 𝑛 𝑗=𝑘+1 𝑡𝑗 −𝑡 Since 𝑊𝑡𝑗 −𝑡 ∼  (0, 𝑡𝑗 − 𝑡) and because the sum of normal variates is also normal, we can therefore deduce 2 log 𝐺𝑛 (𝜏, 𝑇 ) ∼  (𝜇𝐺 , 𝜎𝐺 )

[ ] [ ] 2 = Varℚ log 𝐺 (𝜏, 𝑇 )| 𝑆 . with 𝜇𝐺 = 𝔼ℚ log 𝐺𝑛 (𝜏, 𝑇 )|| 𝑆𝑡 and 𝜎𝐺 𝑛 | 𝑡 Since 𝑡 = 𝑡𝑘 + 𝜖Δ𝑡, 𝑡𝑗 = 𝜏 + 𝑗Δ𝑡 and 𝑡𝑘 = 𝜏 + 𝑘Δ𝑡, [ ] 𝜇𝐺 = 𝔼ℚ log 𝐺𝑛 (𝜏, 𝑇 )|| 𝑆𝑡 ) 1𝑛 ⎡( 𝑘 ⎤ 𝑛 ) ∑ 𝑛−𝑘 ⎢ ∏ ⎥ 1( 1 𝑛 𝑟 − 𝐷 − 𝜎2 = log ⎢ 𝑆 𝑡𝑗 𝑆𝑡 ⎥ + (𝑡𝑖 − 𝑡) 2 ⎢ 𝑗=1 ⎥ 𝑛 𝑗=𝑘+1 ⎣ ⎦ ) 1𝑛 ⎤ ⎡( 𝑘 𝑛 ) ∑ 𝑛−𝑘 ⎥ 1( ⎢ ∏ 1 𝑟 − 𝐷 − 𝜎2 𝑆 𝑡𝑗 𝑆𝑡 𝑛 ⎥ + (𝑖 − 𝑘 − 𝜖)Δ𝑡 = log ⎢ 2 ⎥ 𝑛 ⎢ 𝑗=1 𝑗=𝑘+1 ⎦ ⎣ ) 1𝑛 ⎤ ⎡( 𝑘 𝑛−𝑘 ⎥ ⎢ ∏ 𝑛 𝑆 𝑡𝑗 𝑆𝑡 ⎥ = log ⎢ ⎥ ⎢ 𝑗=1 ⎦ ⎣ ) )( ( 𝑛−𝑘 1 2 (𝑛 − 𝑘 + 1 − 2𝜖) Δ𝑡 + 𝑟−𝐷− 𝜎 2 𝑛 ) 1𝑛 ⎡( 𝑘 ⎤ 𝑛−𝑘 ⎢ ∏ ⎥ = log ⎢ 𝑆 𝑡𝑗 𝑆𝑡 𝑛 ⎥ ⎢ 𝑗=1 ⎥ ⎣ ⎦ ) [ (𝑛 − 𝑘 − 1)(𝑛 − 𝑘) (1 − 𝜖)(𝑛 − 𝑘) ] ( 1 + 𝑟 − 𝐷 − 𝜎2 + Δ𝑡 2 2𝑛 𝑛 and because 𝑛 ∑ 𝑘=1

𝑘2 =

1 𝑛(𝑛 + 1)(2𝑛 + 1) 6

5.2.1 Discrete Sampling

473

the variance is [ ] 2 = Varℚ log 𝐺𝑛 (𝜏, 𝑇 )|| 𝑆𝑡 𝜎𝐺 (

= Cov

=

ℚ

𝑛 𝑛 𝜎 ∑ 𝜎 ∑ 𝑊𝑡𝑝 −𝑡 , 𝑊 𝑛 𝑝=𝑘+1 𝑛 𝑞=𝑘+1 𝑡𝑞 −𝑡

)

𝑛 𝑛 ( ) 𝜎2 ∑ ∑ ℚ 𝑊 Cov , 𝑊 𝑡𝑝 −𝑡 𝑡𝑞 −𝑡 𝑛2 𝑝=𝑘+1 𝑞=𝑘+1

𝑛 𝑛 𝜎2 ∑ ∑ = 2 min{𝑡𝑝 − 𝑡, 𝑡𝑞 − 𝑡} 𝑛 𝑝=𝑘+1 𝑞=𝑘+1 [ 𝑝 𝑛 ∑ 𝜎2 ∑ = 2 min{𝑡𝑝 − 𝑡, 𝑡𝑞 − 𝑡} 𝑛 𝑝=𝑘+1 𝑞=𝑘+1 ] 𝑛 ∑ + min{𝑡𝑝 − 𝑡, 𝑡𝑞 − 𝑡} 𝑞=𝑝+1

𝜎2 = 2 𝑛 𝜎2 = 2 𝑛 𝜎2 = 2 𝑛 +

[ [ [

𝑝 ∑

𝑛 ∑

𝑝=𝑘+1 𝑞=𝑘+1 𝑝 ∑

𝑛 ∑

(𝑡𝑞 − 𝑡) +

(

𝑝=𝑘+1 𝑛 ∑

𝑝−𝑘 2

]

𝑛 ∑

𝑝=𝑘+1 𝑞=𝑝+1

(𝑞 − 𝑘 − 𝜖)Δ𝑡 +

𝑝=𝑘+1 𝑞=𝑘+1 𝑛 ∑

𝑛 ∑

(𝑡𝑝 − 𝑡) 𝑛 ∑

𝑛 ∑

] (𝑝 − 𝑘 − 𝜖)Δ𝑡

𝑝=𝑘+1 𝑞=𝑝+1

)

(𝑝 − 𝑘 + 1 − 2𝜖)Δ𝑡 ]

(𝑛 − 𝑝)(𝑝 − 𝑘 − 𝜖)Δ𝑡

𝑝=𝑘+1

𝜎2 = 2 2𝑛 −

[

𝑛 ∑ 𝑝=𝑘+1

𝑛 ∑

(2(𝑛 − 𝑘) + 1) (𝑝 − 𝑘) − ]

𝑛 ∑

(𝑝 − 𝑘)2

𝑝=𝑘+1

2(𝑛 − 𝑘)𝜖 Δ𝑡

𝑝=𝑘+1

=

[ 𝜎 2 (2(𝑛 − 𝑘) + 1)(𝑛 − 𝑘)(𝑛 − 𝑘 + 1) 2 2𝑛2

] (2(𝑛 − 𝑘) + 1)(𝑛 − 𝑘 + 1)(𝑛 − 𝑘) 2 − − 2(𝑛 − 𝑘) 𝜖 Δ𝑡 6 [ ] (2(𝑛 − 𝑘) + 1)(𝑛 − 𝑘)(𝑛 − 𝑘 + 1) (𝑛 − 𝑘)2 𝜖 = 𝜎2 − Δ𝑡 6𝑛2 𝑛2 [ ] (1 − 𝜖)(𝑛 − 𝑘)2 2 (𝑛 − 𝑘)(2(𝑛 − 𝑘) − 1)(𝑛 − 𝑘 − 1) =𝜎 + Δ𝑡. 6𝑛2 𝑛2

474

5.2.1 Discrete Sampling

To find the covariance of the random variables (log 𝑆𝑡𝑖 , log 𝐺𝑛 (𝜏, 𝑇 )), by definition ( ) 𝑛 ) ( 1∑ log 𝑆𝑡𝑗 Cov log 𝑆𝑡𝑖 , log 𝐺𝑛 (𝜏, 𝑇 ) = Cov log 𝑆𝑡𝑖 , 𝑛 𝑗=1 ( ) 𝑛 1 ∑ = Cov log 𝑆𝑡𝑖 , log 𝑆𝑡𝑗 𝑛 𝑗=𝑘+1 𝑛 ) ( 1 ∑ Cov log 𝑆𝑡𝑖 , log 𝑆𝑡𝑗 𝑛 𝑗=𝑘+1 [ 𝑖 ) ( 1 ∑ = Cov log 𝑆𝑡𝑖 , log 𝑆𝑡𝑗 𝑛 𝑗=𝑘+1 ] 𝑛 ) ( ∑ Cov log 𝑆𝑡𝑖 , log 𝑆𝑡𝑗 +

=

𝑗=𝑖+1

[ 1 = 𝑛 +

𝑖 ∑ 𝑗=𝑘+1

𝑛 ∑ 𝑗=𝑖+1

𝜎 2 min{𝑡𝑖 − 𝑡, 𝑡𝑗 − 𝑡} ]

𝜎 min{𝑡𝑖 − 𝑡, 𝑡𝑗 − 𝑡} 2

] [ 𝑖 ∑ 1 2 2 (𝑡𝑗 − 𝑡) + 𝜎 (𝑛 − 𝑖)(𝑡𝑖 − 𝑡) 𝜎 = 𝑛 𝑗=𝑘+1 [( ) ] 𝑖−𝑘 𝜎2 (𝑡𝑘+1 + 𝑡𝑖 − 2𝑡) + (𝑛 − 𝑖)(𝑡𝑖 − 𝑡) = 𝑛 2 [( ) 𝑖−𝑘 𝜎2 (𝑡𝑘+1 + 𝑡𝑖 − 2(𝑡𝑘 + 𝜖Δ𝑡)) = 𝑛 2 ] +(𝑛 − 𝑖)(𝑡𝑖 − 𝑡𝑘 − 𝜖Δ𝑡) [( ) ] 𝑖−𝑘 𝜎2 (𝑖 − 𝑘 + 1 − 2𝜖) + (𝑛 − 𝑖)(𝑖 − 𝑘 − 𝜖) Δ𝑡 = 𝑛 2 2 𝜎 = [(𝑖 − 𝑘 − 𝜖)(2𝑛 − 𝑖 − 𝑘) + (1 − 𝜖)(𝑖 − 𝑘)] Δ𝑡. 2𝑛 To show that the pair (log 𝑆𝑡𝑖 , log 𝐺𝑛 (𝜏, 𝑇 )) follows a bivariate normal distribution, let log 𝑆𝑡𝑖 = 𝜇𝑖 + 𝜎𝑖 𝑍1 ( ) √ log 𝐺𝑛 (𝜏, 𝑇 ) = 𝜇𝐺 + 𝜎𝐺 𝜌𝑍1 + 1 − 𝜌2 𝑍2 where 𝑍1 , 𝑍2 ∼  (0, 1), 𝑍1 ⟂ ⟂ 𝑍2 and 𝜌 =

𝜎𝑖𝐺 . 𝜎𝑖 𝜎𝐺

5.2.1 Discrete Sampling

475

From the definition of the moment-generating function, for 𝜃1 , 𝜃2 ∈ ℝ [ ] [ ] √ 2 𝔼ℚ 𝑒𝜃1 log 𝑆𝑡𝑖 +𝜃2 log 𝐺𝑛 (𝜏,𝑇 ) = 𝔼ℚ 𝑒𝜃1 (𝜇𝑖 +𝜎𝑖 𝑍1 )+𝜃2 (𝜇𝐺 +𝜎𝐺 (𝜌𝑍1 + 1−𝜌 𝑍2 )) [ ] √ 2 = 𝑒𝜃1 𝜇𝑖 +𝜃2 𝜇𝐺 𝔼ℚ 𝑒(𝜃1 𝜎𝑖 +𝜃2 𝜌𝜎𝑖𝐺 )𝑍1 +(𝜃2 𝜎𝑖𝐺 1−𝜌 )𝑍2 [ ] √ [ ] 2 = 𝑒𝜃1 𝜇𝑖 +𝜃2 𝜇𝐺 𝔼ℚ 𝑒(𝜃1 𝜎𝑖 +𝜃2 𝜌𝜎𝑖𝐺 )𝑍1 𝔼ℚ 𝑒(𝜃2 𝜎𝑖𝐺 1−𝜌 )𝑍2 1

1 2 2

= 𝑒𝜃1 𝜇𝑖 +𝜃2 𝜇𝐺 𝑒 2 (𝜃1 𝜎𝑖 +𝜃2 𝜌𝜎𝑖𝐺 ) 𝑒 2 𝜃2 𝜎𝑖𝐺 (1−𝜌 1

2

2)

= 𝑒𝜃1 𝜇𝑖 +𝜃2 𝜇𝐺 + 2 (𝜃1 𝜎𝑖 +2𝜌𝜃1 𝜃2 𝜎𝑖 𝜎𝐺 +𝜃2 𝜎𝐺 ) 2 2

2 2

which is the moment-generating function of a bivariate normal distribution. Hence [ ] ⎡ 𝜎𝑖2 𝜎𝑖𝐺 ⎤ 𝜇𝑖 ⎥. (log 𝑆𝑡𝑖 , log 𝐺𝑛 (𝜏, 𝑇 )) ∼ 2 (𝝁, 𝚺) where 𝝁 = and 𝚺 = ⎢ 𝜇𝐺 ⎢ 2 ⎥ 𝜎 𝜎 ⎣ 𝑖𝐺 𝐺 ⎦ 𝜎𝑖𝐺 By setting 𝜌 = and following the result of the conditional distribution of normal 𝜎𝑖 𝜎𝐺 distribution, we have ( log 𝑆𝑡𝑖 | log 𝐺𝑛 (𝜏, 𝑇 ) = 𝑦 ∼ 

𝜎𝑖𝐺

𝜇𝑖 + (𝑦 − 𝜇𝐺 ) 2 , 𝜎𝑖2 𝜎𝐺

−

2 𝜎𝑖𝐺

)

2 𝜎𝐺

.

To show that for any random variable 𝑍 under the filtration 𝒢 0 ≤ 𝔼 ( max{𝑍, 0}| 𝒢 ) − max {𝔼 ( 𝑍| 𝒢 ) , 0} ≤

1√ Var ( 𝑍| 𝒢 ) 2

we note that since max{𝑥, 0} is a non-negative convex function, from the conditional Jensen’s inequality (see Problem 1.2.3.14 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus) 𝔼 ( max{𝑍, 0}| 𝒢 ) ≥ max {𝔼 ( 𝑍| 𝒢 ) , 0} . For a random variable 𝑈 we can set |𝑈 | = max{𝑈 , 0} − min{𝑈 , 0} 𝑈 = max{𝑈 , 0} + min{𝑈 , 0}. Thus, we can write 𝔼 ( |𝑍|| 𝒢 ) = 𝔼 ( max{𝑍, 0}| 𝒢 ) − 𝔼 ( min{𝑍, 0}| 𝒢 ) = 𝔼 ( max{𝑍, 0}| 𝒢 ) − [𝔼 ( 𝑍| 𝒢 ) − 𝔼 ( max{𝑍, 0}| 𝒢 )] = 2𝔼 ( max{𝑍, 0}| 𝒢 ) − 𝔼 ( 𝑍| 𝒢 )

476

5.2.1 Discrete Sampling

or 𝔼 ( max{𝑍, 0}| 𝒢 ) =

1 {𝔼 ( |𝑍|| 𝒢 ) + 𝔼 ( 𝑍| 𝒢 )} . 2

Further, we can write |𝔼 ( 𝑍| 𝒢 )| = max {𝔼 ( 𝑍| 𝒢 ) , 0} − min {𝔼 ( 𝑍| 𝒢 ) , 0} 𝔼 ( 𝑍| 𝒢 ) = max {𝔼 ( 𝑍| 𝒢 ) , 0} + min {𝔼 ( 𝑍| 𝒢 ) , 0} and by adding the two equations, max {𝔼 ( 𝑍| 𝒢 ) , 0} =

1 {|𝔼 ( 𝑍| 𝒢 )| + 𝔼 ( 𝑍| 𝒢 )} . 2

Thus, 1 {𝔼 ( |𝑍|| 𝒢 ) − 𝔼 ( 𝑍| 𝒢 )} 2 1 ≤ 𝔼 ( |𝑍 − 𝔼 ( 𝑍| 𝒢 )|| 𝒢 ) 2 1√ Var ( 𝑍| 𝒢 ) . ≤ 2

𝔼 ( max{𝑍, 0}| 𝒢 ) − max {𝔼 ( 𝑍| 𝒢 ) , 0} =

From the definition of the average rate call option price at time 𝑡, 𝜏 < 𝑡𝑘 < 𝑡 < 𝑇 [ ] 𝐶(𝑆𝑡 , 𝐴𝑛 (𝜏, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐴𝑛 (𝜏, 𝑇 ) − 𝐾, 0}|| ℱ𝑡 [ [ ]| ] = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 𝔼ℚ max{𝐴𝑛 (𝜏, 𝑇 ) − 𝐾, 0}|| 𝐺𝑛 (𝜏, 𝑇 ) | ℱ𝑡 . | Using the inequality max {𝔼 ( 𝑍| 𝒢 ) , 0} ≤ 𝔼 ( max{𝑍, 0}| 𝒢 ) ≤ max {𝔼 ( 𝑍| 𝒢 ) , 0} +

1√ Var ( 𝑍| 𝒢 ) 2

for a random variable 𝑍, 𝐶𝑙𝑜𝑤𝑒𝑟 and 𝐶𝑢𝑝𝑝𝑒𝑟 are [ { [ ] }| ] 𝐶𝑙𝑜𝑤𝑒𝑟 = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 ) − 𝐾 || 𝐺𝑛 (𝜏, 𝑇 ) , 0 | ℱ𝑡 | and [√ ] [ ]| 1 𝐶𝑢𝑝𝑝𝑒𝑟 = 𝐶𝑙𝑜𝑤𝑒𝑟 + 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ Var 𝐴𝑛 (𝜏, 𝑇 ) − 𝐾 || 𝐺𝑛 (𝜏, 𝑇 ) || ℱ𝑡 2 | = 𝐶𝑙𝑜𝑤𝑒𝑟 √ [ ]| 𝑛 ⎡√ | | ⎤ ∑ 1 −𝑟(𝑇 −𝑡) ℚ ⎢ √ 1 | | + 𝑒 𝔼 √Var 𝐴𝑛 (𝜏, 𝑡𝑘 ) + 𝑆𝑡𝑖 − 𝐾 | 𝐺𝑛 (𝜏, 𝑇 ) | ℱ𝑡 ⎥ | | ⎥ ⎢ 2 𝑛 𝑖=𝑘+1 | | ⎦ ⎣ | √ [ ]| 𝑛 ⎡√ ⎤ | | √ 1 1 ∑ | | = 𝐶𝑙𝑜𝑤𝑒𝑟 + 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ ⎢ √Var 𝑆𝑡𝑖 | 𝐺𝑛 (𝜏, 𝑇 ) | ℱ𝑡 ⎥ . | ⎥ ⎢ 2 𝑛 𝑖=𝑘+1 || | ⎦ ⎣ |

5.2.1 Discrete Sampling

477

By setting 𝑔(𝑧) as the pdf of 𝐺𝑛 (𝜏, 𝑇 ) and because 𝐴𝑛 (𝜏, 𝑇 ) ≥ 𝐺𝑛 (𝜏, 𝑇 ) [ { [ ] }| ] 𝐶𝑙𝑜𝑤𝑒𝑟 = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 ) − 𝐾 || 𝐺𝑛 (𝜏, 𝑇 ) , 0 | ℱ𝑡 | ∞ [ { ] } max 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 ) − 𝐾 || 𝐺𝑛 (𝜏, 𝑇 ) = 𝑧 , 0 𝑔(𝑧) 𝑑𝑧 = 𝑒−𝑟(𝑇 −𝑡) ∫0 = 𝑒−𝑟(𝑇 −𝑡)

𝐾

{ [ ] } max 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 ) − 𝐾 || 𝐺𝑛 (𝜏, 𝑇 ) = 𝑧 , 0 𝑔(𝑧) 𝑑𝑧

∫0

∞

+𝑒−𝑟(𝑇 −𝑡) = 𝑒−𝑟(𝑇 −𝑡)

∫𝐾 𝐾

∫0

{ [ ] } max 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 ) − 𝐾 || 𝐺𝑛 (𝜏, 𝑇 ) = 𝑧 , 0 𝑔(𝑧) 𝑑𝑧

∞

+𝑒−𝑟(𝑇 −𝑡) = 𝑒−𝑟(𝑇 −𝑡)

𝐾

∫𝐾̃

+𝑒−𝑟(𝑇 −𝑡) = 𝑒−𝑟(𝑇 −𝑡)

∫𝐾

∞

∫𝐾̃

[ ] 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 ) − 𝐾 || 𝐺𝑛 (𝜏, 𝑇 ) = 𝑧 𝑔(𝑧) 𝑑𝑧

[ ] 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 ) − 𝐾 || 𝐺𝑛 (𝜏, 𝑇 ) = 𝑧 𝑔(𝑧) 𝑑𝑧

∞

∫𝐾

{ [ ] } max 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 ) − 𝐾 || 𝐺𝑛 (𝜏, 𝑇 ) = 𝑧 , 0 𝑔(𝑧) 𝑑𝑧

[ ] 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 ) − 𝐾 || 𝐺𝑛 (𝜏, 𝑇 ) = 𝑧 𝑔(𝑧) 𝑑𝑧

[ ] 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 ) − 𝐾 || 𝐺𝑛 (𝜏, 𝑇 ) = 𝑧 𝑔(𝑧) 𝑑𝑧

where [ ] } { ̃ = 𝑧| 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 )| 𝐺𝑛 (𝜏, 𝑇 ) = 𝑧 = 𝐾 . 𝐾 | ̃ we note that To find 𝐾, 𝔼

[ ℚ

[

]

𝐴𝑛 (𝜏, 𝑇 )|| 𝐺𝑛 (𝜏, 𝑇 ) = 𝑧 = 𝐴𝑛 (𝜏, 𝑡𝑘 ) + 𝔼ℚ [ ℚ

= 𝐴𝑛 (𝜏, 𝑡𝑘 ) + 𝔼

𝑛 | 1 ∑ | 𝑆𝑡𝑖 | 𝐺𝑛 (𝜏, 𝑇 ) = 𝑧 𝑛 𝑖=𝑘+1 ||

]

] 𝑛 | 1 ∑ | 𝑆 | log 𝐺𝑛 (𝜏, 𝑇 ) = log 𝑧 . 𝑛 𝑖=𝑘+1 𝑡𝑖 ||

Since ( log 𝑆𝑡𝑖 | log 𝐺𝑛 (𝜏, 𝑇 ) = 𝑦 ∼ 

𝜎𝑖𝐺

𝜇𝑖 + (𝑦 − 𝜇𝐺 ) 2 , 𝜎𝑖2 𝜎𝐺

−

2 𝜎𝑖𝐺

)

2 𝜎𝐺

then ( 𝑆𝑡𝑖 | log 𝐺𝑛 (𝜏, 𝑇 ) = 𝑦 ∼ log −

𝜎𝑖𝐺

𝜇𝑖 + (𝑦 − 𝜇𝐺 ) 2 , 𝜎𝑖2 𝜎𝐺

−

2 𝜎𝑖𝐺 2 𝜎𝐺

) .

478

5.2.1 Discrete Sampling

Hence, [ 𝔼

ℚ

] 𝑛 𝑛 | 2 ∕𝜎 2 ) 1 ∑ 𝜇𝑖 +(log 𝑧−𝜇𝐺 )𝜎𝑖𝐺 ∕𝜎𝐺2 + 21 (𝜎𝑖2 −𝜎𝑖𝐺 1 ∑ | 𝐺 𝑆𝑡𝑖 | log 𝐺𝑛 (𝜏, 𝑇 ) = log 𝑧 = 𝑒 𝑛 𝑖=𝑘+1 || 𝑛 𝑖=𝑘+1

and by finding the root of the equation

𝐴𝑛 (𝜏, 𝑡𝑘 ) +

𝑛 2 ∕𝜎 2 ) 1 ∑ 𝜇𝑖 +(log 𝑧−𝜇𝐺 )𝜎𝑖𝐺 ∕𝜎𝐺2 + 21 (𝜎𝑖2 −𝜎𝑖𝐺 𝐺 = 𝐾 𝑒 𝑛 𝑖=𝑘+1

̃ we can obtain a value for 𝐾. Therefore, 𝐶𝑙𝑜𝑤𝑒𝑟 = 𝑒−𝑟(𝑇 −𝑡) = 𝑒−𝑟(𝑇 −𝑡)

∞

∫𝐾̃

∞

∫𝐾̃

−𝐾𝑒−𝑟(𝑇 −𝑡) =𝑒

−𝑟(𝑇 −𝑡)

[ ] 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 ) − 𝐾 || 𝐺𝑛 (𝜏, 𝑇 ) = 𝑧 𝑔(𝑧) 𝑑𝑧 [ ] 𝔼ℚ 𝐴𝑛 (𝜏, 𝑇 )|| 𝐺𝑛 (𝜏, 𝑇 ) = 𝑧 𝑔(𝑧) 𝑑𝑧 ∞

∫𝐾̃ {

∞

𝑔(𝑧) 𝑑𝑧 [ ℚ

𝐴𝑛 (𝜏, 𝑡𝑘 ) + 𝔼

∫𝐾̃

]} 𝑛 | 1 ∑ | 𝑆 | 𝐺 (𝜏, 𝑇 ) = 𝑧 𝑔(𝑧) 𝑑𝑧 𝑛 𝑖=𝑘+1 𝑡𝑖 || 𝑛

∞

𝑔(𝑧) 𝑑𝑧 ] [ 𝑛 ∞ | ∑ | −𝑟(𝑇 −𝑡) ℚ 1 =𝑒 𝔼 𝑆 | 𝐺 (𝜏, 𝑇 ) = 𝑧 𝑔(𝑧) 𝑑𝑧 ∫𝐾̃ 𝑛 𝑖=𝑘+1 𝑡𝑖 || 𝑛 −𝐾𝑒−𝑟(𝑇 −𝑡)

∫𝐾̃

( ) +𝑒−𝑟(𝑇 −𝑡) 𝐴𝑛 (𝜏, 𝑡𝑘 ) − 𝐾

∞

∫𝐾̃

𝑔(𝑧) 𝑑𝑧.

Since −1 1 𝑒 2 𝑔(𝑧) = √ 2𝜋𝜎𝐺 𝑧

(

log 𝑧−𝜇𝐺 𝜎𝐺

)2

therefore ∞

∫𝐾̃

( 𝑔(𝑧) 𝑑𝑧 = Φ

̃ 𝜇𝐺 − log 𝐾 𝜎𝐺

)

5.2.1 Discrete Sampling

479

and ∞

∫𝐾̃

[ ] | 𝔼ℚ 𝑆𝑡𝑖 | 𝐺𝑛 (𝜏, 𝑇 ) = 𝑧 𝑔(𝑧) 𝑑𝑧 | ∞

=

∫𝐾̃ ∞

=

∫𝐾̃ ∞

=

∫𝐾̃

[ ] | 𝔼ℚ 𝑆𝑡𝑖 | log 𝐺𝑛 (𝜏, 𝑇 ) = log 𝑧 𝑔(𝑧) 𝑑𝑧 | 1

𝑒𝜇𝑖 +(log 𝑧−𝜇𝐺 )𝜎𝑖𝐺 ∕𝜎𝐺 + 2 (𝜎𝑖 −𝜎𝑖𝐺 ∕𝜎𝐺 ) 𝑔(𝑧) 𝑑𝑧 2

𝑒

2

2

2

2 + 1 (𝜎 2 −𝜎 2 ∕𝜎 2 ) 𝜇𝑖 +(log 𝑧−𝜇𝐺 )𝜎𝑖𝐺 ∕𝜎𝐺 𝑖𝐺 𝐺 2 𝑖

×√

1 2𝜋𝜎𝐺 𝑧

( ∞

1 2

= 𝑒𝜇𝑖 + 2 𝜎𝑖

1 𝑒 √ 2𝜋𝜎𝐺 𝑧

∫𝐾̃

2 −1 (log 𝑧−𝜇𝐺 )𝜎𝑖𝐺 ∕𝜎𝐺 2

𝑒

− 21

(

(log 𝑧−𝜇𝐺 )2 +𝜎 2 𝑖𝐺 𝜎2 𝐺

log 𝑧−𝜇𝐺 𝜎𝐺

)2

)

𝑑𝑧.

By setting 𝑣=

log 𝑧 − 𝜇𝐺 − 𝜎𝑖𝐺 𝑑𝑣 1 ⇒ = 𝜎𝐺 𝑑𝑧 𝜎𝐺 𝑧

therefore ( ∞

1 2

𝑒𝜇 𝑖 + 2 𝜎 𝑖

∫𝐾̃

1 𝑒 √ 2𝜋𝜎𝐺 𝑧

2 −1 (log 𝑧−𝜇𝐺 )𝜎𝑖𝐺 ∕𝜎𝐺 2

(log 𝑧−𝜇𝐺 )2 +𝜎 2 𝑖𝐺 𝜎2 𝐺

)

𝑑𝑧

∞

1 − 1 𝑣2 √ 𝑒 2 𝑑𝑣 2𝜋 𝜎𝐺 ( ) ̃ + 𝜎𝑖𝐺 𝜇𝐺 − log 𝐾 𝜇𝑖 + 21 𝜎𝑖2 =𝑒 Φ . 𝜎𝐺 1 2

= 𝑒 𝜇𝑖 + 2 𝜎𝑖

̃ ∫ log 𝐾−𝜇 𝐺 −𝜎𝑖𝐺

Thus, the lower bound of the arithmetic average rate call option is ( ) 𝑛 ̃ + 𝜎𝑖𝐺 ∑ 1 2 𝜇 − log 𝐾 1 𝐺 𝜇 + 𝜎 = 𝑒−𝑟(𝑇 −𝑡) 𝑒 𝑖 2 𝑖Φ 𝑛 𝑖=𝑘+1 𝜎𝐺 ( )} ̃ ( ) 𝜇𝐺 − log 𝐾 + 𝐴𝑛 (𝜏, 𝑡𝑘 ) − 𝐾 Φ . 𝜎𝐺 {

𝐶𝑙𝑜𝑤𝑒𝑟

𝑑𝑧

480

5.2.2 Continuous Sampling

5.2.2

Continuous Sampling

1. Black–Scholes Equation for Asian Option. Let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter and 𝑟 denotes the risk-free interest rate. Let 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) be the exotic path-dependent Asian option price at time 𝑡 which depends on 𝑆𝑡 and 𝐼𝑡 =

𝑡

∫𝜏

𝑓 (𝑆𝑢 , 𝑢) 𝑑 𝑢

where the function 𝑓 is a function of 𝑆𝑡 and 𝑡, so that 𝐼𝑡 is the integral of the asset price function from initial time 𝜏 up to time 𝑡. By considering a hedging portfolio involving both an option 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) which cannot be exercised before its expiry time 𝑇 , 𝜏 < 𝑡 < 𝑇 and an asset price 𝑆𝑡 , show that 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) satisfies the PDE 𝜕𝑉 𝜕2𝑉 𝜕𝑉 𝜕𝑉 1 + 𝑓 (𝑆𝑡 , 𝑡) − 𝑟𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) = 0. + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝐼𝑡 𝜕𝑆𝑡 𝑡 Solution: At time 𝑡 we let the value of a portfolio Π𝑡 be Π𝑡 = 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) − Δ𝑆𝑡 where it involves buying one unit of option 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) and selling Δ units of 𝑆𝑡 . Since we receive 𝐷𝑆𝑡 𝑑𝑡 for every asset held, the change in portfolio Π𝑡 is therefore 𝑑Π𝑡 = 𝑑𝑉 − Δ(𝑑𝑆𝑡 + 𝐷𝑆𝑡 𝑑𝑡) = 𝑑𝑉 − Δ𝑑𝑆𝑡 − Δ𝐷𝑆𝑡 𝑑𝑡. Expanding 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) using Taylor’s series 𝑑𝑉 =

𝜕𝑉 1 𝜕2𝑉 1 𝜕2 𝑉 𝜕𝑉 𝜕𝑉 𝑑𝑆𝑡 + 𝑑𝐼𝑡 + (𝑑𝑆𝑡 )2 + (𝑑𝐼𝑡 )2 𝑑𝑡 + 2 𝜕𝑡 𝜕𝑆𝑡 𝜕𝐼𝑡 2 𝜕𝑆𝑡 2 𝜕𝐼𝑡2 +

𝜕2𝑉 (𝑑𝑆𝑡 𝑑𝐼𝑡 ) + … 𝜕𝑆𝑡 𝜕𝐼𝑡

and by substituting 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 , 𝑑𝐼𝑡 = 𝑓 (𝑆𝑡 , 𝑡)𝑑𝑡 and applying It¯o’s lemma we have ) ( 𝜕𝑉 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 𝜕𝑉 + (𝜇 − 𝐷)𝑆𝑡 + 𝑓 (𝑆𝑡 , 𝑡) 𝑑𝑊𝑡 . 𝑑𝑡 + 𝜎𝑆𝑡 + 𝜎 𝑆𝑡 𝑑𝑉 = 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝐼𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡2

5.2.2 Continuous Sampling

481

Substituting the above expression into 𝑑Π𝑡 and rearranging terms, we have (

) 𝜕𝑉 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 + (𝜇 − 𝐷)𝑆𝑡 + 𝑓 (𝑆𝑡 , 𝑡) 𝑑𝑡 + 𝜎 𝑆𝑡 𝑑Π𝑡 = 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝐼𝑡 𝜕𝑆𝑡2 [ ] 𝜕𝑉 𝑑𝑊𝑡 − Δ (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 + 𝐷𝑆𝑡 𝑑𝑡 +𝜎𝑆𝑡 𝜕𝑆𝑡 ) ( 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 𝜕𝑉 + (𝜇 − 𝐷)𝑆𝑡 + 𝑓 (𝑆𝑡 , 𝑡) − 𝜇Δ𝑆𝑡 𝑑𝑡 + 𝜎 𝑆𝑡 = 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝐼𝑡 𝜕𝑆𝑡2 ) ( 𝜕𝑉 +𝜎𝑆𝑡 − Δ 𝑑𝑊𝑡 . 𝜕𝑆𝑡 To eliminate the random component we choose Δ=

𝜕𝑉 𝜕𝑆𝑡

which leads to ( 𝑑Π𝑡 =

𝜕𝑉 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 + 𝑓 (𝑆𝑡 , 𝑡) + 𝜎 2 𝑆𝑡2 2 − 𝐷𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝐼𝑡 𝜕𝑆𝑡

) 𝑑𝑡.

Under the no-arbitrage condition the return on the amount 𝐼𝑡 invested in a risk-free interest rate would see a growth of 𝑑Π𝑡 = 𝑟Π𝑡 𝑑𝑡 and hence we can set ( 𝑟Π𝑡 𝑑𝑡 =

𝜕𝑉 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 + 𝑓 (𝑆𝑡 , 𝑡) + 𝜎 2 𝑆𝑡2 2 − 𝐷𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝐼𝑡 𝜕𝑆𝑡 𝑡

) 𝜕𝑉 ( + 𝑟 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) − Δ𝑆𝑡 = 𝜕𝑡 ( ) 𝜕𝑉 𝜕𝑉 𝑟 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) − 𝑆𝑡 + = 𝜕𝑆𝑡 𝜕𝑡

) 𝑑𝑡

𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 − 𝐷𝑆𝑡 + 𝑓 (𝑆𝑡 , 𝑡) 𝜎 𝑆𝑡 2 2 𝜕𝑆 𝜕𝐼𝑡 𝜕𝑆𝑡 𝑡 𝜕𝑉 𝜕𝑉 1 2 2 𝜕2𝑉 − 𝐷𝑆𝑡 + 𝑓 (𝑆𝑡 , 𝑡) . 𝜎 𝑆𝑡 2 𝜕𝑆𝑡 𝜕𝐼𝑡 𝜕𝑆𝑡2

Rearranging terms, we finally have the Black–Scholes equation for Asian options given by 𝜕𝑉 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 + 𝑓 (𝑆𝑡 , 𝑡) − 𝑟𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) = 0. + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝐼𝑡 𝜕𝑆𝑡

482

5.2.2 Continuous Sampling

2. Similarity Reduction I. Let {𝑊𝑡 : 𝑡 ≥ 0} be the ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter and let 𝑟 be the risk-free interest rate from the money-market account. Let 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) be a path-dependent Asian option price at time 𝑡, 𝜏 < 𝑡 < 𝑇 which depends on 𝑆𝑡 and 𝐼𝑡 =

𝑡

∫𝜏

𝑆𝑢 𝑑𝑢

where 𝜏 is the initial time and the option can only be exercised at expiry time 𝑇 . Assume that 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) satisfies the following PDE 𝜕2𝑉 𝜕𝑉 𝜕𝑉 1 𝜕𝑉 + 𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝐼𝑡 𝜕𝑆𝑡 with payoff ) ( Ψ(𝑆𝑇 , 𝐼𝑇 ) = 𝑆𝑇𝛼 𝐹 𝐼𝑇 ∕𝑆𝑇 for some constant 𝛼 and function 𝐹 . By considering the change of variables 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) = 𝑆𝑡𝛼 𝐻(𝑥, 𝑡),

𝑥 = 𝐼𝑡 ∕𝑆𝑡

show that 𝐻(𝑥, 𝑡) satisfies { } ] 𝜕𝐻 𝜕𝐻 1 2 2 𝜕 2 𝐻 [ + 1 + (1 − 𝛼)𝜎 2 − (𝑟 − 𝐷) 𝑥 + 𝜎 𝑥 2 𝜕𝑡 2 𝜕𝑥 𝜕𝑥 [ ( ) ] 1 2 − (1 − 𝛼) 𝛼𝜎 + 𝑟 + 𝛼𝐷 𝐻(𝑥, 𝑡) = 0 2 with payoff 𝐻(𝑥, 𝑇 ) = 𝐹 (𝑥). Solution: Based on the change of variables 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) = 𝑆𝑡𝛼 𝐻(𝑥, 𝑡),

𝑥 = 𝐼𝑡 ∕𝑆𝑡

we have 𝜕𝐻 𝜕𝑉 𝜕𝐻 𝜕𝑥 𝜕𝐻 𝜕𝑉 = 𝑆𝑡𝛼 = 𝑆𝑡𝛼−1 = 𝑆𝑡𝛼 ; 𝜕𝑡 𝜕𝑡 𝜕𝐼𝑡 𝜕𝑥 𝜕𝐼𝑡 𝜕𝑥

5.2.2 Continuous Sampling

483

) ( 𝜕𝐻 𝜕𝑉 𝜕𝐻 𝜕𝑥 = 𝛼𝑆𝑡𝛼−1 𝐻(𝑥, 𝑡) + 𝑆𝑡𝛼 = 𝑆𝑡𝛼−1 𝛼𝐻(𝑥, 𝑡) − 𝑥 𝜕𝑆𝑡 𝜕𝑥 𝜕𝑆𝑡 𝜕𝑥 )] [ ( 2 𝜕𝐻 𝜕 𝑉 𝜕 𝑆 𝛼−1 𝛼𝐻(𝑥, 𝑡) − 𝑥 = 𝜕𝑆𝑡 𝑡 𝜕𝑥 𝜕𝑆𝑡2 ) ( 𝜕𝐻 = (𝛼 − 1)𝑆𝑡𝛼−2 𝛼𝐻(𝑥, 𝑡) − 𝑥 𝜕𝑥 ( ) 𝜕𝑥 𝜕𝐻 𝜕𝑥 𝜕 2 𝐻 𝜕𝑥 𝜕𝐻 +𝑆𝑡𝛼−1 𝛼 − −𝑥 2 𝜕𝑥 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑥 𝜕𝑥 𝜕𝑆𝑡 ) ( 𝜕𝐻 = (𝛼 − 1)𝑆𝑡𝛼−2 𝛼𝐻(𝑥, 𝑡) − 𝑥 𝜕𝑥 ) ( 2𝐻 𝜕𝐻 𝜕 +𝑆𝑡𝛼−2 𝑥2 2 − (𝛼 − 1)𝑥 𝜕𝑥 𝜕𝑥 ) ( 2 𝜕𝐻 𝛼−2 2𝜕 𝐻 − 2(𝛼 − 1)𝑥 = 𝑆𝑡 + 𝛼(𝛼 − 1)𝐻(𝑥, 𝑡) . 𝑥 𝜕𝑥 𝜕𝑥2 By substituting the above expressions into the PDE, we have ) ( 2 𝜕𝐻 1 2 𝛼 2𝜕 𝐻 − 2(𝛼 − 1)𝑥 + 𝜎 𝑆𝑡 𝑥 + 𝛼(𝛼 − 1)𝐻(𝑥, 𝑡) 2 𝜕𝑥 𝜕𝑥2 ) ( 𝜕𝐻 𝜕𝐻 + 𝑆𝑡𝛼 − 𝑟𝑆𝑡𝛼 𝐻(𝑥, 𝑡) = 0 +(𝑟 − 𝐷)𝑆𝑡𝛼 𝛼𝐻(𝑥, 𝑡) − 𝑥 𝜕𝑥 𝜕𝑥

𝜕𝐻 𝑆𝑡𝛼 𝜕𝑡

and taking out 𝑆𝑡𝛼 and rearranging terms we eventually have { } ] 𝜕𝐻 𝜕𝐻 1 2 2 𝜕 2 𝐻 [ + 1 + (1 − 𝛼)𝜎 2 − (𝑟 − 𝐷) 𝑥 + 𝜎 𝑥 2 𝜕𝑡 2 𝜕𝑥 𝜕𝑥 [ ( ) ] 1 2 − (1 − 𝛼) 𝛼𝜎 + 𝑟 + 𝛼𝐷 𝐻(𝑥, 𝑡) = 0 2 with payoff 𝐻(𝑥, 𝑇 ) = 𝐹 (𝑥) since 𝑉 (𝑆𝑇 , 𝐼𝑇 , 𝑇 ) = 𝑆𝑇𝛼 𝐹 (𝐼𝑇 ∕𝑆𝑇 ) = 𝑆𝑇𝛼 𝐻(𝐼𝑇 ∕𝑆𝑇 , 𝑇 ) and 𝑥 = 𝐼𝑇 ∕𝑆𝑇 . 3. Similarity Reduction II. Let {𝑊𝑡 : 𝑡 ≥ 0} be the ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter and let 𝑟 be the risk-free interest rate from the money-market account.

484

5.2.2 Continuous Sampling

Let 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) be a path-dependent Asian option price at time 𝑡, 𝜏 < 𝑡 < 𝑇 which depends on 𝑆𝑡 and 𝐼𝑡 =

𝑡

∫𝜏

log 𝑆𝑢 𝑑𝑢

where 𝜏 is the initial time and the option can only be exercised at expiry time 𝑇 . Assume that 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) satisfies the following PDE 𝜕𝑉 𝜕2𝑉 𝜕𝑉 𝜕𝑉 1 + log 𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝐼𝑡 𝜕𝑆𝑡 with payoff Ψ(𝑆𝑇 , 𝐼𝑇 ) = 𝐹 (𝐼𝑇 ) where 𝐹 is only a function of 𝐼𝑇 . By considering the change of variables 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) = 𝐻(𝑥, 𝑡),

𝑥=

𝐼𝑡 + (𝑇 − 𝑡) log 𝑆𝑡 𝑇 −𝜏

show that 𝐻(𝑥, 𝑡) satisfies 𝜕𝐻 1 + 𝜕𝑡 2

(

𝜎(𝑇 − 𝑡) 𝑇 −𝜏

)2

) )( ( 𝑇 − 𝑡 𝜕𝐻 1 2 𝜕2𝐻 + 𝑟 − 𝐷 − 𝜎 − 𝑟𝐻(𝑥, 𝑡) = 0 2 𝑇 − 𝜏 𝜕𝑥 𝜕𝑥2

with payoff 𝐻(𝑥, 𝑇 ) = 𝐹 (𝑥(𝑇 − 𝜏)). Solution: Using the technique of changing variables 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) = 𝐻(𝑥, 𝑡),

𝑥=

𝐼𝑡 + (𝑇 − 𝑡) log 𝑆𝑡 𝑇 −𝜏

we have 𝜕𝐻 𝜕𝑥 𝜕𝐻 𝜕𝐻 log 𝑆𝑡 𝜕𝐻 𝜕𝑉 = + = − 𝜕𝑡 𝜕𝑥 𝜕𝑡 𝜕𝑡 𝜕𝑡 𝑇 − 𝜏 𝜕𝑥

𝜕𝐻 𝜕𝑥 1 𝜕𝐻 𝜕𝑉 = = 𝜕𝐼𝑡 𝜕𝑥 𝜕𝐼𝑡 𝑇 − 𝜏 𝜕𝑥

5.2.2 Continuous Sampling

485

) ( 𝜕𝐻 𝜕𝑥 1 𝑇 − 𝑡 𝜕𝐻 𝜕𝑉 = = 𝜕𝑆𝑡 𝜕𝑥 𝜕𝑆𝑡 𝑆 𝑇 − 𝜏 𝜕𝑥 ] [ ( 𝑡 ) 2 𝜕 𝑉 𝜕 1 𝑇 − 𝑡 𝜕𝐻 = 𝜕𝑆𝑡 𝑆𝑡 𝑇 − 𝜏 𝜕𝑥 𝜕𝑆𝑡2 ( ) ) ( 𝑇 −𝑡 1 𝜕𝐻 1 𝜕 2 𝐻 𝜕𝑥 = − 2 + 𝑇 −𝜏 𝑆𝑡 𝜕𝑥2 𝜕𝑆𝑡 𝑆𝑡 𝜕𝑥 [ ] ) ) ( ( 𝑇 −𝑡 1 𝜕𝐻 1 𝑇 − 𝑡 𝜕2𝐻 = − 2 𝑇 −𝜏 𝑆𝑡2 𝑇 − 𝜏 𝜕𝑥2 𝑆𝑡 𝜕𝑥 ] ) [( ) 2 ( 𝑇 − 𝑡 𝜕 𝐻 𝜕𝐻 1 𝑇 −𝑡 = 2 − . 𝑇 − 𝜏 𝜕𝑥2 𝜕𝑥 𝑆𝑡 𝑇 − 𝜏 By substituting the above expressions into the PDE, we have ] ) [( ) ( 𝑇 − 𝑡 𝜕 2 𝐻 𝜕𝐻 𝜕𝐻 log 𝑆𝑡 𝜕𝐻 1 2 𝑇 − 𝑡 − − + 𝜎 𝜕𝑡 𝑇 − 𝜏 𝜕𝑥 2 𝑇 −𝜏 𝑇 − 𝜏 𝜕𝑥2 𝜕𝑥 +(𝑟 − 𝐷)

(

𝑇 −𝑡 𝑇 −𝜏

)

𝜕𝐻 log 𝑆𝑡 𝜕𝐻 + − 𝑟𝐻(𝑥, 𝑡) = 0 𝜕𝑥 𝑇 − 𝜏 𝜕𝑥

or 𝜕𝐻 1 + 𝜕𝑡 2

(

𝜎(𝑇 − 𝑡) 𝑇 −𝜏

)2

) )( ( 𝑇 − 𝑡 𝜕𝐻 1 2 𝜕2𝐻 + 𝑟 − 𝐷 − 𝜎 − 𝑟𝐻(𝑥, 𝑡) = 0 2 𝑇 − 𝜏 𝜕𝑥 𝜕𝑥2

with payoff 𝐻(𝑥, 𝑇 ) = 𝐹 (𝑥(𝑇 − 𝜏)) since 𝑉 (𝑆𝑇 , 𝐼𝑇 , 𝑇 ) = 𝐹 (𝐼𝑇 ) = 𝐻(𝑥, 𝑇 ) and 𝑥 = 𝐼𝑇 ∕(𝑇 − 𝜏). 4. Similarity Reduction III. Let {𝑊𝑡 : 𝑡 ≥ 0} be the ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter and let 𝑟 be the risk-free interest rate from the money-market account. Let 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) be a path-dependent Asian option price at time 𝑡, 𝜏 < 𝑡 < 𝑇 which depends on 𝑆𝑡

and

𝐼𝑡 =

𝑡

∫𝜏

log 𝑆𝑢 𝑑𝑢

486

5.2.2 Continuous Sampling

where 𝜏 ≥ 0 is the initial time and the option can only be exercised at expiry time 𝑇 . Assume that 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) satisfies the following PDE 𝜕2𝑉 𝜕𝑉 𝜕𝑉 𝜕𝑉 1 + log 𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝐼𝑡 𝜕𝑆𝑡 𝑡 with payoff Ψ(𝑆𝑇 , 𝐼𝑇 ) = 𝑆𝑇𝛼 𝐹

(

𝑒𝐼𝑇 ∕(𝑇 −𝜏) 𝑆𝑇

)

for some constant 𝛼 and function 𝐹 . By considering the change of variables 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) = 𝑆𝑡𝛼 𝐻(𝑥, 𝑡),

𝑥=

𝐼𝑡 − (𝑡 − 𝜏) log 𝑆𝑡 𝑇 −𝜏

show that 𝐻(𝑥, 𝑡) satisfies ) )] ( ( [ 𝜕𝐻 𝑡−𝜏 𝜕𝐻 1 2 𝑡 − 𝜏 2 𝜕 2 𝐻 1 2 − 𝑟 − 𝐷 + + 𝜎 (2𝛼 − 1)𝜎 2 𝜕𝑡 2 𝑇 −𝜏 2 𝑇 − 𝜏 𝜕𝑥 𝜕𝑥 ] [ 1 + 𝛼(𝛼 − 1)𝜎 2 + 𝛼(𝑟 − 𝐷) − 𝑟 𝐻(𝑥, 𝑡) = 0 2 with payoff 𝐻(𝑥, 𝑇 ) = 𝐹 (𝑒𝑥 ). Solution: Using the change of variables 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡) = 𝑆𝑡𝛼 𝐻(𝑥, 𝑡),

𝑥=

𝐼𝑡 − (𝑡 − 𝜏) log 𝑆𝑡 𝑇 −𝜏

we have ) ( ) ( 𝜕𝐻 𝜕𝑥 𝜕𝐻 𝜕𝐻 log 𝑆𝑡 𝜕𝐻 𝜕𝑉 = 𝑆𝑡𝛼 = 𝑆𝑡𝛼 + − 𝜕𝑡 𝜕𝑥 𝜕𝑡 𝜕𝑡 𝜕𝑡 𝑇 − 𝜏 𝜕𝑥

𝛼

𝑆𝑡 𝜕𝐻 𝜕𝐻 𝜕𝑥 𝜕𝑉 = 𝑆𝑡𝛼 = 𝜕𝐼𝑡 𝜕𝑥 𝜕𝐼𝑡 𝑇 − 𝜏 𝜕𝑥 ) ( 𝑡 − 𝜏 𝜕𝐻 𝜕𝐻 𝜕𝑥 𝜕𝑉 = 𝛼𝑆𝑡𝛼−1 𝐻(𝑥, 𝑡) + 𝑆𝑡𝛼 = 𝛼𝑆𝑡𝛼−1 𝐻(𝑥, 𝑡) − 𝑆𝑡𝛼−1 𝜕𝑆𝑡 𝜕𝑥 𝜕𝑆𝑡 𝑇 − 𝜏 𝜕𝑥

5.2.2 Continuous Sampling

487

and 𝜕2𝑉 𝜕𝐻 𝜕𝑥 = 𝛼(𝛼 − 1)𝑆𝑡𝛼−2 𝐻(𝑥, 𝑡) + 𝛼𝑆𝑡𝛼−1 2 𝜕𝑥 𝜕𝑆𝑡 𝜕𝑆𝑡 ) ) ( ( 𝑡 − 𝜏 𝜕𝐻 𝑡 − 𝜏 𝜕 2 𝐻 𝜕𝑥 −(𝛼 − 1)𝑆𝑡𝛼−2 − 𝑆𝑡𝛼−1 𝑇 − 𝜏 𝜕𝑥 𝑇 − 𝜏 𝜕𝑥2 𝜕𝑆𝑡 ) [ ( 𝑡 − 𝜏 𝜕𝐻 = 𝑆𝑡𝛼−2 𝛼(𝛼 − 1)𝐻(𝑥, 𝑡) − 𝛼 𝑇 − 𝜏 𝜕𝑥 ] ) ) ( ( 𝑡 − 𝜏 2 𝜕2𝐻 𝑡 − 𝜏 𝜕𝐻 + −(𝛼 − 1) 𝑇 − 𝜏 𝜕𝑥 𝑇 −𝜏 𝜕𝑥2 ] [( )2 2 ) ( 𝑡−𝜏 𝑡 − 𝜏 𝜕𝐻 𝜕 𝐻 − (2𝛼 − 1) = 𝑆𝑡𝛼−2 + 𝛼(𝛼 − 1)𝐻(𝑥, 𝑡) . 𝑇 −𝜏 𝑇 − 𝜏 𝜕𝑥 𝜕𝑥2 By substituting the above expressions into the PDE, we have ) [( ) 𝑡 − 𝜏 2 𝜕2𝐻 1 𝜕𝐻 log 𝑆𝑡 𝜕𝐻 − + 𝜎 2 𝑆𝑡𝛼 𝜕𝑡 𝑇 𝜕𝑥 2 𝑇 −𝜏 𝜕𝑥2 ) ( ] 𝑡 − 𝜏 𝜕𝐻 −(2𝛼 − 1) + 𝛼(𝛼 − 1)𝐻(𝑥, 𝑡) + 𝛼(𝑟 − 𝐷)𝑆𝑡𝛼 𝐻(𝑥, 𝑡) 𝑇 − 𝜏 𝜕𝑥 ) ( log 𝑆𝑡 𝜕𝐻 𝑡 − 𝜏 𝜕𝐻 −(𝑟 − 𝐷)𝑆𝑡𝛼 + 𝑆𝑡𝛼 − 𝑟𝑆𝑡𝛼 𝐻(𝑥, 𝑡) = 0. 𝑇 − 𝜏 𝜕𝑥 𝑇 − 𝜏 𝜕𝑥 𝑆𝑡𝛼

(

By removing 𝑆𝑡𝛼 and rearranging terms, we have ) )] ( ( [ 𝜕𝐻 𝜕𝐻 1 2 𝑡 − 𝜏 2 𝜕 2 𝐻 1 2 𝑡−𝜏 − 𝑟 − 𝐷 + + 𝜎 (2𝛼 − 1)𝜎 𝜕𝑡 2 𝑇 −𝜏 2 𝑇 −𝜏 𝜕𝑥 𝜕𝑥2 ] [ 1 + 𝛼(𝛼 − 1)𝜎 2 + 𝛼(𝑟 − 𝐷) − 𝑟 𝐻(𝑥, 𝑡) = 0 2 with payoff 𝐻(𝑥, 𝑇 ) = 𝐹 (𝑒𝑥 ) since 𝑉 (𝑆𝑇 , 𝐼𝑇 , 𝑇 ) =

𝑆𝑇𝛼 𝐹

(

𝑒𝐼𝑇 ∕(𝑇 −𝜏) 𝑆𝑇

)

= 𝑆𝑇𝛼 𝐻(𝑥, 𝑇 ) and 𝑥 =

𝐼𝑇 − log 𝑆𝑇 . 𝑇 −𝜏

5. Arithmetic Average Rate (Fixed Strike) Asian Option (PDE Approach). Let {𝑊𝑡 : 𝑡 ≥ 0} be the ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter and let 𝑟 be the risk-free interest rate from the money-market account.

488

5.2.2 Continuous Sampling

We define 𝐼𝑡 =

𝑡

∫𝜏

𝑆𝑢 𝑑𝑢

to be the asset running sum from initial time 𝜏 ≥ 0 until time 𝑡 and consider the arithmetic average rate call option with payoff { Ψ(𝑆𝑇 , 𝐼𝑇 ) = max

} 𝐼𝑇 − 𝐾, 0 𝑇 −𝜏

where 𝐾 > 0 is the strike price and 𝑇 is the option expiry time. Assume that the arithmetic (𝑎) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 ) satisfies the PDE average rate call option price at time 𝑡 < 𝑇 , 𝐶𝑎𝑟 (𝑎) (𝑎) (𝑎) (𝑎) 𝜕𝐶𝑎𝑟 𝜕 2 𝐶𝑎𝑟 𝜕𝐶𝑎𝑟 𝜕𝐶𝑎𝑟 1 (𝑎) + (𝑟 − 𝐷)𝑆 + 𝑆 − 𝑟𝐶𝑎𝑟 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 2 𝑆𝑡2 𝑡 𝑡 2 𝜕𝑡 2 𝜕𝑆 𝜕𝐼 𝜕𝑆𝑡 𝑡 𝑡

with boundary condition { (𝑎) 𝐶𝑎𝑟 (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max

} 𝐼𝑇 − 𝐾, 0 . 𝑇 −𝜏

By considering the change of variables (𝑎) 𝐶𝑎𝑟 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑢(𝑥, 𝑡),

𝑥=

𝐼𝑡 − 𝐾(𝑇 − 𝜏) 𝑆𝑡 (𝑇 − 𝜏)

show that 𝑢(𝑥, 𝑡) satisfies [ ] 𝜕𝑢 1 2 2 𝜕 2 𝑢 1 𝜕𝑢 + + 𝜎 𝑥 − (𝑟 − 𝐷)𝑥 − 𝐷𝑢(𝑥, 𝑡) = 0 2 𝜕𝑡 2 𝑇 −𝜏 𝜕𝑥 𝜕𝑥 with boundary condition 𝑢(𝑥, 𝑇 ) = max{𝑥, 0}. By setting 𝑢(𝑥, 𝑡) = 𝑎(𝑡)𝑥 + 𝑏(𝑡) show that 𝑎(𝑡) and 𝑏(𝑡) satisfy the ODEs 𝑑𝑎 − 𝑟 𝑎(𝑡) = 0 and 𝑑𝑡

𝑎(𝑡) 𝑑𝑏 − 𝐷𝑏(𝑡) = − . 𝑑𝑡 𝑇 −𝜏

5.2.2 Continuous Sampling

489

Hence, show that for 𝑥 ≥ 0 and 𝑟 ≠ 𝐷 the arithmetic average rate call option price at time 𝑡, 𝜏 < 𝑡 < 𝑇 is ( (𝑎) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐶𝑎𝑟

) ( −𝐷(𝑇 −𝑡) ) 𝑡 1 𝑒 − 𝑒−𝑟(𝑇 −𝑡) 𝑆𝑢 𝑑𝑢 − 𝐾 𝑒−𝑟(𝑇 −𝑡) + 𝑆𝑡 . 𝑇 − 𝜏 ∫𝜏 (𝑟 − 𝐷)(𝑇 − 𝜏)

What is the solution if 𝑟 → 𝐷? Solution: From the change of variables (𝑎) 𝐶𝑎𝑟 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑢(𝑥, 𝑡),

𝑥=

𝐼𝑡 − 𝐾(𝑇 − 𝜏) 𝑆𝑡 (𝑇 − 𝜏)

we have (𝑎) 𝜕𝐶𝑎𝑟 𝜕𝑢 = 𝑆𝑡 𝜕𝑡 𝜕𝑡 (𝑎) 𝜕𝐶𝑎𝑟 𝜕𝑢 𝜕𝑥 𝜕𝑢 1 1 𝜕𝑢 = 𝑆𝑡 = 𝑆𝑡 = 𝜕𝐼𝑡 𝜕𝑥 𝜕𝐼𝑡 𝜕𝑥 𝑆𝑡 (𝑇 − 𝜏) 𝑇 − 𝜏 𝜕𝑥 (𝑎) 𝜕𝐶𝑎𝑟 𝜕𝑢 𝜕𝑢 𝜕𝑥 = 𝑢(𝑥, 𝑡) + 𝑆𝑡 = 𝑢(𝑥, 𝑡) − 𝑥 𝜕𝑆𝑡 𝜕𝑥 𝜕𝑆𝑡 𝜕𝑥 (𝑎) 𝜕 2 𝐶𝑎𝑟

𝜕𝑆𝑡2

𝜕𝑢 𝜕𝑥 𝜕 2 𝑢 𝜕𝑥 𝜕𝑢 𝜕𝑥 𝜕2𝑢 = −𝑥 2 − =𝑥 2 𝜕𝑥 𝜕𝑆𝑡 𝜕𝑥 𝜕𝑆𝑡 𝜕𝑥 𝜕𝑆𝑡 𝜕𝑥

(

𝐼𝑡 − 𝐾(𝑇 − 𝜏) 𝑆𝑡2 (𝑇

)

− 𝜏)

=

𝑥2 𝜕 2 𝑢 . 𝑆𝑡 𝜕𝑥2

By substituting the above results into the PDE, we have 𝜕𝑢 1 𝑆𝑡 + 𝜎 2 𝑆𝑡2 𝜕𝑡 2

(

𝑥2 𝑆𝑡

)

) ( 𝑆𝑡 𝜕𝑢 𝜕𝑢 𝜕2𝑢 + 𝑢(𝑥, 𝑡) − 𝑥 + (𝑟 − 𝐷)𝑆 − 𝑟𝑆𝑡 𝑢(𝑥, 𝑡) = 0 𝑡 2 𝜕𝑥 𝑇 − 𝜏 𝜕𝑥 𝜕𝑥

or [ ] 𝜕𝑢 1 2 2 𝜕 2 𝑢 1 𝜕𝑢 + + 𝜎 𝑥 − (𝑟 − 𝐷)𝑥 − 𝐷𝑢(𝑥, 𝑡) = 0 𝜕𝑡 2 𝑇 −𝜏 𝜕𝑥 𝜕𝑥2 with boundary condition 𝑢(𝑥, 𝑇 ) = max{𝑥, 0} since {

(𝑎) (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝐾, 𝑇 ) 𝐶𝑎𝑟

} 𝐼𝑇 = max − 𝐾, 0 𝑇 −𝜏 { } 𝐼𝑇 − 𝐾(𝑇 − 𝜏) = 𝑆𝑇 max ,0 𝑆𝑇 (𝑇 − 𝜏) = 𝑆𝑇 max{𝑥, 0}.

490

5.2.2 Continuous Sampling

Let 𝑢(𝑥, 𝑡) = 𝑎(𝑡)𝑥 + 𝑏(𝑡) so that 𝑑𝑎 𝑑𝑏 𝜕𝑢 =𝑥 + ; 𝜕𝑡 𝑑𝑡 𝑑𝑡

𝜕𝑢 = 𝑎(𝑡); 𝜕𝑥

𝜕2𝑢 =0 𝜕𝑥2

and by substituting the results into the PDE satisfied by 𝑢(𝑥, 𝑡) 𝑥

( ) 1 𝑑𝑎 𝑑𝑏 + + − (𝑟 − 𝐷)𝑥 𝑎(𝑡) − 𝐷𝑎(𝑡)𝑥 − 𝐷𝑏(𝑡) = 0 𝑑𝑡 𝑑𝑡 𝑇 −𝜏

or 𝑥

(

) 𝑎(𝑡) 𝑑𝑏 𝑑𝑎 − 𝑟 𝑎(𝑡) + − 𝐷𝑏(𝑡) + = 0. 𝑑𝑡 𝑑𝑡 𝑇 −𝜏

Hence, we can set 𝑑𝑎 − 𝑟 𝑎(𝑡) = 0 and 𝑑𝑡 To solve

𝑎(𝑡) 𝑑𝑏 − 𝐷𝑏(𝑡) = − . 𝑑𝑡 𝑇 −𝜏

𝑑𝑎 − 𝑟𝑎(𝑡) = 0 we let the integrating factor 𝐼𝑎 = 𝑒− ∫ 𝑑𝑡

𝑟𝑑𝑡

= 𝑒−𝑟𝑡 and therefore

) 𝑑 ( 𝑎(𝑡)𝑒−𝑟𝑡 = 0 or 𝑎(𝑡)𝑒−𝑟𝑡 = 𝐶1 𝑑𝑡 where 𝐶1 is a constant. Hence, 𝑎(𝑡) = 𝐶1 𝑒𝑟𝑡 . 𝑎(𝑡) 𝑑𝑏 − 𝐷𝑏(𝑡) = − we let the integrating factor 𝐼𝑏 = 𝑒− ∫ In contrast, for 𝑑𝑡 𝑇 −𝜏 𝑒−𝐷𝑡 . Therefore, ) 𝑎(𝑡) −𝐷𝑡 𝑑 ( 𝑏(𝑡)𝑒−𝐷𝑡 = − 𝑒 𝑑𝑡 𝑇 −𝜏 1 𝑏(𝑡)𝑒−𝐷𝑡 = − 𝑎(𝑡)𝑒−𝐷𝑡 𝑑𝑡. 𝑇 −𝜏 ∫ Substituting 𝑎(𝑡) = 𝐶1 𝑒𝑟𝑡 we have 𝐶1 𝑒(𝑟−𝐷)𝑡 𝑑𝑡 𝑇 −𝜏 ∫ 𝐶1 𝑒(𝑟−𝐷)𝑡 + 𝐶2 =− (𝑟 − 𝐷)(𝑇 − 𝜏)

𝑏(𝑡)𝑒−𝐷𝑡 = −

where 𝐶2 is a constant. Thus, 𝑢(𝑥, 𝑡) = 𝐶1 𝑒𝑟𝑡 𝑥 −

𝐶1 𝑒𝑟𝑡 + 𝐶2 𝑒𝐷𝑡 . (𝑟 − 𝐷)(𝑇 − 𝜏)

𝐷𝑑𝑡

=

5.2.2 Continuous Sampling

491

At the boundary condition for 𝑥 ≥ 0, 𝑢(𝑥, 𝑇 ) = 𝑥 we therefore have 𝐶1 𝑒𝑟𝑇 𝑥 −

𝐶1 𝑒𝑟𝑇 + 𝐶2 𝑒𝐷𝑇 = 𝑥 (𝑟 − 𝐷)(𝑇 − 𝜏)

and by equating coefficients 𝐶1 = 𝑒−𝑟𝑇

and

𝐶2 =

𝑒−𝐷𝑇 (𝑟 − 𝐷)(𝑇 − 𝜏)

which leads to 𝑢(𝑥, 𝑡) = 𝑥𝑒−𝑟(𝑇 −𝑡) + Thus, for 𝑥 ≥ 0 or

𝑒−𝐷(𝑇 −𝑡) − 𝑒−𝑟(𝑇 −𝑡) . (𝑟 − 𝐷)(𝑇 − 𝜏)

𝑡

1 𝑆 𝑑𝑢 ≥ 𝐾 𝑇 − 𝜏 ∫𝜏 𝑢

(𝑎) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑢(𝑥, 𝑡) 𝐶𝑎𝑟 ( −𝐷(𝑇 −𝑡) ) ) ( 𝐼𝑡 − 𝐾(𝑇 − 𝜏) −𝑟(𝑇 −𝑡) 𝑒 − 𝑒−𝑟(𝑇 −𝑡) + 𝑒 𝑆𝑡 = 𝑆𝑡 𝑆𝑡 (𝑇 − 𝜏) (𝑟 − 𝐷)(𝑇 − 𝜏) ( ) ( −𝐷(𝑇 −𝑡) ) 𝑡 1 𝑒 − 𝑒−𝑟(𝑇 −𝑡) = 𝑆𝑢 𝑑𝑢 − 𝐾 𝑒−𝑟(𝑇 −𝑡) + 𝑆𝑡 . 𝑇 − 𝜏 ∫𝜏 (𝑟 − 𝐷)(𝑇 − 𝜏)

Finally, for the case 𝑟 → 𝐷 and from the application of L’Ĥopital’s rule ) 𝑡 1 = lim 𝑆𝑢 𝑑𝑢 − 𝐾 𝑒−𝑟(𝑇 −𝑡) 𝑟→𝐷 𝑇 − 𝜏 ∫0 ) ( −𝐷(𝑇 −𝑡) − 𝑒−𝑟(𝑇 −𝑡) 𝑒 + lim 𝑆𝑡 𝑟→𝐷 (𝑟 − 𝐷)(𝑇 − 𝜏) ( ) 𝑡 1 −𝑟 𝑒−𝑟(𝑇 −𝑡) = 𝑆𝑢 𝑑𝑢 − 𝐾 𝑒−𝐷(𝑇 −𝑡) + lim 𝑆𝑡 ∫ 𝑟→𝐷 𝑇 −𝜏 𝜏 𝑇 −𝜏 ( ) 𝑡 𝐷𝑆𝑡 −𝐷(𝑇 −𝑡) 1 = 𝑆𝑢 𝑑𝑢 − 𝐾 𝑒−𝐷(𝑇 −𝑡) − 𝑒 𝑇 − 𝜏 ∫𝜏 𝑇 −𝜏 [ ( 𝑡 ) ] 1 = 𝑆 𝑑𝑢 − 𝐷𝑆𝑡 − 𝐾 𝑒−𝐷(𝑇 −𝑡) . 𝑇 − 𝜏 ∫𝜏 𝑢 (

lim 𝐶 (𝑎) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 ) 𝑟→𝐷 𝑎𝑟

6. Arithmetic Average Strike (Floating Strike) Asian Option (PDE Approach). Let {𝑊𝑡 : 𝑡 ≥ 0} be the ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡 𝑆𝑡

492

5.2.2 Continuous Sampling

where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter and let 𝑟 be the risk-free interest rate from the money-market account. We define 𝐼𝑡 =

𝑡

∫𝜏

𝑆𝑢 𝑑𝑢

to be the asset running sum from initial time 𝜏 ≥ 0 until time 𝑡 and consider the arithmetic average strike put option with payoff { Ψ(𝑆𝑇 , 𝐼𝑇 ) = max

} 𝐼𝑇 − 𝑆𝑇 , 0 𝑇 −𝜏

where 𝑇 > 𝑡 is the option expiry time. Assume that the arithmetic average strike put option (𝑎) price at time 𝑡, 𝜏 < 𝑡 < 𝑇 , 𝑃𝑎𝑠 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) satisfies the PDE (𝑎) (𝑎) (𝑎) (𝑎) 𝜕𝑃𝑎𝑠 𝜕 2 𝑃𝑎𝑠 𝜕𝑃𝑎𝑠 𝜕𝑃𝑎𝑠 1 (𝑎) + (𝑟 − 𝐷)𝑆 + 𝑆 − 𝑟𝑃𝑎𝑠 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) = 0 + 𝜎 2 𝑆𝑡2 𝑡 𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝐼𝑡 𝜕𝑆𝑡2

with boundary condition { (𝑎) (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝑇 ) 𝑃𝑎𝑠

= max

} 𝐼𝑇 − 𝑆𝑇 , 0 . 𝑇 −𝜏

By considering the change of variables (𝑎) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) = 𝑆𝑡 𝑢(𝑥, 𝑡), 𝑃𝑎𝑠

𝑥=

𝐼𝑡 − 𝑆𝑡 (𝑇 − 𝜏) 𝑆𝑡 (𝑇 − 𝜏)

show that 𝑢(𝑥, 𝑡) satisfies [ ] 1 𝜕2𝑢 𝜕𝑢 𝜕𝑢 1 2 + 𝜎 (𝑥 + 1)2 2 + − (𝑟 − 𝐷)(𝑥 + 1) − 𝐷𝑢(𝑥, 𝑡) = 0 𝜕𝑡 2 𝑇 −𝜏 𝜕𝑥 𝜕𝑥 with boundary condition 𝑢(𝑥, 𝑇 ) = max{𝑥, 0}. By setting 𝑢(𝑥, 𝑡) = 𝑎(𝑡)𝑥 + 𝑏(𝑡) show that 𝑎(𝑡) and 𝑏(𝑡) satisfy the ODEs 𝑑𝑎 − 𝑟 𝑎(𝑡) = 0 𝑑𝑡

and

[ ] 𝑑𝑏 1 − 𝐷𝑏(𝑡) = − − (𝑟 − 𝐷) 𝑎(𝑡). 𝑑𝑡 𝑇 −𝜏

5.2.2 Continuous Sampling

493

Hence, show that for 𝑥 ≥ 0 and 𝑟 ≠ 𝐷 the arithmetic average strike put option price at time 𝑡, 𝜏 < 𝑡 < 𝑇 is (

(𝑎) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) 𝑃𝑎𝑠

) 𝑡 1 = 𝑆 𝑑𝑢 − 𝑆𝑡 𝑒−𝑟(𝑇 −𝑡) 𝑇 − 𝜏 ∫𝜏 𝑢 ( ) ) 1 − (𝑟 − 𝐷)(𝑇 − 𝜏) ( −𝐷(𝑇 −𝑡) 𝑒 + − 𝑒−𝑟(𝑇 −𝑡) 𝑆𝑡 . (𝑟 − 𝐷)(𝑇 − 𝜏)

What is the solution if 𝑟 → 𝐷? Solution: From the change of variables (𝑎) 𝑃𝑎𝑠 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) = 𝑆𝑡 𝑢(𝑥, 𝑡),

𝑥=

𝐼𝑡 − 𝑆𝑡 (𝑇 − 𝜏) 𝑆𝑡 (𝑇 − 𝜏)

we have (𝑎) 𝜕𝑃𝑎𝑠 𝜕𝑢 = 𝑆𝑡 𝜕𝑡 𝜕𝑡 (𝑎) 𝜕𝑃𝑎𝑠 𝜕𝑢 𝜕𝑥 𝜕𝑢 1 1 𝜕𝑢 = 𝑆𝑡 = 𝑆𝑡 = 𝜕𝐼𝑡 𝜕𝑥 𝜕𝐼𝑡 𝜕𝑥 𝑆𝑡 (𝑇 − 𝜏) 𝑇 − 𝜏 𝜕𝑥

(𝑎) 𝜕𝑃𝑎𝑠 𝜕𝑢 𝜕𝑥 = 𝑢(𝑥, 𝑡) + 𝑆𝑡 𝜕𝑆𝑡 𝜕𝑥 𝜕𝑆𝑡 ( ) 𝐼𝑡 𝜕𝑢 = 𝑢(𝑥, 𝑡) − 𝑆𝑡 𝜕𝑥 𝑆𝑡2 (𝑇 − 𝜏)

= 𝑢(𝑥, 𝑡) − (𝑥 + 1)

𝜕𝑢 𝜕𝑥

and (𝑎) 𝜕 2 𝑃𝑎𝑠

𝜕𝑆𝑡2

𝜕𝑢 𝜕𝑥 𝜕 2 𝑢 𝜕𝑥 𝜕𝑢 𝜕𝑥 − (𝑥 + 1) 2 − 𝜕𝑥 𝜕𝑆𝑡 𝜕𝑥 𝜕𝑆𝑡 𝜕𝑥 𝜕𝑆𝑡 ( ) 𝐼𝑡 𝜕2 𝑢 = (𝑥 + 1) 2 𝜕𝑥 𝑆𝑡2 (𝑇 − 𝜏) =

=

(𝑥 + 1)2 𝜕 2 𝑢 . 𝑆𝑡 𝜕𝑥2

494

5.2.2 Continuous Sampling

By substituting the above results into the PDE, we have 𝜕𝑢 + 𝜕𝑡 𝑆 + 𝑡 𝑇 −𝜏

𝑆𝑡

1 2 2 𝜎 𝑆𝑡 2

(

(𝑥 + 1)2 𝑆𝑡

)

) ( 𝜕𝑢 𝜕2𝑢 𝑢(𝑥, 𝑡) − (𝑥 + 1) + (𝑟 − 𝐷)𝑆 𝑡 𝜕𝑥 𝜕𝑥2

𝜕𝑢 − 𝑟𝑆𝑡 𝑢(𝑥, 𝑡) = 0 𝜕𝑥

or [ ] 𝜕𝑢 1 2 1 𝜕2𝑢 𝜕𝑢 + 𝜎 (𝑥 + 1)2 2 + − (𝑟 − 𝐷)(𝑥 + 1) − 𝐷𝑢(𝑥, 𝑡) = 0 𝜕𝑡 2 𝑇 𝜕𝑥 𝜕𝑥 with boundary condition 𝑢(𝑥, 𝑇 ) = max{𝑥, 0} since {

(𝑎) 𝑃𝑎𝑠 (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝑇 )

} 𝐼𝑇 = max − 𝑆𝑇 , 0 𝑇 −𝜏 { } 𝐼𝑇 − 𝑆𝑇 (𝑇 − 𝜏) = 𝑆𝑇 max ,0 𝑆𝑇 (𝑇 − 𝜏) = 𝑆𝑇 max{𝑥, 0}.

Let 𝑢(𝑥, 𝑡) = 𝑎(𝑡)𝑥 + 𝑏(𝑡) so that 𝜕𝑢 𝑑𝑎 𝑑𝑏 =𝑥 + ; 𝜕𝑡 𝑑𝑡 𝑑𝑡

𝜕𝑢 = 𝑎(𝑡); 𝜕𝑥

𝜕2𝑢 =0 𝜕𝑥2

and by substituting the results into the PDE satisfied by 𝑢(𝑥, 𝑡) we have 𝑥

( ) 𝑑𝑎 𝑑𝑏 1 + + − (𝑟 − 𝐷)(𝑥 + 1) 𝑎(𝑡) − 𝐷𝑎(𝑡)𝑥 − 𝐷𝑏(𝑡) = 0 𝑑𝑡 𝑑𝑡 𝑇 −𝜏

or 𝑥

(

) ( ) 𝑑𝑎 𝑑𝑏 1 − 𝑟𝑎(𝑡) + − 𝐷𝑏(𝑡) + − (𝑟 − 𝐷) 𝑎(𝑡) = 0. 𝑑𝑡 𝑑𝑡 𝑇 −𝜏

Hence, we can set 𝑑𝑎 − 𝑟 𝑎(𝑡) = 0 and 𝑑𝑡 To solve

( ) 𝑑𝑏 1 − 𝐷𝑏(𝑡) = − − (𝑟 − 𝐷) 𝑎(𝑡). 𝑑𝑡 𝑇 −𝜏

𝑑𝑎 − 𝑟 𝑎(𝑡) = 0 we let the integrating factor 𝐼𝑎 = 𝑒− ∫ 𝑑𝑡 ) 𝑑 ( 𝑎(𝑡)𝑒−𝑟𝑡 = 0 or 𝑎(𝑡)𝑒−𝑟𝑡 = 𝐶1 𝑑𝑡

𝑟𝑑𝑡

= 𝑒−𝑟𝑡 and therefore

5.2.2 Continuous Sampling

495

where 𝐶1 is a constant. Hence, 𝑎(𝑡) ( = 𝐶1 𝑒𝑟𝑡 . ) 𝑑𝑏 1 In contrast, for − 𝐷𝑏(𝑡) = − − (𝑟 − 𝐷) 𝑎(𝑡) we let the integrating factor 𝑑𝑡 𝑇 −𝜏 𝐼𝑏 = 𝑒− ∫ 𝐷𝑑𝑡 = 𝑒−𝐷 𝑡 . Therefore, ( ) 𝑑 ( 𝑏(𝑡)𝑒−𝐷𝑡 = − 𝑑𝑡 𝑇 ( 𝑏(𝑡)𝑒−𝐷𝑡 = − 𝑇

) 1 − (𝑟 − 𝐷) 𝑎(𝑡)𝑒−𝐷𝑡 −𝜏 ) 1 − (𝑟 − 𝐷) 𝑎(𝑡)𝑒−𝐷𝑡 𝑑𝑡. ∫ −𝜏

Substituting 𝑎(𝑡) = 𝐶1 𝑒𝑟𝑡 we have (

) 1 − (𝑟 − 𝐷) 𝑒(𝑟−𝐷)𝑡 𝑑𝑡 ∫ 𝑇 −𝜏 ) ( 1 − (𝑟 − 𝐷)(𝑇 − 𝜏) (𝑟−𝐷)𝑡 = −𝐶1 + 𝐶2 𝑒 (𝑟 − 𝐷)(𝑇 − 𝜏) ) ( 1 − (𝑟 − 𝐷)(𝑇 − 𝜏) 𝑟𝑡 = −𝐶1 𝑒 + 𝐶2 𝑒𝐷𝑡 (𝑟 − 𝐷)(𝑇 − 𝜏)

𝑏(𝑡)𝑒−𝐷𝑡 = −𝐶1

where 𝐶2 is a constant. Thus, 𝑢(𝑥, 𝑡) = 𝐶1 𝑒𝑟𝑡 𝑥 − 𝐶1

(

1 − (𝑟 − 𝐷)(𝑇 − 𝜏) (𝑟 − 𝐷)(𝑇 − 𝜏)

)

𝑒𝑟𝑡 + 𝐶2 𝑒𝐷𝑡 .

At the boundary condition for 𝑥 ≥ 0, 𝑢(𝑥, 𝑇 ) = 𝑥 we therefore have 𝐶1 𝑒𝑟𝑇 𝑥 − 𝐶1

(

1 − (𝑟 − 𝐷)(𝑇 − 𝜏) (𝑟 − 𝐷)(𝑇 − 𝜏)

)

𝑒𝑟𝑇 + 𝐶2 𝑒𝐷𝑇 = 𝑥

and by equating coefficients ( 𝐶1 = 𝑒−𝑟𝑇

and

𝐶2 =

1 − (𝑟 − 𝐷)(𝑇 − 𝜏) (𝑟 − 𝐷)(𝑇 − 𝜏)

) 𝑒−𝐷𝑇

which leads to ( 𝑢(𝑥, 𝑡) = 𝑥𝑒−𝑟(𝑇 −𝑡) +

1 − (𝑟 − 𝐷)(𝑇 − 𝜏) (𝑟 − 𝐷)(𝑇 − 𝜏)

)

(

𝑒−𝐷(𝑇 −𝑡) − 𝑒−𝑟(𝑇 −𝑡)

)

and hence (𝑎) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) = 𝑆𝑡 𝑢(𝑥, 𝑡) 𝑃𝑎𝑠 ( ) 𝑡 1 = 𝑆 𝑑𝑢 − 𝑆𝑡 𝑒−𝑟(𝑇 −𝑡) 𝑇 − 𝜏 ∫𝜏 𝑢 ( ) ) 1 − (𝑟 − 𝐷)(𝑇 − 𝜏) ( −𝐷(𝑇 −𝑡) 𝑒 + − 𝑒−𝑟(𝑇 −𝑡) 𝑆𝑡 . (𝑟 − 𝐷)(𝑇 − 𝜏)

496

5.2.2 Continuous Sampling

Finally, for the case 𝑟 → 𝐷 and using L’Ĥopital’s rule (

lim 𝑃 (𝑎) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) 𝑟→𝐷 𝑎𝑠

=

= = =

) 𝑡 1 lim 𝑆𝑢 𝑑𝑢 − 𝑆𝑡 𝑒−𝑟(𝑇 −𝑡) 𝑟→𝐷 𝑇 − 𝜏 ∫0 ) ( −𝐷(𝑇 −𝑡) − 𝑒−𝑟(𝑇 −𝑡) 𝑒 + lim 𝑆𝑡 𝑟→𝐷 (𝑟 − 𝐷)(𝑇 − 𝜏) ( ) − lim 𝑒−𝐷(𝑇 −𝑡) − 𝑒−𝑟(𝑇 −𝑡) 𝑆𝑡 𝑟→𝐷 ( ) 𝑡 1 𝑟𝑒−𝑟(𝑇 −𝑡) 𝑆𝑢 𝑑𝑢 − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − lim 𝑆𝑡 𝑟→𝐷 𝑇 − 𝜏 𝑇 − 𝜏 ∫0 ( ) 𝑡 𝐷𝑆𝑡 −𝐷(𝑇 −𝑡) 1 𝑆 𝑑𝑢 − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝑒 𝑇 − 𝜏 ∫𝜏 𝑢 𝑇 −𝜏 [ ( 𝑡 ) ] 1 𝑆 𝑑𝑢 − 𝐷𝑆𝑡 − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) . 𝑇 − 𝜏 ∫𝜏 𝑢

7. Put–Call Parity for Arithmetic Average Options (Probabilistic Approach). Let {𝑊𝑡 : 𝑡 ≥ 0} be the ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter and let 𝑟 be the risk-free interest rate from the money-market account. We define 𝐼𝑡 =

𝑡

∫𝜏

𝑆𝑢 𝑑𝑢

to be the asset running sum within the time period [𝜏, 𝑡], 0 ≤ 𝜏 < 𝑡 and consider the arithmetic average option with zero-strike call payoff Ψ(𝑆𝑇 , 𝐼𝑇 ) =

𝐼𝑇 𝑇 −𝜏

where 𝑇 is the option expiry time. Using the risk-neutral valuation method find the arbitrage-free arithmetic average option price at time 𝑡, 𝜏 < 𝑡 < 𝑇 . Consider the payoffs of arithmetic average rate options {

} 𝐼𝑇 = max − 𝐾, 0 𝑇 −𝜏 { } 𝐼𝑇 (𝑎) 𝑃𝑎𝑟 (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max 𝐾 − ,0 𝑇 −𝜏

(𝑎) (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝐾, 𝑇 ) 𝐶𝑎𝑟

5.2.2 Continuous Sampling

497

and arithmetic average strike options { (𝑎) (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝑇 ) = max 𝐶𝑎𝑠

𝑆𝑇 − {

(𝑎) 𝑃𝑎𝑠 (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝑇 )

= max

} 𝐼𝑇 ,0 𝑇 −𝜏

} 𝐼𝑇 − 𝑆𝑇 , 0 . 𝑇 −𝜏

Show that the put–call relation for the arithmetic average rate and arithmetic average strike options are (𝑎) (𝑎) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑎𝑟 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 ) 𝐶𝑎𝑟

( =

) ( −𝐷(𝑇 −𝑡) ) 𝑡 𝑒 − 𝑒−𝑟(𝑇 −𝑡) 1 −𝑟(𝑇 −𝑡) 𝑆 𝑑𝑢 − 𝐾 𝑒 + 𝑆𝑡 𝑇 − 𝜏 ∫𝜏 𝑢 (𝑟 − 𝐷)(𝑇 − 𝜏)

and (𝑎) (𝑎) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) − 𝑃𝑎𝑠 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) 𝐶𝑎𝑠

=

( 𝑆𝑡 −

) ( ) 𝑡 ) (𝑟 − 𝐷)(𝑇 − 𝜏) − 1 ( −𝐷(𝑇 −𝑡) 1 𝑒 𝑆𝑢 𝑑𝑢 𝑒−𝑟(𝑇 −𝑡) + − 𝑒−𝑟(𝑇 −𝑡) 𝑆𝑡 𝑇 − 𝜏 ∫𝜏 (𝑟 − 𝐷)(𝑇 − 𝜏)

respectively. Solution: From Girsanov’s theorem, under the risk-neutral measure ℚ, 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡ℚ 𝑆𝑡 (𝜇 − 𝑟) 𝑡 is a ℚ-standard Wiener process. where 𝑊𝑡ℚ = 𝑊𝑡 + 𝜎 Hence, using the risk-neutral valuation method, the price of the option at time 𝑡 < 𝑇 with payoff

Ψ(𝑆𝑇 , 𝐼𝑇 ) =

𝑇

1 𝑆𝑢 𝑑𝑢 𝑇 − 𝜏 ∫𝜏

498

5.2.2 Continuous Sampling

is [ ] 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ Ψ(𝑆𝑇 , 𝐼𝑇 )|| ℱ𝑡 [ ] 𝑇 | 1 | = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 𝑆𝑢 𝑑𝑢| ℱ𝑡 | 𝑇 − 𝜏 ∫𝜏 | [ ] 𝑡 𝑇 | 1 −𝑟(𝑇 −𝑡) ℚ | = 𝔼 𝑆 𝑑𝑢 + 𝑆𝑢 𝑑𝑢| ℱ𝑡 𝑒 | ∫𝑡 ∫𝜏 𝑢 𝑇 −𝜏 | [ ] ( ) 𝑡 𝑇 | 1 1 −𝑟(𝑇 −𝑡) ℚ | −𝑟(𝑇 −𝑡) = 𝑆 𝑑𝑢 𝑒 + 𝔼 𝑆𝑢 𝑑𝑢| ℱ𝑡 𝑒 | ∫𝑡 𝑇 − 𝜏 ∫𝜏 𝑢 𝑇 −𝜏 | ) ( 𝑡 𝑇 [ ] 1 −𝑟(𝑇 −𝑡) 1 𝑆𝑢 𝑑𝑢 𝑒−𝑟(𝑇 −𝑡) + 𝔼ℚ 𝑆𝑢 || ℱ𝑡 𝑑𝑢. 𝑒 = ∫𝑡 𝑇 − 𝜏 ∫𝜏 𝑇 −𝜏 Using It¯o’s lemma we can easily show that for 𝑢 > 𝑡

1 2 ℚ )(𝑢−𝑡)+𝜎𝑊𝑢−𝑡

𝑆𝑢 = 𝑆𝑡 𝑒(𝑟−𝐷− 2 𝜎

,

ℚ 𝑊𝑢−𝑡 ∼  (0, 𝑢 − 𝑡).

[ ] Hence, 𝔼ℚ 𝑆𝑢 || ℱ𝑡 = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑢−𝑡) and the option price becomes (

) 𝑡 𝑇 1 1 −𝑟(𝑇 −𝑡) 𝑆𝑢 𝑑𝑢 𝑒−𝑟(𝑇 −𝑡) + 𝑆𝑡 𝑒(𝑟−𝐷)(𝑢−𝑡) 𝑑𝑢 𝑒 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) = ∫𝑡 𝑇 − 𝜏 ∫𝜏 𝑇 −𝜏 ) [ (𝑟−𝐷)𝑢 ]𝑇 ( 𝑡 1 𝑒 1 𝑆𝑢 𝑑𝑢 𝑒−𝑟(𝑇 −𝑡) + 𝑆𝑡 𝑒−𝑟(𝑇 −𝑡)−(𝑟−𝐷)𝑡 = 𝑇 − 𝜏 ∫𝜏 𝑇 −𝜏 𝑟−𝐷 𝑡 ( ) ( −𝐷(𝑇 −𝑡) ) 𝑡 −𝑟(𝑇 −𝑡) 1 𝑒 −𝑒 = 𝑆 𝑑𝑢 𝑒−𝑟(𝑇 −𝑡) + 𝑆𝑡 . 𝑇 − 𝜏 ∫𝜏 𝑢 (𝑟 − 𝐷)(𝑇 − 𝜏) At expiry time 𝑇 , the put–call parity for arithmetic average rate options is

(𝑎) (𝑎) 𝐶𝑎𝑟 (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝐾, 𝑇 ) − 𝑃𝑎𝑟 (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝐾, 𝑇 )

⎧ ⎪ ⎪𝑇 =⎨ ⎪ ⎪ ⎩𝑇

𝑇

1 𝑆𝑢 𝑑𝑢 − 𝐾 − 𝜏 ∫𝜏

𝑇

if

1 𝑆𝑢 𝑑𝑢 > 𝐾 𝑇 − 𝜏 ∫𝜏

if

1 𝑆𝑢 𝑑𝑢 ≤ 𝐾. 𝑇 − 𝜏 ∫𝜏

𝑇

1 𝑆𝑢 𝑑𝑢 − 𝐾 − 𝜏 ∫𝜏

𝑇

5.2.2 Continuous Sampling

499

By discounting the payoffs back to time 𝑡 under the risk-neutral measure ℚ, we have [

(𝑎) (𝑎) 𝐶𝑎𝑟 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑎𝑟 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 )

] 𝑇 | 1 | =𝑒 𝔼 𝑆𝑢 𝑑𝑢 − 𝐾 | ℱ𝑡 | 𝑇 − 𝜏 ∫𝜏 | [ ] 𝑇 | 1 | = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 𝑆𝑢 𝑑𝑢| ℱ𝑡 | 𝑇 − 𝜏 ∫𝜏 | −𝑟(𝑇 −𝑡) −𝐾𝑒 ( ) 𝑡 1 = 𝑆 𝑑𝑢 𝑒−𝑟(𝑇 −𝑡) 𝑇 − 𝜏 ∫𝜏 𝑢 ( −𝐷(𝑇 −𝑡) ) 𝑒 − 𝑒−𝑟(𝑇 −𝑡) + 𝑆𝑡 − 𝐾𝑒−𝑟(𝑇 −𝑡) (𝑟 − 𝐷)(𝑇 − 𝜏) ( ) 𝑡 1 = 𝑆𝑢 𝑑𝑢 − 𝐾 𝑒−𝑟(𝑇 −𝑡) 𝑇 − 𝜏 ∫𝜏 ( −𝐷(𝑇 −𝑡) ) 𝑒 − 𝑒−𝑟(𝑇 −𝑡) + 𝑆𝑡 . (𝑟 − 𝐷)(𝑇 − 𝜏) −𝑟(𝑇 −𝑡) ℚ

In contrast, at expiry time 𝑇 the put–call parity for arithmetic average strike options is (𝑎) (𝑎) 𝐶𝑎𝑠 (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝑇 ) − 𝑃𝑎𝑠 (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝑇 ) 𝑇 ⎧ 1 𝑆𝑢 𝑑𝑢 ⎪ 𝑆𝑇 − 𝑇 − 𝜏 ∫𝜏 ⎪ =⎨ 𝑇 ⎪ 1 𝑆𝑢 𝑑𝑢 ⎪ 𝑆𝑇 − 𝑇 − 𝜏 ∫𝜏 ⎩

𝑇

if 𝑆𝑇 >

1 𝑆𝑢 𝑑𝑢 𝑇 − 𝜏 ∫𝜏

if 𝑆𝑇 ≤

1 𝑆𝑢 𝑑𝑢. 𝑇 − 𝜏 ∫𝜏

𝑇

By discounting the payoffs back to time 𝑡 under the risk-neutral measure ℚ, we have [

] 𝑇 | 1 | (𝑎) (𝑎) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) − 𝑃𝑎𝑠 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 𝑆𝑇 − 𝑆𝑢 𝑑𝑢| ℱ𝑡 𝐶𝑎𝑠 | 𝑇 − 𝜏 ∫𝜏 | [ ] −𝑟(𝑇 −𝑡) ℚ | =𝑒 𝔼 𝑆𝑇 | ℱ𝑡 [ ] 𝑇 | 1 | −𝑟(𝑇 −𝑡) ℚ −𝑒 𝔼 𝑆𝑢 𝑑𝑢| ℱ𝑡 | 𝑇 − 𝜏 ∫𝜏 | ( ) 𝑡 1 𝑆 𝑑𝑢 𝑒−𝑟(𝑇 −𝑡) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝑇 − 𝜏 ∫𝜏 𝑢 ( −𝐷(𝑇 −𝑡) ) 𝑒 − 𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 (𝑟 − 𝐷)(𝑇 − 𝜏)

500

5.2.2 Continuous Sampling

(

) 𝑡 1 = 𝑆𝑡 − 𝑆 𝑑𝑢 𝑒−𝑟(𝑇 −𝑡) 𝑇 − 𝜏 ∫𝜏 𝑢 ( ) (𝑟 − 𝐷)(𝑇 − 𝜏) − 1 + (𝑟 − 𝐷)(𝑇 − 𝜏) ) ( −𝐷(𝑇 −𝑡) × 𝑒 − 𝑒−𝑟(𝑇 −𝑡) 𝑆𝑡 .

8. Geometric Average Rate (Fixed Strike) Asian Option (PDE Approach). Let {𝑊𝑡 : 𝑡 ≥ 0} be the ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter and let 𝑟 be the risk-free interest rate from the money-market account. We define 𝐼𝑡 =

𝑡

∫𝜏

log 𝑆𝑢 𝑑𝑢

to be the geometric sum of an asset from initial time 𝜏 ≥ 0 until time 𝑡 and consider the geometric average rate call option with payoff { Ψ(𝑆𝑇 , 𝐼𝑇 ) = max

𝑒

} − 𝐾, 0

𝐼𝑇 𝑇 −𝜏

where 𝐾 > 0 is the strike price and 𝑇 is the option expiry time. Assume that the geometric (𝑔) average rate call option price at time 𝑡, 𝜏 < 𝑡 < 𝑇 , 𝐶𝑎𝑟 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 ) satisfies the PDE (𝑔) (𝑔) (𝑔) (𝑔) 𝜕 2 𝐶𝑎𝑟 𝜕𝐶𝑎𝑟 𝜕𝐶𝑎𝑟 𝜕𝐶𝑎𝑟 1 (𝑔) + (𝑟 − 𝐷)𝑆 + log 𝑆 − 𝑟𝐶𝑎𝑟 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎 2 𝑆𝑡2 𝑡 𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝐼𝑡 𝜕𝑆𝑡2

with boundary condition { (𝑔) (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max 𝐶𝑎𝑟

} 𝐼𝑇 𝑒 𝑇 −𝜏 − 𝐾, 0 .

By considering the change of variables (𝑔) 𝐶𝑎𝑟 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐻(𝑥, 𝑡),

𝑥=

𝐼𝑡 + (𝑇 − 𝑡) log 𝑆𝑡 𝑇 −𝜏

5.2.2 Continuous Sampling

501

show that 𝐻(𝑥, 𝑡) satisfies ) ) ( )( ( 𝑇 − 𝑡 𝜕𝐻 𝜕𝐻 1 2 𝑇 − 𝑡 2 𝜕 2 𝐻 1 + 𝑟 − 𝐷 − 𝜎2 + 𝜎 − 𝑟𝐻(𝑥, 𝑡) = 0 2 𝜕𝑡 2 𝑇 −𝜏 2 𝑇 − 𝜏 𝜕𝑥 𝜕𝑥 with boundary condition 𝐻(𝑥, 𝑇 ) = max{𝑒𝑥 − 𝐾, 0}. By setting 𝑆̂𝑡 = 𝑒𝑥 show that the PDE can be transformed into ) )] ( ) ( ( [ 𝑇 − 𝑡 ̂ 𝜕𝐻 𝑡−𝜏 𝜕𝐻 1 2 𝑇 − 𝑡 2 ̂2 𝜕 2 𝐻 1 + 𝑟 − 𝐷 − 𝜎2 + 𝜎 𝑆𝑡 𝑆𝑡 𝜕𝑡 2 𝑇 −𝜏 2 𝑇 −𝜏 𝑇 −𝜏 𝜕 𝑆̂2 𝜕 𝑆̂ −𝑟𝐻(𝑆̂𝑡 , 𝑡) = 0

𝑡

𝑡

with terminal payoff 𝐻(𝑆̂𝑇 , 𝑇 ) = max{𝑆̂𝑇 − 𝐾, 0}. Hence, deduce from the European option price formula that the geometric average rate call option price at time 𝑡, 𝜏 < 𝑡 < 𝑇 is 𝐼𝑡

𝑇 −𝑡

(𝑔) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒 𝑇 −𝜏 𝑆𝑡𝑇 −𝜏 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑 + ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑 − ) 𝐶𝑎𝑟

where ) ) ( ( 𝐼𝑡 1 𝑇 −𝑡 log 𝑆𝑡 − log 𝐾 + 𝑟 − 𝐷 ± 𝜎 2 (𝑇 − 𝑡) + 𝑇 −𝜏 2 𝑑± = 𝑇 − 𝜏 √ 𝜎 𝑇 −𝑡 ( ) ( )( ) ( ) 𝜎2 𝑇 − 𝑡 𝑇 + 2𝑡 − 3𝜏 𝐷 𝑇 −𝑡 𝑟 𝑇 + 𝑡 − 2𝜏 + + 𝐷= 2 𝑇 −𝜏 12 𝑇 − 𝜏 𝑇 −𝜏 2 𝑇 −𝜏 ( ) 𝜎2 𝑇 − 𝑡 2 2 𝜎 = 3 𝑇 −𝜏 and Φ(⋅) is the cdf of a standard normal. (𝑔) Solution: To obtain the PDEs satisfied by 𝐶𝑎𝑟 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝐻(𝑥, 𝑡), see Problem 5.2.2.3 (page 483). By setting 𝑆̂𝑡 = 𝑒𝑥 we can express

𝜕𝐻 𝜕𝐻 𝜕𝐻 𝜕 𝑆̂𝑡 𝜕𝐻 = 𝑆̂𝑡 = = 𝑒𝑥 𝜕𝑥 𝜕𝑆𝑡 𝜕𝑥 𝜕 𝑆̂ 𝜕 𝑆̂ 𝑡

𝑡

502

5.2.2 Continuous Sampling

𝜕 𝑆̂𝑡 𝜕𝐻 𝜕2𝐻 𝜕 2 𝐻 𝜕 𝑆̂𝑡 𝜕𝐻 𝜕2𝐻 𝜕𝐻 𝜕2𝐻 = + 𝑆̂𝑡 + 𝑒𝑥 𝑆̂𝑡 = 𝑆̂𝑡 + 𝑆̂𝑡2 . = 𝑒𝑥 2 𝜕𝑥 𝜕 𝑆̂ 𝜕𝑥 𝜕 𝑆̂𝑡2 𝜕𝑥 𝜕 𝑆̂𝑡 𝜕 𝑆̂𝑡2 𝜕 𝑆̂𝑡 𝜕 𝑆̂𝑡2 𝑡 By substituting the above results into the PDE satisfied by 𝐻(𝑥, 𝑡) we have ) ( 𝜕𝐻 1 2 𝑇 − 𝑡 2 + 𝜎 𝜕𝑡 2 𝑇 −𝜏

(

𝜕𝐻 𝜕2𝐻 + 𝑆̂𝑡2 𝑆̂𝑡 𝜕 𝑆̂𝑡 𝜕 𝑆̂2 𝑡

−𝑟𝐻(𝑆̂𝑡 , 𝑡) = 0

)

) )( ( 𝑇 − 𝑡 ̂ 𝜕𝐻 1 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎2 2 𝑇 −𝜏 𝜕 𝑆̂

𝑡

or ) )] ( ) ( ( [ 𝑇 − 𝑡 ̂ 𝜕𝐻 𝑡−𝜏 1 𝜕𝐻 1 2 𝑇 − 𝑡 2 ̂2 𝜕 2 𝐻 + 𝑟 − 𝐷 − 𝜎2 + 𝜎 𝑆𝑡 𝑆𝑡 𝜕𝑡 2 𝑇 −𝜏 2 𝑇 −𝜏 𝑇 −𝜏 𝜕 𝑆̂2 𝜕 𝑆̂ −𝑟𝐻(𝑆̂𝑡 , 𝑡) = 0

𝑡

𝑡

with payoff 𝐻(𝑆̂𝑇 , 𝑇 ) = max{𝑆̂𝑇 − 𝐾, 0} which is a Black–Scholes equation with time-dependent dividend yield )] ( ) ( ) ( [ 𝑡−𝜏 𝑇 −𝑡 ̂ = 𝑟 + 1 𝜎2 𝑇 − 𝑡 +𝐷 𝐷(𝑡) 2 𝑇 −𝜏 𝑇 −𝜏 𝑇 −𝜏 and volatility 𝜎 ̂(𝑡) = 𝜎

(

) 𝑇 −𝑡 . 𝑇 −𝜏

Hence, we can deduce that 𝐻(𝑆̂𝑡 , 𝑡) = 𝑆̂𝑡 𝑒− ∫𝑡

𝑇

̂ 𝐷(𝑢)𝑑𝑢

Φ(𝑑̂+ ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑̂− )

where 𝑇

𝑑̂± =

̂ ± 1𝜎 ̂(𝑢)2 ) 𝑑𝑢 log(𝑆̂𝑡 ∕𝐾) + ∫𝑡 (𝑟 − 𝐷(𝑢) 2 √ 𝑇 ∫𝑡 𝜎 ̂(𝑢)2 𝑑𝑢

5.2.2 Continuous Sampling

503

with 𝑇

∫𝑡

)] ( ) ( )} ( 𝑢−𝜏 𝑇 −𝑢 1 𝑇 −𝑢 +𝐷 𝑑𝑢 𝑟 + 𝜎2 ∫𝑡 2 𝑇 −𝜏 𝑇 −𝜏 𝑇 −𝜏 ( )} 𝑇 { 𝑟(𝑢 − 𝜏) 1 2 ( 𝑇 − 𝑢 ) ( 𝑢 − 𝜏 ) 𝑇 −𝑢 +𝐷 = + 𝜎 𝑑𝑢 ∫𝑡 𝑇 −𝜏 2 𝑇 −𝜏 𝑇 −𝜏 𝑇 −𝜏 ( ) 𝑇 𝑟(𝑢 − 𝜏)2 𝜎 2 (𝑇 − 𝑢)(𝑢 − 𝜏)2 (𝑢 − 𝜏)3 (𝑇 − 𝑢)2 || = + + −𝐷 | 2(𝑇 − 𝜏) 2 2 6 2(𝑇 − 𝜏) ||𝑡

̂ 𝑑𝑢 = 𝐷(𝑢)

𝑇

{[

𝑟(𝑇 + 𝑡 − 2𝜏)(𝑇 − 𝑡) 2(𝑇 − 𝜏) [ 𝜎 2 (𝑇 − 𝑡)((𝑇 − 𝜏)2 + (𝑡 − 𝜏)(𝑇 − 𝜏) + (𝑡 − 𝜏)2 ) + 12 (𝑇 − 𝜏)2 ] 2 2(𝑇 − 𝑡)(𝑡 − 𝜏) 𝐷(𝑇 − 𝑡)2 − + 2(𝑇 − 𝜏) (𝑇 − 𝜏)2 [ ( ) ( )( ) 2 𝜎 𝑇 −𝑡 𝑇 + 2𝑡 − 3𝜏 𝑟 𝑇 + 𝑡 − 2𝜏 + = 2 𝑇 −𝜏 12 𝑇 − 𝜏 𝑇 −𝜏 ( )] 𝐷 𝑇 −𝑡 (𝑇 − 𝑡) + 2 𝑇 −𝜏 =

and 𝑇

∫𝑡

) 𝑇 −𝑢 2 𝑑𝑢 ∫𝑡 𝑇 −𝜏 𝑇 𝜎 2 (𝑇 − 𝑢)3 || =− | 3(𝑇 − 𝜏)2 ||𝑡 ( ) 𝜎2 𝑇 − 𝑡 2 = (𝑇 − 𝑡). 3 𝑇 −𝜏

𝜎 ̂(𝑢)2 𝑑𝑢 =

𝑇

𝜎2

(

Thus, we can rewrite 𝐻(𝑆̂𝑡 , 𝑡) as 𝐻(𝑆̂𝑡 , 𝑡) = 𝑆̂𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑 + ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑 − ) where

𝑑± =

log(𝑆̂𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡 𝐷=

1 𝑇 − 𝑡 ∫𝑡

𝑇

̂ 𝑑𝑢 𝐷(𝑢)

504

5.2.2 Continuous Sampling

and 𝜎2 = By substituting 𝑆̂𝑡 = 𝑒𝑥 = 𝑒 price is

𝐼𝑡 +(𝑇 −𝑡) log 𝑆𝑡 𝑇 −𝜏

1 𝑇 − 𝑡 ∫𝑡

𝑇

𝜎 ̂(𝑢)2 𝑑𝑢. 𝑇 −𝑡

𝐼𝑡

= 𝑒 𝑇 −𝜏 𝑆𝑡𝑇 −𝜏 , the geometric average rate call option

𝐼𝑡

𝑇 −𝑡

(𝑔) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒 𝑇 −𝜏 𝑆𝑡𝑇 −𝜏 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑 + ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑 − ) 𝐶𝑎𝑟

where ) ) ( ( 𝐼𝑡 1 𝑇 −𝑡 log 𝑆𝑡 − log 𝐾 + 𝑟 − 𝐷 ± 𝜎 2 (𝑇 − 𝑡) + 𝑇 −𝜏 2 . 𝑑± = 𝑇 − 𝜏 √ 𝜎 𝑇 −𝑡

9. Geometric Average Strike (Floating Strike) Asian Option (PDE Approach). Let {𝑊𝑡 : 𝑡 ≥ 0} be the ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter and let 𝑟 be the risk-free interest rate from the money-market account. We define 𝐼𝑡 =

𝑡

∫𝜏

log 𝑆𝑢 𝑑𝑢

to be the geometric sum of an asset in the time period [𝜏, 𝑡], 𝜏 ≥ 0 and consider the geometric average strike put option with payoff { Ψ(𝑆𝑇 , 𝐼𝑇 ) = max

𝑒

𝐼𝑇 𝑇 −𝜏

} − 𝑆𝑇 , 0

where 𝑇 is the option expiry time. Assume that the geometric average strike put option (𝑔) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) satisfies the PDE price at time 𝑡, 𝜏 < 𝑡 < 𝑇 , 𝑃𝑎𝑠 (𝑔) (𝑔) (𝑔) (𝑔) 𝜕 2 𝑃𝑎𝑠 𝜕𝑃𝑎𝑠 𝜕𝑃𝑎𝑠 𝜕𝑃𝑎𝑠 1 (𝑔) + (𝑟 − 𝐷)𝑆 + log 𝑆 − 𝑟𝑃𝑎𝑠 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) = 0 + 𝜎 2 𝑆𝑡2 𝑡 𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝐼𝑡 𝜕𝑆𝑡2

5.2.2 Continuous Sampling

505

with boundary condition { (𝑔) (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝑇 ) = max 𝑃𝑎𝑠

} 𝐼𝑇 𝑒 𝑇 −𝜏 − 𝑆𝑇 , 0 .

By considering the change of variables

(𝑔) 𝑃𝑎𝑠 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) = 𝑆𝑡 𝐻(𝑥, 𝑡),

𝑥=

𝐼𝑡 − (𝑡 − 𝜏) log 𝑆𝑡 𝑇 −𝜏

show that 𝐻(𝑥, 𝑡) satisfies ) ) ( )( ( 𝑡 − 𝜏 𝜕𝐻 1 2 𝜕𝐻 1 2 𝑡 − 𝜏 2 𝜕 2 𝐻 − 𝑟 − 𝐷 + + 𝜎 𝜎 − 𝐷𝐻(𝑥, 𝑡) = 0 𝜕𝑡 2 𝑇 −𝜏 2 𝑇 − 𝜏 𝜕𝑥 𝜕𝑥2 with boundary condition 𝐻(𝑥, 𝑇 ) = max{𝑒𝑥 − 1, 0}. By setting 𝑆̂𝑡 = 𝑒𝑥 show that the PDE can be transformed into ) )] ( ) ( ( [ 𝑡 − 𝜏 ̂ 𝜕𝐻 𝑇 −𝑡 1 𝜕𝐻 1 2 𝑡 − 𝜏 2 ̂2 𝜕 2 𝐻 − 𝑟 − 𝐷 + 𝜎2 + 𝜎 𝑆𝑡 𝑆𝑡 𝜕𝑡 2 𝑇 −𝜏 2 𝑇 −𝜏 𝑇 −𝜏 𝜕 𝑆̂2 𝜕 𝑆̂𝑡 𝑡

−𝐷𝐻(𝑆̂𝑡 , 𝑡) = 0 with terminal payoff

𝐻(𝑆̂𝑇 , 𝑇 ) = max{𝑆̂𝑇 − 1, 0}. Hence, deduce from the European option price formula that the geometric average strike put option price at time 𝑡, 𝜏 < 𝑡 < 𝑇 is 𝐼𝑡

𝑇 −𝑡

(𝑔) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) = 𝑒 𝑇 −𝜏 𝑆𝑡𝑇 −𝜏 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑 + ) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑 − ) 𝑃𝑎𝑠

506

5.2.2 Continuous Sampling

where ) ( 𝐼𝑡 1 𝑡−𝜏 − log 𝑆𝑡 + 𝑟 − 𝐷 ± 𝜎 2 (𝑇 − 𝑡) 2 𝑑± = 𝑇 − 𝜏 𝑇 − 𝜏 √ 𝜎 𝑇 −𝑡 ( ) ( ) ( )( ) 𝐷 𝑇 −𝑡 𝜎2 𝑇 − 𝑡 𝑇 + 2𝑡 − 3𝜏 𝑟 𝑇 + 𝑡 − 2𝜏 + + 𝐷= 2 𝑇 −𝜏 2 𝑇 −𝜏 12 𝑇 − 𝜏 𝑇 −𝜏 ( ) 𝜎 2 (𝑇 − 𝜏)2 + (𝑡 − 𝜏)(𝑇 − 𝜏) + (𝑡 − 𝜏)2 2 𝜎 = 3 (𝑇 − 𝜏)2 and Φ(⋅) is the cdf of a standard normal. (𝑔) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) and 𝐻(𝑥, 𝑡), see Problem 5.2.2.4 Solution: To obtain the PDEs satisfied by 𝑃𝑎𝑠 (page 485). By setting 𝑆̂𝑡 = 𝑒𝑥 we can express

𝜕𝐻 𝜕𝐻 𝜕𝐻 𝜕𝐻 𝜕 𝑆̂𝑡 = 𝑆̂𝑡 = = 𝑒𝑥 𝜕𝑥 𝜕𝑆𝑡 𝜕𝑥 ̂ 𝜕 𝑆𝑡 𝜕 𝑆̂𝑡

2 2 2 𝜕 𝑆̂𝑡 𝜕𝐻 𝜕 𝑆̂ 𝜕2𝐻 ̂𝑡 𝜕 𝐻 𝑡 = 𝑒𝑥 𝜕𝐻 + 𝑒𝑥 𝑆̂𝑡 𝜕 𝐻 = 𝑆̂𝑡 𝜕𝐻 + 𝑆̂2 𝜕 𝐻 . = + 𝑆 𝑡 𝜕𝑥 𝜕 𝑆̂ 𝜕𝑥2 𝜕 𝑆̂2 𝜕𝑥 𝜕 𝑆̂𝑡 𝜕 𝑆̂2 𝜕 𝑆̂𝑡 𝜕 𝑆̂2 𝑡 𝑡

𝑡

𝑡

By substituting the above results into the PDE satisfied by 𝐻(𝑥, 𝑡) we can write ) ( 𝜕𝐻 1 2 𝑡 − 𝜏 2 + 𝜎 𝜕𝑡 2 𝑇 −𝜏

(

𝜕𝐻 𝜕2𝐻 + 𝑆̂𝑡2 𝑆̂𝑡 𝜕 𝑆̂𝑡 𝜕 𝑆̂2

)

𝑡

−𝐷𝐻(𝑆̂𝑡 , 𝑡) = 0

) )( ( 𝑡 − 𝜏 ̂ 𝜕𝐻 1 − 𝑟 − 𝐷 + 𝜎2 𝑆𝑡 2 𝑇 −𝜏 𝜕 𝑆̂

𝑡

or ) )] ( ) ( ( [ 𝑡 − 𝜏 ̂ 𝜕𝐻 𝑇 −𝑡 𝜕𝐻 1 2 𝑡 − 𝜏 2 ̂2 𝜕 2 𝐻 1 − 𝑟 − 𝐷 + 𝜎2 + 𝜎 𝑆𝑡 𝑆𝑡 𝜕𝑡 2 𝑇 −𝜏 2 𝑇 −𝜏 𝑇 −𝜏 𝜕 𝑆̂2 𝜕 𝑆̂𝑡 𝑡

−𝐷𝐻(𝑆̂𝑡 , 𝑡) = 0 with payoff

𝐻(𝑆̂𝑇 , 𝑇 ) = max{𝑆̂𝑇 − 1, 0} which is a Black–Scholes equation with interest rate 𝐷, time-dependent dividend yield ̂ =𝑟 𝐷(𝑡)

(

𝑡−𝜏 𝑇 −𝜏

)

+𝐷

(

𝑇 −𝑡 𝑇 −𝜏

)

)( ) ( 1 𝑡−𝜏 𝑇 −𝑡 + 𝜎2 2 𝑇 −𝜏 𝑇 −𝜏

5.2.2 Continuous Sampling

507

and volatility 𝜎 ̂(𝑡) =

𝜎(𝑡 − 𝜏) . 𝑇 −𝜏

Hence, we can deduce that 𝐻(𝑆̂𝑡 , 𝑡) = 𝑆̂𝑡 𝑒− ∫𝑡

𝑇

̂ 𝐷(𝑢)𝑑𝑢

Φ(𝑑̂+ ) − 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑̂− )

with 𝑇

𝑑̂± =

̂ ± 1𝜎 ̂(𝑢)2 ) 𝑑𝑢 log 𝑆̂𝑡 + ∫𝑡 (𝐷 − 𝐷(𝑢) 2 √ 𝑇 ∫𝑡 𝜎 ̂(𝑢)2 𝑑𝑢

where 𝑇

∫𝑡

) )( ) ( 𝑇 −𝑢 1 𝑢−𝜏 𝑇 −𝑢 + 𝜎2 𝑑𝑢 ∫𝑡 𝑇 −𝜏 2 𝑇 −𝜏 𝑇 −𝜏 ( )𝑇 𝑟(𝑢 − 𝜏)2 (𝑇 − 𝑢)2 𝜎 2 (𝑇 − 𝑢)(𝑢 − 𝜏)2 (𝑢 − 𝜏)3 || = −𝐷 + + | | 2(𝑇 − 𝜏) 2(𝑇 − 𝜏) 2 2 6 |𝑡

̂ 𝑑𝑢 = 𝐷(𝑢)

𝑇

𝑟

(

𝑢−𝜏 𝑇 −𝜏

)

+𝐷

(

𝑟(𝑇 + 𝑡 − 2𝜏)(𝑇 − 𝑡) 𝐷(𝑇 − 𝑡)2 + 2(𝑇 − 𝜏) 2(𝑇 − 𝜏) [ 𝜎 2 (𝑇 − 𝑡)((𝑇 − 𝜏)2 + (𝑡 − 𝜏)(𝑇 − 𝜏) + (𝑡 − 𝜏)2 ) + 12 (𝑇 − 𝜏)2 ] 2 2(𝑇 − 𝑡)(𝑡 − 𝜏) − (𝑇 − 𝜏)2 ) ( ) [ ( 𝐷 𝑇 −𝑡 𝑟 𝑇 + 𝑡 − 2𝜏 + = 2 𝑇 −𝜏 2 𝑇 − 𝜏] ( )( ) 2 𝑇 −𝑡 𝑇 + 2𝑡 − 3𝜏 𝜎 + (𝑇 − 𝑡) 12 𝑇 − 𝜏 𝑇 −𝜏

=

and 𝑇

∫𝑡

𝑇

𝜎 2 (𝑢 − 𝜏)2 𝑑𝑢 ∫𝑡 (𝑇 − 𝜏)2 𝑇 𝜎 2 (𝑢 − 𝜏)3 || = | 3 (𝑇 − 𝜏)2 ||𝑡 ( ) 𝜎 2 (𝑇 − 𝜏)3 − (𝑡 − 𝜏)3 = 3 (𝑇 − 𝜏)2 ) ( 2 2 𝜎 2 (𝑇 − 𝜏) + (𝑡 − 𝜏)(𝑇 − 𝜏) + (𝑡 − 𝜏) (𝑇 − 𝑡) = . 3 (𝑇 − 𝜏)2

𝜎 ̂(𝑢)2 𝑑𝑢 =

508

5.2.2 Continuous Sampling

Hence, we can rewrite 𝐻(𝑆̂𝑡 , 𝑡) as 𝐻(𝑆̂𝑡 , 𝑡) = 𝑆̂𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑 + ) − 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑 − ) where ) ( log(𝑆̂𝑡 ∕𝐾) + 𝐷 − 𝐷 ± 12 𝜎 2 (𝑇 − 𝑡) 𝑑± = √ 𝜎 𝑇 −𝑡 𝐷=

1 𝑇 − 𝑡 ∫𝑡

𝜎2 =

1 𝑇 − 𝑡 ∫𝑡

𝑇

̂ 𝑑𝑢 𝐷(𝑢)

and

By substituting 𝑆̂𝑡 = 𝑒𝑥 = 𝑒 time 𝑡 < 𝑇 is

𝐼𝑡 −𝑡 log 𝑆𝑡 𝑇

𝐼𝑡

− 𝑇𝑡

= 𝑒 𝑇 𝑆𝑡

𝑇

𝜎 ̂(𝑢)2 𝑑𝑢.

, the geometric average strike put option at

(𝑔) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) = 𝑆𝑡 𝐻(𝑥, 𝑡) 𝑃𝑎𝑠 𝐼𝑡

𝑇 −𝑡

= 𝑒 𝑇 𝑆𝑡 𝑇 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑 + ) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑 − ) where ) ( 𝐼𝑡 1 𝑡−𝜏 − log 𝑆𝑡 + 𝑟 − 𝐷 ± 𝜎 2 (𝑇 − 𝑡) 2 𝑑± = 𝑇 − 𝜏 𝑇 − 𝜏 √ 𝜎 𝑇 −𝑡 ( ) ( ) ( )( ) 𝐷 𝑇 −𝑡 𝜎2 𝑇 − 𝑡 𝑇 + 2𝑡 − 3𝜏 𝑟 𝑇 + 𝑡 − 2𝜏 + + 𝐷= 2 𝑇 −𝜏 2 𝑇 −𝜏 12 𝑇 − 𝜏 𝑇 −𝜏 ( ) 𝜎 2 (𝑇 − 𝜏)2 + (𝑡 − 𝜏)(𝑇 − 𝜏) + (𝑡 − 𝜏)2 2 𝜎 = . 3 (𝑇 − 𝜏)2

10. Geometric Average Rate (Fixed Strike) Asian Option (Probabilistic Approach). Let {𝑊𝑡 : 𝑡 ≥ 0} be the ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter and let 𝑟 be the risk-free interest rate from the money-market account.

5.2.2 Continuous Sampling

509

We define 𝐼𝑡 =

𝑡

∫𝜏

log 𝑆𝑢 𝑑𝑢

to be the geometric sum of the asset in the time period [𝜏, 𝑡], where 𝜏 ≥ 0 and consider the geometric average rate put option with payoff { Ψ(𝑆𝑇 , 𝐼𝑇 ) = max

} 𝐼𝑇 𝐾 − 𝑒 𝑇 −𝜏 , 0

where 𝐾 > 0 is the strike price and 𝑇 is the option expiry time. Show that if 𝑋 ∼  (𝜇𝑥 , 𝜎𝑥2 ) then for a constant 𝐾 > 0 1 2 ] 𝔼 max{𝛿(𝑒 − 𝐾, 0)} = 𝛿𝑒𝜇𝑥 + 2 𝜎𝑥 Φ

[

𝑋

(

𝛿(𝜇𝑥 + 𝜎𝑥2 − log 𝐾)

)

( − 𝛿𝐾Φ

𝜎𝑥

𝛿(𝜇𝑥 − log 𝐾) 𝜎𝑥

)

where 𝛿 ∈ {−1, 1} and Φ(⋅) denotes the cdf of a standard normal. Given that 𝑊𝑢−𝑡 ∼ 𝑁(0, 𝑢 − 𝑡), 𝑢 > 𝑡 show, using integration by parts, that 𝑇

∫𝑡

𝑊𝑢−𝑡 𝑑𝑢 =

𝑇

∫𝑡

(𝑇 − 𝑢) 𝑑𝑊𝑢

and deduce that 𝑇

∫𝑡

[ ] (𝑇 − 𝑡)3 𝑊𝑢−𝑡 𝑑𝑢 ∼  0, . 3

𝐼𝑇 𝐼𝑡 𝐼𝑇 By considering 𝑒 𝑇 −𝜏 and 𝑒 𝑡−𝜏 show under the risk-neutral measure ℚ that follows 𝑇 −𝜏 a normal distribution with mean

𝔼ℚ

(

𝐼𝑇 || ℱ 𝑇 || 𝑡

) =

1 𝑇 −𝜏

{

𝑡

∫𝜏

log 𝑆𝑢 𝑑𝑢 + (𝑇 − 𝑡) log 𝑆𝑡 +

} ( ) 1 1 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡)2 2 2

and variance Var

ℚ

(

𝐼𝑇 || ℱ 𝑇 − 𝜏 || 𝑡

) =

𝜎 2 (𝑇 − 𝑡)3 . 3 (𝑇 − 𝜏)2

Hence, deduce that under the risk-neutral measure ℚ the geometric average rate put option at time 𝑡 < 𝑇 is 𝐼𝑡

𝑇 −𝑡

(𝑔) 𝑃𝑎𝑟 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑 − ) − 𝑒 𝑇 𝑆𝑡 𝑇 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑 + )

510

5.2.2 Continuous Sampling

where ) ) ( ( 𝐼𝑡 1 𝑇 −𝑡 log 𝑆𝑡 − log 𝐾 + 𝑟 − 𝐷 ± 𝜎 2 (𝑇 − 𝑡) + 𝑇 −𝜏 2 𝑑± = 𝑇 − 𝜏 √ 𝜎 𝑇 −𝑡 ( ) ( ) ( )( ) 𝐷 𝑇 −𝑡 𝜎2 𝑇 − 𝑡 𝑇 + 2𝑡 − 3𝜏 𝑟 𝑇 + 𝑡 − 2𝜏 + + 𝐷= 2 𝑇 −𝜏 2 𝑇 −𝜏 12 𝑇 − 𝜏 𝑇 −𝜏 2 ( 𝑇 − 𝑡 )2 𝜎 𝜎2 = . 3 𝑇 −𝜏 Solution: For the first part of the results refer to Problem 1.2.2.7 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus. From the stationary increment property of a standard Wiener process 𝑇

∫𝑡

𝑊𝑢−𝑡 𝑑𝑢 = = =

𝑇

∫𝑡 ∫𝑡

𝑇

𝑇

∫𝑡

(𝑊𝑢 − 𝑊𝑡 ) 𝑑𝑢 𝑊𝑢 𝑑𝑢 − 𝑊𝑡

𝑇

∫𝑡

𝑑𝑢

𝑊𝑢 𝑑𝑢 − 𝑊𝑡 (𝑇 − 𝑡).

Using integration by parts 𝑇

∫𝑡

𝑇 𝑊𝑢 𝑑𝑢 = 𝑢𝑊𝑢 ||𝑡 −

𝑇

𝑢 𝑑𝑊𝑢

∫𝑡

= 𝑇 𝑊𝑇 − 𝑡𝑊𝑡 −

𝑇

∫𝑡

𝑢 𝑑𝑊𝑢

and therefore 𝑇

∫𝑡

𝑊𝑢−𝑡 𝑑𝑢 = 𝑇 𝑊𝑇 − 𝑡𝑊𝑡 − = 𝑇 𝑊𝑇 −𝑡 − 𝑇

=

∫𝑡

𝑇

∫𝑡

𝑇

∫𝑡

𝑢 𝑑𝑊𝑢 − 𝑊𝑡 (𝑇 − 𝑡)

𝑢 𝑑𝑊𝑢

(𝑇 − 𝑢) 𝑑𝑊𝑢 .

Using the properties of the standard Wiener process we have [ 𝔼

𝑇

∫𝑡

] 𝑊𝑢−𝑡 𝑑𝑢 = 𝔼

[

𝑇

∫𝑡

] (𝑇 − 𝑢) 𝑑𝑊𝑢 = 0

5.2.2 Continuous Sampling

511

and [( 𝔼

𝑇

∫𝑡

)2 ]

[

𝑊𝑢−𝑡 𝑑𝑢

=𝔼

𝑇

∫𝑡

] 𝑇 (𝑇 − 𝑢)3 || (𝑇 − 𝑡)3 (𝑇 − 𝑢)2 𝑑𝑢 = − . | = | 3 3 |𝑡

Since we can write 𝑇

∫𝑡

𝑊𝑢−𝑡 𝑑𝑢 =

𝑇

∫𝑡

(𝑇 − 𝑢) 𝑑𝑊𝑢 = lim

𝑛→∞

𝑛−1 ∑ (𝑇 − 𝑡𝑖 )(𝑊𝑡𝑖+1 − 𝑊𝑡𝑖 ) 𝑖=0

where 𝑡𝑖 = 𝑡 + 𝑖(𝑇 − 𝑡)∕𝑛, we can see that each term of 𝑊𝑡𝑖+1 − 𝑊𝑡𝑖 is multiplied by a deterministic term. Thus, the product is normal and given that the sum of normal variables [ ] 𝑇 (𝑇 − 𝑡)3 𝑊𝑢−𝑡 𝑑𝑢 ∼  0, is also normal, we can deduce that . ∫𝑡 3 Under the risk-neutral measure ℚ, 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡ℚ 𝑆𝑡 where 𝑊𝑡ℚ = 𝑊𝑡 +

(𝜇 − 𝑟) 𝜎

𝑡 is a ℚ-standard Wiener process. Using It¯o’s lemma

𝑑 log 𝑆𝑡 =

𝑑𝑆𝑡 1 − 𝑆𝑡 2

(

𝑑𝑆𝑡 𝑆𝑡

)2 +…

1 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡ℚ − 𝜎 2 𝑑𝑡 2 ) ( 1 2 = 𝑟 − 𝐷 − 𝜎 𝑑𝑡 + 𝜎 𝑑𝑊𝑡ℚ 2 and taking integrals for 𝑡 < 𝑢 < 𝑇 we have 𝑢

∫𝑡

𝑑 log 𝑆𝑡 =

𝑢(

∫𝑡

) 𝑢 1 𝑟 − 𝐷 − 𝜎 2 𝑑𝑡 + 𝜎 𝑑𝑊𝑡ℚ ∫𝑡 2

or ) ( 1 ℚ log 𝑆𝑢 = log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑢 − 𝑡) + 𝜎𝑊𝑢−𝑡 . 2 1

𝑡

For 𝑡 < 𝑇 , let 𝐺𝑡 = 𝑒 𝑡−𝜏 ∫𝜏

log 𝑆𝑢 𝑑𝑢

𝐼𝑡

1

𝑇

= 𝑒 𝑡−𝜏 and 𝐺𝑇 = 𝑒 𝑇 −𝜏 ∫𝜏

(𝑡 − 𝜏) log 𝐺𝑡 =

𝑡

∫𝜏

log 𝑆𝑢 𝑑𝑢

log 𝑆𝑢 𝑑𝑢

𝐼𝑇

= 𝑒 𝑇 −𝜏 so that

512

5.2.2 Continuous Sampling

and

(𝑇 − 𝜏) log 𝐺𝑇 =

𝑇

∫𝜏

log 𝑆𝑢 𝑑𝑢.

By subtraction

(𝑇 − 𝜏) log 𝐺𝑇 − (𝑡 − 𝜏) log 𝐺𝑡 = =

𝑇

∫𝜏 ∫𝑡

𝑇

log 𝑆𝑢 𝑑𝑢 −

𝑡

∫𝜏

log 𝑆𝑢 𝑑𝑢

log 𝑆𝑢 𝑑𝑢

ℚ we have and substituting log 𝑆𝑢 = log 𝑆𝑡 + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑢 − 𝑡) + 𝜎𝑊𝑢−𝑡

(𝑇 − 𝜏) log 𝐺𝑇 − (𝑡 − 𝜏) log 𝐺𝑡 =

𝑇

[

∫𝑡

) ] ( 1 ℚ log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑢 − 𝑡) + 𝜎𝑊𝑢−𝑡 𝑑𝑢 2

= (𝑇 − 𝑡) log 𝑆𝑡 + 𝑇

+𝜎

∫𝑡 𝑇

Since log 𝐺𝑇 =

∫𝑡

∫𝑡

1 (𝑟 − 𝐷 − 𝜎 2 )(𝑢 − 𝑡) 𝑑𝑢 2

ℚ 𝑊𝑢−𝑡 𝑑𝑢

= (𝑇 − 𝑡) log 𝑆𝑡 + +𝜎

𝑇

( ) 1 1 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡)2 2 2

ℚ (𝑇 − 𝑢) 𝑑𝑊𝑢−𝑡 .

𝐼 𝐼𝑇 and log 𝐺𝑡 = 𝑡 , therefore 𝑇 −𝜏 𝑡−𝜏

𝐼𝑇 = 𝐼𝑡 + (𝑇 − 𝑡) log 𝑆𝑡 +

( ) 𝑇 1 1 ℚ 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡)2 + 𝜎 (𝑇 − 𝑢) 𝑑𝑊𝑢−𝑡 ∫𝑡 2 2

or { ( ) 𝐼𝑇 1 1 1 𝐼𝑡 + (𝑇 − 𝑡) log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡)2 = 𝑇 −𝜏 𝑇 −𝜏 2 2 } 𝑇 ℚ +𝜎 (𝑇 − 𝑢) 𝑑𝑊𝑢−𝑡 . ∫𝑡

5.2.2 Continuous Sampling 𝑇

Since

∫𝑡

513

ℚ (𝑇 − 𝑢) 𝑑𝑊𝑢−𝑡 is normally distributed, then conditional on the filtration ℱ𝑡

{ ( ) } [ 𝐼𝑇 || 1 1 1 𝐼𝑡 + (𝑇 − 𝑡) log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡)2 , ℱ𝑡 ∼  | 𝑇 −𝜏| 𝑇 −𝜏 2 2 ] 3 2 𝜎 (𝑇 − 𝑡) . 3 (𝑇 − 𝜏)2 Using the identity ( ) ( ) (𝜇𝑥 + 𝜎𝑥2 − log 𝐾) (𝜇𝑥 − log 𝐾) 𝜇𝑥 + 12 𝜎𝑥2 Φ − 𝔼 max{𝐾 − 𝑒 , 0} = 𝐾Φ − −𝑒 𝜎𝑥 𝜎𝑥 [

𝑋

]

for 𝑋 ∼  (𝜇𝑥 , 𝜎𝑥2 ) and 𝛿 = −1, under the risk-neutral measure ℚ the geometric average rate put option price at time 𝑡 is [ (𝑔) 𝑃𝑎𝑟 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 )

=𝑒

−𝑟(𝑇 −𝑡) ℚ

𝔼

{ max

𝐾 −𝑒

𝐼𝑇 𝑇 −𝜏

1 2

= 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑 − ) − 𝑒𝑚+ 2 𝑠

}| ] | , 0 | ℱ𝑡 | |

−𝑟(𝑇 −𝑡)

Φ(−𝑑 + )

where { ( ) } 1 1 1 𝐼𝑡 + (𝑇 − 𝑡) log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡)2 𝑇 −𝜏 2 2 3 2 (𝑇 − 𝑡) 𝜎 𝑠2 = 3 (𝑇 − 𝜏)2 𝑚=

𝑚 + 𝑠2 − log 𝐾 𝑠 𝑑 − = 𝑑 + − 𝑠. 𝑑+ =

Hence, 𝑚+ 12 𝑠2 −𝑟(𝑇 −𝑡)

𝑒

=𝑒 =𝑒

where 𝐷 =

1 𝑇 −𝜏 𝐼𝑡 𝑇 −𝜏

{

} 2 3 𝐼𝑡 +(𝑇 −𝑡) log 𝑆𝑡 + 21 (𝑟−𝐷− 12 𝜎 2 )(𝑇 −𝑡)2 + 𝜎6 (𝑇 −𝑡) 2 −𝑟(𝑇 −𝑡) (𝑇 −𝜏)

𝑇 −𝑡 𝑇 −𝜏

𝑆𝑡

𝑒−𝐷(𝑇 −𝑡)

( ) ( ) ( )( ) 𝐷 𝑇 −𝑡 𝜎2 𝑇 − 𝑡 𝑇 + 2𝑡 − 3𝜏 𝑟 𝑇 + 𝑡 − 2𝜏 + + . 2 𝑇 −𝜏 2 𝑇 −𝜏 12 𝑇 − 𝜏 𝑇 −𝜏

514

5.2.2 Continuous Sampling

In contrast, 𝑚 + 𝑠2 − log 𝐾 𝑠 { ( ) } 1 1 ⎡ 1 𝐼𝑡 + (𝑇 − 𝑡) log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡)2 ⎤ ⎢𝑇 −𝜏 ⎥ 2 2 ⎢ ⎥ ⎢ ⎥ 𝜎 2 (𝑇 − 𝑡)3 ⎢ ⎥ − log 𝐾 + ⎣ ⎦ 3 (𝑇 − 𝜏)2 = √ 𝜎 2 (𝑇 − 𝑡)3 3 (𝑇 − 𝜏)2 ) ) ( ( 𝐼𝑡 1 𝑇 −𝑡 log 𝑆𝑡 − log 𝐾 + 𝑟 − 𝐷 + 𝜎 2 (𝑇 − 𝑡) + 𝑇 −𝜏 2 = 𝑇 −𝜏 √ 𝜎 𝑇 −𝑡

𝑑+ =

and 𝑑− = 𝑑+ − 𝑠

√ = 𝑑+ − 𝜎 𝑇 − 𝑡 ) ) ( ( 𝐼𝑡 1 𝑇 −𝑡 log 𝑆𝑡 − log 𝐾 + 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡) + 𝑇 −𝜏 2 = 𝑇 −𝜏 √ 𝜎 𝑇 −𝑡

𝜎2 where 𝜎 2 = 3 Hence,

(

𝑇 −𝑡 𝑇 −𝜏

)2 .

𝐼𝑡

𝑇 −𝑡

(𝑔) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑 − ) − 𝑒 𝑇 𝑆𝑡 𝑇 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑 + ) 𝑃𝑎𝑟

) ) ( ( 𝐼𝑡 1 𝑇 −𝑡 log 𝑆𝑡 − log 𝐾 + 𝑟 − 𝐷 ± 𝜎 2 (𝑇 − 𝑡) + 𝑇 −𝜏 2 √ . where 𝑑 ± = 𝑇 − 𝜏 𝜎 𝑇 −𝑡 11. Geometric Average Strike (Floating Strike) Asian Option (Probabilistic Approach). Let {𝑊𝑡 : 𝑡 ≥ 0} be the ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter and let 𝑟 be the risk-free interest rate from the money-market account.

5.2.2 Continuous Sampling

515

We define 𝐼𝑡 =

𝑡

∫𝜏

log 𝑆𝑢 𝑑𝑢

to be the geometric sum of the asset in the time period [𝜏, 𝑡], where 𝜏 ≥ 0 and consider the geometric average strike call option with payoff Ψ(𝑆𝑇 , 𝐼𝑇 ) = max

{ } 𝐼𝑇 𝑆𝑇 − 𝑒 𝑇 −𝜏 , 0

where 𝑇 is the option expiry time. Show that if 𝑋 ∼  (𝜇𝑥 , 𝜎𝑥2 ) and 𝑌 ∼  (𝜇𝑦 , 𝜎𝑦2 ) have a joint bivariate normal distribution with correlation coefficient 𝜌𝑥𝑦 ∈ (−1, 1), then [ ] 𝔼 max{𝑒𝑋 − 𝑒𝑌 , 0}

=𝑒

𝜇𝑥 + 21 𝜎𝑥2

⎞ ⎞ ⎛ ⎛ 𝜇 − 𝜇𝑦 − 𝜎𝑦 (𝜎𝑦 − 𝜌𝑥𝑦 𝜎𝑥 ) ⎟ ⎜ 𝜇𝑥 − 𝜇𝑦 + 𝜎𝑥 (𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 ) ⎟ 𝜇𝑦 + 12 𝜎𝑦2 ⎜ 𝑥 Φ⎜ √ Φ⎜ √ ⎟−𝑒 ⎟ ⎜ ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟ ⎠ ⎠ ⎝ ⎝

where Φ(⋅) is the cdf of a standard normal. ) ( 𝑇 Show that the pair of random variables 𝑊𝑇 −𝑡 , ∫𝑡 𝑊𝑢−𝑡 𝑑𝑢 has the covariance matrix ⎡ 𝑇 −𝑡 ⎢ 𝚺=⎢ ⎢ 1 (𝑇 − 𝑡)2 ⎣2

− 𝑡)2 ⎤ ⎥ ⎥ 1 3⎥ (𝑇 − 𝑡) ⎦ 3 1 (𝑇 2

√ ) ( 3 𝑇 and 𝑊𝑇 −𝑡 , ∫𝑡 𝑊𝑢−𝑡 𝑑𝑢 follows a bivariate normal diswith correlation coefficient 2 tribution. Using It¯o’s formula show that under the risk-neutral measure, for 𝑡 < 𝑇 ) ( 1 log 𝑆𝑇 = log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡) + 𝜎𝑊𝑇ℚ−𝑡 2 ( ) { ( ) 𝐼𝑇 1 1 1 𝐼𝑡 + (𝑇 − 𝑡) log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡)2 log = 𝑇 −𝜏 𝑇 −𝜏 2 2 } 𝑇 ℚ +𝜎 𝑊𝑢−𝑡 𝑑𝑢 ∫𝑡 where 𝑊𝑡ℚ is a ℚ-standard Wiener process.

516

5.2.2 Continuous Sampling

Finally, deduce that under the risk-neutral measure ℚ the geometric average strike call option at time 𝑡, 𝜏 < 𝑡 < 𝑇 is 𝑇 −𝑡

𝐼𝑡

(𝑔) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑 − ) − 𝑒 𝑇 −𝜏 𝑆𝑡𝑇 −𝜏 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑 + ) 𝐶𝑎𝑠

where ) ( 𝐼𝑡 1 𝑡 − log 𝑆𝑡 + 𝐷 − 𝐷 ± 𝜎 2 (𝑇 − 𝑡) 2 𝑑± = 𝑇 − 𝜏 𝑇 − 𝜏 √ 𝜎 𝑇 −𝑡 ( ) ( ) ( )( ) 𝐷 𝑇 −𝑡 𝜎2 𝑇 − 𝑡 𝑇 + 2𝑡 − 3𝜏 𝑟 𝑇 + 𝑡 − 2𝜏 + + 𝐷= 2 𝑇 2 𝑇 −𝜏 12 𝑇 − 𝜏 𝑇 −𝜏 ( ) 𝜎 2 (𝑇 − 𝜏)2 + (𝑡 − 𝜏)(𝑇 − 𝜏) + (𝑡 − 𝜏)2 𝜎2 = . 3 (𝑇 − 𝜏)2 Solution: For the first part of the results see Problem 1.2.2.17 in Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus. By definition, the covariance matrix for the pair of random variables (𝑊𝑇 −𝑡 , 𝑇 ∫𝑡 𝑊𝑢−𝑡 𝑑𝑢) is ( ) 𝑇 ⎡ Var(𝑊𝑇 −𝑡 ) Cov 𝑊𝑇 −𝑡 , ∫𝑡 𝑊𝑢−𝑡 𝑑𝑢 ⎤ ⎢ ⎥ 𝚺=⎢ ⎥. ) ( ) ( 𝑇 ⎢ Cov 𝑊 , ∫ 𝑇 𝑊 𝑑𝑢 ⎥ ∫ Var 𝑊 𝑑𝑢 𝑇 −𝑡 𝑡 𝑢−𝑡 𝑢−𝑡 ⎣ ⎦ 𝑡 Given that 𝑊𝑇 −𝑡 ∼  (0, 𝑇 − 𝑡) and from Problem 5.2.2.10 (page 508), we have Var(𝑊𝑇 −𝑡 ) = 𝑇 − 𝑡 and ( Var

𝑇

∫𝑡

) 𝑊𝑢−𝑡 𝑑𝑢

( = Var

∫𝑡

( =𝔼 =

𝑇

𝑇

∫𝑡

) (𝑇 − 𝑢) 𝑑𝑊𝑢 )

(𝑇 − 𝑢) 𝑑𝑢 − 𝔼

1 (𝑇 − 𝑡)3 . 3

2

(

𝑇

∫𝑡

)2 (𝑇 − 𝑢) 𝑑𝑢

5.2.2 Continuous Sampling

517

( ) 𝑇 For the case Cov 𝑊𝑇 −𝑡 , ∫𝑡 𝑊𝑢−𝑡 𝑑𝑢 we can write ( Cov 𝑊𝑇 −𝑡 ,

𝑇

∫𝑡

) 𝑊𝑢−𝑡 𝑑𝑢

( = 𝔼 𝑊𝑇 −𝑡

∫𝑡

( =𝔼 = =

𝑇

∫𝑡

∫𝑡

𝑇

∫𝑡

𝑊𝑢−𝑡 𝑑𝑢

(

−𝔼(𝑊𝑇 −𝑡 )𝔼 ( = 𝔼 𝑊𝑇 −𝑡

)

𝑇

𝑇

∫𝑡

) 𝑊𝑢−𝑡 𝑑𝑢 )

𝑊𝑢−𝑡 𝑑𝑢 )

𝑊𝑇 −𝑡 𝑊𝑢−𝑡 𝑑𝑢

𝑇

) ( 𝔼 𝑊𝑇 −𝑡 𝑊𝑢−𝑡 𝑑𝑢

𝑇

] [ 2 𝑑𝑢. 𝔼 𝑊𝑢−𝑡 (𝑊𝑇 −𝑡 − 𝑊𝑢−𝑡 ) + 𝑊𝑢−𝑡

∫𝑡

From the independent increment property of a standard Wiener process we have ( Cov 𝑊𝑇 −𝑡 ,

𝑇

∫𝑡

) 𝑊𝑢−𝑡 𝑑𝑢

𝑇

=

∫𝑡 +

𝑇

∫𝑡 𝑇

=

∫𝑡

𝔼(𝑊𝑢−𝑡 )𝔼(𝑊𝑇 −𝑡 − 𝑊𝑢−𝑡 ) 𝑑𝑢 2 𝔼(𝑊𝑢−𝑡 ) 𝑑𝑠

2 𝔼(𝑊𝑢−𝑡 ) 𝑑𝑠

𝑇

(𝑢 − 𝑡) 𝑑𝑢 ∫𝑡 1 = (𝑇 − 𝑡)2 . 2 =

Therefore, the covariance matrix is ⎡ 𝑇 −𝑡 ⎢ 𝚺=⎢ ⎢ 1 (𝑇 − 𝑡)2 ⎣2

− 𝑡)2 ⎤ ⎥ ⎥ 1 3⎥ (𝑇 − 𝑡) ⎦ 3 1 (𝑇 2

with correlation coefficient ( ) 𝑇 Cov 𝑊𝑇 −𝑡 , ∫𝑡 𝑊𝑢−𝑡 𝑑𝑢

1 (𝑇 2

− 𝑡)2

√

3 𝜌= √ = . ( ) = √1 2 4 𝑇 (𝑇 − 𝑡) Var(𝑊𝑇 −𝑡 )Var ∫𝑡 𝑊𝑢−𝑡 𝑑𝑢 3

518

5.2.2 Continuous Sampling

By expressing √ 𝑇 − 𝑡𝑍1 √ ) √ (𝑇 − 𝑡)3 ( 𝜌𝑍1 + 1 − 𝜌2 𝑍2 𝑊𝑢−𝑡 𝑑𝑢 = 3 𝑊𝑇 −𝑡 =

𝑇

∫𝑡

where 𝑍1 , 𝑍2 ∼  (0, 1) and 𝑍1 ⟂ ⟂ 𝑍2 , for constants 𝜃1 and 𝜃2 ) [ √ ] ( √ √ 𝑇 3 2 𝔼 𝑒𝜃𝑊𝑇 −𝑡 +𝜃2 ∫𝑡 𝑊𝑢−𝑡 𝑑𝑢 = 𝔼 𝑒𝜃1 𝑇 −𝑡𝑍1 +𝜃2 (𝑇 −𝑡) ∕3(𝜌𝑍1 + 1−𝜌 𝑍2 ) ] [ √ √ √ 3 3 2 = 𝔼 𝑒(𝜃1 𝑇 −𝑡+𝜌𝜃2 (𝑇 −𝑡) ∕3)𝑍1 +𝜃2 (𝑇 −𝑡) (1−𝜌 )∕3𝑍2 ] [ √ ] [ √ √ 3 3 2 = 𝔼 𝑒(𝜃1 𝑇 −𝑡+𝜌𝜃2 (𝑇 −𝑡) ∕3)𝑍1 ⋅ 𝔼 𝑒𝜃2 (𝑇 −𝑡) (1−𝜌 )∕3𝑍2 1

= 𝑒 2 (𝜃1

√

√ 𝑇 −𝑡+𝜌𝜃2 (𝑇 −𝑡)3 ∕3)2

1 2

⋅ 𝑒 2 𝜃2 (𝑇 −𝑡)

3 (1−𝜌2 )∕3

.

√

3 and setting 𝜽 = (𝜃1 , 𝜃2 )𝑇 we therefore have 2

By substituting 𝜌 =

) ( 3 1 2 1 𝑇 2 1 2 (𝑇 −𝑡) 𝔼 𝑒𝜃𝑊𝑇 −𝑡 +𝜃2 ∫𝑡 𝑊𝑢−𝑡 𝑑𝑢 = 𝑒 2 𝜃1 (𝑇 −𝑡)+ 2 𝜃1 𝜃2 (𝑇 −𝑡) + 2 𝜃2 3 1 𝑇 = 𝑒 2 𝜽 𝚺𝜽

which is the moment-generating function of a bivariate normal distribution. Hence, ( ) 𝑇 𝑊𝑇 −𝑡 , ∫𝑡 𝑊𝑢−𝑡 𝑑𝑢 follows a bivariate normal distribution. From Girsanov’s theorem, under the risk-neutral measure 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡ℚ 𝑆𝑡 where 𝑊𝑡ℚ = 𝑊𝑡 +

(𝜇 − 𝑟) 𝜎

𝑡 is a ℚ-standard Wiener process. From It¯o’s formula

𝑑𝑆𝑡 1 𝑑 log 𝑆𝑡 = − 𝑆𝑡 2

(

𝑑𝑆𝑡 𝑆𝑡

)2 +…

1 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡ℚ − 𝜎 2 𝑑𝑡 2 ) ( 1 2 = 𝑟 − 𝐷 − 𝜎 𝑑𝑡 + 𝜎 𝑑𝑊𝑡ℚ 2 and taking integrals for 𝑡 < 𝑢 < 𝑇 we have 𝑢

∫𝑡

𝑑 log 𝑆𝑡 =

𝑢(

∫𝑡

) 𝑢 1 𝑟 − 𝐷 − 𝜎 2 𝑑𝑡 + 𝜎 𝑑𝑊𝑡ℚ ∫𝑡 2

5.2.2 Continuous Sampling

519

or ) ( 1 ℚ log 𝑆𝑢 = log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑢 − 𝑡) + 𝜎𝑊𝑢−𝑡 . 2 1

𝑡

For 𝑡 < 𝑇 , let 𝐺𝑡 = 𝑒 𝑡−𝜏 ∫𝜏

log 𝑆𝑢 𝑑𝑢

𝐼𝑡

𝑇

1

= 𝑒 𝑡−𝜏 and 𝐺𝑇 = 𝑒 𝑇 −𝜏 ∫𝜏

(𝑡 − 𝜏) log 𝐺𝑡 =

𝑡

∫𝜏

log 𝑆𝑢 𝑑𝑢

𝐼𝑇

= 𝑒 𝑇 −𝜏 so that

log 𝑆𝑢 𝑑𝑢

and (𝑇 − 𝜏) log 𝐺𝑇 =

𝑇

∫𝜏

log 𝑆𝑢 𝑑𝑢.

By subtraction (𝑇 − 𝜏) log 𝐺𝑇 − (𝑡 − 𝜏) log 𝐺𝑡 =

𝑇

∫𝜏

log 𝑆𝑢 𝑑𝑢 −

𝑡

∫𝜏

log 𝑆𝑢 𝑑𝑢 =

𝑇

∫𝑡

log 𝑆𝑢 𝑑𝑢

ℚ we have and substituting log 𝑆𝑢 = log 𝑆𝑡 + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑢 − 𝑡) + 𝜎𝑊𝑢−𝑡

(𝑇 − 𝜏) log 𝐺𝑇 − (𝑡 − 𝜏) log 𝐺𝑡 =

𝑇

∫𝑡

[ ) ] ( 1 ℚ log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑢 − 𝑡) + 𝜎𝑊𝑢−𝑡 𝑑𝑢 2

= (𝑇 − 𝑡) log 𝑆𝑡 + 𝑇

+𝜎

∫𝑡 𝑇

Since log 𝐺𝑇 =

∫𝑡

∫𝑡

1 (𝑟 − 𝐷 − 𝜎 2 )(𝑢 − 𝑡) 𝑑𝑢 2

ℚ 𝑊𝑢−𝑡 𝑑𝑢

= (𝑇 − 𝑡) log 𝑆𝑡 + +𝜎

𝑇

( ) 1 1 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡)2 2 2

ℚ 𝑊𝑢−𝑡 𝑑𝑢.

𝐼 𝐼𝑇 and log 𝐺𝑡 = 𝑡 , therefore 𝑇 −𝜏 𝑡−𝜏

𝐼𝑇 = 𝐼𝑡 + (𝑇 − 𝑡) log 𝑆𝑡 +

( ) 1 1 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡)2 + 𝜎 ∫𝑡 2 2

𝑇

ℚ 𝑊𝑢−𝑡 𝑑𝑢

or { ( ) 𝐼𝑇 1 1 1 𝐼𝑡 + (𝑇 − 𝑡) log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡)2 = 𝑇 −𝜏 𝑇 −𝜏 2 2 } 𝑇 ℚ +𝜎 𝑊𝑢−𝑡 𝑑𝑢 . ∫𝑡

520

5.2.2 Continuous Sampling

Since 𝑊𝑇ℚ−𝑡 ∼  (0, 𝑇 − 𝑡) and tration ℱ𝑡

𝑇

∫𝑡

] [ 1 ℚ 𝑊𝑢−𝑡 𝑑𝑢 ∼  0, (𝑇 − 𝑡)3 , conditional on the fil3

) [ ( ] 1 log 𝑆𝑇 || ℱ𝑡 ∼  log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡), 𝜎 2 (𝑇 − 𝑡) 2 and { ( ) } [ 𝐼𝑇 || 1 1 1 ℱ𝑡 ∼  𝐼𝑡 + (𝑇 − 𝑡) log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡)2 , | 𝑇 −𝜏| 𝑇 −𝜏 2 2 ] 3 2 𝜎 (𝑇 − 𝑡) . 3 (𝑇 − 𝜏)2 ) ( 𝑇 In addition, since 𝑊𝑇 −𝑡 , ∫𝑡 𝑊𝑢−𝑡 𝑑𝑢 follows a bivariate normal we can also deduce that ) ( 𝐼𝑇 also follows a bivariate normal distribution. log 𝑆𝑇 , 𝑇 −𝜏 From the identity we have ] [ 𝔼 max{𝑒𝑋 − 𝑒𝑌 , 0}

=𝑒

𝜇𝑥 + 12 𝜎𝑥2

⎞ ⎞ ⎛ ⎛ 𝜇 − 𝜇𝑦 − 𝜎𝑦 (𝜎𝑦 − 𝜌𝑥𝑦 𝜎𝑥 ) ⎟ ⎜ 𝜇𝑥 − 𝜇𝑦 + 𝜎𝑥 (𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 ) ⎟ 𝜇𝑦 + 21 𝜎𝑦2 ⎜ 𝑥 Φ⎜ √ Φ⎜ √ ⎟−𝑒 ⎟ ⎜ ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟ ⎠ ⎠ ⎝ ⎝

for a bivariate normal distribution such that 𝑋 ∼  (𝜇𝑥 , 𝜎𝑥2 ) and 𝑌 ∼  (𝜇𝑦 , 𝜎𝑦2 ) with correlation coefficient 𝜌𝑥𝑦 ∈ (−1, 1). By setting

𝑋 = log 𝑆𝑇 ,

𝑌 =

𝐼𝑇 , 𝑇 −𝜏

) ( 1 𝜇𝑥 = log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡), 2

𝜇𝑦 =

𝜎𝑥2 = 𝜎 2 (𝑇 − 𝑡)

[ ( ) ] 1 1 1 𝐼𝑡 + (𝑇 − 𝑡) log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡)2 𝑇 −𝜏 2 2

𝜎𝑦2

𝜎 2 (𝑇 − 𝑡)3 = 3 (𝑇 − 𝜏)2

√ and

𝜌𝑥𝑦 =

3 2

5.2.2 Continuous Sampling

521

we have 1 2

1 2

1 2 (𝑇 −𝑡)

𝑒𝜇𝑥 + 2 𝜎𝑥 = 𝑒(𝑟−𝐷− 2 𝜎 )(𝑇 −𝑡)+ 2 𝜎 = 𝑒(𝑟−𝐷)(𝑇 −𝑡) 1

1 2

𝑒𝜇𝑦 + 2 𝜎𝑦 = 𝑒 𝑇 −𝜏 =𝑒

𝐼𝑡 𝑇 −𝜏 𝐼𝑡

3 2 [𝐼𝑡 +(𝑇 −𝑡) log 𝑆𝑡 + 12 (𝑟−𝐷− 21 𝜎 2 )(𝑇 −𝑡)2 ]+ 𝜎6 (𝑇 −𝑡) 2 𝑇 −𝑡 𝑇 −𝜏

𝑆𝑡

𝑇 −𝑡 𝑇 −𝜏

= 𝑒 𝑇 −𝜏 𝑆𝑡

𝑒

( ) 𝑟−𝐷 (𝑇 −𝑡)2 𝜎 2 𝑇 +2𝑡−3𝜏 (𝑇 −𝑡)2 − 12 2 𝑇 −𝜏 𝑇 −𝜏 𝑇 −𝜏

(𝑇 −𝜏)

𝑒(𝑟−𝐷)(𝑇 −𝑡)

𝜎 2 (𝑇 − 𝑡)2 𝜎 2 (𝑇 − 𝑡)3 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 = 𝜎 2 (𝑇 − 𝑡) − + 𝑇 −𝜏 3 (𝑇 − 𝜏)2 [ ] 2 (𝑇 − 𝑡) (𝑇 − 𝑡)2 𝜎 = 3−3 + (𝑇 − 𝑡) 3 𝑇 −𝜏 (𝑇 − 𝜏)2 ( ) 𝜎 2 (𝑇 − 𝜏)2 + (𝑡 − 𝜏)(𝑇 − 𝜏) + (𝑡 − 𝜏)2 = (𝑇 − 𝑡) 3 (𝑇 − 𝜏)2 = 𝜎 2 (𝑇 − 𝑡) where ( ) ( ) ( )( ) 𝐷 𝑇 −𝑡 𝜎2 𝑇 − 𝑡 𝑇 + 2𝑡 − 3𝜏 𝑟 𝑇 + 𝑡 − 2𝜏 + + 2 𝑇 −𝜏 2 𝑇 −𝜏 12 𝑇 − 𝜏 𝑇 −𝜏 ( ) 2 + (𝑡 − 𝜏)(𝑇 − 𝜏) + (𝑡 − 𝜏)2 2 (𝑇 − 𝜏) 𝜎 𝜎2 = . 3 (𝑇 − 𝜏)2 𝐷=

In addition, 1 1 1 𝜇𝑥 − 𝜇𝑦 + 𝜎𝑥 (𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 ) = 𝜇𝑥 − 𝜇𝑦 + (𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ) + 𝜎𝑥2 − 𝜎𝑦2 2 2 2 ) ( 1 2 = log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 (𝑇 − 𝑡) 2 [ ( ) ] 1 1 1 𝐼𝑡 + (𝑇 − 𝑡) log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡)2 − 𝑇 −𝜏 2 2 1 2 1 2 𝜎 2 (𝑇 − 𝑡)2 + 𝜎 (𝑇 − 𝑡) + 𝜎 (𝑇 − 𝑡) − 2 2 6 (𝑇 − 𝜏)2 𝐼𝑡 𝑡−𝜏 =− + log 𝑆𝑡 𝑇 − 𝜏 𝑇 −𝜏 [ ( ) ( )( ) 𝜎2 𝑇 − 𝑡 𝑇 + 2𝑡 − 3𝜏 𝑟 − 𝐷 𝑇 + 𝑡 − 2𝜏 + + 2 𝑇 −𝜏 12 𝑇 − 𝜏 𝑇 −𝜏 ] 1 + 𝜎 2 (𝑇 − 𝑡) 2 ) ( 𝐼 1 𝑡−𝜏 =− 𝑡 + log 𝑆𝑡 − 𝐷 − 𝐷 − 𝜎 2 (𝑇 − 𝑡) 𝑇 −𝜏 𝑇 −𝜏 2

522

5.2.2 Continuous Sampling

and 1 1 1 𝜇𝑥 − 𝜇𝑦 − 𝜎𝑦 (𝜎𝑦 − 𝜌𝑥𝑦 𝜎𝑥 ) = 𝜇𝑥 − 𝜇𝑦 − (𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ) + 𝜎𝑥2 − 𝜎𝑦2 2 2 2 ) ( 1 2 = log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 (𝑇 − 𝑡) 2 [ ( ) ] 1 1 1 𝐼𝑡 + (𝑇 − 𝑡) log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡)2 − 𝑇 −𝜏 2 2 2 2 1 1 𝜎 (𝑇 − 𝑡) − 𝜎 2 (𝑇 − 𝑡) + 𝜎 2 (𝑇 − 𝑡) − 2 2 6 (𝑇 − 𝜏)2 𝐼 𝑡−𝜏 =− 𝑡 + log 𝑆𝑡 𝑇 − 𝜏 𝑇 −𝜏 [ ( ) ( )( ) 𝜎2 𝑇 − 𝑡 𝑇 + 2𝑡 − 3𝜏 𝑟 − 𝐷 𝑇 + 𝑡 − 2𝜏 + + 2 𝑇 −𝜏 12 𝑇 − 𝜏 𝑇 −𝜏 ] 1 − 𝜎 2 (𝑇 − 𝑡) 2 ) ( 𝐼 1 𝑡−𝜏 =− 𝑡 + log 𝑆𝑡 − 𝐷 − 𝐷 + 𝜎 2 (𝑇 − 𝑡). 𝑇 −𝜏 𝑇 −𝜏 2 Hence, under the risk-neutral measure ℚ, the geometric average strike call option price at time 𝑡 is [ }| ] { 𝐼𝑇 | (𝑔) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝑆𝑇 − 𝑒 𝑇 −𝜏 , 0 | ℱ𝑡 𝐶𝑎𝑠 | | 𝐼𝑡

𝑇 −𝑡

= 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑 − ) − 𝑒 𝑇 −𝜏 𝑆𝑡𝑇 −𝜏 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑 + ) ) ) ( ( 𝐼𝑡 1 𝑡−𝜏 log 𝑆𝑡 + 𝐷 − 𝐷 ± 𝜎 2 (𝑇 − 𝑡) − 𝑇 −𝜏 2 √ . where 𝑑 ± = 𝑇 − 𝜏 𝜎 𝑇 −𝑡 N.B. The same result can also be obtained if we use the exchange option formula (see Problem 6.2.1.6, page 543). 12. Put–Call Parity for Geometric Average Options (Probabilistic Approach). Let {𝑊𝑡 : 𝑡 ≥ 0} be the ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter and let 𝑟 be the risk-free interest rate from the money-market account. We define 𝐼𝑡 =

𝑡

∫𝜏

𝑆𝑢 𝑑𝑢

5.2.2 Continuous Sampling

523

to be the asset running sum within the time period [𝜏, 𝑡], 𝜏 ≥ 0 and consider the geometric average rate option with zero-strike call payoff 𝐼𝑇

Ψ(𝑆𝑇 , 𝐼𝑇 ) = 𝑒 𝑇 −𝜏 where 𝑇 is the option expiry time. Using the risk-neutral valuation method find the arbitrage-free geometric average option price at time 𝑡, 𝜏 < 𝑡 < 𝑇 . Consider the payoffs of geometric average rate options {

} = max 𝑒 − 𝐾, 0 { } 𝐼𝑇 (𝑔) 𝑇 −𝜏 𝑃𝑎𝑟 (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max 𝐾 − 𝑒 , 0 𝐼𝑇 𝑇 −𝜏

(𝑔) (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝐾, 𝑇 ) 𝐶𝑎𝑟

and geometric average strike options {

} 𝐼𝑇 𝑆𝑇 − 𝑒 𝑇 −𝜏 , 0 { 𝐼 } 𝑇 (𝑔) 𝑃𝑎𝑠 (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝑇 ) = max 𝑒 𝑇 −𝜏 − 𝑆𝑇 , 0 .

(𝑔) 𝐶𝑎𝑠 (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝑇 ) = max

Show that the put–call relation for the geometric average rate and geometric average strike options are 𝐼𝑡

𝑇 −𝑡

(𝑔) (𝑔) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑎𝑟 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒 𝑇 −𝜏 𝑆𝑡𝑇 −𝜏 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) 𝐶𝑎𝑟

and 𝐼𝑡

𝑇 −𝑡

(𝑔) (𝑔) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) − 𝑃𝑎𝑠 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝑒 𝑇 −𝜏 𝑆𝑡𝑇 −𝜏 𝑒−𝐷(𝑇 −𝑡) 𝐶𝑎𝑠

respectively where 𝐷=

( ) ( ) ( )( ) 𝐷 𝑇 −𝑡 𝜎2 𝑇 − 𝑡 𝑇 + 2𝑡 − 3𝜏 𝑟 𝑇 + 𝑡 − 2𝜏 + + . 2 𝑇 −𝜏 2 𝑇 −𝜏 12 𝑇 − 𝜏 𝑇 −𝜏

Solution: From Girsanov’s theorem, under the risk-neutral measure ℚ, 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡ℚ 𝑆𝑡 where 𝑊𝑡ℚ = 𝑊𝑡 +

(𝜇 − 𝑟) 𝜎

𝑡 is a ℚ-standard Wiener process.

524

5.2.2 Continuous Sampling

From Problem 5.2.2.11 (page 514), under the ℚ measure { ( ) } [ 𝐼𝑇 || 1 1 1 ℱ𝑡 ∼  𝐼𝑡 + (𝑇 − 𝑡) log 𝑆𝑡 + 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡)2 , | 𝑇 −𝜏| 𝑇 −𝜏 2 2 ] 3 2 𝜎 (𝑇 − 𝑡) . 3 (𝑇 − 𝜏)2 𝐼𝑇

Hence, the price of the option at time 𝑡, 𝜏 < 𝑡 < 𝑇 with payoff Ψ(𝑆𝑇 , 𝐼𝑇 ) = 𝑒 𝑇 −𝜏 is [ ] 𝑉 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ Ψ(𝑆𝑇 , 𝐼𝑇 )|| ℱ𝑡 [ 𝐼 ] 𝑇 | = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 𝑒 𝑇 −𝜏 || ℱ𝑡 | =𝑒

1 𝑇 −𝜏 𝐼𝑡

{

} 2 3 𝐼𝑡 +(𝑇 −𝑡) log 𝑆𝑡 + 21 (𝑟−𝐷− 12 𝜎 2 )(𝑇 −𝑡)2 + 𝜎6 (𝑇 −𝑡) 2 −𝑟(𝑇 −𝑡) (𝑇 −𝜏) 𝑇 −𝑡

= 𝑒 𝑇 −𝜏 𝑆𝑡𝑇 −𝜏 𝑒−𝐷(𝑇 −𝑡) where

𝐷=

( ) ( ) ( )( ) 𝐷 𝑇 −𝑡 𝜎2 𝑇 − 𝑡 𝑇 + 2𝑡 − 3𝜏 𝑟 𝑇 + 𝑡 − 2𝜏 + + . 2 𝑇 −𝜏 2 𝑇 −𝜏 12 𝑇 − 𝜏 𝑇 −𝜏

At option expiry time 𝑇 , the put–call parity for the geometric average rate option is 𝑇 ⎧ 𝑒 𝑇𝐼−𝜏 −𝐾 ⎪ (𝑔) (𝑔) 𝐶𝑎𝑟 (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝐾, 𝑇 ) − 𝑃𝑎𝑟 (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝐾, 𝑇 ) = ⎨ 𝑇 ⎪ 𝑇𝐼−𝜏 −𝐾 ⎩𝑒

𝐼𝑇

if 𝑒 𝐾 > 𝐾 𝐼𝑇

if 𝑒 𝐾 ≤ 𝐾

𝐼𝑇

= 𝑒 𝑇 −𝜏 − 𝐾. By discounting the payoff back to time 𝑡, under the risk-neutral measure ℚ we have

(𝑔) (𝑔) 𝐶𝑎𝑟 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝑃𝑎𝑟 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝐾, 𝑇 )

] | | =𝑒 𝔼 𝑒 − 𝐾 | ℱ𝑡 | [ 𝐼 ] 𝑇 | = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 𝑒 𝑇 −𝜏 || ℱ𝑡 − 𝐾𝑒−𝑟(𝑇 −𝑡) | −𝑟(𝑇 −𝑡) ℚ

𝐼𝑡

𝑇 −𝑡

[

𝐼𝑇 𝑇 −𝜏

= 𝑒 𝑇 −𝜏 𝑆𝑡𝑇 −𝜏 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) .

5.2.2 Continuous Sampling

525

In contrast, at expiry time 𝑇 the put–call parity for geometric average strike options is 𝑇 ⎧ 𝑆 − 𝑒 𝑇𝐼−𝜏 𝑇 ⎪ (𝑔) (𝑔) (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝑇 ) − 𝑃𝑎𝑠 (𝑆𝑇 , 𝐼𝑇 , 𝑇 ; 𝑇 ) = ⎨ 𝐶𝑎𝑠 𝐼𝑇 ⎪ ⎩ 𝑆𝑇 − 𝑒 𝑇 −𝜏

𝐼𝑇

if 𝑆𝑇 > 𝑒 𝑇 −𝜏 𝐼𝑇

if 𝑆𝑇 ≤ 𝑒 𝑇 −𝜏

𝐼𝑇

= 𝑆𝑇 − 𝑒 𝑇 −𝜏 . By discounting the payoffs [back to time ℚ, and taking ( 𝑡 < 𝑇 under)the risk-neutral measure ] 1 2 2 | note that log 𝑆𝑇 | ℱ𝑡 ∼  log 𝑆𝑡 + 𝑟 − 𝐷 − 2 𝜎 (𝑇 − 𝑡), 𝜎 (𝑇 − 𝑡) , we have [ ] 𝐼𝑇 | (𝑔) (𝑔) (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) − 𝑃𝑎𝑠 (𝑆𝑡 , 𝐼𝑡 , 𝑡; 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 𝑆𝑇 − 𝑒 𝑇 −𝜏 || ℱ𝑡 𝐶𝑎𝑠 | [ 𝐼 ] [ ] 𝑇 | = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 𝑆𝑇 || ℱ𝑡 − 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ 𝑒 𝑇 −𝜏 || ℱ𝑡 | 𝐼𝑡

𝑇 −𝑡

= 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝑒 𝑇 −𝜏 𝑆𝑡𝑇 −𝜏 𝑒−𝐷(𝑇 −𝑡) .

13. Symmetry of Average Strike (Floating) and Average Rate (Fixed) Strike in Arithmetic Average Options. Let {𝑊𝑡 : 𝑡 ≥ 0} be the ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow the SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter and let 𝑟 be the risk-free interest rate from the money-market account. Under the risk-neutral measure ℚ, we consider the arithmetic average option prices at time 𝑡 < 𝑇 [ ] (𝑎) (𝑆𝑡 , 𝑡; 𝑟, 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐴(𝑡, 𝑇 ) − 𝐾, 0}| ℱ𝑡 𝐶𝑎𝑟 [ ] (𝑎) (𝑆𝑡 , 𝑡; 𝑟, 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐾 − 𝐴(𝑡, 𝑇 ), 0}| ℱ𝑡 𝑃𝑎𝑟 [ ] (𝑎) (𝑆𝑡 , 𝑡; 𝑟, 𝜆, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝜆𝑆𝑇 − 𝐴(𝑡, 𝑇 ), 0}|| ℱ𝑡 𝐶𝑎𝑠 [ ] (𝑎) (𝑆𝑡 , 𝑡; 𝑟, 𝜆, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐴(𝑡, 𝑇 ) − 𝜆𝑆𝑇 , 0}|| ℱ𝑡 𝑃𝑎𝑠 where 𝜆 > 0 is a scaling factor, 𝑇 is the option expiry and arithmetic average 𝐴(𝑡, 𝑇 ) is defined as 𝐴(𝑡, 𝑇 ) =

1 𝑇 − 𝑡 ∫𝑡

𝑇

𝑆𝑢 𝑑𝑢.

526

5.2.2 Continuous Sampling

Show that the following symmetry results (𝑎) (𝑎) (𝑆𝑡 , 𝑡; 𝑟, 𝜆, 𝑇 ) = 𝑃𝑎𝑟 (𝑆𝑡 , 𝑡; 𝐷, 𝜆𝑆𝑡 , 𝑇 ) 𝐶𝑎𝑠 (𝑎) (𝑎) 𝐶𝑎𝑟 (𝑆𝑡 , 𝑡; 𝑟, 𝐾, 𝑇 ) = 𝑃𝑎𝑠 (𝑆𝑡 , 𝑡; 𝐷, 𝐾∕𝑆𝑡 , 𝑇 )

hold. Do the above results hold if the arithmetic average takes the form

𝐴(𝜏, 𝑇 ) =

𝑇

1 𝑆𝑢 𝑑𝑢 𝑇 − 𝜏 ∫𝜏

where 𝜏 < 𝑡 < 𝑇 ? Solution: From Girsanov’s theorem, under the risk-neutral measure ℚ, 𝑆𝑡 takes the form 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡ℚ 𝑆𝑡 𝜇−𝑟 𝑡 is a ℚ-standard Wiener process. 𝜎 For a payoff Ψ(𝑆𝑇 ), under the change of num´eraire

where 𝑊𝑡ℚ = 𝑊𝑡 +

[ (1) 𝑁𝑡(1) 𝔼ℚ

[ | ] | ] Ψ(𝑆𝑇 ) || Ψ(𝑆𝑇 ) || (2) ℚ(2) | ℱ𝑡 = 𝑁𝑡 𝔼 | ℱ𝑡 𝑁𝑇(1) || 𝑁𝑇(2) ||

where 𝑁 (1) and 𝑁 (2) are num´eraires (positive non-dividend-paying assets) and ℚ(1) and ℚ(2) are the measures under which the asset prices discounted by 𝑁 (1) and 𝑁 (2) are ℚ(1) and ℚ(2) -martingales, respectively. For the first result we have [ ] (𝑎) (𝑆𝑡 , 𝑡; 𝑟, 𝜆, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝜆𝑆𝑇 − 𝐴(𝑡, 𝑇 ), 0}|| ℱ𝑡 . 𝐶𝑎𝑠 Under the risk-neutral measure ℚ, we set 𝑁𝑡(1) = 𝑒𝑟𝑡

and

𝑁𝑇(1) = 𝑒𝑟𝑇

and under the ℚ𝑆 measure where 𝑆𝑡 𝑒𝐷𝑡 is a non-dividend-paying asset we have 𝑁𝑡(2) = 𝑆𝑡 𝑒𝐷𝑡

and

𝑁𝑇(2) = 𝑆𝑇 𝑒𝐷𝑇 .

5.2.2 Continuous Sampling

527

Hence, using a change of num´eraire [ ] (𝑎) (𝑆𝑡 , 𝑡; 𝑟, 𝜆, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝜆𝑆𝑇 − 𝐴(𝑡, 𝑇 ), 0}|| ℱ𝑡 𝐶𝑎𝑠 ] max{𝜆𝑆𝑇 − 𝐴(𝑡, 𝑇 ), 0} || ℱ | 𝑡 𝑒𝑟𝑇 | [ ] | max{𝜆𝑆 − 𝐴(𝑡, 𝑇 ), 0} | 𝑇 = 𝑆𝑡 𝑒𝐷𝑡 𝔼ℚ𝑆 | ℱ𝑡 | 𝑆𝑇 𝑒𝐷𝑇 | [ }| ] { 𝑆 𝐴(𝑡, 𝑇 ) | 𝑡 = 𝑒−𝐷(𝑇 −𝑡) 𝔼ℚ𝑆 max 𝜆𝑆𝑡 − , 0 | ℱ𝑡 | 𝑆𝑇 | [ ] { } | = 𝑒−𝐷(𝑇 −𝑡) 𝔼ℚ𝑆 max 𝜆𝑆𝑡 − 𝐴∗ (𝑡, 𝑇 ), 0 | ℱ𝑡 |

= 𝑒𝑟𝑡 𝔼ℚ

[

𝑆𝑡 𝐴(𝑡, 𝑇 ) . 𝑆𝑇 Under the ℚ𝑆 measure, the discounted money-market account

where 𝐴∗ (𝑡, 𝑇 ) =

)−1 𝑟𝑡 ( 𝑋𝑡 = 𝑆𝑡 𝑒𝐷𝑡 𝑒 = 𝑆𝑡−1 𝑒(𝑟−𝐷)𝑡 is a ℚ𝑆 -martingale. From It¯o’s lemma, ) ( 𝑑𝑋𝑡 = 𝑑 𝑆𝑡−1 𝑒(𝑟−𝐷)𝑡 = (𝑟 − 𝐷)𝑒(𝑟−𝐷)𝑡 𝑆𝑡−1 𝑑𝑡 + 𝑆𝑡−1 𝑒(𝑟−𝐷)𝑡 𝑑(𝑆𝑡−1 ) ( = (𝑟 − 𝐷)𝑋𝑡 𝑑𝑡 + 𝑋𝑡

𝑑𝑆 − 𝑡+ 𝑆𝑡

(

𝑑𝑆𝑡 𝑆𝑡

)

)2

+…

( ) = (𝑟 − 𝐷)𝑋𝑡 𝑑𝑡 + 𝑋𝑡 −(𝑟 − 𝐷)𝑑𝑡 − 𝜎 𝑑𝑊𝑡ℚ + 𝜎 2 𝑑𝑡 = 𝜎 2 𝑋𝑡 𝑑𝑡 − 𝜎𝑋𝑡 𝑑𝑊𝑡ℚ = −𝜎𝑋𝑡 𝑑𝑊𝑡𝑆 where 𝑊𝑡𝑆 = 𝑊𝑡ℚ − 𝜎𝑡 is a ℚ𝑆 -standard Wiener process. Thus, under ℚ𝑆 ( ) 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎 𝑑𝑊𝑡𝑆 + 𝜎𝑑𝑡 𝑆𝑡 = (𝑟 − 𝐷 + 𝜎 2 )𝑑𝑡 + 𝜎 𝑑𝑊𝑡𝑆 .

528

5.2.2 Continuous Sampling

To find the solution of the SDE, we first expand 𝑑 log 𝑆𝑡 using Taylor’s theorem and by applying It¯o’s lemma 𝑑 log 𝑆𝑡 =

𝑑𝑆𝑡 1 − 𝑆𝑡 2

(

𝑑𝑆𝑡 𝑆𝑡

)2 +…

1 = (𝑟 − 𝐷 + 𝜎 2 )𝑑𝑡 + 𝜎 𝑑𝑊𝑡𝑆 − 𝜎 2 𝑑𝑡 2 ) ( 1 2 = 𝑟 − 𝐷 + 𝜎 𝑑𝑡 + 𝜎 𝑑𝑊𝑡𝑆 . 2 Taking integrals, ) 𝑇 ( 𝑇 1 𝑟 − 𝐷 + 𝜎 2 𝑑𝑡 + 𝑑 log 𝑆𝑡 = 𝜎 𝑑𝑊𝑡𝑆 ∫𝑢 ∫𝑢 ∫𝑢 2 ( ) ( ) 𝑆𝑇 1 log = 𝑟 − 𝐷 + 𝜎 2 (𝑇 − 𝑢) + 𝜎𝑊𝑇𝑆−𝑢 𝑆𝑢 2 𝑇

or 1 2 )(𝑇 −𝑢)+𝜎𝑊𝑇𝑆−𝑢

𝑆𝑇 = 𝑆𝑢 𝑒(𝑟−𝐷+ 2 𝜎

.

By substituting 𝑆𝑇 into 𝐴∗ (𝑡, 𝑇 ), 𝐴∗ (𝑡, 𝑇 ) =

𝑆𝑡 𝑇 − 𝑡 ∫𝑡

=

𝑆𝑡 𝑇 − 𝑡 ∫𝑡

=

1 𝑇 − 𝑡 ∫𝑡

=

1 𝑇 − 𝑡 ∫𝑡

𝑇

𝑇

𝑇

𝑇

𝑆𝑢 𝑑𝑢 𝑆𝑇 1 2 )(𝑇 −𝑢)−𝜎𝑊𝑇𝑆−𝑢

𝑒−(𝑟−𝐷+ 2 𝜎

𝑑𝑢

1 2 )(𝑇 −𝑢)−𝜎𝑊𝑇𝑆−𝑢

𝑑𝑢

1 2 ̃𝑆 )(𝑇 −𝑢)+𝜎 𝑊 𝑇 −𝑢

𝑑𝑢

𝑆𝑡 𝑒(𝐷−𝑟− 2 𝜎 𝑆𝑡 𝑒(𝐷−𝑟− 2 𝜎

̃ 𝑆 = −𝑊 𝑆 is also a ℚ𝑆 -standard Wiener since the reflected standard Wiener process 𝑊 𝑡 𝑡 process. Given that 𝐴∗ (𝑡, 𝑇 ) is an arithmetic average of the price process of lognormal variates with drift 𝐷 − 𝑟, therefore [ ] (𝑎) (𝑆𝑡 , 𝑡; 𝑟, 𝜆, 𝑇 ) = 𝑒−𝐷(𝑇 −𝑡) 𝔼ℚ𝑆 max{𝜆𝑆𝑡 − 𝐴∗ (𝑡, 𝑇 ), 0}|| ℱ𝑡 𝐶𝑎𝑠 (𝑎) (𝑆𝑡 , 𝑡; 𝐷, 𝜆𝑆𝑡 , 𝑇 ). = 𝑃𝑎𝑟

For the second result, we note that using the same steps we can also show (𝑎) (𝑎) (𝑆𝑡 , 𝑡; 𝑟, 𝜆, 𝑇 ) = 𝐶𝑎𝑟 (𝑆𝑡 , 𝑡; 𝐷, 𝜆𝑆𝑡 , 𝑇 ). 𝑃𝑎𝑠

5.2.2 Continuous Sampling

529

By substituting 𝐷 with 𝑟 and 𝜆𝑆𝑡 with 𝐾 so that 𝜆 = 𝐾∕𝑆𝑡 , we can deduce (𝑎) (𝑎) 𝐶𝑎𝑟 (𝑆𝑡 , 𝑡; 𝑟, 𝐾, 𝑇 ) = 𝑃𝑎𝑠 (𝑆𝑡 , 𝑡; 𝐷, 𝐾∕𝑆𝑡 , 𝑇 ).

Note that the symmetry results do not hold if the arithmetic average takes the form 𝐴(𝜏, 𝑇 ) =

𝑇

1 𝑆𝑢 𝑑𝑢 𝑇 − 𝜏 ∫𝜏

where 𝜏 < 𝑡 < 𝑇 . For example, for an average strike (floating strike) call option payoff { max{𝜆𝑆𝑇 − 𝐴(𝜏, 𝑇 ), 0} = max

𝜆𝑆𝑇 −

and when we divide it by 𝑆𝑇 , the known term value.

𝑡

1 1 𝑆 𝑑𝑢 − 𝑇 − 𝜏 ∫𝜏 𝑢 𝑇 − 𝜏 ∫𝑡

𝑇

} 𝑆𝑢 𝑑𝑢, 0

𝑡 𝑆 1 𝑢 𝑑𝑢 will no longer be a constant 𝑇 − 𝜏 ∫ 𝜏 𝑆𝑇

6 Exotic Options In the last two chapters we came across barrier and Asian options, which are the most popular exotic options traded in the OTC market. In this chapter we continue to discuss other types of exotic options, which are highly customised options that can be used for hedging or speculative purposes. Although these options are less traded and some might just be a two-party transaction, their payoffs have unique features which make them a very interesting subject in their own right. There are many reasons why exotic options are developed in the first place. Most notably, exotic options enable investors or speculators to focus their view on future market behaviour (such as the exchange rate risk or corporate/sovereign credit risk rating). In this chapter, different types of exotic options are presented and their valuations are discussed in detail. Throughout this chapter, unless otherwise stated, we assume a continuous dividend yield and the volatility of the asset price and the risk-free interest rate are assumed to be constants within the life of the option contract. As the majority of the payoffs discussed in this chapter are based on European-style payoffs, the formulas of many exotic options are analytical.

6.1 INTRODUCTION Within the family of exotic options, the options can be either path-independent or pathdependent-type options.

Path-Independent Options For exotic options which are path-independent, their payoffs only depend on the terminal value of the underlying asset price, irrespective of the route taken. Examples of these are the simple digital and asset-or-nothing options, as the payoffs depend exclusively on the asset price at option expiry time. More complicated path-independent options may involve two or more assets, such as exchange options (exchange one asset for another), spread options (difference between asset prices) and rainbow options (best or worst of asset prices).

Path-Dependent Options For exotic options which are path-dependent, their payoffs may depend on the whole path of the underlying asset price rather than just the terminal value. A great majority of exotic options are path-dependent, such as forward start options (where the option starts at a specified future date), compound options (where the underlying is another option), chooser options (which allow the holder of the option to choose at a specified time whether the contract becomes a European call or a European put) and lookback options (where the maximum or minimum of the underlying asset is attained over a certain period of time).

532

6.2.1 Path-Independent Options

6.2 PROBLEMS AND SOLUTIONS 6.2.1

Path-Independent Options { } 1. Capped Option. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter and let 𝑟 be the risk-free interest-rate parameter from the money-market account. We consider a capped call option with terminal payoff Ψ(𝑆𝑇 ) = min{max{𝑆𝑇 − 𝐾, 0}, 𝑀} at time 𝑇 ≥ 𝑡 with strike price 𝐾 > 0 and 𝑀 the cap value. Show that the capped call option price at time 𝑡 is 𝐶𝑐𝑎𝑝 (𝑆𝑡 , 𝑡; 𝐾, 𝑀, 𝑇 ) = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾 + 𝑀, 𝑇 ) where 𝐶𝑏𝑠 (𝑋, 𝑡; 𝑌 , 𝑇 ) is the vanilla (or European) call option price defined as ( ) ( ) 𝐶𝑏𝑠 (𝑋, 𝑡; 𝑌 , 𝑇 ) = 𝑋𝑒−𝐷(𝑇 −𝑡) Φ 𝑑+ − 𝑌 𝑒−𝑟(𝑇 −𝑡) Φ 𝑑− 𝑥 log(𝑋∕𝑌 ) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝑡) 1 − 1 𝑢2 and Φ(𝑥) = √ 𝑒 2 𝑑𝑢 is the √ ∫−∞ 2𝜋 𝜎 𝑇 −𝑡 cdf of a standard normal.

such that 𝑑± =

Solution: From the definition of a capped call option payoff we can write ⎧0 ⎪ Ψ(𝑆𝑇 ) = ⎨ 𝑆𝑇 − 𝐾 ⎪𝑀 ⎩

𝑆𝑇 < 𝐾 𝐾 ≤ 𝑆𝑇 < 𝐾 + 𝑀 𝑆𝑇 ≥ 𝐾 + 𝑀

so that the payoff diagram for the capped call option 𝐶𝑐𝑎𝑝 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑀, 𝑇 ) can be constructed with the portfolios in Figure 6.1. Thus, at expiry time 𝑇 we can write the capped call option price as ( ( ) ) 𝐶𝑐𝑎𝑝 𝑆𝑇 , 𝑇 ; 𝐾, 𝑀, 𝑇 = 𝐶𝑏𝑠 𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 − 𝐶𝑏𝑠 (𝑆𝑇 , 𝑇 ; 𝐾 + 𝑀, 𝑇 ) and by discounting the entire payoff under the risk-neutral measure ℚ, we can write the solution for the capped call option at time 𝑡 as ( ( ) ) 𝐶𝑐𝑎𝑝 𝑆𝑡 , 𝑡; 𝐾, 𝑀, 𝑇 = 𝐶𝑏𝑠 𝑆𝑡 , 𝑡; 𝐾, 𝑇 − 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾 + 𝑀, 𝑇 ).

6.2.1 Path-Independent Options

533

Figure 6.1

Capped call payoff diagram.

{ } 2. Corridor Option. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter and let 𝑟 be the risk-free interest-rate parameter from the money-market account. We consider a corridor call option with terminal payoff { Ψ(𝑆𝑇 ) =

1 if 𝐾1 < 𝑆𝑇 < 𝐾2 0 otherwise

which pays $1 at expiry time 𝑇 ≥ 𝑡 if 𝐾1 < 𝑆𝑇 < 𝐾2 and nothing if 𝑆𝑇 ≥ 𝐾2 or 𝑆𝑇 ≤ 𝐾1 , 𝐾1 < 𝐾2 . Show that the corridor call option price at time 𝑡 is 𝐶𝑐𝑜𝑟 (𝑆𝑡 , 𝑡; 𝐾1 , 𝐾2 , 𝑇 ) = 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 ) − 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ) where 𝐶𝑑 (𝑋, 𝑡; 𝑌 , 𝑇 ) is the European digital call option price defined as ( ) 𝐶𝑑 (𝑋, 𝑡; 𝑌 , 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) Φ 𝑑− 𝑥 log (𝑋∕𝑌 ) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡) 1 − 1 𝑢2 such that 𝑑− = and Φ(𝑥) = √ 𝑒 2 𝑑𝑢 is the √ ∫−∞ 2𝜋 𝜎 𝑇 −𝑡 cdf of a standard normal.

534

6.2.1 Path-Independent Options

Figure 6.2

Corridor payoff diagram.

Solution: From the corridor option payoff { Ψ(𝑆𝑇 ) =

1 if 𝐾1 < 𝑆𝑇 < 𝐾2 0 otherwise

the payoff diagram for 𝐶𝑐𝑜𝑟 (𝑆𝑇 , 𝑇 ; 𝐾1 , 𝐾2 , 𝑇 ) can be constructed with the portfolios in Figure 6.2. Thus, at expiry time 𝑇 we can write the corridor option price as ( ( ) ) 𝐶𝑐𝑜𝑟 𝑆𝑇 , 𝑇 ; 𝐾1 , 𝐾2 , 𝑇 = 𝐶𝑑 𝑆𝑇 , 𝑇 ; 𝐾1 , 𝑇 − 𝐶𝑑 (𝑆𝑇 , 𝑇 ; 𝐾2 , 𝑇 ) and by discounting the entire payoff under the risk-neutral measure ℚ, we can write the solution for the corridor option at time 𝑡 as ( ( ) ) 𝐶𝑐𝑜𝑟 𝑆𝑡 , 𝑡; 𝐾1 , 𝐾2 , 𝑇 = 𝐶𝑑 𝑆𝑡 , 𝑡; 𝐾1 , 𝑇 − 𝐶𝑑 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇 ). { } 3. Power Option. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the stock price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter and let 𝑟 be the risk-free interest rate from a money-market account.

6.2.1 Path-Independent Options

535

We define 𝑌𝑡 = 𝑆𝑡𝑛 for 𝑛 ≥ 1. Using It¯o’s lemma, find the SDE satisfied by 𝑌𝑡 and show that under the risk-neutral measure ℚ ( log

𝑌𝑇 𝑌𝑡

)

) ] [ ( 1 ∼  𝑛 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡) , 𝑛2 𝜎 2 (𝑇 − 𝑡) 2

for 𝑡 ≤ 𝑇 . By using the European call price formula, deduce that the call option of a power option payoff Ψ(𝑆𝑇𝑛 ) = max{𝑆𝑇𝑛 − 𝐾, 0} for strike price 𝐾 > 0 is 1

𝐶(𝑆𝑡𝑛 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡𝑛 𝑒−𝑛𝐷(𝑇 −𝑡) Φ(𝑑+∗ ) − 𝐾𝑒−𝑛(𝑟+ 2 (𝑛−1)𝜎

2 )(𝑇 −𝑡)

Φ(𝑑−∗ )

1

where

𝑑+∗

log(𝑆𝑡 ∕𝐾 𝑛 ) + (𝑟 − 𝐷 + (𝑛 − 12 )𝜎 2 )(𝑇 − 𝑡) √ = and 𝑑−∗ = 𝑑+∗ − 𝑛𝜎 𝑇 − 𝑡. √ 𝜎 𝑇 −𝑡

Solution: From Girsanov’s theorem, under the risk-neutral measure ℚ we can write 𝑑𝑆𝑡 = (𝑟 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡 (𝜇 − 𝑟) 𝑡 is a ℚ-standard Wiener process. By expanding 𝑑(𝑆𝑡𝑛 ) such that 𝑊𝑡ℚ = 𝑊𝑡 + 𝜎 using Taylor’s theorem and subsequently with the application of It¯o’s lemma, we can write ( )2 ( ) 1 𝑑 𝑆𝑡𝑛 = 𝑛𝑆𝑡𝑛−1 𝑑𝑆𝑡 + 𝑛(𝑛 − 1)𝑆𝑡𝑛−2 𝑑𝑆𝑡 + … 2 ( ) ( ) 𝑑𝑆𝑡 𝑑𝑆𝑡 2 1 = 𝑛𝑆𝑡𝑛 + 𝑛(𝑛 − 1)𝑆𝑡𝑛 𝑆𝑡 2 𝑆𝑡 [ ] 1 = 𝑛𝑆𝑡𝑛 (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ + 𝑛(𝑛 − 1)𝑆𝑡𝑛 𝜎 2 𝑑𝑡 2 [( ) ] 1 𝑛 2 = 𝑛𝑆𝑡 𝑟 − 𝐷 + (𝑛 − 1)𝜎 𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 2 or ) ( 𝑑𝑌𝑡 1 = 𝑛 𝑟 − 𝐷 + (𝑛 − 1)𝜎 2 𝑑𝑡 + 𝑛𝜎𝑑𝑊𝑡ℚ . 𝑌𝑡 2

536

6.2.1 Path-Independent Options

) ( Using It̄o’s lemma on 𝑑 log 𝑌𝑡 we can write ( ) ) 𝑑𝑌𝑡 1 𝑑𝑌𝑡 2 ( 𝑑 log𝑌𝑡 = − +… 𝑌𝑡 2 𝑌𝑡 ) ( 1 1 = 𝑛 𝑟 − 𝐷 + (𝑛 − 1)𝜎 2 𝑑𝑡 + 𝑛𝜎𝑑𝑊𝑡ℚ − 𝑛2 𝜎 2 𝑑𝑡 2 2 ) ( 1 1 2 2 2 = 𝑛(𝑟 − 𝐷) + 𝑛(𝑛 − 1)𝜎 − 𝑛 𝜎 𝑑𝑡 + 𝑛𝜎𝑑𝑊𝑡ℚ 2 2 ) ( 1 2 = 𝑛(𝑟 − 𝐷) − 𝑛𝜎 𝑑𝑡 + 𝑛𝜎𝑑𝑊𝑡ℚ 2 ) ( 1 2 = 𝑛 𝑟 − 𝐷 − 𝜎 𝑑𝑡 + 𝑛𝜎𝑑𝑊𝑡ℚ . 2 Integrating over time we can write 𝑇

∫𝑡

) 𝑇 ( 𝑇 ) ( 1 𝑑 log𝑌𝑢 𝑑𝑢 = 𝑛 𝑟 − 𝐷 − 𝜎 2 𝑑𝑢 + 𝑛𝜎 𝑑𝑊𝑢ℚ ∫𝑡 ∫𝑡 2 ( ) ) ( ) ( 𝑌𝑇 1 log = 𝑛 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡) + 𝑛𝜎 𝑊𝑇ℚ − 𝑊𝑡ℚ 𝑌𝑡 2

where 𝑊𝑇ℚ − 𝑊𝑡ℚ = 𝑊𝑇ℚ−𝑡 ∼  (0, 𝑇 − 𝑡). The expectation and variance are given as 𝔼

ℚ

[

(

log

𝑌𝑇 𝑌𝑡

)]

) 1 = 𝑛 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡) 2 (

and

Var

ℚ

[

(

log

𝑌𝑇 𝑌𝑡

)]

= 𝑛2 𝜎 2 (𝑇 − 𝑡)

or ( log

𝑌𝑇 𝑌𝑡

)

) ] [ ( 1 ∼  𝑛 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡) , 𝑛2 𝜎 2 (𝑇 − 𝑡) . 2

To find the European call price for a power option with payoff Ψ(𝑆𝑇𝑛 ) = max{𝑆𝑇𝑛 − 𝐾, 0} by comparing the SDEs of 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡ℚ and ) ( 1 𝑑𝑌𝑡 = 𝑛 𝑟 − 𝐷 + (𝑛 − 1)𝜎 2 𝑌𝑡 𝑑𝑡 + 𝑛𝜎𝑌𝑡 𝑑𝑊𝑡ℚ 2 we can set ) 𝑑𝑌𝑡 ( ∗ = 𝑟 − 𝐷∗ 𝑑𝑡 + 𝜎 ∗ 𝑑𝑊𝑡ℚ 𝑌𝑡 ) ( where 𝑟∗ = 𝑛 𝑟 + 12 (𝑛 − 1)𝜎 2 , 𝐷∗ = 𝑛𝐷 and 𝜎 ∗ = 𝑛𝜎. Hence, we can deduce that the call option price at time 𝑡 ≤ 𝑇 is ( ) ( ) ∗ ∗ 𝐶(𝑌𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑌𝑡 𝑒−𝐷 (𝑇 −𝑡) Φ 𝑑+∗ − 𝐾𝑒−𝑟 (𝑇 −𝑡) Φ 𝑑−∗

6.2.1 Path-Independent Options

537

log(𝑌𝑡 ∕𝐾) + (𝑟∗ − 𝐷∗ ± 12 (𝜎 ∗ )2 )(𝑇 − 𝑡) where = . √ 𝜎∗ 𝑇 − 𝑡 By substituting back 𝑌𝑡 = 𝑆𝑡𝑛 we have 𝑑±∗

𝐶(𝑆𝑡𝑛 , 𝑡; 𝐾, 𝑇 )

=

𝑆𝑡𝑛 𝑒−𝑛𝐷(𝑇 −𝑡) Φ

( ∗) −𝑛 𝑑+ − 𝐾𝑒

) ( 𝑟+ 21 (𝑛−1)𝜎 2 (𝑇 −𝑡)

( ) Φ 𝑑−∗

where 𝑑+∗

log(𝑆𝑡𝑛 ∕𝐾) + 𝑛(𝑟 − 𝐷 + 12 (𝑛 − 1)𝜎 2 + 12 𝑛𝜎 2 )(𝑇 − 𝑡) = √ 𝑛𝜎 𝑇 − 𝑡 1

log(𝑆𝑡 ∕𝐾 𝑛 ) + (𝑟 − 𝐷 + 12 𝑛𝜎 2 − 12 𝜎 2 + 12 𝑛𝜎 2 )(𝑇 − 𝑡) = √ 𝜎 𝑇 −𝑡 1

log(𝑆𝑡 ∕𝐾 𝑛 ) + (𝑟 − 𝐷 + (𝑛 − 12 )𝜎 2 )(𝑇 − 𝑡) = √ 𝜎 𝑇 −𝑡 and √ 𝑑−∗ = 𝑑+∗ − 𝑛𝜎 𝑇 − 𝑡.

4. Let {𝑊𝑡𝑥 : 𝑡 ≥ 0} and {𝑊𝑡𝑦 : 𝑡 ≥ 0} be ℙ-standard Wiener processes on the probability space (Ω, ℱ, ℙ) and let the asset prices 𝑋𝑡 and 𝑌𝑡 satisfy the following diffusion processes 𝑑𝑋𝑡 = (𝜇𝑥 − 𝐷𝑥 )𝑋𝑡 𝑑𝑡 + 𝜎𝑥 𝑋𝑡 𝑑𝑊𝑡𝑥 𝑑 𝑌𝑡 = (𝜇𝑦 − 𝐷𝑦 )𝑌𝑡 𝑑𝑡 + 𝜎𝑦 𝑌𝑡 𝑑𝑊𝑡𝑦

𝑑𝑊𝑡𝑥 ⋅ 𝑑𝑊𝑡𝑦 = 𝜌𝑥𝑦 𝑑𝑡

where 𝜇𝑥 and 𝜇𝑦 are the drifts, 𝐷𝑥 and 𝐷𝑦 are the continuous dividend yields, 𝜎𝑥 and 𝜎𝑦 are the volatilities, 𝜌𝑥𝑦 is the correlation coefficient such that 𝜌𝑥𝑦 ∈ (−1, 1) and let 𝑟 be the risk-free interest rate from a money-market account. Using a hedging portfolio, show that a European-style option price 𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡) which depends on both 𝑋𝑡 and 𝑌𝑡 satisfies the following two-dimensional Black–Scholes equation 1 2 2 𝜕2𝑉 𝜕2𝑉 𝜕2 𝑉 𝜕𝑉 1 𝜕𝑉 + 𝑌 + 𝜌 𝜎 𝜎 + (𝑟 − 𝐷𝑥 )𝑋𝑡 + 𝜎𝑥2 𝑋𝑡2 𝜎 𝑥𝑦 𝑥 𝑦 𝑦 𝑡 2 2 𝜕𝑡 2 2 𝜕𝑋 𝜕𝑌 𝜕𝑋 𝜕𝑋𝑡 𝜕𝑌𝑡 𝑡 𝑡 𝑡 +(𝑟 − 𝐷𝑦 )𝑌𝑡

𝜕𝑉 − 𝑟𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡) = 0. 𝜕𝑌𝑡

538

6.2.1 Path-Independent Options

Solution: At time 𝑡 we let the portfolio Π𝑡 be Π𝑡 = 𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡) − Δ𝑥 𝑋𝑡 − Δ𝑦 𝑌𝑡 where the investor is long one unit of option 𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡) and short Δ𝑥 and Δ𝑦 units of 𝑋𝑡 and 𝑌𝑡 , respectively. Let the change of portfolio be 𝑑Π𝑡 = 𝑑𝑉 − Δ𝑥 (𝑑𝑋𝑡 + 𝐷𝑥 𝑋𝑡 𝑑𝑡) − Δ𝑦 (𝑑𝑌𝑡 + 𝐷𝑦 𝑌𝑡 𝑑𝑡) where the investor receives 𝐷𝑥 𝑋𝑡 𝑑𝑡 and 𝐷𝑦 𝑌𝑡 𝑑𝑡 for holding assets 𝑋𝑡 and 𝑌𝑡 , respectively. From Taylor’s theorem 𝑑𝑉 =

𝜕𝑉 1 𝜕2𝑉 1 𝜕2 𝑉 𝜕𝑉 𝜕𝑉 2 𝑑𝑋𝑡 + 𝑑𝑌𝑡 + (𝑋 ) + (𝑌 )2 𝑑𝑡 + 𝑡 𝜕𝑡 𝜕𝑋𝑡 𝜕𝑌𝑡 2 𝜕𝑋𝑡2 2 𝜕𝑌𝑡2 𝑡 +

𝜕2𝑉 (𝑑𝑋𝑡 𝑑𝑌𝑡 ) + … 𝜕𝑋𝑡 𝜕𝑌𝑡

and by substituting 𝑑𝑋𝑡 = (𝜇𝑥 − 𝐷𝑥 )𝑋𝑡 𝑑𝑡 + 𝜎𝑥 𝑋𝑡 𝑑𝑊𝑡𝑥 , 𝜎𝑦 𝑌𝑡 𝑑𝑊𝑡𝑦 , 𝑑𝑊𝑡𝑥 ⋅ 𝑑𝑊𝑡𝑦 = 𝜌𝑥𝑦 𝑑𝑡 and applying It¯o’s lemma

𝑑𝑌𝑡 = (𝜇𝑦 − 𝐷𝑦 )𝑌𝑡 𝑑𝑡 +

( 𝑑𝑉 =

𝜕𝑉 1 𝜕2 𝑉 𝜕2𝑉 𝜕2𝑉 𝜕𝑉 1 + 𝜎𝑦2 𝑌𝑡2 2 + 𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + (𝜇𝑥 − 𝐷𝑥 )𝑋𝑡 + 𝜎𝑥2 𝑋𝑡2 2 𝜕𝑡 2 2 𝜕𝑋𝑡 𝜕𝑌𝑡 𝜕𝑋𝑡 𝜕𝑋𝑡 𝜕𝑌𝑡 ) 𝜕𝑉 𝜕𝑉 𝜕𝑉 +(𝜇𝑦 − 𝐷𝑦 )𝑌𝑡 𝑑𝑊𝑡𝑥 + 𝜎𝑦 𝑌𝑡 𝑑𝑊𝑡𝑦 . 𝑑𝑡 + 𝜎𝑥 𝑋𝑡 𝜕𝑌𝑡 𝜕𝑋𝑡 𝜕𝑌𝑡

Substituting 𝑑𝑉 back into 𝑑Π𝑡 we eventually have ( 𝑑Π𝑡 =

𝜕𝑉 1 𝜕2𝑉 𝜕2𝑉 𝜕2𝑉 1 + 𝜎𝑦2 𝑌𝑡2 2 + 𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑥2 𝑋𝑡2 2 𝜕𝑡 2 2 𝜕𝑋𝑡 𝜕𝑌𝑡 𝜕𝑋𝑡 𝜕𝑌𝑡

) 𝜕𝑉 𝜕𝑉 +(𝜇𝑥 − 𝐷𝑥 )𝑋𝑡 − 𝜇𝑥 Δ𝑥 𝑋𝑡 + (𝜇𝑦 − 𝐷𝑦 )𝑌𝑡 − 𝜇𝑦 Δ𝑦 𝑌𝑡 𝑑𝑡 𝜕𝑋𝑡 𝜕𝑌𝑡 ( ) ( ) 𝜕𝑉 𝜕𝑉 − Δ𝑥 𝑑𝑊𝑡𝑥 + 𝜎𝑦 𝑌𝑡 − Δ𝑦 𝑑𝑊𝑡𝑦 . +𝜎𝑥 𝑋𝑡 𝜕𝑋𝑡 𝜕𝑌𝑡 To eliminate the random components we set

Δ𝑥 =

𝜕𝑉 𝜕𝑋𝑡

and

Δ𝑦 =

𝜕𝑉 𝜕𝑌𝑡

6.2.1 Path-Independent Options

539

which leads to ( 𝑑Π𝑡 =

𝜕𝑉 1 2 2 𝜕2𝑉 𝜕2 𝑉 𝜕2𝑉 𝜕𝑉 1 + 𝑌 + 𝜌 𝜎 𝜎 − 𝐷𝑥 𝑋𝑡 + 𝜎𝑥2 𝑋𝑡2 𝜎 𝑥𝑦 𝑥 𝑦 𝑦 𝑡 2 2 𝜕𝑡 2 2 𝜕𝑋 𝜕𝑌 𝜕𝑋 𝜕𝑋𝑡 𝜕𝑌𝑡 𝑡 𝑡 𝑡 ) 𝜕𝑉 −𝐷𝑦 𝑌𝑡 𝑑𝑡. 𝜕𝑌𝑡

Under the no-arbitrage condition the return on the amount Π𝑡 invested in a risk-free interest rate 𝑟 would see a growth of 𝑑Π𝑡 = 𝑟Π𝑡 𝑑𝑡 and therefore we have ( 𝜕𝑉 1 2 2 𝜕2𝑉 𝜕2𝑉 𝜕2𝑉 𝜕𝑉 1 + 𝑌 + 𝜌 𝜎 𝜎 − 𝐷𝑥 𝑋𝑡 + 𝜎𝑥2 𝑋𝑡2 𝜎 𝑟Π𝑡 𝑑𝑡 = 𝑥𝑦 𝑥 𝑦 𝑦 𝑡 2 2 𝜕𝑡 2 2 𝜕𝑋𝑡 𝜕𝑌𝑡 𝜕𝑋𝑡 𝜕𝑋𝑡 𝜕𝑌𝑡 ) 𝜕𝑉 −𝐷𝑦 𝑌𝑡 𝑑𝑡 𝜕𝑌𝑡 or ) ( 𝑟 𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡) − Δ𝑥 𝑋𝑡 − Δ𝑦 𝑌𝑡 𝑑𝑡 =

(

𝜕𝑉 1 2 2 𝜕2 𝑉 𝜕2𝑉 1 + + 𝜎𝑥2 𝑋𝑡2 𝜎 𝑌 𝜕𝑡 2 𝜕𝑋𝑡2 2 𝑦 𝑡 𝜕𝑌𝑡2

𝜕2𝑉 𝜕𝑉 − 𝐷𝑥 𝑋𝑡 𝜕𝑋𝑡 𝜕𝑌𝑡 𝜕𝑋𝑡 ) 𝜕𝑉 −𝐷𝑦 𝑌𝑡 𝑑𝑡 𝜕𝑌𝑡 ( ( ) 1 2 2 𝜕2 𝑉 𝜕𝑉 𝜕𝑉 𝜕2𝑉 1 𝜕𝑉 − 𝑌𝑡 + 𝑟 𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡) − 𝑋𝑡 + 𝜎𝑥2 𝑋𝑡2 𝜎 𝑌 𝑑𝑡 = 𝜕𝑋𝑡 𝜕𝑌𝑡 𝜕𝑡 2 𝜕𝑋𝑡2 2 𝑦 𝑡 𝜕𝑌𝑡2 +𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦

𝜕2𝑉 𝜕𝑉 − 𝐷𝑥 𝑋𝑡 𝜕𝑋𝑡 𝜕𝑌𝑡 𝜕𝑋𝑡 ) 𝜕𝑉 −𝐷𝑦 𝑌𝑡 𝑑𝑡. 𝜕𝑌𝑡

+𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦

Finally, by removing 𝑑𝑡 and rearranging terms we eventually have 𝜕𝑉 1 2 2 𝜕2𝑉 𝜕2𝑉 𝜕2 𝑉 𝜕𝑉 1 + 𝑌 + 𝜌 𝜎 𝜎 + (𝑟 − 𝐷𝑥 )𝑋𝑡 + 𝜎𝑥2 𝑋𝑡2 𝜎 𝑥𝑦 𝑥 𝑦 𝑦 𝑡 2 2 𝜕𝑡 2 2 𝜕𝑋 𝜕𝑌 𝜕𝑋 𝜕𝑋𝑡 𝜕𝑌𝑡 𝑡 𝑡 𝑡 +(𝑟 − 𝐷𝑦 )𝑌𝑡

𝜕𝑉 − 𝑟𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡) = 0. 𝜕𝑌𝑡

540

6.2.1 Path-Independent Options

5. Exchange Option (PDE Approach). Let {𝑊𝑡𝑥 : 𝑡 ≥ 0} and {𝑊𝑡𝑦 : 𝑡 ≥ 0} be ℙ-standard Wiener processes on the probability space (Ω, ℱ, ℙ) and let the asset prices 𝑋𝑡 and 𝑌𝑡 satisfy the following diffusion processes 𝑑𝑋𝑡 = (𝜇𝑥 − 𝐷𝑥 )𝑋𝑡 𝑑𝑡 + 𝜎𝑥 𝑋𝑡 𝑑𝑊𝑡𝑥 𝑑𝑌𝑡 = (𝜇𝑦 − 𝐷𝑦 )𝑌𝑡 𝑑𝑡 + 𝜎𝑦 𝑌𝑡 𝑑𝑊𝑡𝑦

𝑑𝑊𝑡𝑥 ⋅ 𝑑𝑊𝑡𝑦 = 𝜌𝑥𝑦 𝑑𝑡

where 𝜇𝑥 and 𝜇𝑦 are the drifts, 𝐷𝑥 and 𝐷𝑦 are the continuous dividend yields, 𝜎𝑥 and 𝜎𝑦 are the volatilities, 𝜌𝑥𝑦 is the correlation coefficient such that 𝜌𝑥𝑦 ∈ (−1, 1) and let 𝑟 be the risk-free interest rate from a money-market account. We consider the exchange option with payoff Ψ(𝑋𝑇 , 𝑌𝑇 ) = max{𝑋𝑇 − 𝑌𝑇 , 0} at expiry time 𝑇 ≥ 𝑡 where it gives the option holder the right but not the obligation to exchange asset 𝑌𝑇 for another asset 𝑋𝑇 . Show that if 𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡; 𝑇 ) is the price of the exchange option at time 𝑡, then it satisfies the two-dimensional Black–Scholes equation 1 2 2 𝜕2𝑉 𝜕2𝑉 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 + 𝑌 + 𝜌 𝜎 𝜎 + (𝑟 − 𝐷𝑥 )𝑋𝑡 + 𝜎𝑥2 𝑋𝑡2 𝜎 𝑥𝑦 𝑥 𝑦 𝑦 𝑡 2 2 𝜕𝑡 2 2 𝜕𝑋 𝜕𝑌 𝜕𝑋 𝜕𝑋𝑡 𝜕𝑌𝑡 𝑡 𝑡 𝑡 +(𝑟 − 𝐷𝑦 )𝑌𝑡

𝜕𝑉 − 𝑟𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡; 𝑇 ) = 0 𝜕𝑌𝑡

with boundary condition 𝑉 (𝑋𝑇 , 𝑌𝑇 , 𝑇 ; 𝑇 ) = max{𝑋𝑇 − 𝑌𝑇 , 0}. By setting 𝑍𝑡 =

𝑋𝑡 𝑌𝑡

and

𝑈 (𝑍𝑡 , 𝑡; 𝑇 ) =

𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡; 𝑇 ) 𝑌𝑡

show that 𝑈 (𝑍𝑡 , 𝑡) satisfies 𝜕𝑈 1 2 2 𝜕 2 𝑈 𝜕𝑈 + (𝐷𝑦 − 𝐷𝑥 )𝑍𝑡 − 𝐷𝑦 𝑈 (𝑍𝑡 , 𝑡; 𝑇 ) = 0 + 𝜎 𝑍𝑡 2 𝜕𝑡 2 𝜕𝑍 𝜕𝑍𝑡 𝑡 with boundary condition 𝑈 (𝑍𝑇 , 𝑇 ; 𝑇 ) = max{𝑍𝑇 − 1, 0} where 𝜎 =

√ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 .

6.2.1 Path-Independent Options

541

Hence, deduce that the exchange option price 𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡; 𝑇 ) is 𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡; 𝑇 ) = 𝑋𝑡 𝑒−𝐷𝑥 (𝑇 −𝑡) Φ(𝑑+ ) − 𝑌𝑡 𝑒−𝐷𝑦 (𝑇 −𝑡) Φ(𝑑− ) 𝑥 log(𝑋𝑡 ∕𝑌𝑡 ) + (𝐷𝑦 − 𝐷𝑥 ± 12 𝜎 2 )(𝑇 − 𝑡) 1 − 1 𝑢2 where 𝑑± = and Φ(𝑥) = √ 𝑒 2 𝑑 𝑢. √ ∫−∞ 2𝜋 𝜎 𝑇 −𝑡

Solution: The first part of the solution can easily be derived following Problem 6.2.1.4 (page 537). As for the second part, we let 𝑍𝑡 = 𝑋𝑡 𝑌𝑡−1

and

𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡; 𝑇 ) = 𝑈 (𝑍𝑡 , 𝑡; 𝑇 )𝑌𝑡

so that 𝜕𝑉 𝜕𝑈 = 𝑌𝑡 𝜕𝑡 𝜕𝑡 𝜕𝑈 𝜕𝑈 𝜕𝑍𝑡 𝜕𝑈 −1 𝜕𝑉 = 𝑌𝑡 = 𝑌𝑡 𝑌 = 𝜕𝑋𝑡 𝜕𝑍𝑡 𝜕𝑋𝑡 𝜕𝑍𝑡 𝑡 𝜕𝑍𝑡 𝜕𝑉 𝜕𝑈 𝜕𝑍𝑡 𝜕𝑈 = 𝑈 (𝑍𝑡 , 𝑡) + 𝑌𝑡 = 𝑈 (𝑍𝑡 , 𝑡) − 𝑋𝑡 𝑌𝑡−1 𝜕𝑌𝑡 𝜕𝑍𝑡 𝜕𝑌𝑡 𝜕𝑍𝑡 ( ) 𝜕 𝜕2𝑉 𝜕2𝑈 𝜕𝑈 𝜕 2 𝑈 𝜕𝑍𝑡 = = 𝑌𝑡−1 = 2 2 𝜕𝑋𝑡 𝜕𝑍𝑡 𝜕𝑋𝑡 𝜕𝑍𝑡 𝜕𝑋𝑡 𝜕𝑍𝑡2

2 𝜕𝑈 𝜕𝑍𝑡 𝜕2𝑉 −2 𝜕𝑈 −1 𝜕 𝑈 𝜕𝑍𝑡 = + 𝑋 𝑌 − 𝑋 𝑌 = 𝑋𝑡2 𝑌𝑡−3 𝑡 𝑡 𝑡 𝜕𝑍 𝑡 𝜕𝑍𝑡 𝜕𝑌𝑡 𝜕𝑌𝑡2 𝜕𝑍𝑡2 𝜕𝑌𝑡 𝑡 ( ) 2 𝜕 𝜕2𝑈 𝜕 2 𝑈 𝜕𝑍𝑡 𝜕𝑈 −2 𝜕 𝑈 = = −𝑋 𝑌 . = 𝑡 𝑡 𝜕𝑋𝑡 𝜕𝑌𝑡 𝜕𝑌𝑡 𝜕𝑍𝑡 𝜕𝑍𝑡2 𝜕𝑌𝑡 𝜕𝑍𝑡2

Hence, by substituting

𝜕𝑉 𝜕𝑉 𝜕𝑉 𝜕 2 𝑉 𝜕 2 𝑉 𝜕2𝑈 , , , and into , 2 2 𝜕𝑡 𝜕𝑋𝑡 𝜕𝑌𝑡 𝜕𝑋𝑡 𝜕𝑌𝑡 𝜕𝑋𝑡 𝜕𝑌𝑡

𝜕𝑉 1 2 2 𝜕2𝑉 𝜕2𝑉 𝜕2𝑉 1 + 𝑌 + 𝜌 𝜎 𝜎 + 𝜎𝑥2 𝑋𝑡2 𝜎 𝑥𝑦 𝑥 𝑦 𝑦 𝑡 𝜕𝑡 2 𝜕𝑋𝑡 𝜕𝑌𝑡 𝜕𝑋𝑡2 2 𝜕𝑌𝑡2 +(𝑟 − 𝐷𝑥 )𝑋𝑡

𝜕𝑉 𝜕𝑉 + (𝑟 − 𝐷𝑦 )𝑌𝑡 − 𝑟𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡; 𝑇 ) = 0 𝜕𝑋𝑡 𝜕𝑌𝑡

in terms of 𝑈 (𝑍𝑡 , 𝑡; 𝑇 ) and 𝑍𝑡 we eventually have [

𝜕2𝑈 𝜕𝑈 1 2 + (𝜎𝑥 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 )𝑋𝑡2 𝑌𝑡−2 𝜕𝑡 2 𝜕𝑍𝑡2 ] 𝜕𝑈 +(𝐷𝑦 − 𝐷𝑥 )𝑋𝑡 𝑌𝑡−1 − 𝐷𝑦 𝑈 (𝑍𝑡 , 𝑡; 𝑇 ) = 0 𝜕𝑍𝑡

𝑌𝑡

542

6.2.1 Path-Independent Options

or 𝜕𝑈 𝜕𝑈 1 2 2 𝜕 2 𝑈 + (𝐷𝑦 − 𝐷𝑥 )𝑍𝑡 − 𝐷𝑦 𝑈 (𝑍𝑡 , 𝑡; 𝑇 ) = 0 + 𝜎 𝑍𝑡 2 𝜕𝑡 2 𝜕𝑍𝑡 𝜕𝑍𝑡 where 𝜎 =

√ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 , which is equivalent to a Black–Scholes equation with volatility = 𝜎 continuous dividend yield = 𝐷𝑥 risk-free interest rate = 𝐷𝑦

and boundary condition 𝑈 (𝑍𝑇 , 𝑇 ; 𝑇 ) = max{𝑋𝑇 − 𝑌𝑇 , 0}𝑌𝑇−1 = max{𝑋𝑇 ∕𝑌𝑇 − 1, 0} = max{𝑍𝑇 − 1, 0} which is the payoff of a European call option with strike price 𝐾 = 1. Hence, following Problem 2.2.2.4 (page 95) we can deduce that 𝑈 (𝑍𝑡 , 𝑡; 𝑇 ) = 𝑍𝑡 𝑒−𝐷𝑥 (𝑇 −𝑡) Φ(𝑑+ ) − 𝑒−𝐷𝑦 (𝑇 −𝑡) Φ(𝑑− ) where 𝑑± =

log 𝑍𝑡 + (𝐷𝑦 − 𝐷𝑥 ± 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

and substituting 𝑈 (𝑍𝑡 , 𝑡; 𝑇 ) = 𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡)𝑌𝑡−1

and

𝑍𝑡 = 𝑋𝑡 𝑌𝑡−1

we have 𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡; 𝑇 ) = 𝑌𝑡

(

𝑋𝑡 𝑌𝑡

) 𝑒−𝐷𝑥 (𝑇 −𝑡) Φ(𝑑+ ) − 𝑒−𝐷𝑦 (𝑇 −𝑡) Φ(𝑑− )

or 𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡; 𝑇 ) = 𝑋𝑡 𝑒−𝐷𝑥 (𝑇 −𝑡) Φ(𝑑+ ) − 𝑌𝑡 𝑒−𝐷𝑦 (𝑇 −𝑡) Φ(𝑑− ) where 𝑑± =

log(𝑋𝑡 ∕𝑌𝑡 ) + (𝐷𝑦 − 𝐷𝑥 ± 12 𝜎 2 )(𝑇 − 𝑡) . √ 𝜎 𝑇 −𝑡

6.2.1 Path-Independent Options

543

6. Exchange Option (Probabilistic Approach). Let {𝑊𝑡𝑥 : 𝑡 ≥ 0} and {𝑊𝑡𝑦 : 𝑡 ≥ 0} be ℙ-standard Wiener processes on the probability space (Ω, ℱ, ℙ) and let the asset prices 𝑋𝑡 and 𝑌𝑡 have the following diffusion processes 𝑑𝑋𝑡 = (𝜇𝑥 − 𝐷𝑥 )𝑋𝑡 𝑑𝑡 + 𝜎𝑥 𝑋𝑡 𝑑𝑊𝑡𝑥 𝑑𝑌𝑡 = (𝜇𝑦 − 𝐷𝑦 )𝑌𝑡 𝑑𝑡 + 𝜎𝑦 𝑌𝑡 𝑑𝑊𝑡𝑦

𝑑𝑊𝑡𝑥 ⋅ 𝑑𝑊𝑡𝑦 = 𝜌𝑥𝑦 𝑑𝑡

where 𝜇𝑥 and 𝜇𝑦 are the drifts, 𝐷𝑥 and 𝐷𝑦 are the continuous dividend yields, 𝜎𝑥 and 𝜎𝑦 are the volatilities, 𝜌𝑥𝑦 is the correlation coefficient such that 𝜌𝑥𝑦 ∈ (−1, 1) and let 𝑟 be the risk-free interest rate from a money-market account. ⊥ ̂ 𝑦 such that ̂ 𝑦 and 𝑊 By defining two new independent standard Wiener processes 𝑊 𝑡 𝑡 ̃ 𝑥 = 𝜌𝑥𝑦 𝑑 𝑊 ̂𝑦 + 𝑑𝑊 𝑡 𝑡

√ ⊥ ̂𝑦 1 − 𝜌2𝑥𝑦 𝑑 𝑊 𝑡

̃𝑦 = 𝑑𝑊 ̂𝑦 𝑑𝑊 𝑡 𝑡 ⊥

̂𝑦 ⋅ 𝑑𝑊 ̂𝑦 = 0 𝑑𝑊 𝑡 𝑡 ( ) 𝜇𝑥 − 𝑟 𝑥 𝑥 ̃ ̃𝑦 = 𝑊 𝑦 + where, under the risk-neutral measure ℚ, 𝑊𝑡 = 𝑊𝑡 + 𝑡 and 𝑊 𝑡 𝑡 𝜎 𝑥 ( ) 𝜇𝑦 − 𝑟 𝑡 are ℚ-standard Wiener processes, we define the price of an exchange option 𝜎𝑦 at time 𝑡 with payoff Ψ(𝑋𝑇 , 𝑌𝑇 ) = max{𝑋𝑇 − 𝑌𝑇 , 0} at expiry time 𝑇 ≥ 𝑡 as [ ] 𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡; 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑋𝑇 − 𝑌𝑇 , 0}|| ℱ𝑡 . By denoting ℚ𝑌 as the new measure where 𝑌𝑡 𝑒𝐷𝑦 𝑡 is the num´eraire and taking note that the discounted money-market account ( 𝐷 𝑡 )−1 𝑟𝑡 𝑌𝑡 𝑒 𝑦 𝑒 is a ℚ𝑌 -martingale, show that under ℚ𝑌 , 𝑋𝑡 and 𝑌𝑡 have the following diffusion processes √ ⊥ 𝑑𝑋𝑡 𝑦 ̂𝑦 = (𝑟 − 𝐷𝑥 + 𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 )𝑑𝑡 + 𝜌𝑥𝑦 𝜎𝑥 𝑑𝑊 𝑡 + 𝜎𝑥 1 − 𝜌2𝑥𝑦 𝑑 𝑊 𝑡 𝑋𝑡 𝑑𝑌𝑡 𝑦 = (𝑟 − 𝐷𝑦 + 𝜎𝑦2 )𝑑𝑡 + 𝜎𝑦 𝑑𝑊 𝑡 𝑌𝑡

544

6.2.1 Path-Independent Options 𝑦 ̂ 𝑦 − 𝜎𝑦 𝑡. Hence, by finding the diffusion process for 𝑋𝑡 ∕𝑌𝑡 under ℚ𝑌 show where 𝑊 𝑡 = 𝑊 𝑡 that

( log

where 𝜎 =

𝑋𝑇 𝑌𝑇

)

] [ ( ) ( ) 𝑋𝑡 1 ∼  log + 𝐷𝑦 − 𝐷𝑥 − 𝜎 2 (𝑇 − 𝑡), 𝜎 2 (𝑇 − 𝑡) 𝑌𝑡 2

√ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 and finally deduce Margrabe’s formula 𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡; 𝑇 ) = 𝑋𝑡 𝑒−𝐷𝑥 (𝑇 −𝑡) Φ(𝑑+ ) − 𝑌𝑡 𝑒−𝐷𝑦 (𝑇 −𝑡) Φ(𝑑− )

where 𝑑± =

𝑥 log(𝑋𝑡 ∕𝑌𝑡 ) + (𝐷𝑦 − 𝐷𝑥 ± 12 𝜎 2 )(𝑇 − 𝑡) 1 − 1 𝑢2 and Φ(𝑥) = √ 𝑒 2 𝑑𝑢. √ ∫−∞ 2𝜋 𝜎 𝑇 −𝑡

Solution: From Girsanov’s theorem, under the risk-neutral measure ℚ 𝑑𝑋𝑡 ̃𝑥 = (𝑟 − 𝐷𝑥 )𝑑𝑡 + 𝜎𝑥 𝑑 𝑊 𝑡 𝑋𝑡 𝑑𝑌𝑡 ̃𝑦 = (𝑟 − 𝐷𝑦 )𝑑𝑡 + 𝜎𝑦 𝑑 𝑊 𝑡 𝑌𝑡 ̃𝑥 ⋅ 𝑑𝑊 ̃ 𝑦 = 𝜌𝑥𝑦 𝑑𝑡 𝑑𝑊 𝑡 𝑡 ̃𝑥 = 𝑊 𝑥 + where 𝑊 𝑡 𝑡

(

processes. By defining

𝜇𝑥 − 𝑟 𝜎𝑥

)

̃𝑦 = 𝑊 𝑦 + 𝑡 and 𝑊 𝑡 𝑡

̂𝑦 + ̃ 𝑥 = 𝜌𝑥𝑦 𝑑 𝑊 𝑑𝑊 𝑡 𝑡

(

𝜇𝑦 − 𝑟 𝜎𝑦

) 𝑡 are ℚ-standard Wiener

√ ⊥ ̂𝑦 1 − 𝜌2𝑥𝑦 𝑑 𝑊 𝑡

̃𝑦 = 𝑑𝑊 ̂𝑦 𝑑𝑊 𝑡 𝑡 ⊥

̂𝑦 ⋅ 𝑑𝑊 ̂𝑦 = 0 𝑑𝑊 𝑡 𝑡 we have √ ( ) ⊥ 𝑑𝑋𝑡 ̂ 𝑦 + 1 − 𝜌2 𝑑 𝑊 ̂𝑦 = (𝑟 − 𝐷𝑥 )𝑑𝑡 + 𝜎𝑥 𝜌𝑥𝑦 𝑑 𝑊 𝑡 𝑡 𝑥𝑦 𝑋𝑡 𝑑𝑌𝑡 ̂𝑦 = (𝑟 − 𝐷𝑦 )𝑑𝑡 + 𝜎𝑦 𝑑 𝑊 𝑡 𝑌𝑡 such that (

̂𝑦 + 𝜌𝑥𝑦 𝑑 𝑊 𝑡

√ ) ⊥ ̂ 𝑦 = 𝜌𝑥𝑦 𝑑𝑡. ̂𝑦 ⋅ 𝑑𝑊 1 − 𝜌2𝑥𝑦 𝑑 𝑊 𝑡 𝑡

6.2.1 Path-Independent Options

545

For a payoff Ψ(𝑋𝑇 , 𝑌𝑇 ) under the change of num´eraire, 𝑁 𝑡 𝔼ℚ

[

] [ ] Ψ(𝑋𝑇 , 𝑌𝑇 ) || Ψ(𝑋𝑇 , 𝑌𝑇 ) || ℚ𝑌 ℱ ℱ 𝔼 = 𝑀 𝑡 | 𝑡 | 𝑡 𝑁𝑇 𝑀𝑇 | |

where 𝑁 and 𝑀 are num´eraires (positive non-dividend-paying assets) and ℚ and ℚ𝑌 are the measures under which the asset prices discounted by 𝑁 and 𝑀 are ℚ and ℚ𝑌 martingales, respectively. Under the risk-neutral measure ℚ we have 𝑁𝑡 = 𝑒𝑟𝑡 and 𝑁𝑇 = 𝑒𝑟𝑇 and under the ℚ𝑌 measure, where 𝑌𝑡 𝑒𝐷𝑦 𝑡 is a non-dividend-paying asset we have 𝑀𝑡 = 𝑌𝑡 𝑒𝐷𝑦 𝑡 and 𝑀𝑇 = 𝑌𝑇 𝑒𝐷𝑦 𝑇 . Using a change of num´eraire, the exchange option at time 𝑡 is [ ] 𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡; 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑋𝑇 − 𝑌𝑇 , 0}|| ℱ𝑡 [ ] | 𝑟𝑡 ℚ max{𝑋𝑇 ∕𝑌𝑇 − 1, 0}𝑌𝑇 | =𝑒 𝔼 | ℱ𝑡 𝑒𝑟𝑇 | [ ] | max{𝑋 ∕𝑌 − 1, 0}𝑌 | 𝑇 𝑇 𝑇 = 𝑌𝑡 𝑒𝐷𝑦 𝑡 𝔼ℚ𝑌 | ℱ𝑡 | 𝑌𝑇 𝑒𝐷𝑦 𝑇 | [ ] −𝐷𝑦 (𝑇 −𝑡) ℚ𝑌 max{𝑋𝑇 ∕𝑌𝑇 − 1, 0}|| ℱ𝑡 . = 𝑌𝑡 𝑒 𝔼 Under ℚ𝑌 the discounted money-market account )−1 𝑟𝑡 ( 𝑍𝑡 = 𝑌𝑡 𝑒𝐷𝑦 𝑡 𝑒 = 𝑌𝑡−1 𝑒(𝑟−𝐷𝑦 )𝑡 is a ℚ𝑌 -martingale and from It¯o’s lemma (

(

𝑑𝑌𝑡 𝑑𝑍𝑡 = (𝑟 − 𝐷𝑦 )𝑒 𝑌𝑡 ( ) ̂ 𝑦 + 𝜎 2 𝑑𝑡 = (𝑟 − 𝐷𝑦 )𝑍𝑡 𝑑𝑡 + 𝑍𝑡 −(𝑟 − 𝐷𝑦 )𝑑𝑡 − 𝜎𝑦 𝑑 𝑊 𝑡 𝑦 (𝑟−𝐷𝑦 )𝑡

𝑌𝑡−1 𝑑𝑡 + 𝑌𝑡−1 𝑒(𝑟−𝐷𝑦 )𝑡

𝑑𝑌 − 𝑡+ 𝑌𝑡

)

)2 +…

̂𝑦 = 𝜎𝑦2 𝑍𝑡 𝑑𝑡 − 𝜎𝑦 𝑍𝑡 𝑑 𝑊 𝑡 ̂ 𝑦 − 𝜎𝑦 𝑑𝑡) = −𝜎𝑦 𝑍𝑡 (𝑑 𝑊 𝑡 𝑦

= −𝜎𝑦 𝑍𝑡 𝑑𝑊 𝑡 𝑦

⊥

̂ 𝑦 − 𝜎𝑦 𝑡 is a ℚ𝑌 -standard Wiener process. Because 𝑊 ̂𝑦 ⟂ ̂ 𝑦 , then 𝑊 ̂𝑦 where 𝑊 𝑡 = 𝑊 ⟂𝑊 𝑡 𝑡 𝑡 𝑡 is also a ℚ𝑌 -standard Wiener process.

⊥

546

6.2.1 Path-Independent Options

Thus, under ℚ𝑌 √ ( ) ⊥ 𝑑𝑋𝑡 ̂ 𝑦 + 1 − 𝜌2 𝑑 𝑊 ̂𝑦 = (𝑟 − 𝐷𝑥 )𝑑𝑡 + 𝜎𝑥 𝜌𝑥𝑦 𝑑 𝑊 𝑡 𝑡 𝑥𝑦 𝑋𝑡 √ ( ) ⊥ 𝑦 ̂𝑦 = (𝑟 − 𝐷𝑥 )𝑑𝑡 + 𝜎𝑥 𝜌𝑥𝑦 𝑑𝑊 𝑡 + 𝜌𝑥𝑦 𝜎𝑦 𝑑𝑡 + 1 − 𝜌2𝑥𝑦 𝑑 𝑊 𝑡 √ ⊥ 𝑦 ̂𝑦 = (𝑟 − 𝐷𝑥 + 𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 )𝑑𝑡 + 𝜌𝑥𝑦 𝜎𝑥 𝑑𝑊 𝑡 + 𝜎𝑥 1 − 𝜌2𝑥𝑦 𝑑 𝑊 𝑡 and 𝑑𝑌𝑡 ̂𝑦 = (𝑟 − 𝐷𝑦 )𝑑𝑡 + 𝜎𝑦 𝑑 𝑊 𝑡 𝑌𝑡 ( ) 𝑦 = (𝑟 − 𝐷𝑦 )𝑑𝑡 + 𝜎𝑦 𝑑𝑊 𝑡 + 𝜎𝑦 𝑑𝑡 𝑦

= (𝑟 − 𝐷𝑦 + 𝜎𝑦2 )𝑑𝑡 + 𝜎𝑦 𝑑𝑊 𝑡 . From Taylor’s theorem (

) 𝑋𝑡 ] ( [ ) 𝑌𝑡 𝑌𝑡 𝑑𝑋𝑡 𝑋𝑡 𝑑𝑋𝑡 𝑑𝑌𝑡 1 2𝑋𝑡 2 − 2 𝑑𝑌𝑡 + +⋯ (𝑑𝑌𝑡 ) − ( ) = 𝑋𝑡 𝑌𝑡 2! 𝑌𝑡3 𝑋𝑡 𝑌𝑡 𝑌𝑡2 𝑌𝑡 ( ( ) )( ) 𝑑𝑌𝑡 2 𝑑𝑋𝑡 𝑑𝑋𝑡 𝑑𝑌𝑡 𝑑𝑌𝑡 − + − = +⋯ 𝑋𝑡 𝑌𝑡 𝑌𝑡 𝑋𝑡 𝑌𝑡

𝑑

and by substituting 𝑑𝑋𝑡 ∕𝑋𝑡 , 𝑑𝑌𝑡 ∕𝑌𝑡 and using It¯o’s lemma (

) 𝑋𝑡 𝑑 √ 𝑌𝑡 ⊥ 𝑦 ̂𝑦 ( ) = (𝑟 − 𝐷𝑥 + 𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 )𝑑𝑡 + 𝜌𝑥𝑦 𝜎𝑥 𝑑𝑊 𝑡 + 𝜎𝑥 1 − 𝜌2𝑥𝑦 𝑑 𝑊 𝑡 𝑋𝑡 𝑌𝑡 𝑦

−(𝑟 − 𝐷𝑦 + 𝜎𝑦2 )𝑑𝑡 − 𝜎𝑦 𝑑𝑊 𝑡 + 𝜎𝑦2 𝑑𝑡 − 𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 𝑑𝑡 √ ⊥ 𝑦 ̂𝑦 . = (𝐷𝑦 − 𝐷𝑥 )𝑑𝑡 + (𝜌𝑥𝑦 𝜎𝑥 − 𝜎𝑦 )𝑑𝑊 𝑡 + 𝜎𝑥 1 − 𝜌2𝑥𝑦 𝑑 𝑊 𝑡 To find the distribution of 𝑋𝑡 ∕𝑌𝑡 we note that ( ) 2 ) 𝑋𝑡 ⎤ 𝑋𝑡 ⎡ 𝑑 ( ( )) ⎢ 𝑌𝑡 𝑌𝑡 ⎥ 𝑋𝑡 1 𝑑 log = ( ) − ⎢ ( ) ⎥ +… 𝑌𝑡 2! ⎢ 𝑋𝑡 ⎥ 𝑋𝑡 ⎥ ⎢ 𝑌𝑡 ⎦ ⎣ 𝑌𝑡 √ ) ( ⊥ 𝑦 1 2 ̂𝑦 = 𝐷𝑦 − 𝐷𝑥 − 𝜎 𝑑𝑡 + (𝜌𝑥𝑦 𝜎𝑥 − 𝜎𝑦 )𝑑𝑊 𝑡 + 𝜎𝑥 1 − 𝜌2𝑥𝑦 𝑑 𝑊 𝑡 2 (

𝑑

6.2.1 Path-Independent Options

547

√ where 𝜎 = 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 . Taking integrals 𝑇

∫𝑡

(

(

𝑑 log

𝑋𝑢 𝑌𝑢

))

𝑇

=

∫𝑡 +

∫𝑡

(

) 𝑇 𝑦 1 𝐷𝑦 − 𝐷𝑥 − 𝜎 2 𝑑𝑢 + (𝜌𝑥𝑦 𝜎𝑥 − 𝜎𝑦 )𝑑𝑊 𝑢 ∫𝑡 2 √ 𝑇 ̂ 𝑦⊥ 𝜎𝑥 1 − 𝜌2𝑥𝑦 𝑑 𝑊 𝑢

or ( log

𝑋𝑇 𝑌𝑇

)

(

) ( ) 𝑋𝑡 𝑦 1 + 𝐷𝑦 − 𝐷𝑥 − 𝜎 2 (𝑇 − 𝑡) + (𝜌𝑥𝑦 𝜎𝑥 − 𝜎𝑦 )𝑊 𝑇 −𝑡 𝑌𝑡 2 √ ⊥ 𝑦 ̂ . +𝜎𝑥 1 − 𝜌2𝑥𝑦 𝑊 𝑇 −𝑡

= log

𝑦 ̂ 𝑦 ∼  (0, 𝑇 − 𝑡) and 𝑊 𝑦 ⟂ ̂𝑦 Because 𝑊 𝑇 −𝑡 , 𝑊 𝑇 −𝑡 ⟂ 𝑊𝑇 −𝑡 , hence 𝑇 −𝑡 ⊥

( log

𝑋𝑇 𝑌𝑇

)

⊥

[

(

∼  log

𝑋𝑡 𝑌𝑡

)

] ) 1 2 2 + 𝐷𝑦 − 𝐷𝑥 − 𝜎 (𝑇 − 𝑡), 𝜎 (𝑇 − 𝑡) . 2 (

Finally, by analogy with a European call price formula, we can deduce [ 𝑉 (𝑋𝑡 , 𝑌𝑡 , 𝑡; 𝑇 ) = 𝑌𝑡 𝑒

−𝐷𝑦 (𝑇 −𝑡) ℚ

𝔼

[ = 𝑌𝑡 𝑒

−𝐷𝑦 (𝑇 −𝑡)

{ max

}| ] 𝑋𝑇 | − 1, 0 | ℱ𝑡 | 𝑌𝑇 |

] 𝑋𝑡 −(𝐷 −𝐷 )(𝑇 −𝑡) 𝑥 𝑦 𝑒 Φ(𝑑+ ) − Φ(𝑑− ) 𝑌𝑡

= 𝑋𝑡 𝑒−𝐷𝑥 (𝑇 −𝑡) Φ(𝑑+ ) − 𝑌𝑡 𝑒−𝐷𝑦 (𝑇 −𝑡) Φ(𝑑− ) where 𝑑± =

log(𝑋𝑡 ∕𝑌𝑡 ) + (𝐷𝑦 − 𝐷𝑥 ± 12 𝜎 2 )(𝑇 − 𝑡) . √ 𝜎 𝑇 −𝑡

7. Spread Option I. Let {𝑊𝑡𝑥 : 𝑡 ≥ 0} and {𝑊𝑡𝑦 : 𝑡 ≥ 0} be ℙ-standard Wiener processes on the probability space (Ω, ℱ, ℙ) and let the asset prices 𝑋𝑡 and 𝑌𝑡 have the following diffusion processes 𝑑𝑋𝑡 = (𝜇𝑥 − 𝐷𝑥 )𝑋𝑡 𝑑𝑡 + 𝜎𝑥 𝑋𝑡 𝑑𝑊𝑡𝑥 𝑑𝑌𝑡 = (𝜇𝑦 − 𝐷𝑦 )𝑌𝑡 𝑑𝑡 + 𝜎𝑦 𝑌𝑡 𝑑𝑊𝑡𝑦

𝑑𝑊𝑡𝑥 ⋅ 𝑑𝑊𝑡𝑦 = 𝜌𝑥𝑦 𝑑𝑡

where 𝜇𝑥 and 𝜇𝑦 are the drifts, 𝐷𝑥 and 𝐷𝑦 are the continuous dividend yields, 𝜎𝑥 and 𝜎𝑦 are the volatilities, 𝜌𝑥𝑦 is the correlation coefficient such that 𝜌𝑥𝑦 ∈ (−1, 1) and let 𝑟 be the risk-free interest rate from a money-market account.

548

6.2.1 Path-Independent Options ⊥

̂ 𝑦 and 𝑊 ̂ 𝑦 such that By setting two new independent standard Wiener processes 𝑊 𝑡 𝑡 ̃ 𝑥 = 𝜌𝑥𝑦 𝑑 𝑊 ̂𝑦 + 𝑑𝑊 𝑡 𝑡

√ ⊥ ̂𝑦 1 − 𝜌2𝑥𝑦 𝑑 𝑊 𝑡

̃𝑦 = 𝑑𝑊 ̂𝑦 𝑑𝑊 𝑡 𝑡 ⊥

̂𝑦 ⋅ 𝑑𝑊 ̂𝑦 = 0 𝑑𝑊 𝑡 𝑡 ( ) 𝜇𝑥 − 𝑟 ̃𝑦 = 𝑊 𝑦 + ̃𝑥 = 𝑊 𝑥 + 𝑡 and 𝑊 where, under the risk-neutral measure ℚ, 𝑊 𝑡 𝑡 𝑡 𝑡 𝜎 𝑥 ( ) 𝜇𝑦 − 𝑟 𝑡 are ℚ-standard Wiener processes, for a strike price 𝐾 > 0 we define the price 𝜎𝑦 of a call spread option at time 𝑡 with payoff Ψ(𝑋𝑇 , 𝑌𝑇 ) = max{𝑋𝑇 − 𝑌𝑇 − 𝐾, 0} at expiry time 𝑇 ≥ 𝑡 as [ ] 𝐶𝑠𝑝 (𝑋𝑡 , 𝑌𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑋𝑇 − 𝑌𝑇 − 𝐾, 0}|| ℱ𝑡 . For a small strike 𝐾 ≪ 𝑌𝑡 , show that under the ℚ measure, the process 𝑍𝑡 = 𝑌𝑡 + 𝐾𝑒−𝑟(𝑇 −𝑡) is approximately log normal with the following diffusion 𝑑𝑍𝑡 ̂𝑦 = (𝑟 − 𝐷𝑧 )𝑑𝑡 + 𝜎𝑧 𝑑 𝑊 𝑡 𝑍𝑡 where 𝐷𝑧 =

𝐷𝑦 𝑌𝑡

and 𝜎𝑧 =

𝜎𝑦 𝑌𝑡

are assumed to be constants. 𝑌𝑡 + 𝐾𝑒−𝑟(𝑇 −𝑡) 𝑌𝑡 + 𝐾𝑒−𝑟(𝑇 −𝑡) By denoting ℚ𝑍 as the new measure where 𝑍𝑡 𝑒𝐷𝑧 𝑡 is the num´eraire and taking note that the discounted money-market account )−1 𝑟𝑡 ( 𝑍𝑡 𝑒 𝐷 𝑧 𝑡 𝑒

is a ℚ𝑍 -martingale, show that under ℚ𝑍 , 𝑋𝑡 and 𝑍𝑡 have the following diffusion processes √ ⊥ 𝑑𝑋𝑡 𝑦 ̂𝑦 = (𝑟 − 𝐷𝑥 + 𝜌𝑥𝑦 𝜎𝑥 𝜎𝑧 )𝑑𝑡 + 𝜌𝑥𝑦 𝜎𝑥 𝑑𝑊 𝑡 + 𝜎𝑥 1 − 𝜌2𝑥𝑦 𝑑 𝑊 𝑡 𝑋𝑡 𝑑𝑍𝑡 𝑦 = (𝑟 − 𝐷𝑧 + 𝜎𝑧2 )𝑑𝑡 + 𝜎𝑧 𝑑𝑊 𝑡 𝑍𝑡

6.2.1 Path-Independent Options

549

𝑦 ̂ 𝑦 − 𝜎𝑧 𝑡. Therefore, by finding the diffusion process for 𝑋𝑡 ∕𝑍𝑡 under ℚ𝑍 where 𝑊 𝑡 = 𝑊 𝑡 show that

( log

𝑋𝑇 𝑌𝑇

)

[ ( ∻  log

)

𝑋𝑡

( +

𝑌𝑡 + 𝐾𝑒−𝑟(𝑇 −𝑡) ]

𝐷𝑦 𝑌𝑡 𝑌𝑡 + 𝐾𝑒−𝑟(𝑇 −𝑡)

) 1 − 𝐷𝑥 − 𝜎 2 (𝑇 − 𝑡), 2

𝜎 (𝑇 − 𝑡) 2

√ where

𝜎=

(

)

𝑌𝑡

𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦

𝑌𝑡 + 𝐾𝑒−𝑟(𝑇 −𝑡) deduce Kirk’s approximation formula

𝐶𝑠𝑝 (𝑋𝑡 , 𝑌𝑡 , 𝑡; 𝐾) ≈ 𝑋𝑡 𝑒

−𝐷𝑥 (𝑇 −𝑡)

( +

𝜎𝑦 𝑌𝑡

)2

𝑌𝑡 + 𝐾𝑒−𝑟(𝑇 −𝑡)

) − ( Φ(𝑑+ ) − 𝑌𝑡 + 𝐾𝑒−𝑟(𝑇 −𝑡) 𝑒

(

𝐷𝑦 𝑌𝑡 𝑌𝑡 +𝐾𝑒−𝑟(𝑇 −𝑡)

and

) (𝑇 −𝑡)

finally

Φ(𝑑− )

where ( log 𝑑± =

)

𝑋𝑡 𝑌𝑡 + 𝐾𝑒−𝑟(𝑇 −𝑡)

( +

𝐷𝑦 𝑌𝑡 𝑌𝑡 + 𝐾𝑒−𝑟(𝑇 −𝑡) √ 𝜎 𝑇 −𝑡

) 1 − 𝐷𝑥 ± 𝜎 2 (𝑇 − 𝑡) 2

and 𝑥

Φ(𝑥) =

∫−∞

1 − 1 𝑢2 √ 𝑒 2 𝑑 𝑢. 2𝜋

Solution: From Girsanov’s theorem, under the risk-neutral measure ℚ we can write 𝑑𝑋𝑡 ̃𝑥 = (𝑟 − 𝐷𝑥 )𝑑𝑡 + 𝜎𝑥 𝑑 𝑊 𝑡 𝑋𝑡 𝑑𝑌𝑡 ̃𝑦 = (𝑟 − 𝐷𝑦 )𝑑𝑡 + 𝜎𝑦 𝑑 𝑊 𝑡 𝑌𝑡 ̃𝑥 ⋅ 𝑑𝑊 ̃ 𝑦 = 𝜌𝑥𝑦 𝑑𝑡 𝑑𝑊 𝑡 𝑡 ̃𝑥 = 𝑊 𝑥 + where 𝑊 𝑡 𝑡

(

processes. By defining

𝜇𝑥 − 𝑟 𝜎𝑥

)

̃𝑦 = 𝑊 𝑦 + 𝑡 and 𝑊 𝑡 𝑡

̂𝑦 + ̃ 𝑥 = 𝜌𝑥𝑦 𝑑 𝑊 𝑑𝑊 𝑡 𝑡 ̃𝑦 = 𝑑𝑊 ̂𝑦 𝑑𝑊 𝑡 𝑡 ⊥

̂𝑦 ⋅ 𝑑𝑊 ̂𝑦 = 0 𝑑𝑊 𝑡 𝑡

(

𝜇𝑦 − 𝑟 𝜎𝑦

) 𝑡 are ℚ-standard Wiener

√ ⊥ ̂𝑦 1 − 𝜌2𝑥𝑦 𝑑 𝑊 𝑡

550

6.2.1 Path-Independent Options

we have √ ( ) ⊥ 𝑑𝑋𝑡 ̂ 𝑦 + 1 − 𝜌2 𝑑 𝑊 ̂𝑦 = (𝑟 − 𝐷𝑥 )𝑑𝑡 + 𝜎𝑥 𝜌𝑥𝑦 𝑑 𝑊 𝑡 𝑡 𝑥𝑦 𝑋𝑡 𝑑𝑌𝑡 ̂𝑦 = (𝑟 − 𝐷𝑦 )𝑑𝑡 + 𝜎𝑦 𝑑 𝑊 𝑡 𝑌𝑡 such that (

̂𝑦 + 𝜌𝑥𝑦 𝑑 𝑊 𝑡

√ ) ⊥ ̂ 𝑦 = 𝜌𝑥𝑦 𝑑𝑡. ̂𝑦 ⋅ 𝑑𝑊 1 − 𝜌2𝑥𝑦 𝑑 𝑊 𝑡 𝑡

For a small strike 𝐾 ≪ 𝑌𝑡 , the diffusion process for 𝑍𝑡 = 𝑌𝑡 + 𝐾𝑒−𝑟(𝑇 −𝑡) is 𝑑𝑍𝑡 = 𝑑𝑌𝑡 + 𝑟𝐾𝑒−𝑟(𝑇 −𝑡) 𝑑𝑡

̂ 𝑦 + 𝑟𝐾𝑒−𝑟(𝑇 −𝑡) 𝑑𝑡 = (𝑟 − 𝐷𝑦 )𝑌𝑡 𝑑𝑡 + 𝜎𝑦 𝑌𝑡 𝑑 𝑊 𝑡 ) ( −𝑟(𝑇 −𝑡) ̂𝑦 𝑑𝑡 − 𝐷𝑦 𝑌𝑡 𝑑𝑡 + 𝜎𝑦 𝑌𝑡 𝑑 𝑊 = 𝑟 𝑌𝑡 + 𝐾𝑒 𝑡 ̂𝑦 = 𝑟𝑍𝑡 𝑑𝑡 − 𝐷𝑦 𝑌𝑡 𝑑𝑡 + 𝜎𝑦 𝑌𝑡 𝑑 𝑊 𝑡

or 𝑑𝑍𝑡 ̂𝑦 = (𝑟 − 𝐷𝑧 )𝑑𝑡 + 𝜎𝑧 𝑑 𝑊 𝑡 𝑍𝑡 where 𝐷𝑧 =

𝑌𝑡 lemma we have

𝐷𝑦 𝑌𝑡 + 𝐾𝑒−𝑟(𝑇 −𝑡)

and 𝜎𝑧 =

𝜎𝑦 𝑌𝑡 𝑌𝑡 + 𝐾𝑒−𝑟(𝑇 −𝑡)

. By expanding 𝑑(log 𝑍𝑡 ) using It¯o’s

( )2 𝑑𝑍𝑡 1 𝑑𝑍𝑡 𝑑(log 𝑍𝑡 ) = − +… 𝑍𝑡 2! 𝑍𝑡 ) ( 1 ̂ 𝑦. = 𝑟 − 𝐷𝑧 − 𝜎𝑧2 𝑑𝑡 + 𝜎𝑧 𝑑 𝑊 𝑡 2 By assuming both 𝐷𝑧 and 𝜎𝑧 to be constants and taking integrals ) 𝑇 1 ̂𝑦 𝑟 − 𝐷𝑧 − 𝜎𝑧2 𝑑𝑢 + 𝜎𝑧 𝑑 𝑊 𝑢 ∫𝑡 ∫𝑡 ∫𝑡 2 ( ) ( ) 𝑇 𝑍𝑇 1 ̂𝑦 log 𝜎𝑧 𝑑 𝑊 = 𝑟 − 𝐷𝑧 − 𝜎𝑧2 (𝑇 − 𝑡) + 𝑢 ∫𝑡 𝑍𝑡 2 𝑇

𝑑(log 𝑍𝑢 ) =

𝑇

(

6.2.1 Path-Independent Options

551

we can therefore deduce that ( log

𝑌𝑇 + 𝐾 𝑌𝑡 + 𝐾𝑒−𝑟(𝑇 −𝑡)

) ∻

[(

) ] 1 𝑟 − 𝐷𝑧 − 𝜎𝑦2 (𝑇 − 𝑡), 𝜎𝑧2 (𝑇 − 𝑡) . 2

For a payoff Ψ(𝑋𝑇 , 𝑌𝑇 ) under the change of num´eraire 𝑁𝑡 𝔼

ℚ

[

] [ ] | Ψ(𝑋𝑇 , 𝑌𝑇 ) || ℚ𝑍 Ψ(𝑋𝑇 , 𝑌𝑇 ) | | ℱ𝑡 = 𝑀𝑡 𝔼 | ℱ𝑡 𝑁𝑇 𝑀𝑇 | |

where 𝑁 and 𝑀 are num´eraires (positive non-dividend-paying assets) and ℚ and ℚ𝑍 are the measures under which the asset prices discounted by 𝑁 and 𝑀 are ℚ and ℚ𝑍 martingales, respectively. Under the risk-neutral, measure ℚ we have 𝑁𝑡 = 𝑒𝑟𝑡 and 𝑁𝑇 = 𝑒𝑟𝑇 and under the ℚ𝑍 measure, where 𝑍𝑡 𝑒𝐷𝑧 𝑡 is a non-dividend-paying asset, we have 𝑀𝑡 = 𝑍𝑡 𝑒𝐷𝑧 𝑡 and 𝑀𝑇 = 𝑍𝑇 𝑒𝐷𝑧 𝑇 . Using a change of num´eraire, the call spread option at time 𝑡 is [ ] 𝐶𝑠𝑝 (𝑋𝑡 , 𝑌𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑋𝑇 − 𝑌𝑇 − 𝐾, 0}|| ℱ𝑡 } { 𝑋𝑇 | ⎤ ⎡ − 1, 0 (𝑌𝑇 + 𝐾) || ⎥ max ⎢ 𝑌 + 𝐾 | ⎥ 𝑇 = 𝑒𝑟𝑡 𝔼ℚ ⎢ | ℱ𝑡 | ⎥ ⎢ 𝑒𝑟𝑇 | ⎥ ⎢ | | ⎦ ⎣ = (𝑌𝑡 + 𝐾𝑒−𝑟(𝑇 −𝑡) )𝑒𝐷𝑧 𝑡 } { 𝑋𝑇 | ⎤ ⎡ − 1, 0 (𝑌𝑇 + 𝐾) || ⎥ max ⎢ 𝑌𝑇 + 𝐾 | ⎥ ×𝔼ℚ𝑍 ⎢ | ℱ𝑡 𝐷 𝑇 | ⎥ ⎢ 𝑧 (𝑌𝑇 + 𝐾)𝑒 | ⎥ ⎢ | | ⎦ ⎣ = (𝑌𝑡 + 𝐾𝑒−𝑟(𝑇 −𝑡) )𝑒−𝐷𝑧 (𝑇 −𝑡) [ { }| ] 𝑋𝑇 | ℚ𝑍 ×𝔼 max − 1, 0 | ℱ𝑡 | 𝑌𝑇 + 𝐾 | [ }| ] { 𝑋𝑇 | −𝐷𝑧 (𝑇 −𝑡) ℚ𝑍 = 𝑍𝑡 𝑒 𝔼 − 1, 0 | ℱ𝑡 . max | 𝑍𝑇 |

552

6.2.1 Path-Independent Options

Under ℚ𝑍 , the discounted money-market account )−1 𝑟𝑡 ( 𝑍 𝑡 = 𝑍𝑡 𝑒 𝐷 𝑧 𝑡 𝑒 = 𝑍𝑡−1 𝑒(𝑟−𝐷𝑧 )𝑡 is a ℚ𝑍 -martingale and from It¯o’s lemma (

(

𝑑𝑍𝑡 𝑑𝑍 𝑡 = (𝑟 − 𝐷𝑧 )𝑒(𝑟−𝐷𝑧 )𝑡 𝑍𝑡−1 𝑑𝑡 + 𝑍𝑡−1 𝑒(𝑟−𝐷𝑧 )𝑡 𝑍𝑡 ( ) ̂ 𝑦 + 𝜎 2 𝑑𝑡 = (𝑟 − 𝐷𝑧 )𝑍 𝑡 𝑑𝑡 + 𝑍 𝑡 −(𝑟 − 𝐷𝑧 )𝑑𝑡 − 𝜎𝑧 𝑑 𝑊 𝑡 𝑧 𝑑𝑍 − 𝑡+ 𝑍𝑡

)

)2 +…

̂𝑦 = 𝜎𝑧2 𝑍 𝑡 𝑑𝑡 − 𝜎𝑧 𝑍 𝑡 𝑑 𝑊 𝑡

̂ 𝑦 − 𝜎𝑧 𝑑𝑡) = −𝜎𝑧 𝑍 𝑡 (𝑑 𝑊 𝑡 𝑦

= −𝜎𝑧 𝑍 𝑡 𝑑𝑊 𝑡

𝑦 ̂ 𝑦 − 𝜎𝑧 𝑡 is a ℚ𝑍 -standard Wiener process. Because 𝑊 ̂𝑦 ⟂ ̂ 𝑦 , then where 𝑊 𝑡 = 𝑊 ⟂𝑊 𝑡 𝑡 𝑡 ⊥ ̂ 𝑦 is also a ℚ𝑍 -standard Wiener process. therefore 𝑊 𝑡 Thus, under ℚ𝑍 ⊥

√ ( ) ⊥ 𝑑𝑋𝑡 ̂ 𝑦 + 1 − 𝜌2 𝑑 𝑊 ̂𝑦 = (𝑟 − 𝐷𝑥 )𝑑𝑡 + 𝜎𝑥 𝜌𝑥𝑦 𝑑 𝑊 𝑡 𝑡 𝑥𝑦 𝑋𝑡 √ ( ) ⊥ 𝑦 ̂𝑦 = (𝑟 − 𝐷𝑥 )𝑑𝑡 + 𝜎𝑥 𝜌𝑥𝑦 𝑑𝑊 𝑡 + 𝜌𝑥𝑦 𝜎𝑧 𝑑𝑡 + 1 − 𝜌2𝑥𝑦 𝑑 𝑊 𝑡 √ ⊥ 𝑦 ̂𝑦 = (𝑟 − 𝐷𝑥 + 𝜌𝑥𝑦 𝜎𝑥 𝜎𝑧 )𝑑𝑡 + 𝜌𝑥𝑦 𝜎𝑥 𝑑𝑊 + 𝜎𝑥 1 − 𝜌2 𝑑 𝑊 𝑥𝑦

𝑡

𝑡

and 𝑑𝑍𝑡 ̂𝑦 = (𝑟 − 𝐷𝑧 )𝑑𝑡 + 𝜎𝑧 𝑑 𝑊 𝑡 𝑍𝑡 ( ) 𝑦 = (𝑟 − 𝐷𝑧 )𝑑𝑡 + 𝜎𝑧 𝑑𝑊 𝑡 + 𝜎𝑧 𝑑𝑡 𝑦

= (𝑟 − 𝐷𝑧 + 𝜎𝑧2 )𝑑𝑡 + 𝜎𝑧 𝑑𝑊 𝑡 . From Taylor’s theorem (

) 𝑋𝑡 ] [ ) ( 𝑑 𝑍𝑡 𝑍𝑡 𝑑𝑋𝑡 𝑋𝑡 𝑑𝑋𝑡 𝑑𝑍𝑡 1 2𝑋𝑡 2 − 2 𝑑𝑍𝑡 + +… (𝑑𝑍𝑡 ) − ( ) = 𝑋𝑡 𝑍𝑡 2! 𝑍𝑡3 𝑋𝑡 𝑍𝑡 𝑍𝑡2 𝑍𝑡 ( ( ) )( ) 𝑑𝑍𝑡 2 𝑑𝑋𝑡 𝑑𝑋𝑡 𝑑𝑍𝑡 𝑑𝑍𝑡 − + − = +… 𝑋𝑡 𝑍𝑡 𝑍𝑡 𝑋𝑡 𝑍𝑡

6.2.1 Path-Independent Options

553

and by substituting 𝑑𝑋𝑡 ∕𝑋𝑡 , 𝑑𝑍𝑡 ∕𝑍𝑡 and using It¯o’s lemma (

) 𝑋𝑡 𝑑 √ 𝑍𝑡 ⊥ 𝑦 ̂𝑦 ( ) = (𝑟 − 𝐷𝑥 + 𝜌𝑥𝑦 𝜎𝑥 𝜎𝑧 )𝑑𝑡 + 𝜌𝑥𝑦 𝜎𝑥 𝑑𝑊 𝑡 + 𝜎𝑥 1 − 𝜌2𝑥𝑦 𝑑 𝑊 𝑡 𝑋𝑡 𝑍𝑡 𝑦

−(𝑟 − 𝐷𝑦 + 𝜎𝑧2 )𝑑𝑡 − 𝜎𝑧 𝑑𝑊 𝑡 + 𝜎𝑧2 𝑑𝑡 − 𝜌𝑥𝑦 𝜎𝑥 𝜎𝑧 𝑑𝑡 √ ⊥ 𝑦 ̂𝑦 . = (𝐷𝑧 − 𝐷𝑥 )𝑑𝑡 + (𝜌𝑥𝑦 𝜎𝑥 − 𝜎𝑧 )𝑑𝑊 𝑡 + 𝜎𝑥 1 − 𝜌2𝑥𝑦 𝑑 𝑊 𝑡 To find the distribution of 𝑋𝑡 ∕𝑍𝑡 we note that ( ) 2 ) 𝑋𝑡 ⎤ 𝑋𝑡 ⎡ 𝑑 ( ( )) 𝑑 ⎢ 𝑍 𝑍𝑡 ⎥ 𝑋𝑡 1 𝑡 𝑑 log = ( ) − ⎢ ( ) ⎥ +… 𝑍𝑡 2! ⎢ 𝑋𝑡 ⎥ 𝑋𝑡 ⎥ ⎢ 𝑍𝑡 ⎣ 𝑍𝑡 ⎦ ( ) 𝑦 1 = 𝐷𝑧 − 𝐷𝑥 − 𝜎 2 𝑑𝑡 + (𝜌𝑥𝑦 𝜎𝑥 − 𝜎𝑧 )𝑑𝑊 𝑡 2 √ ⊥ 2 ̂𝑦 +𝜎𝑥 1 − 𝜌𝑥𝑦 𝑑 𝑊 𝑡 (

√ where 𝜎 = 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑧 + 𝜎𝑧2 . Since we are assuming 𝐷𝑧 and 𝜎𝑧 to be constants and taking integrals 𝑇

∫𝑡

( ( )) 𝑋𝑢 𝑑 log = ∫𝑡 𝑍𝑢 +

𝑇

∫𝑡

(

) 𝑇 𝑦 1 𝐷𝑧 − 𝐷𝑥 − 𝜎 2 𝑑𝑢 + (𝜌𝑥𝑦 𝜎𝑥 − 𝜎𝑧 )𝑑𝑊 𝑢 ∫ 2 𝑡 √ 𝑇 ̂ 𝑦⊥ 𝜎𝑥 1 − 𝜌2 𝑑 𝑊 𝑥𝑦

𝑢

or ( log

𝑋𝑇 𝑍𝑇

)

(

) ( ) 𝑋𝑡 𝑦 1 = log + 𝐷𝑧 − 𝐷𝑥 − 𝜎 2 (𝑇 − 𝑡) + (𝜌𝑥𝑦 𝜎𝑥 − 𝜎𝑧 )𝑊 𝑇 −𝑡 𝑍𝑡 2 √ ⊥ ̂𝑦 . +𝜎𝑥 1 − 𝜌2𝑥𝑦 𝑊 𝑇 −𝑡

𝑦

𝑦

⊥

⊥

̂ 𝑦 ∼  (0, 𝑇 − 𝑡) and 𝑊 ̂ 𝑦 , hence Because 𝑊 𝑇 −𝑡 , 𝑊 ⟂𝑊 𝑇 −𝑡 ⟂ 𝑇 −𝑡 𝑇 −𝑡 ( log

𝑋𝑇 𝑍𝑇

)

[ ∻  log

(

𝑋𝑡 𝑍𝑡

)

] ) 1 2 2 + 𝐷𝑧 − 𝐷𝑥 − 𝜎 (𝑇 − 𝑡), 𝜎 (𝑇 − 𝑡) . 2 (

554

6.2.1 Path-Independent Options

Finally, by analogy with a European call price formula we can deduce Kirk’s approximation formula [ 𝐶𝑠𝑝 (𝑋𝑡 , 𝑌𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑍𝑡 𝑒

−𝐷𝑧 (𝑇 −𝑡) ℚ𝑍

𝔼

[ ≈ 𝑍𝑡 𝑒

−𝐷𝑧 (𝑇 −𝑡)

{ max

𝑋𝑡 −(𝐷 −𝐷 )(𝑇 −𝑡) 𝑒 𝑥 𝑧 Φ(𝑑+ ) − Φ(𝑑− ) 𝑍𝑡

= 𝑋𝑡 𝑒−𝐷𝑥 (𝑇 −𝑡) Φ(𝑑+ ) (

− 𝑌𝑡 + 𝐾𝑒 ( log where 𝑑± =

)

𝑋𝑡 𝑌𝑡 + 𝐾𝑒−𝑟(𝑇 −𝑡)

) −𝑟(𝑇 −𝑡)

( +

}| ] 𝑋𝑇 | − 1, 0 | ℱ𝑡 | 𝑍𝑇 |

𝑒

( −

𝐷𝑦 𝑌𝑡 𝑌𝑡 +𝐾𝑒−𝑟(𝑇 −𝑡)

𝐷𝑦 𝑌𝑡 𝑌𝑡 + 𝐾𝑒−𝑟(𝑇 −𝑡) √ 𝜎 𝑇 −𝑡

) (𝑇 −𝑡)

]

Φ(𝑑− )

) 1 2 − 𝐷𝑥 ± 𝜎 (𝑇 − 𝑡) 2

.

8. Spread Option II. Let {𝑊𝑡(1) : 𝑡 ≥ 0} and {𝑊𝑡(2) : 𝑡 ≥ 0} be ℙ-standard Wiener processes on the probability space (Ω, ℱ, ℙ) and let the asset prices 𝑆𝑡(1) and 𝑆𝑡(2) follow ABMs of the form 𝑑𝑆𝑡(1) = (𝜇1 − 𝐷1 )𝑑𝑡 + 𝜎1 𝑑𝑊𝑡(1) 𝑑𝑆𝑡(2) = (𝜇2 − 𝐷2 )𝑑𝑡 + 𝜎2 𝑑𝑊𝑡(2) 𝑑𝑊𝑡(1) ⋅ 𝑑𝑊𝑡(2) = 𝜌𝑑𝑡 where 𝜇1 and 𝜇2 are the drifts, 𝐷1 and 𝐷2 are the continuous dividend yields, 𝜎1 and 𝜎2 are the volatilities, 𝜌 ∈ (−1, 1) is the correlation coefficient and let 𝑟 be the risk-free interest rate from a money-market account. Using It¯o’s lemma show that under the risk-neutral measure ℚ for 𝑇 > 𝑡, the conditional distribution of 𝑆𝑇(2) − 𝑆𝑇(1) given 𝑆𝑡(1) and 𝑆𝑡(2) is ) ( | 𝑆𝑇(2) − 𝑆𝑇(1) | 𝑆𝑡(1) , 𝑆𝑡(2) ∼  𝑚, 𝑠2 | where 𝑚 = 𝑆𝑡(2) 𝑒−𝐷2 (𝑇 −𝑡) − 𝑆𝑡(1) 𝑒−𝐷1 (𝑇 −𝑡) and 𝑠2 =

𝜎12

[ 2(𝑟−𝐷 )(𝑇 −𝑡) ] 1 𝑒 −1 +

𝜎22

2(𝑟 − 𝐷1 ) 2(𝑟 − 𝐷2 ) [ (2𝑟−𝐷 −𝐷 )(𝑇 −𝑡) ] 2𝜌𝜎1 𝜎2 1 2 𝑒 −1 . − 2𝑟 − 𝐷1 − 𝐷2

[

𝑒2(𝑟−𝐷2 )(𝑇 −𝑡) − 1

]

6.2.1 Path-Independent Options

555

Using the risk-neutral valuation approach show that for a strike price 𝐾 > 0, the price of a call spread option at time 𝑡 < 𝑇 with payoff Ψ(𝑆𝑇(1) , 𝑆𝑇(2) ) = max{𝑆𝑇(2) − 𝑆𝑇(1) − 𝐾, 0} is ) ( 𝐶𝑠𝑝 (𝑆𝑡(1) , 𝑆𝑡(2) , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡(2) 𝑒−𝐷2 (𝑇 −𝑡) − 𝑆𝑡(1) 𝑒−𝐷1 (𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ (𝑑) + 𝜎Φ′ (𝑑) where 𝑆𝑡(2) 𝑒−𝐷2 (𝑇 −𝑡) − 𝑆𝑡(1) 𝑒−𝐷1 (𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡)

𝑑= 𝜎2 =

𝜎 [ −2𝐷 (𝑇 −𝑡) ] 𝑒 1 − 𝑒−2𝑟(𝑇 −𝑡) +

𝜎12

𝜎22

[

2(𝑟 − 𝐷1 ) 2(𝑟 − 𝐷2 ) [ −(𝐷 +𝐷 )(𝑇 −𝑡) ] 2𝜌𝜎1 𝜎2 𝑒 1 2 − 𝑒−2𝑟(𝑇 −𝑡) − 2𝑟 − 𝐷1 − 𝐷2

𝑒−2𝐷2 (𝑇 −𝑡) − 𝑒−2𝑟(𝑇 −𝑡)

]

𝑥

and Φ(𝑥) =

1 − 21 𝑢2 𝑒 𝑑𝑢 is the standard normal cdf. ∫−∞ √2𝜋

Solution: Under the risk-neutral measure ℚ, ̃ (1) 𝑑𝑆𝑡(1) = (𝑟 − 𝐷1 )𝑆𝑡(1) 𝑑𝑡 + 𝜎1 𝑑 𝑊 𝑡 ̃ (2) 𝑑𝑆𝑡(2) = (𝑟 − 𝐷2 )𝑆𝑡(2) 𝑑𝑡 + 𝜎2 𝑑 𝑊 𝑡

( ̃ (1) 𝑊 𝑡

𝑊𝑡(1)

𝜇1 − 𝑟𝑆𝑡(1)

(

) ̃ (2) 𝑊 𝑡

𝑊𝑡(2)

𝜇2 − 𝑟𝑆𝑡(2)

)

= + 𝑡 and 𝑡 are ℚ𝜎1 𝜎2 standard Wiener processes (see Problem 4.2.3.7, Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus). From Problem 2.2.2.8 (page 109) we can easily show that for 𝑡 < 𝑇 ,

where

=

+

𝑆𝑇(1) = 𝑆𝑡(1) 𝑒(𝑟−𝐷1 )(𝑇 −𝑡) + 𝜎1

𝑇

∫𝑡

̃ (1) 𝑒(𝑟−𝐷1 )(𝑇 −𝑢) 𝑑 𝑊 𝑢

and 𝑆𝑇(2) = 𝑆𝑡(2) 𝑒(𝑟−𝐷2 )(𝑇 −𝑡) + 𝜎2

𝑇

∫𝑡

̃ (2) . 𝑒(𝑟−𝐷2 )(𝑇 −𝑢) 𝑑 𝑊 𝑢

556

6.2.1 Path-Independent Options

Thus, ( ) | 𝔼ℚ 𝑆𝑇(2) − 𝑆𝑇(1) | ℱ𝑡 = 𝑆𝑡(2) 𝑒(𝑟−𝐷2 )(𝑇 −𝑡) − 𝑆𝑡(1) 𝑒(𝑟−𝐷1 )(𝑇 −𝑡) | and ( ) | Varℚ 𝑆𝑇(2) − 𝑆𝑇(1) | ℱ𝑡 | ( ) ( ) ( ) | | | = Varℚ 𝑆𝑇(1) | ℱ𝑡 + Varℚ 𝑆𝑇(2) | ℱ𝑡 − 2Covℚ 𝑆𝑇(1) , 𝑆𝑇(2) | ℱ𝑡 | | | )2 | ] | | ℱ𝑡 𝑒 = | ∫𝑡 | | [( ] ) 2| 𝑇 | ̃ (2) | ℱ𝑡 +𝜎22 𝔼ℚ 𝑒(𝑟−𝐷2 )(𝑇 −𝑢) 𝑑 𝑊 | 𝑢 ∫𝑡 | | [ 𝜎12 𝔼ℚ

[(

𝑇

(𝑟−𝐷1 )(𝑇 −𝑢)

𝑇

−2𝜎1 𝜎2 Covℚ [ =

𝜎12 𝔼ℚ

𝑇

∫𝑡

∫𝑡

𝑒

[ −2𝜎1 𝜎2 Cov

̃ (1) , 𝑒(𝑟−𝐷1 )(𝑇 −𝑢) 𝑑 𝑊 𝑢

2(𝑟−𝐷1 )(𝑇 −𝑢)

ℚ

𝑇

∫𝑡

̃ (1) 𝑑𝑊 𝑢

𝑇

∫𝑡

| ̃ (2) || ℱ𝑡 𝑒(𝑟−𝐷2 )(𝑇 −𝑢) 𝑑 𝑊 𝑢 | |

] [ | | 2 ℚ 𝑑𝑢| ℱ𝑡 + 𝜎2 𝔼 | ∫𝑡 |

̃ (1) , 𝑒(𝑟−𝐷1 )(𝑇 −𝑢) 𝑑 𝑊 𝑢

𝑇

∫𝑡

]

] | | 𝑒 𝑑𝑢| ℱ𝑡 | | ] | (𝑟−𝐷2 )(𝑇 −𝑢) ̃ (2) | 𝑒 𝑑 𝑊𝑢 | ℱ𝑡 . | | 𝑇

2(𝑟−𝐷2 )(𝑇 −𝑢)

By setting ̃ (2) = 𝜌𝑑 𝑊 ̃ (1) + 𝑑𝑊 𝑡 𝑡

√

̂ (1) 1 − 𝜌2 𝑑 𝑊 𝑡

̂ (1) ⋅ 𝑑 𝑊 ̃ (1) = 0, then so that 𝑑 𝑊 𝑡 𝑡 ( ) | Varℚ 𝑆𝑇(2) − 𝑆𝑇(1) | ℱ𝑡 | [ =

𝜎12 𝔼ℚ

𝑇

∫𝑡

−2𝜎1 𝜎2 Cov

𝑒 ℚ

2(𝑟−𝐷1 )(𝑇 −𝑢)

[

𝑇

∫𝑡

] [ | | 2 ℚ 𝑑𝑢| ℱ𝑡 + 𝜎2 𝔼 | ∫𝑡 |

̃ (1) , 𝑒(𝑟−𝐷1 )(𝑇 −𝑢) 𝑑 𝑊 𝑢

𝑇

𝑒

2(𝑟−𝐷2 )(𝑇 −𝑢)

| | 𝑑𝑢| ℱ𝑡 | |

]

6.2.1 Path-Independent Options

𝜌

𝑇

∫𝑡

= 𝜎12

𝑇

∫𝑡

𝑒

(𝑟−𝐷2 )(𝑇 −𝑢)

= 𝜎12

∫𝑡

𝑇

∫𝑡

√ + 1 − 𝜌2

𝑇

ℚ

∫𝑡

𝑒

(𝑟−𝐷1 )(𝑇 −𝑢)

𝑒2(𝑟−𝐷1 )(𝑇 −𝑢) 𝑑𝑢 + 𝜎22

−2𝜌𝜎1 𝜎2 𝔼 = 𝜎12

̃ (1) 𝑑𝑊 𝑢

𝑒2(𝑟−𝐷1 )(𝑇 −𝑢) 𝑑𝑢 + 𝜎22 ∫𝑡 [

−2𝜌𝜎1 𝜎2 𝔼 𝑇

557

ℚ

[

𝑇

∫𝑡

𝑒

𝑇

∫𝑡

̃ (1) 𝑑𝑊 𝑢 𝑇

∫𝑡

𝑒

|

̂ (1) || ℱ𝑡 𝑑𝑊 𝑢 |

]

|

⋅

𝑇

∫𝑡

𝑒

(𝑟−𝐷2 )(𝑇 −𝑢)

|

̃ (1) || ℱ𝑡 𝑑𝑊 𝑢 | |

𝑒2(𝑟−𝐷2 )(𝑇 −𝑢) 𝑑𝑢 ]

𝑇

∫𝑡

∫𝑡

(𝑟−𝐷2 )(𝑇 −𝑢)

𝑒2(𝑟−𝐷2 )(𝑇 −𝑢) 𝑑𝑢

(2𝑟−𝐷1 −𝐷2 )(𝑇 −𝑢)

𝑒2(𝑟−𝐷1 )(𝑇 −𝑢) 𝑑𝑢 + 𝜎22

−2𝜌𝜎1 𝜎2

𝑇

𝑇

𝑑𝑢

𝑒2(𝑟−𝐷2 )(𝑇 −𝑢) 𝑑𝑢

𝑒(2𝑟−𝐷1 −𝐷2 )(𝑇 −𝑢) 𝑑𝑢.

Solving the integrals, we have ( ) 𝜎12 [ 2(𝑟−𝐷 )(𝑇 −𝑡) ] | 1 Varℚ 𝑆𝑇(2) − 𝑆𝑇(1) | ℱ𝑡 = 𝑒 −1 | 2(𝑟 − 𝐷1 ) +

𝜎22

[

] 𝑒2(𝑟−𝐷2 )(𝑇 −𝑡) − 1

2(𝑟 − 𝐷2 ) [ (2𝑟−𝐷 −𝐷 )(𝑇 −𝑡) ] 2𝜌𝜎1 𝜎2 1 2 𝑒 −1 . − 2𝑟 − 𝐷1 − 𝐷2 Since | 𝑆𝑇(1) | 𝑆𝑡(1) |

( ∼

and | 𝑆𝑇(2) | 𝑆𝑡(2) |

𝑆𝑡(1) 𝑒(𝑟−𝐷1 )(𝑇 −𝑡) ,

( ∼

𝑆𝑡(2) 𝑒(𝑟−𝐷2 )(𝑇 −𝑡) ,

𝜎12

[

]

[

]

𝑒2(𝑟−𝐷1 )(𝑇 −𝑡) − 1

2(𝑟 − 𝐷1 )

𝜎22 2(𝑟 − 𝐷2 )

𝑒2(𝑟−𝐷2 )(𝑇 −𝑡) − 1

therefore | 𝑆𝑇(2) − 𝑆𝑇(1) | 𝑆𝑡(1) , 𝑆𝑡(2) ∼  (𝑚, 𝑠2 ) | where 𝑚 = 𝑆𝑡(2) 𝑒(𝑟−𝐷2 )(𝑇 −𝑡) − 𝑆𝑡(1) 𝑒(𝑟−𝐷1 )(𝑇 −𝑡)

)

)

]

558

6.2.1 Path-Independent Options

and 𝑠2 =

𝜎12

[ 2(𝑟−𝐷 )(𝑇 −𝑡) ] 1 𝑒 −1 +

𝜎22

[

2(𝑟 − 𝐷1 ) 2(𝑟 − 𝐷2 ) [ (2𝑟−𝐷 −𝐷 )(𝑇 −𝑡) ] 2𝜌𝜎1 𝜎2 1 2 𝑒 −1 . − 2𝑟 − 𝐷1 − 𝐷2

𝑒2(𝑟−𝐷2 )(𝑇 −𝑡) − 1

]

Using the same steps as described in Problem 2.2.2.8 (page 109), we can easily show that the call spread option price at time 𝑡 < 𝑇 is [ ] | 𝐶𝑠𝑝 (𝑆𝑡(1) , 𝑆𝑡(2) , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑆𝑇(2) − 𝑆𝑇(1) − 𝐾, 0}| ℱ𝑡 | ) )] [ ( ( 𝑚−𝐾 −𝑟(𝑇 −𝑡) ′ 𝑚−𝐾 + 𝑠Φ (𝑚 − 𝐾)Φ =𝑒 𝑠 𝑠) ( (2) −𝐷2 (𝑇 −𝑡) (1) −𝐷1 (𝑇 −𝑡) −𝑟(𝑇 −𝑡) Φ (𝑑) = 𝑆𝑡 𝑒 − 𝑆𝑡 𝑒 − 𝐾𝑒 +𝜎Φ′ (𝑑) where 𝑑=

𝑆𝑡(2) 𝑒−𝐷2 (𝑇 −𝑡) − 𝑆𝑡(1) 𝑒−𝐷1 (𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) 𝜎

and 𝜎 = 2

𝜎12

[ −2𝐷 (𝑇 −𝑡) ] 𝑒 1 − 𝑒−2𝑟(𝑇 −𝑡) +

𝜎22

[

𝑒−2𝐷2 (𝑇 −𝑡) − 𝑒−2𝑟(𝑇 −𝑡)

2(𝑟 − 𝐷1 ) 2(𝑟 − 𝐷2 ) [ ] 2𝜌𝜎1 𝜎2 𝑒−(𝐷1 +𝐷2 )(𝑇 −𝑡) − 𝑒−2𝑟(𝑇 −𝑡) . − 2𝑟 − 𝐷1 − 𝐷2

]

9. Rainbow Option I. Let 𝑋 ∼  (𝜇𝑥 , 𝜎𝑥2 ) and 𝑌 ∼  (𝜇𝑦 , 𝜎𝑦2 ) be jointly normally distributed with correlation coefficient 𝜌𝑥𝑦 ∈ (−1, 1). If 𝑍 = min{𝑋, 𝑌 } show that the probability density function (pdf) of 𝑍 is 𝜌𝑥𝑦 𝜎𝑥 𝜌 𝜎 ⎛ ⎞ ⎛ −𝑧 + 𝜇𝑦 + 𝑥𝑦 𝑦 (𝑧 − 𝜇𝑥 ) ⎞ (𝑧 − 𝜇𝑦 ) ⎟ −𝑧 + 𝜇𝑥 + ⎜ 𝜎𝑦 𝜎𝑥 ⎟ ⎜ ⎟ 𝑓 (𝑧) 𝑓𝑍 (𝑧) = Φ ⎜ √ √ ⎟ 𝑓𝑋 (𝑧) + Φ ⎜⎜ ⎟ 𝑌 2 2 ⎟ ⎜ 𝜎𝑦 1 − 𝜌𝑥𝑦 𝜎 1 − 𝜌 𝑥 𝑥𝑦 ⎜ ⎟ ⎠ ⎝ ⎝ ⎠ − 21

(

𝑧−𝜇𝑥 𝜎𝑥

)2

( 1

− 1 𝑒 , 𝑓𝑌 (𝑧) = √ 𝑒 2 where 𝑓𝑋 (𝑧) = 𝜎𝑥 2𝜋 𝜎𝑦 2𝜋 cumulative distribution function of a standard normal.

1 √

𝑧−𝜇𝑦 𝜎𝑦

)2

and Φ denotes the

6.2.1 Path-Independent Options

559

Hence, show that }] [ { 𝔼 max 𝑒𝑍 − 𝐾, 0 =𝑒

𝜇𝑥 + 21 𝜎𝑥2

+𝑒

⎞ ⎛ 𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 ⎟ ⎜ 𝜇𝑥 + 𝜎𝑥2 − log 𝐾 𝜇𝑦 − 𝜇𝑥 − 𝜎𝑥 (𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 ) 𝚽⎜ , √ ,√ ⎟ 𝜎 𝑥 ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟ ⎠ ⎝

𝜇𝑦 + 12 𝜎𝑦2

⎞ ⎛ 2 𝜎𝑦 − 𝜌𝑥𝑦 𝜎𝑥 ⎟ ⎜ 𝜇𝑦 + 𝜎𝑦 − log 𝐾 𝜇𝑥 − 𝜇𝑦 − 𝜎𝑦 (𝜎𝑦 − 𝜌𝑥𝑦 𝜎𝑥 ) 𝚽⎜ , √ ,√ ⎟ 𝜎𝑦 ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟ ⎠ ⎝

⎞ ⎡ ⎛ 𝜇𝑦 − 𝜇𝑥 𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 ⎟ ⎢ ⎜ 𝜇𝑥 − log 𝐾 ,√ ,√ −𝐾 ⎢𝚽 ⎜ ⎟ 𝜎𝑥 ⎢ ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟ ⎣ ⎝ ⎠ ⎛ ⎞⎤ 𝜇𝑥 − 𝜇𝑦 𝜎𝑦 − 𝜌𝑥𝑦 𝜎𝑥 ⎜ 𝜇𝑦 − log 𝐾 ⎟⎥ + 𝚽⎜ ,√ ,√ ⎟⎥ 𝜎 𝑦 ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟⎥ ⎝ ⎠⎦ where 𝐾 > 0, 𝔼 denotes the expectation with respect to 𝑒𝑍 distribution and 𝚽 denotes the cumulative distribution function of a standard bivariate normal distribution given as 𝚽(𝑢, 𝑣, 𝜌𝑢𝑣 ) =

𝑢

𝑣

∫−∞ ∫−∞

√ 2𝜋

1 1 − 𝜌2𝑢𝑣

𝑒

−

1 (𝑥2 −2𝜌𝑢𝑣 𝑥𝑦+𝑦2 ) 2(1−𝜌2 𝑢𝑣 )

𝑑𝑦𝑑𝑥

where 𝜌𝑢𝑣 ∈ (−1, 1). Let the asset prices 𝑆𝑡(1) and 𝑆𝑡(2) have the following diffusion processes 𝑑𝑆𝑡(1) = (𝜇1 − 𝐷1 )𝑆𝑡(1) 𝑑𝑡 + 𝜎1 𝑆𝑡(1) 𝑑𝑊𝑡(1) 𝑑𝑆𝑡(2) = (𝜇2 − 𝐷2 )𝑆𝑡(2) 𝑑𝑡 + 𝜎2 𝑆𝑡(2) 𝑑𝑊𝑡(2) 𝑑𝑊𝑡(1) ⋅ 𝑑𝑊𝑡(2) = 𝜌𝑑𝑡 where {𝑊𝑡(1) : 𝑡 ≥ 0} and {𝑊𝑡(2) : 𝑡 ≥ 0} are ℙ-standard Wiener processes on the probability space (Ω, ℱ , ℙ), 𝜇1 and 𝜇2 are the drift parameters, 𝐷1 and 𝐷2 are the continuous dividend yields, 𝜎1 and 𝜎2 are the volatilities, 𝜌 ∈ (−1, 1) is the correlation coefficient and let 𝑟 be the risk-free interest rate from a money market account. Show that (𝑊𝑡(1) , 𝑊𝑡(2) ) follows a bivariate normal distribution. Given the payoff of a rainbow call on the minimum option is defined as { } Ψ(𝑆𝑇(1) , 𝑆𝑇(2) ) = max min{𝑆𝑇(1) , 𝑆𝑇(2) } − 𝐾, 0 where 𝐾 > 0 is the strike price, 𝑇 ≥ 𝑡 is the option expiry time, using the above results find the rainbow call on the minimum option price at time 𝑡 under the risk-neutral measure ℚ.

560

6.2.1 Path-Independent Options

Solution: The first part of the results follows from Problem 1.2.2.15 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus. To calculate the closed-form solution of }] [ { 𝔼 max 𝑒𝑍 − 𝐾, 0 where 𝑍 = min{𝑋, 𝑌 }, 𝑋 ∼  (𝜇𝑥 , 𝜎𝑥2 ), 𝑌 ∼  (𝜇𝑦 , 𝜎𝑦2 ), 𝐶𝑜𝑣(𝑋, 𝑌 ) = 𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 , 𝜌𝑥𝑦 ∈ (−1, 1) and 𝐾 > 0, the expectation can be written as [ { }] 𝔼 max 𝑒𝑍 − 𝐾, 0 =

+∞

∫−∞

{ } max 𝑒𝑧 − 𝐾, 0 𝑓𝑍 (𝑧)𝑑𝑧

where 𝑓𝑍 (𝑧) is the pdf of 𝑍. By substituting the pdf of 𝑍 we have }] [ { 𝔼 max 𝑒𝑍 − 𝐾, 0 =

+∞ (

∫log 𝐾

) 𝑒𝑧 − 𝐾 𝑓𝑍 (𝑧) 𝑑𝑧

𝜌 𝜎 ⎛ −𝑧 + 𝜇𝑦 + 𝑥𝑦 𝑦 (𝑧 − 𝜇𝑥 ) ⎞ ) ⎜ 𝜎𝑥 ⎟ 𝑒𝑧 − 𝐾 Φ ⎜ = √ ⎟ 𝑓𝑋 (𝑧) 𝑑𝑧 ∫log 𝐾 ⎟ ⎜ 𝜎𝑦 1 − 𝜌2𝑥𝑦 ⎠ ⎝ 𝜎 𝜌 𝑥𝑦 𝑥 ⎛ ⎞ ⎜ −𝑧 + 𝜇𝑥 + 𝜎 (𝑧 − 𝜇𝑦 ) ⎟ +∞ ( ) 𝑦 ⎟ 𝑓 (𝑧) 𝑑𝑧 𝑒𝑧 − 𝐾 Φ ⎜ + √ ∫log 𝐾 ⎜ ⎟ 𝑌 2 𝜎 1 − 𝜌 𝑥 𝑥𝑦 ⎜ ⎟ ⎝ ⎠ = 𝐴1 + 𝐴2 − 𝐵1 − 𝐵2 +∞ (

where

𝐴1 =

𝐴2 =

𝐵1 =

𝐵2 =

𝜌 𝜎 ⎛ −𝑧 + 𝜇𝑦 + 𝑥𝑦 𝑦 (𝑧 − 𝜇𝑥 ) ⎞ 𝜎𝑥 ⎟ ⎜ 𝑒𝑧 Φ ⎜ √ ⎟ 𝑓𝑋 (𝑧) 𝑑𝑧 ∫log 𝐾 ⎟ ⎜ 𝜎𝑦 1 − 𝜌2𝑥𝑦 ⎠ ⎝ 𝜎 𝜌 𝑥𝑦 𝑥 ⎛ ⎞ ⎜ −𝑧 + 𝜇𝑥 + 𝜎 (𝑧 − 𝜇𝑦 ) ⎟ +∞ 𝑦 ⎟ 𝑓 (𝑧) 𝑑𝑧 𝑒𝑧 Φ ⎜ √ ∫log 𝐾 ⎜ ⎟ 𝑌 2 𝜎 1 − 𝜌 𝑥 𝑥𝑦 ⎜ ⎟ ⎝ ⎠ 𝜌𝑥𝑦 𝜎𝑦 ⎛ −𝑧 + 𝜇𝑦 + (𝑧 − 𝜇𝑥 ) ⎞ +∞ 𝜎𝑥 ⎟ ⎜ 𝐾 Φ √ ⎟ 𝑓𝑋 (𝑧) 𝑑𝑧 ∫log 𝐾 ⎜⎜ ⎟ 𝜎𝑦 1 − 𝜌2𝑥𝑦 ⎠ ⎝ 𝜎 𝜌 𝑥𝑦 𝑥 ⎛ ⎞ ⎜ −𝑧 + 𝜇𝑥 + 𝜎 (𝑧 − 𝜇𝑦 ) ⎟ +∞ 𝑦 ⎟ 𝑓 (𝑧) 𝑑𝑧. 𝐾 Φ⎜ √ ∫log 𝐾 ⎜ ⎟ 𝑌 2 𝜎𝑥 1 − 𝜌𝑥𝑦 ⎜ ⎟ ⎝ ⎠ +∞

6.2.1 Path-Independent Options

561

For the case

𝐴1 =

∫log 𝐾

+∞

∫𝑤= log 𝐾−𝜇𝑥

𝑒

𝜇𝑥 +𝑤𝜎𝑥

𝑒

𝜇𝑥 + 21 𝜎𝑥2

𝜎𝑥

+∞

=

𝜌 𝜎 ⎛ −𝑧 + 𝜇𝑦 + 𝑥𝑦 𝑦 (𝑧 − 𝜇𝑥 ) ⎞ 𝜎𝑥 ⎟ ⎜ 𝑒𝑧 Φ ⎜ √ ⎟ 𝑓𝑋 (𝑧) 𝑑𝑧 ⎟ ⎜ 𝜎𝑦 1 − 𝜌2𝑥𝑦 ⎠ ⎝

𝑧 − 𝜇𝑥 and therefore 𝜎𝑥

we let 𝑤 =

𝐴1 =

+∞

∫𝑤= log 𝐾−𝜇𝑥 𝜎𝑥

⎞ ⎛ ⎜ 𝜇𝑦 − 𝜇𝑥 − 𝑤(𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 ) ⎟ 1 − 1 𝑤2 Φ⎜ √ ⎟ √ 𝑒 2 𝑑𝑤 ⎟ 2𝜋 ⎜ 𝜎𝑦 1 − 𝜌2𝑥𝑦 ⎠ ⎝ ⎛ 𝑦= ⎜ ⎜∫−∞ ⎝

𝜇𝑦 −𝜇𝑥 −𝑤(𝜎𝑥 −𝜌𝑥𝑦 𝜎𝑦 ) 𝜎𝑦

√

1−𝜌2𝑥𝑦

⎞ 1 − 1 (𝑤−𝜎𝑥 )2 1 − 1 𝑦2 𝑑𝑤. √ 𝑒 2 𝑑𝑦⎟ √ 𝑒 2 ⎟ 2𝜋 ⎠ 2𝜋

Let 𝑣 = 𝑤 − 𝜎𝑥 and we then have

𝐴1 =

+∞

∫𝑣= log 𝐾−𝜇𝑥 −𝜎𝑥2

𝑒

𝜇𝑥 + 21 𝜎𝑥2

𝜎𝑥

𝜇𝑥 + 21 𝜎𝑥2

=𝑒

Let 𝑢 = 𝑦 +

⎛ 𝑦= ⎜ ⎜∫−∞ ⎝

𝜇𝑦 −𝜇𝑥 −(𝑣+𝜎𝑥 )(𝜎𝑥 −𝜌𝑥𝑦 𝜎𝑦 ) 𝜎𝑦

√

1−𝜌2 𝑥𝑦

⎞ 1 − 1 𝑣2 1 − 1 𝑦2 √ 𝑒 2 𝑑𝑦⎟ √ 𝑒 2 𝑑𝑣 ⎟ 2𝜋 2𝜋 ⎠

𝜇 −𝜇 −𝜎𝑥 (𝜎𝑥 −𝜌𝑥𝑦 𝜎𝑦 ) 𝑣(𝜎𝑥 −𝜌𝑥𝑦 𝜎𝑦 ) 𝑦= 𝑦 𝑥 √ − √

𝜇 +𝜎 2 −log 𝐾 𝑣= 𝑥 𝑥𝜎 𝑥

∫−∞

𝜎𝑦

∫−∞

1−𝜌2𝑥𝑦

𝜎𝑦

1−𝜌2 𝑥𝑦

1 − 12 (𝑦2 +𝑣2 ) 𝑑𝑦𝑑𝑣. 𝑒 2𝜋

𝑣(𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 ) and by setting 𝜎 2 = 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 we will have √ 2 𝜎𝑦 1 − 𝜌𝑥𝑦 𝜇 +𝜎 2 −log 𝐾 𝑣= 𝑥 𝑥𝜎

𝜇𝑥 + 21 𝜎𝑥2

𝑥

𝐴1 = 𝑒

∫−∞

𝜇 −𝜇 −𝜎𝑥 (𝜎𝑥 −𝜌𝑥𝑦 𝜎𝑦 ) 𝑢= 𝑦 𝑥 √

∫−∞

𝜎𝑦

1−𝜌2 𝑥𝑦

𝑓 (𝑢, 𝑣) 𝑑𝑢𝑑𝑣

such that

𝑓 (𝑢, 𝑣) =

1 2𝜋

) ⎞ ( ⎛ 2(𝜎𝑥 −𝜌𝑥𝑦 𝜎𝑦 ) (𝜎 −𝜌 𝜎 )2 𝑣2 ⎟ − 21 ⎜𝑢2 − √ 𝑢𝑣+ 1+ 𝑥2 𝑥𝑦 2 𝑦 ⎜ ⎟ 𝜎𝑦 (1−𝜌𝑥𝑦 ) 𝜎𝑦 1−𝜌2𝑥𝑦 ⎝ ⎠ 𝑒 ( − 21

=

1 𝑒 2𝜋

⎞ ⎛ √ )⎜ ⎟ 2(𝜎𝑥 −𝜌𝑥𝑦 𝜎𝑦 )𝜎𝑦 1−𝜌2 2 𝑥𝑦 𝑢 ⎜( )− 𝑢𝑣+𝑣2 ⎟ 2 2 2 𝜎 ⎟ 𝜎𝑦 (1−𝜌𝑥𝑦 ) ⎜ 𝜎2 ⎟ ⎜ 𝜎 2 (1−𝜌2 ) 𝑥𝑦 ⎠. ⎝ 𝑦 𝜎2

562

6.2.1 Path-Independent Options

By setting 𝑢̄ =

𝑢 ⎛ ⎞ ⎜ ⎟ 𝜎 ⎜ √ ⎟ ⎜ 𝜎𝑦 1 − 𝜌2𝑥𝑦 ⎟ ⎝ ⎠

𝐴1 = 𝑒

𝜇𝑥 + 21 𝜎𝑥2

and 𝜌̄𝑥𝑦 =

𝜇 +𝜎 2 −log 𝐾 𝑣= 𝑥 𝑥𝜎 𝑥

∫−∞

𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 𝜎

we have

𝜇 −𝜇 −𝜎 (𝜎 −𝜌 𝜎 ) 𝑢= ̄ 𝑦 𝑥 𝑥𝜎 𝑥 𝑥𝑦 𝑦

∫−∞

𝑓 (𝑢, ̄ 𝑣) 𝑑 𝑢̄ 𝑑𝑣

where ⎛

1

= 2𝜋

𝑒 √ 1 − 𝜌̄2𝑥𝑦

−

2

⎞ (

⎞ − 1 ⎜ √𝜎 ⎛ ⎟ ⎟ 2 ⎜⎝ 𝜎𝑦 1−𝜌2𝑥𝑦 ⎟⎠ 𝜎 1 ⎜ 𝑒 𝑓 (𝑢, ̄ 𝑣) = √ 2𝜋 ⎜⎜ 𝜎 1 − 𝜌2 ⎟⎟ 𝑥𝑦 ⎠ ⎝ 𝑦

1 2) (𝑢̄ 2 −2𝜌̄𝑥𝑦 𝑢𝑣+𝑣 ̄ 2(1−𝜌̄2𝑥𝑦 )

𝑢̄ 2 −

2(𝜎𝑥 −𝜌𝑥𝑦 𝜎𝑦 ) 2 𝑢𝑣+𝑣 ̄ 𝜎

)

.

Hence,

𝐴1 = 𝑒

𝜇𝑥 + 12 𝜎𝑥2

⎞ ⎛ 𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 ⎟ ⎜ 𝜇𝑥 + 𝜎𝑥2 − log 𝐾 𝜇𝑦 − 𝜇𝑥 − 𝜎𝑥 (𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 ) 𝚽⎜ , √ ,√ ⎟. 𝜎 𝑥 ⎜ 𝜎𝑥2 + 𝜎𝑦2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟ ⎠ ⎝

Using the same techniques as discussed above we have

𝐴2 = 𝑒

𝜇𝑦 + 12 𝜎𝑦2

⎞ ⎛ 2 𝜎𝑦 − 𝜌𝑥𝑦 𝜎𝑥 ⎟ ⎜ 𝜇𝑦 + 𝜎𝑦 − log 𝐾 𝜇𝑥 − 𝜇𝑦 − 𝜎𝑦 (𝜎𝑦 − 𝜌𝑥𝑦 𝜎𝑥 ) 𝚽⎜ , √ ,√ ⎟. 𝜎 𝑦 ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟ ⎠ ⎝

For the case

𝐵1 = 𝐾

+∞

∫log 𝐾

𝜌 𝜎 ⎛ −𝑧 + 𝜇𝑦 + 𝑥𝑦 𝑦 (𝑧 − 𝜇𝑥 ) ⎞ 𝜎𝑥 ⎟ ⎜ Φ⎜ √ ⎟ 𝑓𝑋 (𝑧) 𝑑𝑧 ⎟ ⎜ 𝜎𝑦 1 − 𝜌2𝑥𝑦 ⎠ ⎝

6.2.1 Path-Independent Options

563

𝑧 − 𝜇𝑥 and therefore 𝜎𝑥

we let 𝑤 =

𝐵1 = 𝐾

+∞

∫𝑤= log 𝐾−𝜇𝑥 𝜎𝑥

=𝐾

+∞

∫𝑤= log 𝐾−𝜇𝑥 𝜎𝑥

=𝐾

Let 𝑢 = 𝑦 +

⎞ ⎛ ⎜ 𝜇𝑦 − 𝜇𝑥 − 𝑤(𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 ) ⎟ 1 − 1 𝑤2 Φ⎜ √ ⎟ √ 𝑒 2 𝑑𝑤 ⎟ 2𝜋 ⎜ 𝜎𝑦 1 − 𝜌2𝑥𝑦 ⎠ ⎝ ⎛ 𝑦= ⎜ ⎜∫−∞ ⎝

𝜇 −log 𝐾 𝑤= 𝑥 𝜎 𝑥

∫−∞

𝑦=

∫−∞

𝜇𝑦 −𝜇𝑥 −𝑤(𝜎𝑥 −𝜌𝑥𝑦 𝜎𝑦 ) 𝜎𝑦

√

𝜇 −𝜇𝑥

𝜎𝑦

⎞ 1 − 1 𝑤2 1 − 1 𝑦2 √ 𝑒 2 𝑑𝑦⎟ √ 𝑒 2 𝑑𝑤 ⎟ 2𝜋 2𝜋 ⎠

1−𝜌2𝑥𝑦

𝑦 √

1−𝜌2𝑥𝑦

−

𝑤(𝜎𝑥 −𝜌𝑥𝑦 𝜎𝑦 ) 𝜎𝑦

√

1−𝜌2 𝑥𝑦

1 − 21 (𝑦2 +𝑤2 ) 𝑑𝑦𝑑𝑤. 𝑒 2𝜋

𝑤(𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 ) and by setting 𝜎 2 = 𝜎𝑥2 + 𝜎𝑦2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 we will have √ 2 𝜎𝑦 1 − 𝜌𝑥𝑦

𝐵1 = 𝐾

𝜇 −log 𝐾 𝑤= 𝑥 𝜎 𝑥

∫−∞

𝑢=

∫−∞

𝜇 −𝜇𝑥

𝜎𝑦

𝑦 √

1−𝜌2 𝑥𝑦

𝑔(𝑢, 𝑤) 𝑑𝑢𝑑𝑤

such that

𝑔(𝑢, 𝑤) =

1 2𝜋

) ⎞ ( ⎛ 2(𝜎𝑥 −𝜌𝑥𝑦 𝜎𝑦 ) (𝜎 −𝜌 𝜎 )2 𝑤2 ⎟ − 21 ⎜𝑢2 − √ 𝑢𝑤+ 1+ 𝑥2 𝑥𝑦 2 𝑦 ⎜ ⎟ 𝜎𝑦 (1−𝜌𝑥𝑦 ) 𝜎𝑦 1−𝜌2𝑥𝑦 ⎝ ⎠ 𝑒 ( − 21

=

By setting 𝑢̄ =

1 𝑒 2𝜋 𝑢

⎛ ⎞ ⎜ ⎟ 𝜎 ⎜ √ ⎟ ⎜ 𝜎𝑦 1 − 𝜌2𝑥𝑦 ⎟ ⎝ ⎠

𝐵1 = 𝐾

𝜎2 𝜎𝑦2 (1−𝜌2𝑥𝑦 )

⎞ ⎛ √ )⎜ ⎟ 2(𝜎𝑥 −𝜌𝑥𝑦 𝜎𝑦 )𝜎𝑦 1−𝜌2 2 𝑥𝑦 𝑢 2 ⎜( )− 𝑢𝑤+𝑤 ⎟ 2 𝜎 ⎟ ⎜ 2 𝜎 ⎟ ⎜ 𝜎 2 (1−𝜌2 ) 𝑥𝑦 ⎠. ⎝ 𝑦

and 𝜌̄𝑥𝑦 =

𝜇 −log 𝐾 𝑤= 𝑥 𝜎 𝑥

∫−∞

𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 𝜎

𝜇 −𝜇 𝑢= ̄ 𝑦𝜎 𝑥

∫−∞

we have

𝑔(𝑢, ̄ 𝑤) 𝑑 𝑢̄ 𝑑𝑤

564

6.2.1 Path-Independent Options

where ⎛

2

⎞ ( ) 2(𝜎 −𝜌 𝜎 ) 2 𝑢̄ 2 − 𝑥 𝜎 𝑥𝑦 𝑦 𝑢𝑤+𝑤 ̄

⎞ − 1 ⎜ √𝜎 ⎛ ⎟ ⎟ 2 ⎜⎝ 𝜎𝑦 1−𝜌2𝑥𝑦 ⎟⎠ 𝜎 1 ⎜ 𝑒 𝑔(𝑢, ̄ 𝑤) = √ 2𝜋 ⎜⎜ 𝜎 1 − 𝜌2 ⎟⎟ 𝑥𝑦 ⎠ ⎝ 𝑦 1

= 2𝜋

√ 1 − 𝜌̄2𝑥𝑦

𝑒

−

1 2) (𝑢̄ 2 −2𝜌̄𝑥𝑦 𝑢𝑤+𝑤 ̄ 2(1−𝜌̄2𝑥𝑦 )

.

Hence, ⎞ ⎛ 𝜇𝑦 − 𝜇𝑥 𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 ⎟ ⎜ 𝜇𝑥 − log 𝐾 𝐵1 = 𝐾𝚽 ⎜ ,√ ,√ ⎟. 𝜎 𝑥 ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟ ⎠ ⎝ In the same vein, we can also show ⎞ ⎛ 𝜇𝑥 − 𝜇𝑦 𝜎𝑦 − 𝜌𝑥𝑦 𝜎𝑥 ⎟ ⎜ 𝜇𝑦 − log 𝐾 𝐵2 = 𝐾𝚽 ⎜ ,√ ,√ ⎟. 𝜎 𝑦 ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟ ⎠ ⎝ [ { }] Hence, by substituting 𝐴1 , 𝐴2 , 𝐵1 and 𝐵2 back to 𝔼 max 𝑒𝑍 − 𝐾, 0 we have [ { }] 𝔼 max 𝑒𝑍 − 𝐾, 0 =𝑒

𝜇𝑥 + 12 𝜎𝑥2

+𝑒

⎞ ⎛ 𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 ⎟ ⎜ 𝜇𝑥 + 𝜎𝑥2 − log 𝐾 𝜇𝑦 − 𝜇𝑥 − 𝜎𝑥 (𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 ) 𝚽⎜ , √ ,√ ⎟ 𝜎 𝑥 ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟ ⎠ ⎝

𝜇𝑦 + 21 𝜎𝑦2

⎞ ⎛ 2 𝜎𝑦 − 𝜌𝑥𝑦 𝜎𝑥 ⎟ ⎜ 𝜇𝑦 + 𝜎𝑦 − log 𝐾 𝜇𝑥 − 𝜇𝑦 − 𝜎𝑦 (𝜎𝑦 − 𝜌𝑥𝑦 𝜎𝑥 ) 𝚽⎜ , √ ,√ ⎟ 𝜎𝑦 ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟ ⎠ ⎝

⎞ ⎡ ⎛ 𝜇𝑦 − 𝜇𝑥 𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 ⎟ ⎢ ⎜ 𝜇𝑥 − log 𝐾 ,√ ,√ −𝐾 ⎢𝚽 ⎜ ⎟ 𝜎 𝑥 ⎢ ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟ ⎣ ⎝ ⎠ ⎛ ⎞⎤ 𝜇𝑥 − 𝜇𝑦 𝜎𝑦 − 𝜌𝑥𝑦 𝜎𝑥 ⎜ 𝜇𝑦 − log 𝐾 ⎟⎥ + 𝚽⎜ ,√ ,√ ⎟⎥ . 𝜎𝑦 ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟⎥ ⎝ ⎠⎦

6.2.1 Path-Independent Options

565

Since 𝑊𝑡(1) , 𝑊𝑡(2) ∼  (0, 𝑡) with Cov(𝑊𝑡(1) , 𝑊𝑡(2) ) = 𝜌𝑡 then by expressing √ 𝑡𝑍1 ) √ √ ( = 𝑡 𝜌𝑍1 + 1 − 𝜌2 𝑍2

𝑊𝑡(1) = 𝑊𝑡(2)

where 𝑍1 , 𝑍2 ∼  (0, 1) and 𝑍1 ⟂ ⟂ 𝑍2 , therefore for constants 𝜃1 and 𝜃2 , ) [ √ ] ( √ √ (1) (2) 2 = 𝔼 𝑒𝜃1 𝑡𝑍1 +𝜃2 𝑡(𝜌𝑍1 + 1−𝜌 𝑍2 ) 𝔼 𝑒𝜃𝑊𝑡 +𝜃2 𝑊𝑡 ] [ √ √ √ 2 = 𝔼 𝑒(𝜃1 𝑡+𝜌𝜃2 𝑡)𝑍1 +𝜃2 𝑡(1−𝜌 )𝑍2 ] [ √ ] [ √ √ 2 = 𝔼 𝑒(𝜃1 𝑡+𝜌𝜃2 𝑡)𝑍1 ⋅ 𝔼 𝑒𝜃2 𝑡(1−𝜌 )𝑍2 1

= 𝑒 2 (𝜃1

√ √ 𝑡+𝜌𝜃2 𝑡)2

⎡ 𝑡 𝑇 By setting 𝜽 = (𝜃1 , 𝜃2 ) and 𝚺 = ⎢ ⎢1 ⎣ 2 𝜌𝑡

1 2

⋅ 𝑒 2 𝜃2 𝑡(1−𝜌 ) . 2

1 𝜌𝑡 2 ⎤

⎥ we therefore have ⎥ 𝑡 ⎦

( ) 1 2 1 2 (1) (2) 𝔼 𝑒𝜃𝑊𝑡 +𝜃2 𝑊𝑡 = 𝑒 2 𝜃1 𝑡+𝜌𝜃1 𝜃2 𝑡+ 2 𝜃2 𝑡 1 𝑇 = 𝑒 2 𝜽 𝚺𝜽

which is the moment generating function of a bivariate normal distribution. Hence, (𝑊𝑡(1) , 𝑊𝑡(2) ) follows a bivariate normal distribution. From the definition of a rainbow call on the minimum option at time 𝑡 under risk-neutral measure ℚ, we have [ }| ] { 𝐶min (𝑆𝑡(1) , 𝑆𝑡(2) , 𝑡; 𝑇 , 𝐾) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max min{𝑆𝑇(1) , 𝑆𝑇(2) } − 𝐾, 0 || ℱ𝑡 | where ̃ (1) 𝑑𝑆𝑡(1) = (𝑟 − 𝐷1 )𝑆𝑡(1) 𝑑𝑡 + 𝜎1 𝑆𝑡(1) 𝑑 𝑊 𝑡

̃ (2) 𝑑𝑆𝑡(2) = (𝑟 − 𝐷2 )𝑆𝑡(2) 𝑑𝑡 + 𝜎2 𝑆𝑡(2) 𝑑 𝑊 𝑡

̃ (1) ⋅ 𝑑 𝑊 ̃ (2) = 𝜌𝑑𝑡 𝑑𝑊 𝑡 𝑡 ̃ (1) = 𝑊 (1) + such that 𝑊 𝑡 𝑡 Wiener processes.

(

𝜇1 − 𝑟 𝜎1

)

̃ (2) = 𝑊 (2) + 𝑡 and 𝑊 𝑡 𝑡

(

𝜇2 − 𝑟 𝜎2

) 𝑡 are ℚ-standard

566

6.2.1 Path-Independent Options (1)

(2)

By setting 𝑆𝑇(1) = 𝑒log 𝑆𝑇 and 𝑆𝑇(2) = 𝑒log 𝑆𝑇 we can write 𝐶min (𝑆𝑡(1) , 𝑆𝑡(2) , 𝑡; 𝐾, 𝑇 )

}| ] { { (1) (2) } log 𝑆𝑇 log 𝑆𝑇 − 𝐾, 0 || ℱ𝑡 =𝑒 𝔼 max min 𝑒 ,𝑒 | [ } { }| ] { (1) (2) min log 𝑆𝑇 ,log 𝑆𝑇 | = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝑒 − 𝐾, 0 | ℱ𝑡 | | [ ] { } | = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝑒𝑍 − 𝐾, 0 | ℱ𝑡 | −𝑟(𝑇 −𝑡) ℚ

(1)

[

(2)

where 𝑍 = min{𝑋, 𝑌 }, 𝑋 = 𝑒log 𝑆𝑇 and 𝑌 = 𝑒log 𝑆𝑇 such that (log 𝑆𝑇(1) , log 𝑆𝑇(2) ) follows a bivariate normal distribution. From It¯o’s formula we can show that ) [ ( ] 1 log 𝑆𝑇(1) ∼  log 𝑆𝑡(1) + 𝑟 − 𝐷1 − 𝜎12 (𝑇 − 𝑡), 𝜎12 (𝑇 − 𝑡) 2 ) [ ( ] 1 2 (2) (2) log 𝑆𝑇 ∼  log 𝑆𝑡 + 𝑟 − 𝐷2 − 𝜎2 (𝑇 − 𝑡), 𝜎22 (𝑇 − 𝑡) . 2 By setting ) ( 1 𝜇𝑥 = log 𝑆𝑡(1) + 𝑟 − 𝐷1 − 𝜎12 (𝑇 − 𝑡) 2 𝜎𝑥2 = 𝜎12 (𝑇 − 𝑡) ) ( 1 𝜇𝑦 = log 𝑆𝑡(2) + 𝑟 − 𝐷2 − 𝜎22 (𝑇 − 𝑡) 2 𝜎𝑦2 = 𝜎22 (𝑇 − 𝑡) and after some algebraic manipulations we will have 1 2

𝑒𝜇𝑥 + 2 𝜎𝑥 = 𝑆𝑡(1) 𝑒(𝑟−𝐷1 )(𝑇 −𝑡) 1 2

𝑒𝜇𝑦 + 2 𝜎𝑦 = 𝑆𝑡(2) 𝑒(𝑟−𝐷2 )(𝑇 −𝑡) 𝜇𝑥 + 𝜎𝑥2 − log 𝐾 𝜎𝑥

𝜇𝑦 + 𝜎𝑦2 − log 𝐾 𝜎𝑦

log(𝑆𝑡(1) ∕𝐾) + (𝑟 − 𝐷1 + 12 𝜎12 )(𝑇 − 𝑡) = √ 𝜎1 𝑇 − 𝑡 =

log(𝑆𝑡(2) ∕𝐾) + (𝑟 − 𝐷2 + 12 𝜎22 )(𝑇 − 𝑡) √ 𝜎2 𝑇 − 𝑡

( ) log(𝑆𝑡(2) ∕𝑆𝑡(1) ) + 𝐷1 − 𝐷2 − 12 (𝜎12 − 2𝜌𝜎1 𝜎2 + 𝜎22 ) (𝑇 − 𝑡) 𝜇𝑦 − 𝜇𝑥 − 𝜎𝑥 (𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 ) = √ √ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 (𝜎 2 − 2𝜌𝜎1 𝜎2 + 𝜎22 )(𝑇 − 𝑡) ( 1 ) (1) (2) 1 2 2 log(𝑆 ∕𝑆 ) + 𝐷 − 𝐷 − (𝜎 − 2𝜌𝜎 𝜎 + 𝜎 ) (𝑇 − 𝑡) 2 1 1 2 𝜇𝑥 − 𝜇𝑦 − 𝜎𝑦 (𝜎𝑦 − 𝜌𝑥𝑦 𝜎𝑥 ) 𝑡 𝑡 2 2 1 = √ √ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 (𝜎12 − 2𝜌𝜎1 𝜎2 + 𝜎22 )(𝑇 − 𝑡)

6.2.1 Path-Independent Options

√ √

567

𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 𝜎𝑦 − 𝜌𝑥𝑦 𝜎𝑥 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2

= √ = √

𝜎1 − 𝜌𝜎2 (𝜎12 − 2𝜌𝜎1 𝜎2 + 𝜎22 )(𝑇 − 𝑡) 𝜎2 − 𝜌𝜎1 (𝜎12 − 2𝜌𝜎1 𝜎2 + 𝜎22 )(𝑇 − 𝑡)

log(𝑆𝑡(1) ∕𝐾) + (𝑟 − 𝐷1 − 12 𝜎12 )(𝑇 − 𝑡) 𝜇𝑥 − log 𝐾 = 𝜎𝑥 𝜎1 (𝑇 − 𝑡) log(𝑆𝑡(2) ∕𝐾) + (𝑟 − 𝐷2 − 12 𝜎22 )(𝑇 − 𝑡) 𝜇𝑦 − log 𝐾 = 𝜎𝑦 𝜎2 (𝑇 − 𝑡) ( ) log(𝑆𝑡(2) ∕𝑆𝑡(1) ) + 𝐷1 − 𝐷2 + 12 (𝜎12 − 𝜎22 ) (𝑇 − 𝑡) 𝜇𝑦 − 𝜇 𝑥 = √ √ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 (𝜎12 − 2𝜌𝜎1 𝜎2 + 𝜎22 )(𝑇 − 𝑡) ( ) (1) (2) 1 2 2 log(𝑆 ∕𝑆 ) + 𝐷 − 𝐷 + (𝜎 − 𝜎 ) (𝑇 − 𝑡) 2 1 𝜇 𝑥 − 𝜇𝑦 𝑡 𝑡 1 2 2 = √ √ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 (𝜎12 − 2𝜌𝜎1 𝜎2 + 𝜎22 )(𝑇 − 𝑡)

By setting 𝜎 =

√ 𝜎12 − 2𝜌𝜎1 𝜎2 + 𝜎22 and substituting the above expressions into

𝐶min (𝑆𝑡(1) , 𝑆𝑡(2) , 𝑡; 𝐾, 𝑇 ), the rainbow call on the minimum option at time 𝑡, 𝑡 < 𝑇 is therefore 𝐶min (𝑆𝑡(1) , 𝑆𝑡(2) , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡(1) 𝑒−𝐷1 (𝑇 −𝑡) 𝚽(𝛼+(1) , 𝛽+ , 𝜌(1) ) + 𝑆𝑡(2) 𝑒−𝐷2 (𝑇 −𝑡) 𝚽(𝛼+(2) , 𝛽− , 𝜌(2) ) −𝐾𝑒−𝑟(𝑇 −𝑡) 𝚽(𝛼−(1) , 𝛾+ , 𝜌(1) ) − 𝐾𝑒−𝑟(𝑇 −𝑡) 𝚽(𝛼−(2) , 𝛾− , 𝜌(2) ) where 𝛼±(1)

log(𝑆𝑡(1) ∕𝐾) + (𝑟 − 𝐷1 ± 12 𝜎12 )(𝑇 − 𝑡) = , √ 𝜎1 𝑇 − 𝑡

± log(𝑆𝑡(2) ∕𝑆𝑡(1) ) ± (𝐷1 − 𝐷2 ∓ 12 𝜎 2 )(𝑇 − 𝑡) , 𝛽± = √ 𝜎 𝑇 −𝑡 𝛼±(2)

log(𝑆𝑡(2) ∕𝐾) + (𝑟 − 𝐷2 ± 12 𝜎22 )(𝑇 − 𝑡) , = √ 𝜎2 𝑇 − 𝑡 ± log(𝑆𝑡(2) ∕𝑆𝑡(1) ) ± (𝐷1 − 𝐷2 ± 12 (𝜎12 − 𝜎22 ))(𝑇 − 𝑡) , √ 𝜎 𝑇 −𝑡 𝜎1 − 𝜌𝜎2 = √ , 𝜎 𝑇 −𝑡 𝜎2 − 𝜌𝜎1 = √ . 𝜎 𝑇 −𝑡

𝛾± = 𝜌(1) 𝜌(2)

568

6.2.1 Path-Independent Options

10. Rainbow Option II. Let 𝑋 ∼  (𝜇𝑥 , 𝜎𝑥2 ) and 𝑌 ∼  (𝜇𝑦 , 𝜎𝑦2 ) be jointly normally distributed with correlation coefficient 𝜌𝑥𝑦 ∈ (−1, 1). If 𝑍 = max{𝑋, 𝑌 } show that the pdf of 𝑍 is 𝜌𝑥𝑦 𝜎𝑥 𝜌 𝜎 ⎛ ⎞ ⎛ 𝑧 − 𝜇𝑦 − 𝑥𝑦 𝑦 (𝑧 − 𝜇𝑥 ) ⎞ − (𝑧 − 𝜇𝑦 ) ⎟ 𝑧 − 𝜇 𝑥 ⎜ 𝜎 𝜎𝑥 ⎟ ⎜ 𝑦 ⎟ 𝑓 (𝑧) 𝑓𝑍 (𝑧) = Φ ⎜ √ √ ⎟ 𝑓𝑋 (𝑧) + Φ ⎜⎜ ⎟ 𝑌 2 ⎟ ⎜ 1 − 𝜌 𝜎𝑦 1 − 𝜌2𝑥𝑦 𝜎 𝑥 𝑥𝑦 ⎜ ⎟ ⎠ ⎝ ⎝ ⎠

1 √ 𝑒 𝜎𝑥 2𝜋 of a standard normal. Hence, show that where 𝑓𝑋 (𝑧) =

− 12

(

𝑧−𝜇𝑥 𝜎𝑥

(

)2

, 𝑓𝑌 (𝑧) =

1 √ 𝑒 𝜎𝑦 2𝜋

− 21

𝑧−𝜇𝑦 𝜎𝑦

)2

and Φ(⋅) denotes the cdf

}] [ { 𝔼 max 𝑒𝑍 − 𝐾, 0 =𝑒

𝜇𝑥 + 21 𝜎𝑥2

⎞ ⎛ −𝜎𝑥 + 𝜌𝑥𝑦 𝜎𝑦 ⎟ ⎜ 𝜇𝑥 + 𝜎𝑥2 − log 𝐾 −𝜇𝑦 + 𝜇𝑥 + 𝜎𝑥 (𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 ) 𝚽⎜ , √ ,√ ⎟ 𝜎 𝑥 ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟ ⎠ ⎝

𝜇𝑦 + 12 𝜎𝑦2

+𝑒

⎞ ⎛ 2 −𝜎𝑦 + 𝜌𝑥𝑦 𝜎𝑥 ⎟ ⎜ 𝜇𝑦 + 𝜎𝑦 − log 𝐾 −𝜇𝑥 + 𝜇𝑦 + 𝜎𝑦 (𝜎𝑦 − 𝜌𝑥𝑦 𝜎𝑥 ) 𝚽⎜ , √ ,√ ⎟ 𝜎𝑦 ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟ ⎠ ⎝

⎞ ⎡ ⎛ −𝜇𝑦 + 𝜇𝑥 −𝜎𝑥 + 𝜌𝑥𝑦 𝜎𝑦 ⎟ ⎢ ⎜ 𝜇𝑥 − log 𝐾 ,√ ,√ −𝐾 ⎢𝚽 ⎜ ⎟ 𝜎 𝑥 ⎢ ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟ ⎣ ⎝ ⎠ ⎞⎤ ⎛ −𝜇𝑥 + 𝜇𝑦 −𝜎𝑦 + 𝜌𝑥𝑦 𝜎𝑥 ⎟⎥ ⎜ 𝜇𝑦 − log 𝐾 ,√ ,√ + 𝚽⎜ ⎟⎥ 𝜎𝑦 ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟⎥ ⎠⎦ ⎝ where 𝐾 > 0, 𝔼 denotes the expectation with respect to the 𝑒𝑍 distribution and 𝚽 denotes the cumulative standard bivariate normal distribution function given as

𝚽(𝑢, 𝑣, 𝜌𝑢𝑣 ) =

where 𝜌𝑢𝑣 ∈ (−1, 1).

𝑢

𝑣

∫−∞ ∫−∞

√ 2𝜋

1 1 − 𝜌2𝑢𝑣

𝑒

−

1 (𝑥2 −2𝜌𝑢𝑣 𝑥𝑦+𝑦2 ) 2(1−𝜌2 𝑢𝑣 )

𝑑𝑦𝑑𝑥

6.2.1 Path-Independent Options

569

Let the asset prices 𝑆𝑡(1) and 𝑆𝑡(2) have the following diffusion processes 𝑑𝑆𝑡(1) = (𝜇1 − 𝐷1 )𝑆𝑡(1) 𝑑𝑡 + 𝜎1 𝑆𝑡(1) 𝑑𝑊𝑡(1) 𝑑𝑆𝑡(2) = (𝜇2 − 𝐷2 )𝑆𝑡(2) 𝑑𝑡 + 𝜎2 𝑆𝑡(2) 𝑑𝑊𝑡(2) 𝑑𝑊𝑡(1) ⋅ 𝑑𝑊𝑡(2) = 𝜌𝑑𝑡 where {𝑊𝑡(1) : 𝑡 ≥ 0} and {𝑊𝑡(2) : 𝑡 ≥ 0} are ℙ-standard Wiener processes on the probability space (Ω, ℱ, ℙ), 𝜇1 and 𝜇2 are the drift parameters, 𝐷1 and 𝐷2 are the continuous dividend yields, 𝜎1 and 𝜎2 are the volatilities, 𝜌 ∈ (−1, 1) is the correlation coefficient and let 𝑟 be the risk-free interest rate from a money-market account. Show that (𝑊𝑡(1) , 𝑊𝑡(2) ) follows a bivariate normal distribution. Given the payoff of a rainbow call on the maximum option is defined as { } Ψ(𝑆𝑇(1) , 𝑆𝑇(2) ) = max max{𝑆𝑇(1) , 𝑆𝑇(2) } − 𝐾, 0 where 𝐾 > 0 is the strike price, 𝑇 ≥ 𝑡 is the option expiry time, using the above results find the rainbow call on the maximum option price at time 𝑡 under the risk-neutral measure ℚ. Solution: The first part of the results follows from Problem 1.2.2.15 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus. Following Problem 6.2.1.9 (page 558), it can easily be shown that for 𝐾 > 0 }] [ { 𝔼 max 𝑒𝑍 − 𝐾, 0 =𝑒

+𝑒

𝜇𝑥 + 21 𝜎𝑥2

𝜇𝑦 + 12 𝜎𝑦2

⎞ ⎛ −𝜎𝑥 + 𝜌𝑥𝑦 𝜎𝑦 ⎟ ⎜ 𝜇𝑥 + 𝜎𝑥2 − log 𝐾 −𝜇𝑦 + 𝜇𝑥 + 𝜎𝑥 (𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 ) 𝚽⎜ , √ ,√ ⎟ 𝜎𝑥 ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟ ⎠ ⎝

⎞ ⎛ 2 −𝜎𝑦 + 𝜌𝑥𝑦 𝜎𝑥 ⎟ ⎜ 𝜇𝑦 + 𝜎𝑦 − log 𝐾 −𝜇𝑥 + 𝜇𝑦 + 𝜎𝑦 (𝜎𝑦 − 𝜌𝑥𝑦 𝜎𝑥 ) 𝚽⎜ , √ ,√ ⎟ 𝜎 𝑦 ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟ ⎠ ⎝

⎞ ⎡ ⎛ −𝜇𝑦 + 𝜇𝑥 −𝜎𝑥 + 𝜌𝑥𝑦 𝜎𝑦 ⎟ ⎢ ⎜ 𝜇𝑥 − log 𝐾 ,√ ,√ −𝐾 ⎢𝚽 ⎜ ⎟ 𝜎𝑥 ⎢ ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟ ⎣ ⎝ ⎠ ⎛ ⎞⎤ −𝜇𝑥 + 𝜇𝑦 −𝜎𝑦 + 𝜌𝑥𝑦 𝜎𝑥 ⎜ 𝜇𝑦 − log 𝐾 ⎟⎥ + 𝚽⎜ ,√ ,√ ⎟⎥ . 𝜎 𝑦 ⎜ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 ⎟⎥ ⎝ ⎠⎦ To show that (𝑊𝑡(1) , 𝑊𝑡(2) ) follows a bivariate normal distribution, see Problem 6.2.1.9 (page 558).

570

6.2.1 Path-Independent Options

The rainbow call on the maximum option at time 𝑡 under the risk-neutral measure ℚ is defined as 𝐶max (𝑆𝑡(1) , 𝑆𝑡(2) , 𝑡; 𝐾, 𝑇 )

=𝑒

−𝑟(𝑇 −𝑡) ℚ

𝔼

[

{ max

}| ] |ℱ | 𝑡 |

max{𝑆𝑇(1) , 𝑆𝑇(2) } − 𝐾, 0

where ̃ (1) 𝑑𝑆𝑡(1) = (𝑟 − 𝐷1 )𝑆𝑡(1) 𝑑𝑡 + 𝜎1 𝑆𝑡(1) 𝑑 𝑊 𝑡 ̃ (2) 𝑑𝑆𝑡(2) = (𝑟 − 𝐷2 )𝑆𝑡(2) 𝑑𝑡 + 𝜎2 𝑆𝑡(2) 𝑑 𝑊 𝑡

̃ (1) ⋅ 𝑑 𝑊 ̃ (2) = 𝜌𝑑𝑡 𝑑𝑊 𝑡 𝑡 ̃ (1) = 𝑊 (1) + such that 𝑊 𝑡 𝑡

(

Wiener processes.

𝜇1 − 𝑟 𝜎1

)

̃ (2) = 𝑊 (2) + 𝑡 and 𝑊 𝑡 𝑡

(1)

(

𝜇2 − 𝑟 𝜎2

) 𝑡 are ℚ-standard

(2)

By setting 𝑆𝑇(1) = 𝑒log 𝑆𝑇 and 𝑆𝑇(2) = 𝑒log 𝑆𝑇 we can write 𝐶max (𝑆𝑡(1) , 𝑆𝑡(2) , 𝑡; 𝐾, 𝑇 )

}| ] { { (1) (2) } log 𝑆𝑇 log 𝑆𝑇 − 𝐾, 0 || ℱ𝑡 =𝑒 𝔼 max max 𝑒 ,𝑒 | [ } { }| ] { (1) (2) max log 𝑆𝑇 ,log 𝑆𝑇 | = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝑒 − 𝐾, 0 | ℱ𝑡 | | [ ] { } | = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝑒𝑍 − 𝐾, 0 | ℱ𝑡 | −𝑟(𝑇 −𝑡) ℚ

[

(1)

(2)

where 𝑍 = max{𝑋, 𝑌 }, 𝑋 = 𝑒log 𝑆𝑇 and 𝑌 = 𝑒log 𝑆𝑇 such that (log 𝑆𝑇(1) , log 𝑆𝑇(2) ) follows a bivariate normal distribution. From It¯o’s formula we can easily show that [ ( log 𝑆𝑇(1) ∼  log 𝑆𝑡(1) + 𝑟 − 𝐷1 − [ ( log 𝑆𝑇(2) ∼  log 𝑆𝑡(2) + 𝑟 − 𝐷2 −

) ] 1 2 𝜎1 (𝑇 − 𝑡), 𝜎12 (𝑇 − 𝑡) 2 ) ] 1 2 𝜎2 (𝑇 − 𝑡), 𝜎22 (𝑇 − 𝑡) . 2

By setting ) ( 1 𝜇𝑥 = log 𝑆𝑡(1) + 𝑟 − 𝐷1 − 𝜎12 (𝑇 − 𝑡) 2 𝜎𝑥2 = 𝜎12 (𝑇 − 𝑡) ) ( 1 𝜇𝑦 = log 𝑆𝑡(2) + 𝑟 − 𝐷2 − 𝜎22 (𝑇 − 𝑡) 2 𝜎𝑦2 = 𝜎22 (𝑇 − 𝑡)

6.2.1 Path-Independent Options

571

and after some algebraic manipulations we have 1 2

𝑒𝜇𝑥 + 2 𝜎𝑥 = 𝑆𝑡(1) 𝑒(𝑟−𝐷1 )(𝑇 −𝑡) 1 2

𝑒𝜇𝑦 + 2 𝜎𝑦 = 𝑆𝑡(2) 𝑒(𝑟−𝐷2 )(𝑇 −𝑡)

𝜇𝑥 + 𝜎𝑥2 − log 𝐾

log(𝑆𝑡(1) ∕𝐾) + (𝑟 − 𝐷1 + 12 𝜎12 )(𝑇 − 𝑡) = √ 𝜎1 𝑇 − 𝑡

𝜇𝑦 + 𝜎𝑦2 − log 𝐾

log(𝑆𝑡(2) ∕𝐾) + (𝑟 − 𝐷2 + 12 𝜎22 )(𝑇 − 𝑡) = √ 𝜎2 𝑇 − 𝑡

𝜎𝑥

𝜎𝑦

( ) log(𝑆𝑡(1) ∕𝑆𝑡(2) ) − 𝐷1 − 𝐷2 − 12 (𝜎12 − 2𝜌𝜎1 𝜎2 + 𝜎22 ) (𝑇 − 𝑡) −𝜇𝑦 + 𝜇𝑥 + 𝜎𝑥 (𝜎𝑥 − 𝜌𝑥𝑦 𝜎𝑦 ) = √ √ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 (𝜎12 − 2𝜌𝜎1 𝜎2 + 𝜎22 )(𝑇 − 𝑡) ( ) (2) (1) 1 2 2 ) (𝑇 − 𝑡) ∕𝑆 ) − 𝐷 − 𝐷 − (𝜎 − 2𝜌𝜎 𝜎 + 𝜎 log(𝑆 2 1 1 2 −𝜇𝑥 + 𝜇𝑦 + 𝜎𝑦 (𝜎𝑦 − 𝜌𝑥𝑦 𝜎𝑥 ) 𝑡 𝑡 2 2 1 = √ √ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 (𝜎12 − 2𝜌𝜎1 𝜎2 + 𝜎22 )(𝑇 − 𝑡) −𝜎𝑥 + 𝜌𝑥𝑦 𝜎𝑦 −𝜎1 + 𝜌𝜎2 = √ √ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 (𝜎12 − 2𝜌𝜎1 𝜎2 + 𝜎22 )(𝑇 − 𝑡) −𝜎𝑦 + 𝜌𝑥𝑦 𝜎𝑥 −𝜎2 + 𝜌𝜎1 = √ √ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 (𝜎12 − 2𝜌𝜎1 𝜎2 + 𝜎22 )(𝑇 − 𝑡) log(𝑆𝑡(1) ∕𝐾) + (𝑟 − 𝐷1 − 12 𝜎12 )(𝑇 − 𝑡) 𝜇𝑥 − log 𝐾 = 𝜎𝑥 𝜎1 (𝑇 − 𝑡) 𝜇𝑦 − log 𝐾 𝜎𝑦

=

log(𝑆𝑡(2) ∕𝐾) + (𝑟 − 𝐷2 − 12 𝜎22 )(𝑇 − 𝑡) 𝜎2 (𝑇 − 𝑡)

( ) log(𝑆𝑡(1) ∕𝑆𝑡(2) ) − 𝐷1 − 𝐷2 + 12 (𝜎12 − 𝜎22 ) (𝑇 − 𝑡) = √ √ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 (𝜎12 − 2𝜌𝜎1 𝜎2 + 𝜎22 )(𝑇 − 𝑡) ( ) log(𝑆𝑡(2) ∕𝑆𝑡(1) ) − 𝐷2 − 𝐷1 + 12 (𝜎22 − 𝜎12 ) (𝑇 − 𝑡) −𝜇𝑥 + 𝜇𝑦 . = √ √ 𝜎𝑥2 − 2𝜌𝑥𝑦 𝜎𝑥 𝜎𝑦 + 𝜎𝑦2 (𝜎12 − 2𝜌𝜎1 𝜎2 + 𝜎22 )(𝑇 − 𝑡) −𝜇𝑦 + 𝜇𝑥

572

6.2.1 Path-Independent Options

By setting 𝜎 =

√ 𝜎12 − 2𝜌𝜎1 𝜎2 + 𝜎22 and substituting the above expressions into the

option price 𝐶max (𝑆𝑡(1) , 𝑆𝑡(2) , 𝑡; 𝐾, 𝑇 ), the rainbow call on the maximum option at time 𝑡 is therefore 𝐶max (𝑆𝑡(1) , 𝑆𝑡(2) , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡(1) 𝑒−𝐷1 (𝑇 −𝑡) 𝚽(𝛼+(1) , −𝛽+ , −𝜌(1) ) +𝑆𝑡(2) 𝑒−𝐷2 (𝑇 −𝑡) 𝚽(𝛼+(2) , −𝛽− , −𝜌(2) ) −𝐾𝑒−𝑟(𝑇 −𝑡) 𝚽(𝛼−(1) , −𝛾+ , −𝜌(1) ) −𝐾𝑒−𝑟(𝑇 −𝑡) 𝚽(𝛼−(2) , −𝛾− , −𝜌(2) ) where 𝛼±(1)

log(𝑆𝑡(1) ∕𝐾) + (𝑟 − 𝐷1 ± 12 𝜎12 )(𝑇 − 𝑡) = √ 𝜎1 𝑇 − 𝑡

± log(𝑆𝑡(2) ∕𝑆𝑡(1) ) ± (𝐷1 − 𝐷2 ∓ 12 𝜎 2 )(𝑇 − 𝑡) 𝛽± = √ 𝜎 𝑇 −𝑡

𝛼±(2) =

log(𝑆𝑡(2) ∕𝐾) + (𝑟 − 𝐷2 ± 12 𝜎22 )(𝑇 − 𝑡) √ 𝜎2 𝑇 − 𝑡

± log(𝑆𝑡(2) ∕𝑆𝑡(1) ) ± (𝐷1 − 𝐷2 ± 12 (𝜎12 − 𝜎22 ))(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡 𝜎1 − 𝜌𝜎2 = √ 𝜎 𝑇 −𝑡 𝜎2 − 𝜌𝜎1 = √ . 𝜎 𝑇 −𝑡

𝛾± = 𝜌(1) 𝜌(2)

11. Black–Scholes Equation for Cross-Currency Option. Let (Ω, ℱ, ℙ) be a probability space and let {𝑊𝑡𝑠 : 𝑡 ≥ 0} and {𝑊𝑡𝑥 : 𝑡 ≥ 0} be ℙ-standard Wiener processes such that 𝑑𝑊𝑡𝑠 ⋅ 𝑑𝑊𝑡𝑥 = 𝜌𝑑𝑡, 𝜌 ∈ (−1, 1). Suppose that 𝑆𝑡 denotes the asset price quoted in foreign currency having the following SDE 𝑑𝑆𝑡 = (𝜇𝑠 − 𝐷𝑠 ) 𝑑𝑡 + 𝜎𝑠 𝑑𝑊𝑡𝑠 𝑆𝑡 where 𝜇𝑠 is the drift parameter, 𝐷𝑠 is the continuous dividend yield, 𝜎𝑠 is the volatility parameter and let 𝑟𝑓 denote the foreign risk-free interest rate. Let 𝑋𝑡 be the foreign-todomestic exchange rate having the SDE 𝑑𝑋𝑡 = 𝜇𝑥 𝑑𝑡 + 𝜎𝑥 𝑑𝑊𝑡𝑥 𝑋𝑡

6.2.1 Path-Independent Options

573

where 𝜇𝑥 is the exchange rate drift, 𝜎𝑥 is the exchange rate volatility parameter and let 𝑟𝑑 be the domestic risk-free interest rate. Here, 𝑋𝑡 𝑆𝑡 is the foreign asset price quoted in domestic currency. By considering a hedging portfolio involving a cross-currency option 𝑉 (𝑆𝑡 , 𝑋𝑡 , 𝑡) denoted in domestic currency that can only be exercised at expiry time 𝑇 , 𝑡 ≤ 𝑇 , with asset price 𝑋𝑡 𝑆𝑡 and exchange rate 𝑋𝑡 , show that 𝑉 (𝑆𝑡 , 𝑋𝑡 , 𝑡) satisfies the two-dimensional PDE 1 𝜕2𝑉 𝜕2𝑉 𝜕2𝑉 1 𝜕𝑉 + 𝜌𝜎 𝜎 𝑋 𝑆 + 𝜎𝑠2 𝑆𝑡2 2 + 𝜎𝑥2 𝑋𝑡2 𝑥 𝑠 𝑡 𝑡 𝜕𝑡 2 2 𝜕𝑋𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑋𝑡2 +(𝑟𝑓 − 𝐷𝑠 − 𝜌𝜎𝑥 𝜎𝑠 )𝑆𝑡

𝜕𝑉 𝜕𝑉 + (𝑟𝑑 − 𝑟𝑓 )𝑋𝑡 − 𝑟𝑑 𝑉 (𝑆𝑡 , 𝑋𝑡 , 𝑡) = 0. 𝜕𝑆𝑡 𝜕𝑋𝑡

Solution: To eliminate both asset risk and exchange rate risk, at time 𝑡 we let the value of a portfolio Π𝑡 be Π𝑡 = 𝑉 (𝑆𝑡 , 𝑋𝑡 , 𝑡) − Δ1 (𝑋𝑡 𝑆𝑡 ) − Δ2 𝑋𝑡 where it involves buying one unit of cross-currency option 𝑉 (𝑆𝑡 , 𝑋𝑡 , 𝑡), selling Δ1 units of foreign assets 𝑋𝑡 𝑆𝑡 converted into domestic currency and selling Δ2 units of 𝑋𝑡 . Since we receive 𝐷𝑠 𝑆𝑡 𝑑𝑡 for every asset held, and because we hold −Δ1 𝑋𝑡 𝑆𝑡 , our portfolio changes by an amount −Δ𝐷𝑠 𝑋𝑡 𝑆𝑡 𝑑𝑡. In addition, given that 𝑋𝑡 will also grow at the foreign riskfree rate 𝑟𝑓 , the change in portfolio Π𝑡 is 𝑑Π𝑡 = 𝑑𝑉 − Δ1 (𝑑(𝑋𝑡 𝑆𝑡 ) + 𝐷𝑠 𝑋𝑡 𝑆𝑡 𝑑𝑡) − Δ2 (𝑑𝑋𝑡 + 𝑟𝑓 𝑋𝑡 𝑑𝑡) = 𝑑𝑉 − Δ1 (𝑆𝑡 𝑑𝑋𝑡 + 𝑋𝑡 𝑑𝑆𝑡 + 𝜌𝜎𝑥 𝜎𝑠 𝑋𝑡 𝑆𝑡 𝑑𝑡 + 𝐷𝑠 𝑋𝑡 𝑆𝑡 𝑑𝑡) −Δ2 (𝑑𝑋𝑡 + 𝑟𝑓 𝑋𝑡 𝑑𝑡) where 𝑑(𝑋𝑡 𝑆𝑡 ) = 𝑆𝑡 𝑑𝑋𝑡 + 𝑋𝑡 𝑑𝑆𝑡 + 𝑑𝑋𝑡 𝑑𝑆𝑡 and from It¯o’s lemma we can write 𝑑𝑋𝑡 𝑑𝑆𝑡 = 𝜌𝜎𝑥 𝜎𝑠 𝑋𝑡 𝑆𝑡 𝑑𝑡. Expanding 𝑉 (𝑆𝑡 , 𝑋𝑡 , 𝑡) using Taylor’s theorem 𝑑𝑉 =

𝜕𝑉 𝜕𝑉 𝜕𝑉 𝑑𝑆 + 𝑑𝑋𝑡 𝑑𝑡 + 𝜕𝑡 𝜕𝑆𝑡 𝑡 𝜕𝑋𝑡 ] [ 2𝑉 2𝑉 1 𝜕2𝑉 𝜕 𝜕 + (𝑑𝑆𝑡 )2 + (𝑑𝑋𝑡 )2 + 2 (𝑑𝑋𝑡 𝑑𝑆𝑡 ) + … 2 𝜕𝑆𝑡2 𝜕𝑋𝑡 𝜕𝑆𝑡 𝜕𝑋𝑡2

) ( and by substituting 𝑑𝑆𝑡 = 𝜇𝑠 − 𝐷𝑠 𝑆𝑡 𝑑𝑡 + 𝜎𝑠 𝑆𝑡 𝑑𝑊𝑡𝑠 , 𝑑𝑋𝑡 = 𝜇𝑥 𝑋𝑡 𝑑𝑡 + 𝜎𝑥 𝑋𝑡 𝑑𝑊𝑡𝑥 and subsequently applying It¯o’s lemma we have [ 𝑑𝑉 =

1 𝜕2𝑉 𝜕2𝑉 𝜕2𝑉 1 𝜕𝑉 + 𝜌𝜎𝑥 𝜎𝑠 𝑋𝑡 𝑆𝑡 + 𝜎𝑠2 𝑆𝑡2 2 + 𝜎𝑥2 𝑋𝑡2 2 𝜕𝑡 2 2 𝜕𝑋𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑋𝑡 ] 𝜕𝑉 𝜕𝑉 𝜕𝑉 𝜕𝑉 +(𝜇𝑠 − 𝐷𝑠 )𝑆𝑡 + 𝜇𝑥 𝑋𝑡 𝑑𝑊𝑡𝑠 + 𝜎𝑥 𝑋𝑡 𝑑𝑊𝑡𝑥 . 𝑑𝑡 + 𝜎𝑠 𝑆𝑡 𝜕𝑆𝑡 𝜕𝑋𝑡 𝜕𝑆𝑡 𝜕𝑋𝑡

574

6.2.1 Path-Independent Options

Substituting back into 𝑑Π𝑡 and rearranging terms, we have [ 𝜕𝑉 1 𝜕2𝑉 𝜕2𝑉 𝜕2 𝑉 1 + 𝜌𝜎𝑥 𝜎𝑠 𝑋𝑡 𝑆𝑡 + 𝜎𝑠2 𝑆𝑡2 2 + + 𝜎𝑥2 𝑋𝑡2 𝑑Π𝑡 = 2 𝜕𝑡 2 2 𝜕𝑋𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑋𝑡

𝜕𝑉 𝜕𝑉 + 𝜇𝑥 𝑋𝑡 − Δ1 (𝜇𝑥 + 𝜇𝑠 + 𝜌𝜎𝑥 𝜎𝑠 )𝑋𝑡 𝑆𝑡 𝜕𝑆𝑡 𝜕𝑋𝑡 ( ) ] 𝜕𝑉 −Δ2 (𝜇𝑥 + 𝑟𝑓 )𝑋𝑡 𝑑𝑡 + 𝜎𝑠 − Δ1 𝑋𝑡 𝑆𝑡 𝑑𝑊𝑡𝑠 𝜕𝑆𝑡 ( ) 𝜕𝑉 +𝜎𝑥 − Δ1 𝑆𝑡 − Δ2 𝑋𝑡 𝑑𝑊𝑡𝑥 . 𝜕𝑋𝑡 +(𝜇𝑠 − 𝐷𝑠 )𝑆𝑡

To eliminate the 𝑑𝑊𝑡𝑠 and 𝑑𝑊𝑡𝑥 terms we have Δ1 = which leads to

1 𝜕𝑉 1 and Δ2 = 𝑋𝑡 𝜕𝑆𝑡 𝑋𝑡

( 𝑋𝑡

𝜕𝑉 𝜕𝑉 − 𝑆𝑡 𝜕𝑋𝑡 𝜕𝑆𝑡

)

[

𝑑Π𝑡 =

𝜕𝑉 1 𝜕2𝑉 𝜕2 𝑉 𝜕2𝑉 1 + 𝜌𝜎𝑥 𝜎𝑠 𝑋𝑡 𝑆𝑡 + 𝜎𝑠2 𝑆𝑡2 2 + 𝜎𝑥2 𝑋𝑡2 2 𝜕𝑡 2 2 𝜕𝑋𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑋𝑡 ] 𝜕𝑉 𝜕𝑉 +(𝑟𝑓 − 𝐷𝑠 − 𝜌𝜎𝑥 𝜎𝑠 )𝑆𝑡 − 𝑟𝑓 𝑋𝑡 𝑑𝑡. 𝜕𝑆𝑡 𝜕𝑋𝑡

Under the no-arbitrage condition the return on the amount Π𝑡 invested in a risk-free interest rate in domestic currency would see a growth of 𝑑Π𝑡 = 𝑟𝑑 Π𝑡 𝑑𝑡 = 𝑟𝑑 [𝑉 (𝑆𝑡 , 𝑋𝑡 , 𝑡) − Δ1 (𝑋𝑡 𝑆𝑡 ) − Δ2 𝑋𝑡 ]𝑑𝑡 [ ] 𝜕𝑉 = 𝑟𝑑 𝑉 (𝑆𝑡 , 𝑋𝑡 , 𝑡) − 𝑋𝑡 𝑑𝑡 𝜕𝑋𝑡 and hence we have

[ 𝑟𝑑 Π𝑡 𝑑𝑡 =

𝜕𝑉 1 𝜕2𝑉 𝜕2𝑉 1 + 𝜎𝑠2 𝑆𝑡2 2 + 𝜎𝑥2 𝑋𝑡2 𝜕𝑡 2 2 𝜕𝑆𝑡 𝜕𝑋𝑡2

𝜕2𝑉 𝜕𝑉 +𝜌𝜎𝑥 𝜎𝑠 𝑋𝑡 𝑆𝑡 + (𝑟𝑓 − 𝐷𝑠 − 𝜌𝜎𝑥 𝜎𝑠 )𝑆𝑡 𝜕𝑋 𝜕𝑆 𝜕𝑆𝑡 ] 𝑡 𝑡 𝜕𝑉 −𝑟𝑓 𝑋𝑡 𝑑𝑡 𝜕𝑋𝑡 [ [ ] 1 𝜕𝑉 𝜕2𝑉 𝜕2𝑉 1 𝜕𝑉 𝑟𝑑 𝑉 (𝑆𝑡 , 𝑋𝑡 , 𝑡) − 𝑋𝑡 + 𝜎𝑠2 𝑆𝑡2 2 + 𝜎𝑥2 𝑋𝑡2 𝑑𝑡 = 𝜕𝑋𝑡 𝜕𝑡 2 2 𝜕𝑆𝑡 𝜕𝑋𝑡2 𝜕2𝑉 𝜕𝑉 +𝜌𝜎𝑥 𝜎𝑠 𝑋𝑡 𝑆𝑡 + (𝑟𝑓 − 𝐷𝑠 − 𝜌𝜎𝑥 𝜎𝑠 )𝑆𝑡 𝜕𝑋𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 ] 𝜕𝑉 −𝑟𝑓 𝑋𝑡 𝑑𝑡 𝜕𝑋𝑡

6.2.1 Path-Independent Options

575

and finally 1 𝜕2𝑉 𝜕2𝑉 𝜕2𝑉 1 𝜕𝑉 + 𝜌𝜎𝑥 𝜎𝑠 𝑋𝑡 𝑆𝑡 + 𝜎𝑠2 𝑆𝑡2 2 + 𝜎𝑥2 𝑋𝑡2 2 𝜕𝑡 2 2 𝜕𝑋𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑋𝑡 +(𝑟𝑓 − 𝐷𝑠 − 𝜌𝜎𝑥 𝜎𝑠 )𝑆𝑡

𝜕𝑉 𝜕𝑉 + (𝑟𝑑 − 𝑟𝑓 )𝑋𝑡 − 𝑟𝑑 𝑉 (𝑆𝑡 , 𝑋𝑡 , 𝑡) = 0 𝜕𝑆𝑡 𝜕𝑋𝑡

which is a two-dimensional PDE. 12. Cross-Currency Option (PDE Approach). Let (Ω, ℱ , ℙ) be a probability space and let {𝑊𝑡𝑠 : 𝑡 ≥ 0} and {𝑊𝑡𝑥 : 𝑡 ≥ 0} be ℙ-standard Wiener processes. Let 𝑆𝑡 and 𝑋𝑡 denote the asset price quoted in foreign currency and the foreign-to-domestic exchange rate respectively each having the following SDEs 𝑑𝑆𝑡 = (𝜇𝑠 − 𝐷𝑠 )𝑑𝑡 + 𝜎𝑠 𝑑𝑊𝑡𝑠 𝑆𝑡 𝑑𝑋𝑡 = 𝜇𝑥 𝑑𝑡 + 𝜎𝑥 𝑑𝑊𝑡𝑥 𝑋𝑡 𝑑𝑊𝑡𝑠 ⋅ 𝑑𝑊𝑡𝑥 = 𝜌𝑑𝑡, 𝜌 ∈ (−1, 1) where 𝜇𝑠 is the asset drift parameter, 𝐷𝑠 is the asset continuous dividend yield, 𝜎𝑠 is the asset volatility parameter, 𝜇𝑥 is exchange rate drift, 𝜎𝑥 is the exchange rate volatility parameter and let 𝑟𝑓 be the foreign risk-free interest rate and 𝑟𝑑 be the domestic risk-free interest rate. For a strike price 𝐾 > 0 and expiry time 𝑇 , let 𝐶(𝑆𝑡 , 𝑋𝑡 , 𝑡; 𝐾, 𝑇 ) be the European-style call option price at time 𝑡 ≤ 𝑇 denoted in domestic currency satisfying the two-dimensional partial differential equation 𝜕2𝐶 𝜕𝐶 1 2 2 𝜕 2 𝐶 1 2 2 𝜕 2 𝐶 + 𝜎𝑥 𝑋𝑡 + 𝜌𝜎𝑥 𝜎𝑠 𝑋𝑡 𝑆𝑡 + 𝜎𝑠 𝑆𝑡 2 2 𝜕𝑡 2 2 𝜕𝑋𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑋𝑡 +(𝑟𝑓 − 𝐷𝑠 − 𝜌𝜎𝑥 𝜎𝑠 )𝑆𝑡

𝜕𝐶 𝜕𝐶 + (𝑟𝑑 − 𝑟𝑓 )𝑋𝑡 − 𝑟𝑑 𝐶(𝑆𝑡 , 𝑋𝑡 , 𝑡; 𝐾, 𝑇 ) = 0. 𝜕𝑆𝑡 𝜕𝑋𝑡

Using the above partial differentiation equation find the call option price at time 𝑡 ≤ 𝑇 for each of the payoffs at expiry time 𝑇 : (a) A Foreign Equity Option Converted to Domestic Currency Ψ(𝑆𝑇 , 𝑋𝑇 ) = 𝑋𝑇 max{𝑆𝑇 − 𝐾𝑠 , 0} (b) A Foreign Equity Option Struck in Domestic Currency (or Compo Option) ̂ 0} Ψ(𝑆𝑇 , 𝑋𝑇 ) = max{𝑋𝑇 𝑆𝑇 − 𝐾, (c) A Foreign Equity Option Struck in Pre-Determined Domestic Currency (or Quanto Option) Ψ(𝑆𝑇 , 𝑋𝑇 ) = 𝑋 max{𝑆𝑇 − 𝐾𝑠 , 0}

576

6.2.1 Path-Independent Options

(d) An FX Option Denoted in Domestic Currency Ψ(𝑆𝑇 , 𝑋𝑇 ) = 𝑆𝑇 max{𝑋𝑇 − 𝐾𝑥 , 0} where at expiry time 𝑇 , 𝑆𝑇 is the asset price in foreign currency, 𝑋𝑇 is the foreign-tô is the strike price domestic exchange rate, 𝐾𝑠 is the strike price in foreign currency, 𝐾 in domestic currency, 𝐾𝑥 is the strike price on the exchange rate and 𝑋 is some predetermined fixed exchange rate. Solution: (a) For the payoff Ψ(𝑆𝑇 , 𝑋𝑇 ) = 𝑋𝑇 max{𝑆𝑇 − 𝐾𝑠 , 0} we can set 𝐶(𝑆𝑡 , 𝑋𝑡 , 𝑡; 𝐾𝑠 , 𝑇 ) = 𝑋𝑡 𝑓 (𝑆𝑡 , 𝑡) where 𝑓 (𝑆𝑇 , 𝑇 ) = max{𝑆𝑇 − 𝐾𝑠 , 0}. With a change of variables we have 𝜕𝑓 𝜕𝐶 = 𝑋𝑡 ; 𝜕𝑡 𝜕𝑡

𝜕𝑓 𝜕𝐶 = 𝑋𝑡 ; 𝜕𝑆𝑡 𝜕𝑆𝑡

𝜕2𝑓 𝜕2𝐶 = 𝑋𝑡 2 ; 2 𝜕𝑆𝑡 𝜕𝑆𝑡

𝜕2𝐶 = 0; 𝜕𝑋𝑡2

𝜕𝐶 = 𝑓 (𝑆𝑡 , 𝑡) 𝜕𝑋𝑡 𝜕𝑓 𝜕2𝐶 = 𝜕𝑋𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡

and by substituting the above results into the two-dimensional partial differential equation we have 𝑋𝑡

𝜕𝑓 1 2 2 𝜕 2 𝑓 𝜕𝑓 𝜕𝑓 + (𝑟𝑓 − 𝐷𝑠 − 𝜌𝜎𝑥 𝜎𝑠 )𝑋𝑡 𝑆𝑡 + 𝜎𝑠 𝑆𝑡 𝑋𝑡 2 + 𝜌𝜎𝑥 𝜎𝑠 𝑋𝑡 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡

+(𝑟𝑑 − 𝑟𝑓 )𝑋𝑡 𝑓 (𝑆𝑡 , 𝑡) − 𝑟𝑑 𝑋𝑡 𝑓 (𝑆𝑡 , 𝑡) = 0 or

𝜕𝑓 𝜕𝑓 1 2 2 𝜕 2 𝑓 + (𝑟𝑓 − 𝐷𝑠 )𝑆𝑡 − 𝑟𝑓 𝑓 (𝑆𝑡 , 𝑡) = 0 + 𝜎𝑠 𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 which is a Black–Scholes equation with volatility 𝜎𝑠 , foreign risk-free interest rate 𝑟𝑓 and continuous dividend yield 𝐷𝑠 . Given that 𝑓 (𝑆𝑇 , 𝑇 ) = max{𝑆𝑇 − 𝐾𝑠 , 0} is a European call option payoff therefore we can deduce that 𝑓 (𝑆𝑡 , 𝑡) = 𝑆𝑡 𝑒−𝐷𝑠 (𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑠 𝑒−𝑟𝑓 (𝑇 −𝑡) Φ(𝑑− ) log(𝑆𝑡 ∕𝐾𝑠 ) + (𝑟𝑓 − 𝐷𝑠 ± 12 𝜎𝑠2 )(𝑇 − 𝑡) and Φ(⋅) is the cumulative distriwhere 𝑑± = √ 𝜎𝑠 𝑇 − 𝑡 bution function of a standard normal. Hence, the call option price at time 𝑡 is [ ] 𝐶(𝑆𝑡 , 𝑋𝑡 , 𝑡; 𝐾𝑠 , 𝑇 ) = 𝑋𝑡 𝑆𝑡 𝑒−𝐷𝑠 (𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑠 𝑒−𝑟𝑓 (𝑇 −𝑡) Φ(𝑑− ) .

6.2.1 Path-Independent Options

577

̂ 0} we define 𝑆̂𝑡 = 𝑋𝑡 𝑆𝑡 and we consider (b) For payoff Ψ(𝑆𝑇 , 𝑋𝑇 ) = max{𝑋𝑇 𝑆𝑇 − 𝐾, the option price at time 𝑡 as ̂ 𝑇 ) = 𝑔(𝑆̂𝑡 , 𝑡) 𝐶(𝑆𝑡 , 𝑋𝑡 , 𝑡; 𝐾, ̂ 0}. From the change of variables we can write where 𝑔(𝑆̂𝑇 , 𝑇 ) = max{𝑋𝑇 𝑆𝑇 − 𝐾, 𝜕𝑔 𝜕𝐶 = ; 𝜕𝑡 𝜕𝑡

𝜕𝑔 𝜕 𝑆̂𝑡 𝜕𝑔 𝜕𝐶 = = 𝑋𝑡 ; 𝜕𝑆𝑡 𝜕𝑆 ̂ 𝜕 𝑆𝑡 𝑡 𝜕 𝑆̂𝑡

𝜕 𝜕2𝐶 = 𝜕𝑆𝑡 𝜕𝑆𝑡2

(

𝜕 𝜕2𝐶 = 𝜕𝑋𝑡 𝜕𝑋𝑡2 𝜕 𝜕2𝐶 = 𝜕𝑋𝑡 𝜕𝑆𝑡 𝜕𝑋𝑡

𝜕𝑔 𝑋𝑡 𝜕 𝑆̂𝑡

(

)

𝜕𝑔 𝑆𝑡 𝜕 𝑆̂𝑡

(

𝜕𝑔 𝑋𝑡 𝜕 𝑆̂𝑡

) =

= 𝑋𝑡

𝜕𝑔 𝜕 𝑆̂𝑡 𝜕𝑔 𝜕𝐶 = = 𝑆𝑡 𝜕𝑋𝑡 𝜕𝑋 ̂ 𝑡 𝜕 𝑆𝑡 𝜕 𝑆̂𝑡

𝜕 2 𝑔 𝜕 𝑆̂𝑡 𝜕2𝑔 = 𝑋𝑡2 ; 𝜕 𝑆̂2 𝜕𝑆𝑡 𝜕 𝑆̂2 𝑡

) = 𝑆𝑡

𝑡

𝜕 2 𝑔 𝜕 𝑆̂ 𝜕2 𝑔 = 𝑆𝑡2 𝜕 𝑆̂2 𝜕𝑋𝑡 𝜕 𝑆̂2 𝑡

𝑡

𝜕𝑔 𝜕𝑔 𝜕 2 𝑔 𝜕 𝑆̂𝑡 𝜕2𝑔 + 𝑋𝑡 = + 𝑋𝑡 𝑆𝑡 . 𝜕 𝑆̂𝑡 𝜕 𝑆̂𝑡2 𝜕𝑋𝑡 𝜕 𝑆̂𝑡 𝜕 𝑆̂𝑡2

By substituting the above results into the two-dimensional partial differentiation equation we have 𝜕2𝑔 𝜕2𝑔 𝜕2𝑔 𝜕𝑔 1 2 1 + 𝜎𝑥2 (𝑋𝑡 𝑆𝑡 )2 + 𝜌𝜎𝑥 𝜎𝑠 (𝑋𝑡 𝑆𝑡 )2 + 𝜎𝑠 (𝑋𝑡 𝑆𝑡 )2 𝜕𝑡 2 𝜕 𝑆̂2 2 𝜕 𝑆̂2 𝜕 𝑆̂2 𝑡

𝑡

𝑡

𝜕𝑔 𝜕𝑔 𝜕𝑔 +(𝑟𝑓 − 𝐷𝑠 − 𝜌𝜎𝑥 𝜎𝑠 )𝑋𝑡 𝑆𝑡 + 𝜌𝜎𝑥 𝜎𝑠 𝑋𝑡 𝑆𝑡 + (𝑟𝑑 − 𝑟𝑓 )𝑋𝑡 𝑆𝑡 ̂ ̂ 𝜕 𝑆𝑡 𝜕 𝑆𝑡 𝜕 𝑆̂𝑡 −𝑟𝑑 𝑔(𝑆̂𝑡 , 𝑡) = 0 or 𝜕2𝑔 𝜕𝑔 𝜕𝑔 1 2 + (𝑟𝑑 − 𝐷𝑠 )𝑆̂𝑡 − 𝑟𝑑 𝑔(𝑆̂𝑡 , 𝑡) = 0 + (𝜎𝑥 + 2𝜌𝜎𝑥 𝜎𝑠 + 𝜎𝑠2 )𝑆̂𝑡2 2 𝜕𝑡 2 𝜕 𝑆̂ 𝜕 𝑆̂ 𝑡

𝑡

√ which is a Black–Scholes equation with volatility 𝜎𝑔 = 𝜎𝑥2 + 2𝜌𝜎𝑥 𝜎𝑠 + 𝜎𝑠2 , domestic risk-free interest rate 𝑟𝑑 and continuous dividend yield 𝐷𝑠 . Since ̂ 0} 𝐶(𝑆𝑇 , 𝑋𝑇 , 𝑇 ; 𝐾𝑥 , 𝑇 ) = 𝑔(𝑆̂𝑇 , 𝑇 ) = max{𝑆̂𝑇 − 𝐾,

578

6.2.1 Path-Independent Options

is the payoff of a European call option therefore the call option at time 𝑡 is ̂ 𝑇 ) = 𝑆̂𝑡 𝑒−𝐷𝑠 (𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑒 ̂ −𝑟𝑑 (𝑇 −𝑡) Φ(𝑑− ) 𝐶(𝑆𝑡 , 𝑋𝑡 , 𝑡; 𝐾, ̂ −𝑟𝑑 (𝑇 −𝑡) Φ(𝑑− ) = 𝑋𝑡 𝑆𝑡 𝑒−𝐷𝑠 (𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑒 ̂ + (𝑟𝑑 − 𝐷𝑠 ± 1 𝜎 2 )(𝑇 − 𝑡) log((𝑋𝑡 𝑆𝑡 )∕𝐾) 2 𝑔 and Φ(⋅) is the cumulative diswhere 𝑑± = √ 𝜎𝑔 𝑇 − 𝑡 tribution function of a standard normal. (c) For the payoff Ψ(𝑆𝑇 , 𝑋𝑇 ) = 𝑋 max{𝑆𝑇 − 𝐾𝑠 , 0} we can write 𝐶(𝑆𝑡 , 𝑋𝑡 , 𝑡; 𝐾𝑠 , 𝑇 ) = 𝑋ℎ(𝑆𝑡 , 𝑡) such that ℎ(𝑆𝑇 , 𝑇 ) = max{𝑆𝑇 − 𝐾𝑠 , 0}. Hence, we can express 𝜕𝐶 𝜕ℎ 𝜕ℎ 𝜕𝐶 𝜕𝐶 =𝑋 ; =0 =𝑋 ; 𝜕𝑡 𝜕𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑋𝑡 𝜕2𝐶 𝜕2𝐶 𝜕2𝐶 𝜕2𝐶 = 𝑋 2; = 0; = 0. 2 2 𝜕𝑋𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑋𝑡 By substituting the above expressions into the two-dimensional partial differentiation equation we have 𝑋

𝜕ℎ 1 2 2 𝜕 2 ℎ 𝜕ℎ − 𝑟𝑑 𝑋ℎ(𝑆𝑡 , 𝑡) = 0 + 𝜎𝑠 𝑆𝑡 𝑋 2 + (𝑟𝑓 − 𝐷𝑠 − 𝜌𝜎𝑥 𝜎𝑠 )𝑆𝑡 𝑋 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡

or 𝜕ℎ 1 2 2 𝜕 2 ℎ 𝜕ℎ + (𝑟𝑓 − 𝐷𝑠 − 𝜌𝜎𝑥 𝜎𝑠 )𝑆𝑡 − 𝑟𝑑 ℎ(𝑆𝑡 , 𝑡) = 0 + 𝜎𝑠 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 which is a Black–Scholes equation with volatility 𝜎𝑠 , domestic risk-free interest rate 𝑟𝑑 and continuous dividend yield 𝐷ℎ = 𝑟𝑑 − 𝑟𝑓 + 𝐷𝑠 + 𝜌𝜎𝑥 𝜎𝑠 . Given that ℎ(𝑆𝑇 , 𝑇 ) = max{𝑆𝑇 − 𝐾𝑠 , 0} is a European call option payoff therefore we can deduce that ℎ(𝑆𝑡 , 𝑡) = 𝑆𝑡 𝑒−𝐷ℎ (𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑠 𝑒−𝑟𝑑 (𝑇 −𝑡) Φ(𝑑− ) log(𝑆𝑡 ∕𝐾𝑠 ) + (𝑟𝑑 − 𝐷ℎ ± 12 𝜎𝑠2 )(𝑇 − 𝑡) and Φ(⋅) is the cumulative distriwhere 𝑑± = √ 𝜎𝑠 𝑇 − 𝑡 bution function of a standard normal. Hence, the call option price at time 𝑡 is [ ] 𝐶(𝑆𝑡 , 𝑋𝑡 , 𝑡; 𝐾𝑠 , 𝑇 ) = 𝑋 𝑆𝑡 𝑒−𝐷ℎ (𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑠 𝑒−𝑟𝑑 (𝑇 −𝑡) Φ(𝑑− ) .

6.2.1 Path-Independent Options

579

(d) For payoff Ψ(𝑆𝑇 , 𝑋𝑇 ) = 𝑆𝑇 max{𝑋𝑇 − 𝐾𝑥 , 0} we can define 𝐶(𝑆𝑡 , 𝑋𝑡 , 𝑡; 𝐾𝑥 , 𝑇 ) = 𝑆𝑡 𝑢(𝑋𝑡 , 𝑡) where 𝑢(𝑋𝑇 , 𝑇 ) = max{𝑋𝑇 − 𝐾𝑥 , 0}. From the change of variables we can set 𝜕𝐶 𝜕𝑢 = 𝑆𝑡 ; 𝜕𝑡 𝜕𝑡 𝜕2𝐶 = 0; 𝜕𝑆𝑡2

𝜕𝐶 = 𝑢(𝑋𝑡 , 𝑡), 𝜕𝑆𝑡 𝜕2 𝐶 𝜕2 𝑢 = 𝑆 , 𝑡 𝜕𝑋𝑡2 𝜕𝑋𝑡2

𝜕𝐶 𝜕𝑢 = 𝑆𝑡 𝜕𝑋𝑡 𝜕𝑋𝑡 𝜕2𝑢 𝜕𝑢 = . 𝜕𝑋𝑡 𝜕𝑆𝑡 𝜕𝑋𝑡

By substituting the above results into the two-dimensional partial differentiation equation we have 𝑆𝑡

𝜕𝑢 1 2 2 𝜕 2 𝑢 𝜕𝑢 + 𝜌𝜎𝑥 𝜎𝑠 𝑋𝑡 𝑆𝑡 + (𝑟𝑓 − 𝐷𝑠 − 𝜌𝜎𝑥 𝜎𝑠 )𝑆𝑡 𝑢(𝑋𝑡 , 𝑡) + 𝜎𝑥 𝑋𝑡 𝑆𝑡 2 𝜕𝑡 2 𝜕𝑋𝑡 𝜕𝑋𝑡

+(𝑟𝑑 − 𝑟𝑓 )𝑋𝑡 𝑆𝑡

𝜕𝑢 − 𝑟𝑑 𝑆𝑡 𝑢(𝑋𝑡 , 𝑡) = 0 𝜕𝑋𝑡

or 𝜕𝑢 𝜕𝑢 1 2 2 𝜕 2 𝑢 + (𝑟𝑑 − 𝑟𝑓 + 𝜌𝜎𝑥 𝜎𝑠 ) − (𝑟𝑑 − 𝑟𝑓 + 𝐷𝑠 + 𝜌𝜎𝑥 𝜎𝑠 )𝑢(𝑋𝑡 , 𝑡) = 0 + 𝜎 𝑋 𝜕𝑡 2 𝑥 𝑡 𝜕𝑋𝑡2 𝜕𝑋𝑡 which is a Black–Scholes equation with volatility 𝜎𝑥 , risk-free interest rate 𝑟𝑢 = 𝑟𝑑 − 𝑟𝑓 + 𝐷𝑠 + 𝜌𝜎𝑥 𝜎𝑠 and continuous dividend yield 𝐷𝑠 . Since 𝑢(𝑋𝑇 , 𝑇 ) = max{𝑋𝑇 − 𝐾𝑥 , 0} is the payoff of a European call option therefore 𝑢(𝑋𝑡 , 𝑡) = 𝑋𝑡 𝑒−𝐷𝑠 (𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑥 𝑒−𝑟𝑢 (𝑇 −𝑡) Φ(𝑑− ) log(𝑋𝑡 ∕𝐾𝑥 ) + (𝑟𝑢 − 𝐷𝑠 ± 12 𝜎𝑥2 )(𝑇 − 𝑡) and Φ(⋅) is the cumulative distriwhere 𝑑± = √ 𝜎𝑥 𝑇 − 𝑡 bution function of a standard normal. Hence, the call option price at time 𝑡 is [ ] 𝐶(𝑆𝑡 , 𝑋𝑡 , 𝑡; 𝐾𝑥 , 𝑇 ) = 𝑆𝑡 𝑋𝑡 𝑒−𝐷𝑠 (𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑥 𝑒−𝑟𝑢 (𝑇 −𝑡) Φ(𝑑− ) .

13. Cross-Currency Option (Probabilistic Approach). Let (Ω, ℱ , ℙ) be a probability space and let {𝑊𝑡𝑠 : 𝑡 ≥ 0} and {𝑊𝑡𝑥 : 𝑡 ≥ 0} be ℙ-standard Wiener processes. Let 𝑆𝑡 and 𝑋𝑡

580

6.2.1 Path-Independent Options

denote the asset price quoted in foreign currency and the foreign-to-domestic exchange rate respectively each having the following SDEs 𝑑𝑆𝑡 = (𝜇𝑠 − 𝐷𝑠 )𝑑𝑡 + 𝜎𝑠 𝑑𝑊𝑡𝑠 𝑆𝑡 𝑑𝑋𝑡 = 𝜇𝑥 𝑑𝑡 + 𝜎𝑥 𝑑𝑊𝑡𝑥 𝑋𝑡 𝑑𝑊𝑡𝑠 ⋅ 𝑑𝑊𝑡𝑥 = 𝜌𝑑𝑡, 𝜌 ∈ (−1, 1) where 𝜇𝑠 is the asset drift parameter, 𝐷𝑠 is the asset continuous dividend yield, 𝜎𝑠 is the asset volatility parameter, 𝜇𝑥 is exchange rate drift, 𝜎𝑥 is the exchange rate volatility parameter and let 𝑟𝑓 be the foreign risk-free interest rate and 𝑟𝑑 be the domestic risk-free interest rate. Show that under the domestic risk-neutral measure ℚ𝑑 , the diffusion processes for the exchange rate 𝑋𝑡 , asset price denominated in foreign currency 𝑆𝑡 , asset price denominated in domestic currency 𝑋𝑡 𝑆𝑡 are 𝑑𝑋𝑡 𝑥 = (𝑟𝑑 − 𝑟𝑓 )𝑑𝑡 + 𝜎𝑥 𝑑𝑊𝑡 𝑑 𝑋𝑡 𝑑𝑆𝑡 𝑠 = (𝑟𝑓 − 𝜌𝜎𝑥 𝜎𝑠 − 𝐷𝑠 )𝑑𝑡 + 𝜎𝑠 𝑑𝑊𝑡 𝑑 𝑆𝑡 √ 𝑑(𝑋𝑡 𝑆𝑡 ) 𝑥𝑠 = (𝑟𝑑 − 𝐷𝑠 )𝑑𝑡 + 𝜎𝑥2 + 2𝜌𝜎𝑥 𝜎𝑠 + 𝜎𝑠2 𝑑𝑊𝑡 𝑑 𝑋𝑡 𝑆𝑡 𝑥

𝑠

𝑥𝑠

where 𝑊𝑡 𝑑 , 𝑊𝑡 𝑑 and 𝑊𝑡 𝑑 are ℚ𝑑 -standard Wiener processes. Find the call option price at time 𝑡 ≤ 𝑇 for each of the payoffs (denominated in domestic currency) at expiry time 𝑇 : (a) A Foreign Equity Option Converted to Domestic Currency Ψ(𝑆𝑇 , 𝑋𝑇 ) = 𝑋𝑇 max{𝑆𝑇 − 𝐾𝑠 , 0} (b) A Foreign Equity Option Struck in Domestic Currency (or Compo Option) ̂ 0} Ψ(𝑆𝑇 , 𝑋𝑇 ) = max{𝑋𝑇 𝑆𝑇 − 𝐾, (c) A Foreign Equity Option Struck in Pre-Determined Domestic Currency (or Quanto Option) Ψ(𝑆𝑇 , 𝑋𝑇 ) = 𝑋 max{𝑆𝑇 − 𝐾𝑠 , 0} (d) An FX Option Denoted in Domestic Currency Ψ(𝑆𝑇 , 𝑋𝑇 ) = 𝑆𝑇 max{𝑋𝑇 − 𝐾𝑥 , 0} where 𝑆𝑇 is the asset price in foreign currency, 𝑋𝑇 is the foreign-to-domestic exchange ̂ is the strike price in domestic currate, 𝐾𝑠 is the strike price in foreign currency, 𝐾 rency, 𝐾𝑥 is the strike price on the exchange rate and 𝑋 is some pre-determined fixed exchange rate.

6.2.1 Path-Independent Options

581

Solution: To show the diffusion processes of 𝑋𝑡 and 𝑋𝑡 𝑆𝑡 under the domestic risk-neutral measure ℚ𝑑 see Problems 4.2.3.15 and 4.2.3.17 of Problems and Solutions of Mathematical Finance, Volume 1: Stochastic Calculus. For the case of asset price, let the diffusion process of 𝑆𝑡 under ℚ𝑑 -measure be 𝑑𝑆𝑡 𝑠 = 𝜇𝑠𝑑 𝑑𝑡 + 𝜎𝑠 𝑑𝑊𝑡 𝑑 𝑆𝑡 𝑠

where 𝜇𝑠𝑑 is the drift and 𝑊𝑡 𝑑 is the ℚ𝑑 -standard Wiener process. Since under the ℚ𝑑 measure 𝑑𝑋𝑡 𝑥 = (𝑟𝑑 − 𝑟𝑓 )𝑑𝑡 + 𝜎𝑥 𝑑𝑊𝑡 𝑑 𝑋𝑡 √ 𝑑(𝑋𝑡 𝑆𝑡 ) 𝑥𝑠 = (𝑟𝑑 − 𝐷𝑠 )𝑑𝑡 + 𝜎𝑥2 + 2𝜌𝜎𝑥 𝜎𝑠 + 𝜎𝑠2 𝑑𝑊𝑡 𝑑 𝑋𝑡 𝑆𝑡 𝑥𝑑

where 𝑊𝑡

𝑥𝑠𝑑

and 𝑊𝑡

are ℚ𝑑 -standard Wiener processes, then from It¯o’s lemma,

𝑑(𝑋𝑡 𝑆𝑡 ) = 𝑋𝑡 𝑑𝑆𝑡 + 𝑆𝑡 𝑑𝑋𝑡 + (𝑑𝑋𝑡 )(𝑑𝑆𝑡 ) 𝑠 = 𝜇𝑠𝑑 𝑋𝑡 𝑆𝑡 𝑑𝑡 + 𝜎𝑠 𝑋𝑡 𝑆𝑡 𝑑𝑊𝑡 𝑑

𝑥𝑑

+(𝑟𝑑 − 𝑟𝑓 )𝑋𝑡 𝑆𝑡 𝑑𝑡 + 𝜎𝑥 𝑋𝑡 𝑆𝑡 𝑑𝑊𝑡

+ 𝜌𝜎𝑥 𝜎𝑠 𝑑𝑡 √ 𝑥𝑠 = (𝜇𝑠𝑑 + 𝑟𝑑 − 𝑟𝑓 + 𝜌𝜎𝑥 𝜎𝑠 )𝑋𝑡 𝑆𝑡 𝑑𝑡 + 𝜎𝑥2 + 2𝜌𝜎𝑥 𝜎𝑠 + 𝜎𝑠2 𝑋𝑡 𝑆𝑡 𝑑𝑊𝑡 𝑑 𝑥

𝑠

𝜎𝑥 𝑊 𝑑 + 𝜎𝑠 𝑊𝑡 𝑑 where =√ 𝑡 is a ℚ𝑑 -standard Wiener process. 𝜎𝑥2 + 2𝜌𝜎𝑥 𝜎𝑠 + 𝜎𝑠2 Hence, by comparing with 𝑥𝑠 𝑊𝑡 𝑑

√ 𝑑(𝑋𝑡 𝑆𝑡 ) 𝑥𝑠 = (𝑟𝑑 − 𝐷𝑠 )𝑑𝑡 + 𝜎𝑥2 + 2𝜌𝜎𝑥 𝜎𝑠 + 𝜎𝑠2 𝑑𝑊𝑡 𝑑 𝑋𝑡 𝑆𝑡 we can deduce 𝜇𝑠𝑑 = 𝑟𝑓 − 𝜌𝜎𝑥 𝜎𝑠 − 𝐷𝑠 . Thus, under the domestic risk-neutral measure ℚ𝑑 𝑑𝑆𝑡 𝑠 = (𝑟𝑓 − 𝜌𝜎𝑥 𝜎𝑠 − 𝐷𝑠 )𝑑𝑡 + 𝜎𝑠 𝑑𝑊𝑡 𝑑 . 𝑆𝑡 Denoting Φ(⋅) as the cumulative distribution function of a standard normal we note: (a) For the payoff Ψ(𝑆𝑇 , 𝑋𝑇 ) = 𝑋𝑇 max{𝑆𝑇 − 𝐾𝑠 , 0}, and since we can write 1 2

𝑥𝑑

𝑋𝑇 = 𝑋𝑡 𝑒(𝑟𝑑 −𝑟𝑓 − 2 𝜎𝑥 )(𝑇 −𝑡)+𝜎𝑥 𝑊𝑇 −𝑡

582

6.2.1 Path-Independent Options

the call option price at time 𝑡 is 𝐶(𝑆𝑡 , 𝑋𝑡 , 𝑡; 𝐾𝑠 , 𝑇 ) [ ] = 𝑒−𝑟𝑑 (𝑇 −𝑡) 𝔼ℚ𝑑 𝑋𝑇 max{𝑆𝑇 − 𝐾𝑠 , 0}|| ℱ𝑡 [ ] 𝑥𝑑 | (𝑟𝑑 −𝑟𝑓 − 12 𝜎𝑥2 )(𝑇 −𝑡)+𝜎𝑥 𝑊𝑇 −𝑡 −𝑟𝑑 (𝑇 −𝑡) ℚ𝑑 | =𝑒 𝔼 max{𝑆𝑇 − 𝐾𝑠 , 0}| ℱ𝑡 𝑋𝑡 𝑒 | [ ] 𝑥𝑑 1 2 | = 𝑋𝑡 𝑒−𝑟𝑓 (𝑇 −𝑡) 𝔼ℚ𝑑 𝑒− 2 𝜎𝑥 (𝑇 −𝑡)+𝜎𝑥 𝑊𝑇 −𝑡 max{𝑆𝑇 − 𝐾𝑠 , 0}|| ℱ𝑡 . | ̃ on the filtration ℱ𝑠 , 0 ≤ 𝑠 ≤ 𝑡 where we set We define a new probability measure ℚ the Radon–Nikod´ym derivative as 𝑥𝑑 1 𝑡 𝑡 ̃ || 2 𝑑ℚ | = 𝑒− ∫0 (−𝜎𝑥 )𝑑𝑊𝑢 − 2 ∫0 (−𝜎𝑥 ) 𝑑𝑢 𝑑ℚ𝑑 ||ℱ 𝑡

𝑥𝑑

̃ − 𝜎𝑥 𝑡 follows a ℚ-standard Wiener process.

̃𝑥 = 𝑊 where the process 𝑊 𝑡 𝑡 Since we can also write

𝑥𝑑

𝑊𝑡

𝑠

= 𝜌𝑊𝑡 𝑑 +

√ 1 − 𝜌2 𝑌𝑡𝑑

𝑠

𝑠𝑑

where 𝑊𝑡 𝑑 and 𝑌𝑡𝑑 are ℚ𝑑 -standard Wiener processes and 𝑊𝑡 Nikod´ym derivative can be expressed as

⟂ ⟂ 𝑌𝑡𝑑 , the Radon–

𝑥𝑑 1 𝑡 𝑡 ̃ || 2 𝑑ℚ | = 𝑒− ∫0 (−𝜎𝑥 )𝑑𝑊𝑢 − 2 ∫0 (−𝜎𝑥 ) 𝑑𝑢 | 𝑑ℚ𝑑 |ℱ 𝑡

𝑠𝑑 √ 𝑡 + 1−𝜌2 𝑑𝑌𝑡𝑑 )− 12 ∫0 (−𝜎𝑥 )2 𝑑𝑢

𝑡

= 𝑒− ∫0 (−𝜎𝑥 )(𝜌𝑑𝑊𝑡 𝑡

𝑠𝑑 1 𝑡 − 2 ∫0 (−𝜌𝜎𝑥 )2 𝑑𝑢

= 𝑒− ∫0 (−𝜌𝜎𝑥 )𝑑𝑊𝑢

𝑡

⋅ 𝑒− ∫0 (−

√

√ 𝑡 1−𝜌2 𝜎𝑥 )𝑑𝑌𝑢𝑑 − 21 ∫0 (− 1−𝜌2 𝜎𝑥 )2 𝑑𝑢

such that from the two-dimension Girsanov’s theorem ̃ 𝑠 = 𝑊 𝑠𝑑 − 𝜌𝜎𝑥 𝑡 𝑊 𝑡 𝑡 and 𝑌̃𝑡 = 𝑌𝑡𝑑 −

√ 1 − 𝜌2 𝜎𝑥 𝑡

̃ ̃𝑠 ⟂ ̃ are ℚ-standard Wiener process and 𝑊 𝑡 ⟂ 𝑌𝑡 . ̃ Thus, under the ℚ-measure, the call option price is ] ̃[ 𝑉 (𝑆𝑡 , 𝑋𝑡 , 𝑡; 𝐾𝑠 , 𝑇 ) = 𝑋𝑡 𝑒−𝑟𝑓 (𝑇 −𝑡) 𝔼ℚ max{𝑆𝑇 − 𝐾𝑠 , 0}|| ℱ𝑡

6.2.1 Path-Independent Options

583

̃ with the dynamics of 𝑆𝑡 under the ℚ-measure being 𝑑𝑆𝑡 𝑠 = (𝑟𝑓 − 𝜌𝜎𝑥 𝜎𝑠 − 𝐷𝑠 )𝑑𝑡 + 𝜎𝑠 𝑑𝑊𝑡 𝑑 𝑆𝑡 ( ) ̃ 𝑠 + 𝜌𝜎𝑥 𝑑𝑡 = (𝑟𝑓 − 𝜌𝜎𝑥 𝜎𝑠 − 𝐷𝑠 )𝑑𝑡 + 𝜎𝑠 𝑑 𝑊 𝑡 ̃ 𝑠. = (𝑟𝑓 − 𝐷𝑠 )𝑑𝑡 + 𝜎𝑠 𝑑 𝑊 𝑡 Hence, the call option price at time 𝑡 ≤ 𝑇 is ] ̃[ 𝐶(𝑆𝑡 , 𝑋𝑡 , 𝑡; 𝐾𝑠 , 𝑇 ) = 𝑋𝑡 𝑒−𝑟𝑓 (𝑇 −𝑡) 𝔼ℚ max{𝑆𝑇 − 𝐾𝑠 , 0}|| ℱ𝑡 [ ] = 𝑋𝑡 𝑆𝑡 𝑒−𝐷𝑠 (𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑠 𝑒−𝑟𝑓 (𝑇 −𝑡) Φ(𝑑− ) where log(𝑆𝑡 ∕𝐾𝑠 ) + (𝑟𝑓 − 𝐷𝑠 ± 12 𝜎𝑠2 )(𝑇 − 𝑡) . 𝑑± = √ 𝜎𝑠 𝑇 − 𝑡 ̂ 0} which is denominated in domestic (b) For the payoff Ψ(𝑆𝑇 , 𝑋𝑇 ) = max{𝑋𝑇 𝑆𝑇 − 𝐾, currency, we note that the diffusion process 𝑋𝑡 𝑆𝑡 under the ℚ𝑑 measure is √ 𝑑(𝑋𝑡 𝑆𝑡 ) 𝑥𝑠 = (𝑟𝑑 − 𝐷𝑠 )𝑑𝑡 + 𝜎𝑥2 + 2𝜌𝜎𝑥 𝜎𝑠 + 𝜎𝑠2 𝑑𝑊𝑡 𝑑 𝑋𝑡 𝑆𝑡 𝑥𝑠

where 𝑊𝑡 𝑑 is a ℚ𝑑 -standard Wiener process. Thus, the call option price of the foreign equity option struck in domestic currency can be easily deduced as [ }| ] { ̂ 𝑇 ) = 𝑒−𝑟𝑑 (𝑇 −𝑡) 𝔼ℚ𝑑 max 𝑋𝑇 𝑆𝑇 − 𝐾, ̂ 0 | ℱ𝑡 𝐶(𝑆𝑡 , 𝑋𝑡 , 𝑡; 𝐾, | | ̂ −𝑟𝑑 (𝑇 −𝑡) Φ(𝑑− ) = 𝑋𝑡 𝑆𝑡 𝑒−𝐷𝑠 (𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑒 where 𝑑± =

̂ + (𝑟𝑑 − 𝐷𝑠 ± 1 (𝜎 2 + 2𝜌𝜎𝑥 𝜎𝑠 + 𝜎 2 ))(𝑇 − 𝑡) log((𝑋𝑡 𝑆𝑡 )∕𝐾) 𝑠 2 𝑥 . √ 2 2 (𝜎𝑥 + 2𝜌𝜎𝑥 𝜎𝑠 + 𝜎𝑠 )(𝑇 − 𝑡)

(c) For the quanto option payoff Ψ(𝑆𝑇 , 𝑋𝑇 ) = 𝑋 max{𝑆𝑇 − 𝐾𝑠 , 0}, under the domestic risk-neutral measure ℚ𝑑 , 𝑆𝑡 follows 𝑑𝑆𝑡 𝑠 = (𝑟𝑓 − 𝜌𝜎𝑥 𝜎𝑠 − 𝐷𝑠 )𝑑𝑡 + 𝜎𝑠 𝑑𝑊𝑡 𝑑 𝑆𝑡

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6.2.1 Path-Independent Options

or 𝑑𝑆𝑡 𝑠 = (𝑟𝑑 − (𝑟𝑑 − 𝑟𝑓 + 𝜌𝜎𝑥 𝜎𝑠 + 𝐷𝑠 ))𝑑𝑡 + 𝜎𝑠 𝑑𝑊𝑡 𝑑 𝑆𝑡 𝑠𝑑

= (𝑟𝑑 − 𝐷𝑠 )𝑑𝑡 + 𝜎𝑠 𝑑𝑊𝑡 𝑠

where 𝐷𝑠 = 𝑟𝑑 − 𝑟𝑓 + 𝜌𝜎𝑥 𝜎𝑠 + 𝐷𝑠 and 𝑊𝑡 𝑑 is a ℚ𝑑 -standard Wiener process. Because 𝑋 is a fixed exchange rate, the call option price at time 𝑡 can be deduced as [ ] 𝐶(𝑆𝑡 , 𝑋𝑡 , 𝑡; 𝐾𝑠 , 𝑇 ) = 𝑋𝑒−𝑟𝑑 (𝑇 −𝑡) 𝔼ℚ𝑑 max{𝑆𝑇 − 𝐾𝑠 , 0}|| ℱ𝑡 [ ] = 𝑋 𝑆𝑡 𝑒−𝐷𝑠 (𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑠 𝑒−𝑟𝑑 (𝑇 −𝑡) Φ(𝑑− ) where

𝑑± =

log(𝑆𝑡 ∕𝐾𝑠 ) + (𝑟𝑑 − 𝐷𝑠 ± 12 𝜎𝑠2 )(𝑇 − 𝑡) . √ 𝜎𝑠 𝑇 − 𝑡

(d) For the terminal payoff Ψ(𝑆𝑇 , 𝑋𝑇 ) = 𝑆𝑇 max{𝑋𝑇 − 𝐾𝑥 , 0} which is denominated in the domestic currency, under the domestic risk-neutral measure ℚ𝑑 , 𝑆𝑡 follows 𝑑𝑆𝑡 𝑠 = (𝑟𝑓 − 𝜌𝜎𝑥 𝜎𝑠 − 𝐷𝑠 )𝑑𝑡 + 𝜎𝑠 𝑑𝑊𝑡 𝑑 𝑆𝑡 𝑠

where 𝑊𝑡 𝑑 is the ℚ𝑑 -standard Wiener process. Thus, by solving the SDE for 𝑆𝑡 we can write 1 2

𝑠𝑑

𝑆𝑇 = 𝑆𝑡 𝑒(𝑟𝑓 −𝜌𝜎𝑥 𝜎𝑠 −𝐷𝑠 − 2 𝜎𝑥 )(𝑇 −𝑡)+𝜎𝑠 𝑊𝑇 −𝑡 and the call option price at time 𝑡 is 𝐶(𝑆𝑡 , 𝑋𝑡 , 𝑡; 𝐾𝑥 , 𝑇 ) [ ] = 𝑒−𝑟𝑑 (𝑇 −𝑡) 𝔼ℚ𝑑 𝑆𝑇 max{𝑋𝑇 − 𝐾𝑥 , 0}|| ℱ𝑡 [ ] 𝑠𝑑 1 2 | = 𝑒−𝑟𝑑 (𝑇 −𝑡) 𝔼ℚ𝑑 𝑆𝑡 𝑒(𝑟𝑓 −𝜌𝜎𝑥 𝜎𝑠 −𝐷𝑠 − 2 𝜎𝑥 )(𝑇 −𝑡)+𝜎𝑠 𝑊𝑇 −𝑡 max{𝑋𝑇 − 𝐾𝑥 , 0}|| ℱ𝑡 | [ ] 𝑠𝑑 1 2 | − 𝜎 (𝑇 −𝑡)+𝜎 𝑊 𝑠 𝑇 −𝑡 max{𝑋 − 𝐾 , 0}| ℱ = 𝑆𝑡 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ𝑑 𝑒 2 𝑠 𝑇 𝑥 | 𝑡 . | where 𝑟 = 𝑟𝑑 − 𝑟𝑓 + 𝜌𝜎𝑥 𝜎𝑠 + 𝐷𝑠 .

6.2.1 Path-Independent Options

585

We define a new probability measure ℚ on the filtration ℱ𝑠 , 0 ≤ 𝑠 ≤ 𝑡 where we set the Radon–Nikod´ym derivative as 𝑠𝑑 1 𝑡 𝑡 2 𝑑ℚ || ∫ ∫ | = 𝑒− 0 (−𝜎𝑠 )𝑑𝑊𝑢 − 2 0 (−𝜎𝑠 ) 𝑑𝑢 𝑑ℚ𝑑 ||ℱ 𝑡

𝑠

𝑠

where the process 𝑊 𝑡 = 𝑊𝑡 𝑑 − 𝜎𝑠 𝑡 follows a ℚ-standard Wiener process. Since we can also write √ 𝑠 𝑥 𝑊𝑡 𝑑 = 𝜌𝑊𝑡 𝑑 + 1 − 𝜌2 𝑍𝑡𝑑 𝑥

𝑥𝑑

where 𝑊𝑡 𝑑 and 𝑍𝑡𝑑 are ℚ𝑑 -standard Wiener processes and 𝑊𝑡 Nikod´ym derivative can be expressed as

⟂ ⟂ 𝑍𝑡𝑑 , the Radon–

𝑠𝑑 1 𝑡 𝑡 2 𝑑ℚ || ∫ ∫ | = 𝑒− 0 (−𝜎𝑠 )𝑑𝑊𝑢 − 2 0 (−𝜎𝑠 ) 𝑑𝑢 𝑑ℚ𝑑 ||ℱ 𝑡

√ 𝑡 + 1−𝜌2 𝑑𝑍𝑡𝑑 )− 12 ∫0 (−𝜎𝑠 )2 𝑑𝑢 √ √ 𝑥 𝑡 𝑡 𝑡 𝑡 − ∫0 (−𝜌𝜎𝑠 )𝑑𝑊𝑢 𝑑 − 12 ∫0 (−𝜌𝜎𝑠 )2 𝑑𝑢 − ∫0 (− 1−𝜌2 𝜎𝑠 )𝑑𝑍𝑢𝑑 − 12 ∫0 (− 1−𝜌2 𝜎𝑠 )2 𝑑𝑢 𝑡

𝑥𝑑

= 𝑒− ∫0 (−𝜎𝑠 )(𝜌𝑑𝑊𝑡 =𝑒

⋅𝑒

such that from the two-dimension Girsanov’s theorem 𝑥

𝑥𝑑

𝑊 𝑡 = 𝑊𝑡

− 𝜌𝜎𝑠 𝑡

and 𝑍 𝑡 = 𝑍𝑡𝑑 −

√

1 − 𝜌2 𝜎𝑠 𝑡

𝑥

are ℚ-standard Wiener process and 𝑊 𝑡 ⟂ ⟂ 𝑍 𝑡. Thus, under the ℚ-measure, the call option price is [ ] 𝑉 (𝑆𝑡 , 𝑋𝑡 , 𝑡; 𝐾𝑥 , 𝑇 ) = 𝑆𝑡 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑋𝑇 − 𝐾𝑥 , 0}|| ℱ𝑡 with the dynamics of 𝑋𝑡 under the ℚ-measure being 𝑑𝑋𝑡 𝑥 = (𝑟𝑑 − 𝑟𝑓 )𝑑𝑡 + 𝜎𝑥 𝑑𝑊𝑡 𝑑 𝑋𝑡 ( ) 𝑥 = (𝑟𝑑 − 𝑟𝑓 )𝑑𝑡 + 𝜎𝑥 𝑑𝑊 𝑡 + 𝜌𝜎𝑠 𝑑𝑡 𝑥

= (𝑟𝑑 − 𝑟𝑓 + 𝜌𝜎𝑥 𝜎𝑠 )𝑑𝑡 + 𝜎𝑥 𝑑𝑊 𝑡 𝑥

= (𝑟 − 𝐷𝑠 )𝑑𝑡 + 𝜎𝑥 𝑑𝑊 𝑡 .

586

6.2.2 Path-Dependent Options

Hence, the call option price at time 𝑡 ≤ 𝑇 is [ ] 𝐶(𝑆𝑡 , 𝑋𝑡 , 𝑡; 𝐾𝑥 , 𝑇 ) = 𝑆𝑡 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑋𝑇 − 𝐾𝑥 , 0}|| ℱ𝑡 [ ] = 𝑆𝑡 𝑋𝑡 𝑒−𝐷𝑠 (𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑥 𝑒−𝑟(𝑇 −𝑡) Φ(𝑑− ) where log(𝑋𝑡 ∕𝐾𝑥 ) + (𝑟 − 𝐷𝑠 ± 12 𝜎𝑥2 )(𝑇 − 𝑡) 𝑑± = √ 𝜎𝑥 𝑇 − 𝑡 N.B. For terminal payoffs in (a) and (d), we can also express them as exchange option payoffs and the solutions follow from applying the formula given in Problem 6.2.1.6 (page 543).

6.2.2

Path-Dependent Options

1. Forward Start Option. Let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. In addition, we let 𝑟 be the risk-free interest rate. A forward start call option gives the holder the right to buy at time 𝑇1 an at-the-money European call option with an expiry date of 𝑇2 > 𝑇1 , where the strike price 𝐾 is set as 𝐾 = 𝑆𝑇1 . Hence, the payoff of this option at 𝑇2 is Ψ(𝑆𝑇1 , 𝑆𝑇2 ) = max{𝑆𝑇2 − 𝑆𝑇1 , 0}. By considering 𝑇1 < 𝑡 ≤ 𝑇2 , 𝑡 = 𝑇1 and 𝑡 < 𝑇1 , show that the forward call option price is ⎧ 𝑆 𝑒−𝐷(𝑇 −𝑡) [𝑒−𝐷(𝑇2 −𝑇1 ) Φ(𝑑̄ ) − 𝑒−𝑟(𝑇2 −𝑇1 ) Φ(𝑑̄ )] + − ⎪ 𝑡 𝐶𝑓 𝑠 (𝑆𝑡 , 𝑡; 𝑇1 , 𝑇2 ) = ⎨ ⎪ 𝑆𝑡 𝑒−𝐷(𝑇2 −𝑡) Φ(𝑑+ ) − 𝑆𝑇 𝑒−𝑟(𝑇2 −𝑡) Φ(𝑑− ) 1 ⎩

𝑡 ≤ 𝑇1 𝑇1 < 𝑡 ≤ 𝑇2

(𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇2 − 𝑇1 ) log(𝑆𝑡 ∕𝑆𝑇1 ) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇2 − 𝑡) and where 𝑑̄± = , 𝑑± = √ √ 𝜎 𝑇2 − 𝑇1 𝜎 𝑇2 − 𝑡 Φ(⋅) is the cdf of a standard normal.

6.2.2 Path-Dependent Options

587

Solution: From Girsanov’s theorem, under the risk-neutral measure ℚ 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡

(𝜇 − 𝑟) 𝑡 is a ℚ-standard Wiener process. where 𝑊𝑡ℚ = 𝑊𝑡 + 𝜎 For 𝑇1 < 𝑡 ≤ 𝑇2 , as 𝑆𝑇1 is known then [ ] ( ) | 𝐶𝑓 𝑠 𝑆𝑡 , 𝑡; 𝑇1 , 𝑇2 = 𝑒−𝑟(𝑇2 −𝑡) 𝔼ℚ max{𝑆𝑇2 − 𝑆𝑇1 , 0}| ℱ𝑡 | = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝑆𝑇1 , 𝑇2 ) = 𝑆𝑡 𝑒−𝐷(𝑇2 −𝑡) Φ(𝑑+ ) − 𝑆𝑇1 𝑒−𝑟(𝑇2 −𝑡) Φ(𝑑− ) which is a regular European call option price 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇2 ) with strike price 𝐾 = 𝑆𝑇1

log(𝑆𝑡 ∕𝑆𝑇1 ) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇2 − 𝑡) . at expiry time 𝑇2 such that 𝑑± = √ 𝜎 𝑇2 − 𝑡 At 𝑡 = 𝑇1 , 𝑆𝑡 = 𝑆𝑇1 and therefore 𝐶𝑓 𝑠 (𝑆𝑡 , 𝑡; 𝑇1 , 𝑇2 ) = 𝐶𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝑆𝑇1 , 𝑇2 )

= 𝑆𝑇1 𝑒−𝐷(𝑇2 −𝑇1 ) Φ(𝑑̄+ ) − 𝑆𝑇1 𝑒−𝑟(𝑇2 −𝑇1 ) Φ(𝑑̄− ) [ ] = 𝑆𝑇1 𝑒−𝐷(𝑇2 −𝑇1 ) Φ(𝑑̄+ ) − 𝑒−𝑟(𝑇2 −𝑇1 ) Φ(𝑑̄− ) where 𝑑̄± =

log(𝑆𝑇1 ∕𝑆𝑇1 ) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇2 − 𝑇1 ) (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇2 − 𝑇1 ) = . √ √ 𝜎 𝑇2 − 𝑇1 𝜎 𝑇2 − 𝑇1

Finally, for 𝑡 < 𝑇1 [ ] | 𝐶𝑓 𝑠 (𝑆𝑡 , 𝑡; 𝑇1 , 𝑇2 ) = 𝑒−𝑟(𝑇1 −𝑡) 𝔼ℚ 𝐶𝑓 𝑠 (𝑆𝑇1 , 𝑇1 ; 𝑇1 , 𝑇2 )| ℱ𝑡 | [ [ −𝐷(𝑇 −𝑇 ) ]| ] −𝑟(𝑇1 −𝑡) ℚ 2 1 Φ(𝑑̄ ) − 𝑒−𝑟(𝑇2 −𝑇1 ) Φ(𝑑̄ ) | ℱ 𝔼 𝑆 𝑇1 𝑒 =𝑒 + − | 𝑡 [ ] [ ] | = 𝑒−𝑟(𝑇1 −𝑡) 𝑒−𝐷(𝑇2 −𝑇1 ) Φ(𝑑̄+ ) − 𝑒−𝑟(𝑇2 −𝑇1 ) Φ(𝑑̄− ) 𝔼ℚ 𝑆𝑇1 | ℱ𝑡 | since 𝑑̄+ and 𝑑̄− do not depend on 𝑆𝑇1 . From It¯o’s lemma 𝑆𝑇 1 = 𝑆 𝑡 𝑒

(𝑟−𝐷− 21 𝜎 2 )(𝑇1 −𝑡)+𝜎𝑊𝑇ℚ −𝑡 1

therefore 𝔼ℚ [𝑆𝑇1 |ℱ𝑡 ] = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇1 −𝑡) and hence [ ] 𝐶𝑓 𝑠 (𝑆𝑡 , 𝑡; 𝑇1 , 𝑇2 ) = 𝑆𝑡 𝑒−𝐷(𝑇1 −𝑡) 𝑒−𝐷(𝑇2 −𝑇1 ) Φ(𝑑̄+ ) − 𝑒−𝑟(𝑇2 −𝑇1 ) Φ(𝑑̄− ) .

588

6.2.2 Path-Dependent Options

Collectively, we therefore have 𝐶𝑓 𝑠 (𝑆𝑡 , 𝑡; 𝑇1 , 𝑇2 ) ⎧ 𝑆 𝑒−𝐷(𝑇 −𝑡) [𝑒−𝐷(𝑇2 −𝑇1 ) Φ(𝑑̄ ) − 𝑒−𝑟(𝑇2 −𝑇1 ) Φ(𝑑̄ )] + − ⎪ 𝑡 =⎨ ⎪ 𝑆𝑡 𝑒−𝐷(𝑇2 −𝑡) Φ(𝑑+ ) − 𝑆𝑇 𝑒−𝑟(𝑇2 −𝑡) Φ(𝑑− ) 1 ⎩

𝑡 ≤ 𝑇1 𝑇 1 < 𝑡 ≤ 𝑇2 .

N.B. Using similar arguments we can also show the forward start put option as 𝑃𝑓 𝑠 (𝑆𝑡 , 𝑡; 𝑇1 , 𝑇2 ) ⎧ 𝑆 𝑒−𝐷(𝑇 −𝑡) [𝑒−𝑟(𝑇2 −𝑇1 ) Φ(−𝑑̄ ) − 𝑒−𝐷(𝑇2 −𝑇1 ) Φ(−𝑑̄ )] 𝑡 ≤ 𝑇 − + 1 ⎪ 𝑡 =⎨ ⎪ 𝑆𝑇1 𝑒−𝑟(𝑇2 −𝑡) Φ(−𝑑− ) − 𝑆𝑡 𝑒−𝐷(𝑇2 −𝑡) Φ(−𝑑+ ) 𝑇 1 < 𝑡 ≤ 𝑇2 . ⎩

2. Rachet/Cliquet Option. Let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. In addition, let 𝑟 be the risk-free interest rate from a money-market account. A rachet or cliquet option is a series of forward start options where at the end of a typical leg, from 𝑇𝑖−1 to 𝑇𝑖 the option allows the holder to “lock in” intermediate profits max{𝑆𝑇𝑖 − 𝐾𝑖−1 , 0} = max{𝑆𝑇𝑖 − 𝑆𝑇𝑖−1 , 0} which will be paid out at expiry time 𝑇 where the strike 𝐾𝑖−1 is reset to the asset price 𝑆𝑇𝑖−1 at time 𝑇𝑖−1 , 𝑇𝑖−1 < 𝑇𝑖 . We consider a 3-leg rachet call option with initial strike 𝐾0 at time 𝑇0 . At time 𝑇1 > 𝑇0 the strike is reset to 𝐾1 = 𝑆𝑇1 , which is the asset price at time 𝑇1 . At time 𝑇2 > 𝑇1 the strike is reset again to 𝐾2 = 𝑆𝑇2 , which is the asset price at time 𝑇2 . Finally, at the option expiry time 𝑇 > 𝑇2 the holder of the call rachet will receive the call payoff with strike 𝐾2 = 𝑆𝑇2 and “locked-in” amounts of max{𝑆𝑇1 − 𝐾0 , 0} and {𝑆𝑇2 − 𝐾1 , 0}. From the above information (a) Write down the overall payoff Ψ(𝑆𝑇 ) noting that the strike is being reset to the spot price at each reset date. (b) By working back from the expiry, determine the option price 𝐶𝑟𝑎 (𝑆𝑡 , 𝑡; 𝐾0 , 𝑇1 , 𝑇2 , 𝑇 ) under the risk-neutral measure ℚ, for 𝑇0 < 𝑡 ≤ 𝑇1 , 𝑇1 < 𝑡 ≤ 𝑇2 and 𝑇2 < 𝑡 ≤ 𝑇 . Solution: (a) The overall payoff at expiry time 𝑇 is given as Ψ(𝑆𝑇 ) = max{𝑆𝑇 − 𝐾2 , 0} + max{𝐾2 − 𝐾1 , 0} + max{𝐾1 − 𝐾0 , 0} = max{𝑆𝑇 − 𝑆𝑇2 , 0} + max{𝑆𝑇2 − 𝑆𝑇1 , 0} + max{𝑆𝑇1 − 𝐾0 , 0}.

6.2.2 Path-Dependent Options

589

(b) Under the risk-neutral measure ℚ, the asset price 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡 (𝜇 − 𝑟) 𝑡 is the ℚ-standard where 𝑟 is the risk-free interest rate and 𝑊𝑡ℚ = 𝑊𝑡 + 𝜎 Wiener process. For 𝑇2 < 𝑡 ≤ 𝑇 , under the filtration ℱ𝑡 the information at 𝑇1 and 𝑇2 is known, therefore we can write the option price as ( [ ] ) 𝐶𝑟𝑎 𝑆𝑡 , 𝑡; 𝐾0 , 𝑇1 , 𝑇2 , 𝑇 = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ Ψ(𝑆𝑇 )|| ℱ𝑡 [ ] | = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑆𝑇 − 𝑆𝑇2 , 0}| ℱ𝑡 | [ ] | −𝑟(𝑇 −𝑡) ℚ 𝔼 max{𝑆𝑇2 − 𝑆𝑇1 , 0}| ℱ𝑡 +𝑒 | [ ] | −𝑟(𝑇 −𝑡) ℚ 𝔼 max{𝑆𝑇1 − 𝐾0 , 0}| ℱ𝑡 +𝑒 | = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝑆𝑇2 , 𝑇 ) + 𝑒−𝑟(𝑇 −𝑡) max{𝑆𝑇2 − 𝑆𝑇1 , 0} +𝑒−𝑟(𝑇 −𝑡) max{𝑆𝑇1 − 𝐾0 , 0} where 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝑆𝑇2 , 𝑇 ) is the European call option price with strike 𝑆𝑇2 and expiry time 𝑇 . For 𝑇1 < 𝑡 ≤ 𝑇2 , under the filtration ℱ𝑡 the information up to 𝑇1 is known, therefore we can write the payoff as Ψ(𝑆𝑇2 ) = 𝐶𝑟𝑎 (𝑆𝑇2 , 𝑇2 ; 𝐾0 , 𝑇1 , 𝑇2 , 𝑇 ) = 𝐶𝑏𝑠 (𝑆𝑇2 , 𝑇2 ; 𝑆𝑇2 , 𝑇 ) + 𝑒−𝑟(𝑇 −𝑇2 ) max{𝑆𝑇2 − 𝑆𝑇1 , 0} +𝑒−𝑟(𝑇 −𝑇2 ) max{𝑆𝑇1 − 𝐾0 , 0} and the option price at time 𝑇1 < 𝑡 ≤ 𝑇2 is ( ) 𝐶𝑟𝑎 𝑆𝑡 , 𝑡; 𝐾0 , 𝑇1 , 𝑇2 , 𝑇 [ ] | = 𝑒−𝑟(𝑇2 −𝑡) 𝔼ℚ Ψ(𝑆𝑇2 )| ℱ𝑡 | [ ] | −𝑟(𝑇2 −𝑡) ℚ 𝔼 𝐶𝑏𝑠 (𝑆𝑇2 , 𝑇2 ; 𝑆𝑇2 , 𝑇 )| ℱ𝑡 =𝑒 | [ ] | −𝑟(𝑇2 −𝑡) ℚ −𝑟(𝑇 −𝑇2 ) 𝔼 𝑒 max{𝑆𝑇2 − 𝑆𝑇1 , 0}| ℱ𝑡 +𝑒 | [ ] | −𝑟(𝑇2 −𝑡) ℚ −𝑟(𝑇 −𝑇2 ) 𝔼 𝑒 max{𝑆𝑇1 − 𝐾0 , 0}| ℱ𝑡 +𝑒 | [ ( )| ] = 𝑒−𝑟(𝑇2 −𝑡) 𝔼ℚ 𝑆𝑇2 𝑒−𝐷(𝑇 −𝑇2 ) Φ(𝑑+ ) − 𝑒−𝑟(𝑇 −𝑇2 ) Φ(𝑑− ) | ℱ𝑡 | [ ] | −𝑟(𝑇 −𝑇2 ) ℚ −𝑟(𝑇2 −𝑡) 𝔼 𝑒 max{𝑆𝑇2 − 𝑆𝑇1 , 0}| ℱ𝑡 +𝑒 | +𝑒−𝑟(𝑇2 −𝑡) 𝑒−𝑟(𝑇 −𝑇2 ) max{𝑆𝑇1 − 𝐾0 , 0}

590

6.2.2 Path-Dependent Options

where log(𝑆𝑇2 ∕𝑆𝑇2 ) + (𝑟 ± 12 𝜎 2 )(𝑇 − 𝑇2 ) (𝑟 ± 12 𝜎 2 )(𝑇 − 𝑇2 ) 𝑑± = = . √ √ 𝜎 𝑇 − 𝑇2 𝜎 𝑇 − 𝑇2 [ ] | Since 𝔼ℚ 𝑆𝑇2 | ℱ𝑡 = 𝑆𝑡 𝑒𝑟(𝑇2 −𝑡) and 𝑑± is independent of the filtration ℱ𝑡 , we there| fore have ( ) 𝐶𝑟𝑎 (𝑆𝑡 , 𝑡; 𝐾0 , 𝑇1 , 𝑇2 , 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑇2 ) Φ(𝑑+ ) − 𝑒−𝑟(𝑇 −𝑇2 ) Φ(𝑑− ) +𝑒−𝑟(𝑇 −𝑇2 ) 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝑆𝑇1 , 𝑇2 ) +𝑒−𝑟(𝑇 −𝑡) max{𝑆𝑇1 − 𝐾0 , 0} where 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝑆𝑇1 , 𝑇2 ) is a European call option price with strike price 𝑆𝑇1 and expiry time 𝑇2 . Finally, for 𝑇0 < 𝑡 ≤ 𝑇1 the payoff is Ψ(𝑆𝑇1 ) = 𝐶𝑟𝑎 (𝑆𝑇1 , 𝑇1 ; 𝐾0 , 𝑇1 , 𝑇2 , 𝑇 ) ( ) = 𝑆𝑇1 𝑒−𝐷(𝑇 −𝑇2 ) Φ(𝑑+ ) − 𝑒−𝑟(𝑇 −𝑇2 ) Φ(𝑑− ) +𝑒−𝑟(𝑇 −𝑇2 ) 𝐶𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝑆𝑇1 , 𝑇2 ) + 𝑒−𝑟(𝑇 −𝑇1 ) max{𝑆𝑇1 − 𝐾0 , 0} and the option price at time 𝑇0 < 𝑡 ≤ 𝑇1 is ( ) 𝐶𝑟𝑎 𝑆𝑡 , 𝑡; 𝐾0 , 𝑇1 , 𝑇2 , 𝑇 [ ] = 𝑒−𝑟(𝑇1 −𝑡) 𝔼ℚ Ψ(𝑆𝑇1 )|ℱ𝑡 [ ( )| ] = 𝑒−𝑟(𝑇1 −𝑡) 𝔼ℚ 𝑆𝑇1 𝑒−𝐷(𝑇 −𝑇2 ) Φ(𝑑+ ) − 𝑒−𝑟(𝑇 −𝑇2 ) Φ(𝑑− ) | ℱ𝑡 | [ ( )| ] +𝑒−𝑟(𝑇1 −𝑡) 𝔼ℚ 𝑆𝑇1 𝑒−𝐷(𝑇2 −𝑇1 ) Φ(𝑑̂+ ) − 𝑒−𝑟(𝑇2 −𝑇1 ) Φ(𝑑̂− ) | ℱ𝑡 | [ ] +𝑒−𝑟(𝑇1 −𝑡) 𝔼ℚ 𝑒−𝑟(𝑇 −𝑇1 ) max{𝑆𝑇1 − 𝐾0 , 0}|ℱ𝑡 where 𝑑̂± =

) ( log(𝑆𝑇1 ∕𝑆𝑇1 ) + (𝑟 ± 12 𝜎 2 )(𝑇2 − 𝑇1 ) (𝑟 ± 12 𝜎 2 ) 𝑇2 − 𝑇1 = . √ √ 𝜎 𝑇2 − 𝑇1 𝜎 𝑇2 − 𝑇1

6.2.2 Path-Dependent Options

591

[ ] | Since both 𝑑± and 𝑑̂± are independent of the filtration ℱ𝑡 , and because 𝔼ℚ 𝑆𝑇1 | ℱ𝑡 = | 𝑆𝑡 𝑒𝑟(𝑇1 −𝑡) , therefore ( ) 𝐶𝑟𝑎 (𝑆𝑡 , 𝑡; 𝐾0 , 𝑇1 , 𝑇2 , 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑇2 ) Φ(𝑑+ ) − 𝑒−𝑟(𝑇 −𝑇2 ) Φ(𝑑− ) ( ) +𝑆𝑡 𝑒−𝐷(𝑇2 −𝑇1 ) Φ(𝑑̂+ ) − 𝑒−𝑟(𝑇2 −𝑇1 ) Φ(𝑑̂− ) [ ] | +𝑒−𝑟(𝑇 −𝑇1 ) 𝔼ℚ 𝑒−𝑟(𝑇1 −𝑡) max{𝑆𝑇1 − 𝐾0 , 0}| ℱ𝑡 | ( ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑇2 ) Φ(𝑑+ ) − 𝑒−𝑟(𝑇 −𝑇2 ) Φ(𝑑− ) ( ) +𝑆𝑡 𝑒−𝐷(𝑇2 −𝑇1 ) Φ(𝑑̂+ ) − 𝑒−𝑟(𝑇2 −𝑇1 ) Φ(𝑑̂− ) +𝑒−𝑟(𝑇 −𝑇1 ) 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾0 , 𝑇1 ) where 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾0 , 𝑇1 ) is a European call option at time 𝑡 with strike price 𝐾0 and expiry time 𝑇1 . 3. Compound Option I. Let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. In addition, we let 𝑟 be the risk-free interest rate. Consider a European call-on-a-call option price with payoff Ψ(𝑆𝑇1 ) = max{𝐶𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) − 𝐾1 , 0} where on the first expiry date 𝑇1 , the option holder has the right to buy the underlying European call option worth 𝐶𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) = 𝑆𝑇1 𝑒−𝐷(𝑇2 −𝑇1 ) Φ(𝑑+ ) − 𝐾2 𝑒−𝑟(𝑇2 −𝑇1 ) Φ(𝑑− ) log(𝑆𝑇1 ∕𝐾2 ) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇2 − 𝑇1 ) 𝑑± = √ 𝜎 𝑇 2 − 𝑇1 by paying the first strike price 𝐾1 . Here the underlying European call option gives the holder the right but not the obligation to buy the underlying asset by paying the second strike price 𝐾2 at expiry date 𝑇2 ≥ 𝑇1 . Using the risk-neutral measure valuation, show that the European call-on-a-call option price at time 𝑡 ≤ 𝑇1 ≤ 𝑇2 is 𝐶𝑐 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇1 , 𝐾2 , 𝑇2 ) = 𝑆𝑡 𝑒−𝐷(𝑇2 −𝑡) 𝚽(𝛼+ , 𝛽+ , 𝜌) − 𝐾2 𝑒−𝑟(𝑇2 −𝑡) 𝚽(𝛼− , 𝛽− , 𝜌) −𝐾1 𝑒−𝑟(𝑇1 −𝑡) Φ(𝛼− )

592

6.2.2 Path-Dependent Options

where

log(𝑆𝑡 ∕𝑆̃𝑇1 ) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇1 − 𝑡) log(𝑆𝑡 ∕𝐾2 ) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇2 − 𝑡) , 𝛽± = , 𝛼± = √ √ 𝜎 𝑇1 − 𝑡 𝜎 𝑇2 − 𝑡 √ 𝑇1 − 𝑡 𝜌= and 𝑆̃𝑇1 satisfies 𝐶𝑏𝑠 (𝑆̃𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) = 𝐾1 , 𝚽(𝑥, 𝑦, 𝜌𝑥𝑦 ) is the cdf of a stan𝑇2 − 𝑡 dard bivariate normal with correlation coefficient 𝜌𝑥𝑦 ∈ (−1, 1) and Φ(𝑥) is the cdf of a standard normal. Finally, deduce the put-on-a-call option price at time 𝑡 ≤ 𝑇1 ≤ 𝑇2 with payoff Ψ(𝑆𝑇1 ) = max{𝐾1 − 𝐶𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ), 0}. Solution: Under the risk-neutral measure ℚ, 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡 (𝜇 − 𝑟) 𝑡 is a ℚ-standard Wiener process. From It¯o’s lemma we can where 𝑊𝑡ℚ = 𝑊𝑡 + 𝜎 easily show for 𝑇 > 𝑡 ( log

𝑆𝑇 𝑆𝑡

) ∼

[(

) ] 1 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡), 𝜎 2 (𝑇 − 𝑡) 2

with density function (

𝑓 (𝑆𝑇 |𝑆𝑡 ) =

√

1

𝑆𝑇 𝜎 2𝜋(𝑇 − 𝑡)

− 21

𝑒

log(𝑆𝑇 ∕𝑆𝑡 )−(𝑟−𝐷− 1 𝜎 2 )(𝑇 −𝑡) 2 √ 𝜎 𝑇 −𝑡

)2

.

By definition, the call-on-a-call option at time 𝑡 ≤ 𝑇1 ≤ 𝑇2 is 𝐶𝑐 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇1 , 𝐾2 , 𝑇2 ) [ }| ] { = 𝑒−𝑟(𝑇1 −𝑡) 𝔼ℚ max 𝐶𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) − 𝐾1 , 0 || ℱ𝑡 | { } ∞ −𝑟(𝑇1 −𝑡) max 𝐶𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) − 𝐾1 , 0 𝑓 (𝑆𝑇1 |𝑆𝑡 ) 𝑑𝑆𝑇1 =𝑒 ∫0 ] ∞[ 𝐶𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) − 𝐾1 𝑓 (𝑆𝑇1 |𝑆𝑡 ) 𝑑𝑆𝑇1 = 𝑒−𝑟(𝑇1 −𝑡) ∫𝑆̃𝑇 1

= 𝐴 1 − 𝐴2 − 𝐴3

6.2.2 Path-Dependent Options

593

where 𝑆̃𝑇1 satisfies the equation 𝐶𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) = 𝐾1 and 𝐴1 = 𝑒−𝑟(𝑇1 −𝑡) 𝐴2 = 𝑒−𝑟(𝑇1 −𝑡)

∞

∫𝑆̃𝑇

1

∞

∫𝑆̃𝑇

𝐴3 = 𝐾1 𝑒−𝑟(𝑇1 −𝑡)

𝑆𝑇1 𝑒−𝐷(𝑇2 −𝑇1 ) Φ(𝑑+ )𝑓 (𝑆𝑇1 |𝑆𝑡 ) 𝑑𝑆𝑇1 𝐾2 𝑒−𝑟(𝑇2 −𝑇1 ) Φ(𝑑− )𝑓 (𝑆𝑇1 |𝑆𝑡 ) 𝑑𝑆𝑇1

1

∞

∫𝑆̃𝑇

𝑓 (𝑆𝑇1 |𝑆𝑡 ) 𝑑𝑆𝑇1 .

1

For the case 𝐴1 = 𝑒−𝑟(𝑇1 −𝑡)−𝐷(𝑇2 −𝑇1 ) ( ) ∞ log(𝑆𝑇1 ∕𝐾2 ) + (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇2 − 𝑇1 ) × 𝑆 Φ 𝑓 (𝑆𝑇1 |𝑆𝑡 ) 𝑑𝑆𝑇1 √ ∫𝑆̃𝑇 𝑇1 𝜎 𝑇 −𝑇 2

1

1

log(𝑆𝑇1 ∕𝑆𝑡 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇1 − 𝑡) we have and by setting 𝑥 = √ 𝜎 𝑇1 − 𝑡 1 2

𝐴1 = 𝑆𝑡 𝑒−𝑟(𝑇1 −𝑡)−𝐷(𝑇2 −𝑇1 )+(𝑟−𝐷− 2 𝜎 )(𝑇1 −𝑡) ( ) √ ∞ √ 𝑚 + 𝑥𝜎 𝑇1 − 𝑡 1 − 1 (𝑥2 −2𝑥𝜎 𝑇2 −𝑡) × Φ 𝑑𝑥 √ √ 𝑒 2 ∫𝛼 𝜎 𝑇 −𝑇 2𝜋 2

1

or

𝐴1 = 𝑆𝑡 𝑒−𝐷(𝑇2 −𝑡)

(

∞

∫𝛼

Φ

) √ √ 𝑚 + 𝑥𝜎 𝑇1 − 𝑡 1 − 1 (𝑥−𝜎 𝑇1 −𝑡)2 𝑑𝑥 √ √ 𝑒 2 𝜎 𝑇2 − 𝑇1 2𝜋

where

𝛼=

log(𝑆̃𝑇1 ∕𝑆𝑡 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇1 − 𝑡) √ 𝜎 𝑇1 − 𝑡

and 1 1 𝑚 = log(𝑆𝑡 ∕𝐾2 ) + (𝑟 − 𝐷 + 𝜎 2 )(𝑇2 − 𝑇1 ) + (𝑟 − 𝐷 − 𝜎 2 )(𝑇1 − 𝑡). 2 2

594

6.2.2 Path-Dependent Options

√ By setting 𝑦 = −(𝑥 − 𝜎 𝑇1 − 𝑡) and from Problem 1.2.2.16 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus we have ) ( √ −∞ 𝑚 + 𝜎 2 (𝑇1 − 𝑡) − 𝑦𝜎 𝑇1 − 𝑡 1 − 1 𝑦2 −𝐷(𝑇2 −𝑡) 𝐴1 = −𝑆𝑡 𝑒 Φ √ √ 𝑒 2 𝑑𝑦 ∫𝛼+ 𝜎 𝑇2 − 𝑇1 2𝜋 ) ( √ 2 𝛼+ 𝑚 + 𝜎 (𝑇1 − 𝑡) − 𝑦𝜎 𝑇1 − 𝑡 1 − 1 𝑦2 Φ = 𝑆𝑡 𝑒−𝐷(𝑇2 −𝑡) √ √ 𝑒 2 𝑑𝑦 ∫−∞ 𝜎 𝑇2 − 𝑇1 2𝜋 ) ( 𝛼+ 𝛽+ − 𝜌𝑦 1 − 1 𝑦2 Φ √ = 𝑆𝑡 𝑒−𝐷(𝑇2 −𝑡) √ 𝑒 2 𝑑𝑦 ∫−∞ 1 − 𝜌2 2𝜋 = 𝑆𝑡 𝑒−𝐷(𝑇2 −𝑡) 𝚽(𝛼+ , 𝛽+ , 𝜌) where

log(𝑆𝑡 ∕𝑆̃𝑇1 ) + (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇1 − 𝑡) log(𝑆𝑡 ∕𝐾2 ) + (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇2 − 𝑡) , 𝛽+ = 𝛼+ = √ √ 𝜎 𝑇1 − 𝑡 𝜎 𝑇2 − 𝑡 √ 𝑇1 − 𝑡 and 𝜌 = . 𝑇2 − 𝑡 In contrast, for 𝐴2 = 𝐾2 𝑒−𝑟(𝑇2 −𝑡) ( ∞

×

∫𝑆̃𝑇

1

Φ

log(𝑆𝑇1 ∕𝐾2 ) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇2 − 𝑇1 ) √ 𝜎 𝑇2 − 𝑇1

) 𝑓 (𝑆𝑇1 |𝑆𝑡 ) 𝑑𝑆𝑇1

log(𝑆𝑇1 ∕𝑆𝑡 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇1 − 𝑡) and by setting 𝑥 = we have √ 𝜎 𝑇1 − 𝑡 𝐴2 = 𝐾2 𝑒−𝑟(𝑇2 −𝑡) ( ∞

×

∫−𝛼−

Φ

√ ) log(𝑆𝑡 ∕𝐾2 ) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇2 − 𝑡) + 𝑥𝜎 𝑇1 − 𝑡 1 − 1 𝑥2 √ √ 𝑒 2 𝑑𝑥 𝜎 𝑇2 − 𝑇1 2𝜋

log(𝑆𝑡 ∕𝑆̃𝑇1 ) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇1 − 𝑡) . √ 𝜎 𝑇1 − 𝑡 Setting 𝑦 = −𝑥 ) ( −∞ − 𝜌𝑦 𝛽 1 − 1 𝑦2 − 𝐴2 = −𝐾2 𝑒−𝑟(𝑇2 −𝑡) Φ √ √ 𝑒 2 𝑑𝑦 ∫𝛼− 2 1−𝜌 2𝜋 ) ( 𝛼− 𝛽 − 𝜌𝑦 1 − 1 𝑦2 Φ √− = 𝐾2 𝑒−𝑟(𝑇2 −𝑡) √ 𝑒 2 𝑑𝑦 ∫−∞ 1 − 𝜌2 2𝜋

where 𝛼− =

= 𝐾2 𝑒−𝑟(𝑇2 −𝑡) 𝚽(𝛼− , 𝛽− , 𝜌) where 𝛽− =

log(𝑆𝑡 ∕𝐾2 ) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇2 − 𝑡) . √ 𝜎 𝑇2 − 𝑡

6.2.2 Path-Dependent Options

595

Finally, because 𝑊𝑇ℚ−𝑡 ∼  (0, 𝑇1 − 𝑡) we have 1

𝐴3 = 𝐾1 𝑒−𝑟(𝑇1 −𝑡)

∞

𝑓 (𝑆𝑇1 |𝑆𝑡 ) 𝑑𝑆𝑇1 ∫𝑆̃𝑇 1 ) ( | = 𝐾1 𝑒−𝑟(𝑇1 −𝑡) ℚ 𝑆𝑇1 > 𝑆̃𝑇1 | 𝑆𝑡 | ( ) 1 2 | (𝑟−𝐷− 2 𝜎 )(𝑇1 −𝑡)+𝜎𝑊𝑇ℚ −𝑡 1 = 𝐾1 𝑒−𝑟(𝑇1 −𝑡) ℚ 𝑆𝑡 𝑒 > 𝑆̃𝑇1 || 𝑆𝑡 | ( ) ̃ log(𝑆𝑇1 ∕𝑆𝑡 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇1 − 𝑡) || ℚ −𝑟(𝑇1 −𝑡) |𝑆 ℚ 𝑊𝑇 −𝑡 > = 𝐾1 𝑒 | 𝑡 1 𝜎 | | = 𝐾1 𝑒−𝑟(𝑇1 −𝑡) Φ(𝛼− ).

Thus, the call-on-a-call option price at time 𝑡 ≤ 𝑇1 ≤ 𝑇2 is 𝐶𝑐 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇1 , 𝐾2 , 𝑇2 ) = 𝑆𝑡 𝑒−𝐷(𝑇2 −𝑡) 𝚽(𝛼+ , 𝛽+ , 𝜌) − 𝐾2 𝑒−𝑟(𝑇2 −𝑡) 𝚽(𝛼− , 𝛽− , 𝜌) −𝐾1 𝑒−𝑟(𝑇1 −𝑡) Φ(𝛼− ). For the case of a put-on-a-call option, the payoff at time 𝑇1 is 𝑃𝑐 (𝑆𝑇1 , 𝑇1 ; 𝐾1 , 𝑇1 ) = max{𝐾1 − 𝐶𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ), 0} whilst the payoff at time 𝑇1 for a call-on-a-call option is 𝐶𝑐 (𝑆𝑇1 , 𝑇1 ; 𝐾1 , 𝑇1 ) = max{𝐶𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) − 𝐾1 , 0}. Hence, 𝐶𝑐 (𝑆𝑇1 , 𝑇1 ; 𝐾1 , 𝑇1 , 𝐾2 , 𝑇2 ) − 𝑃𝑐 (𝑆𝑇1 , 𝑇1 ; 𝐾1 , 𝑇1 , 𝐾2 , 𝑇2 ) ⎧ 𝐶 (𝑆 , 𝑇 ; 𝐾 , 𝑇 ) − 𝐾 1 ⎪ 𝑏𝑠 𝑇1 1 2 2 =⎨ ⎪ 𝐶𝑏𝑠 (𝑆𝑇 , 𝑇1 ; 𝐾2 , 𝑇2 ) − 𝐾1 1 ⎩

if 𝐶𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) > 𝐾1 if 𝐶𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) ≤ 𝐾1

= 𝐶𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) − 𝐾1 . By discounting the payoff back to time 𝑡 under the risk-neutral measure ℚ, we have 𝐶𝑐 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇1 , 𝐾2 , 𝑇2 ) − 𝑃𝑐 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇1 , 𝐾2 , 𝑇2 ) = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇2 ) − 𝐾1 𝑒−𝑟(𝑇1 −𝑡) or 𝑃𝑐 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇1 , 𝐾2 , 𝑇2 ) = 𝐶𝑐 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇1 , 𝐾2 , 𝑇2 ) − 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇2 ) + 𝐾1 𝑒−𝑟(𝑇1 −𝑡) .

596

6.2.2 Path-Dependent Options

4. Compound Option II. Let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. In addition, we let 𝑟 be the risk-free interest rate. Consider a European put-on-a-put option price with payoff Ψ(𝑆𝑇1 ) = max{𝐾1 − 𝑃𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ), 0} where on the first expiry date 𝑇1 , the option holder has the right to sell the underlying European put option 𝑃𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) = 𝐾2 𝑒−𝑟(𝑇2 −𝑇1 ) Φ(−𝑑− ) − 𝑆𝑇1 𝑒−𝐷(𝑇2 −𝑇1 ) Φ(−𝑑+ ) log(𝑆𝑇1 ∕𝐾2 ) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇2 − 𝑇1 ) 𝑑± = √ 𝜎 𝑇2 − 𝑇1 at the first strike price 𝐾1 . Here the underlying European put option gives the holder the right but not the obligation to sell the underlying asset by receiving the second strike price 𝐾2 at expiry date 𝑇2 ≥ 𝑇1 . Using the risk-neutral measure valuation, show that the European put-on-a-put option price at time 𝑡 ≤ 𝑇1 ≤ 𝑇2 is 𝑃𝑝 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇1 , 𝐾2 , 𝑇2 ) = 𝐾1 𝑒−𝑟(𝑇1 −𝑡) Φ(−𝛼− ) − 𝐾2 𝑒−𝑟(𝑇2 −𝑡) 𝚽(−𝛼− , −𝛽− , 𝜌) +𝑆𝑡 𝑒−𝐷(𝑇2 −𝑡) 𝚽(−𝛼+ , −𝛽+ , 𝜌) where 𝛼± = √

log(𝑆𝑡 ∕𝑆̂𝑇1 ) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇1 − 𝑡) log(𝑆𝑡 ∕𝐾2 ) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇2 − 𝑡) , 𝛽 , = √ √ ± 𝜎 𝑇1 − 𝑡 𝜎 𝑇2 − 𝑡

𝑇1 − 𝑡 and 𝑆̂𝑇1 satisfies 𝑃𝑏𝑠 (𝑆̂𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) = 𝐾1 , 𝚽(𝑥, 𝑦, 𝜌𝑥𝑦 ) is the cdf of a stan𝑇2 − 𝑡 dard bivariate normal with correlation coefficient 𝜌𝑥𝑦 ∈ (−1, 1) and Φ(𝑥) is the cdf of a standard normal. Finally, deduce the call-on-a-put option price at time 𝑡 ≤ 𝑇1 ≤ 𝑇2 with option payoff 𝜌=

Ψ(𝑆𝑇1 ) = max{𝑃𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) − 𝐾1 , 0}. Solution: From Girsanov’s theorem, under the risk-neutral measure ℚ, 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡

(𝜇 − 𝑟) 𝑡 is a ℚ-standard Wiener process. Using It¯o’s lemma we can where 𝑊𝑡ℚ = 𝑊𝑡 + 𝜎 easily show for 𝑇 > 𝑡 ( ) ) ] [( 𝑆𝑇 1 log ∼  𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡), 𝜎 2 (𝑇 − 𝑡) 𝑆𝑡 2

6.2.2 Path-Dependent Options

597

with density function (

𝑓 (𝑆𝑇 |𝑆𝑡 ) =

√

− 21

1

𝑆𝑇 𝜎 2𝜋(𝑇 − 𝑡)

𝑒

log(𝑆𝑇 ∕𝑆𝑡 )−(𝑟−𝐷− 1 𝜎 2 )(𝑇 −𝑡) 2 √ 𝜎 𝑇 −𝑡

)2

.

By definition the put-on-a-put option at time 𝑡 ≤ 𝑇1 ≤ 𝑇2 is 𝑃𝑝 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇1 , 𝐾2 , 𝑇2 ) [ }| ] { −𝑟(𝑇1 −𝑡) ℚ 𝔼 max 𝐾1 − 𝑃𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ), 0 || ℱ𝑡 =𝑒 | { } ∞ max 𝐾1 − 𝑃𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ), 0 𝑓 (𝑆𝑇1 |𝑆𝑡 ) 𝑑𝑆𝑇1 = 𝑒−𝑟(𝑇1 −𝑡) ∫0 ] 𝑆̂𝑇 [ 1 −𝑟(𝑇1 −𝑡) 𝐾1 − 𝑃𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) 𝑓 (𝑆𝑇1 |𝑆𝑡 ) 𝑑𝑆𝑇1 =𝑒 ∫0 = 𝐵1 − 𝐵2 + 𝐵3 where 𝑆̂𝑇1 satisfies the equation 𝑃𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) = 𝐾1 and 𝐵1 = 𝐾1 𝑒

−𝑟(𝑇1 −𝑡)

𝐵2 = 𝑒

−𝑟(𝑇1 −𝑡)

𝐵3 = 𝑒−𝑟(𝑇1 −𝑡)

𝑆̂𝑇

∫0

𝑓 (𝑆𝑇1 |𝑆𝑡 ) 𝑑𝑆𝑇1

𝑆̂𝑇

1

𝐾2 𝑒−𝑟(𝑇2 −𝑇1 ) Φ(−𝑑− )𝑓 (𝑆𝑇1 |𝑆𝑡 ) 𝑑𝑆𝑇1

𝑆̂𝑇

1

𝑆𝑇1 𝑒−𝐷(𝑇2 −𝑇1 ) Φ(−𝑑+ )𝑓 (𝑆𝑇1 |𝑆𝑡 ) 𝑑𝑆𝑇1 .

∫0 ∫0

1

Given 𝑊𝑇ℚ−𝑡 ∼  (0, 𝑇1 − 𝑡) we have 1

𝐵1 = 𝐾1 𝑒−𝑟(𝑇1 −𝑡)

𝑆̂𝑇

1

𝑓 (𝑆𝑇1 |𝑆𝑡 ) 𝑑𝑆𝑇1 ∫0 ) ( | = 𝐾1 𝑒−𝑟(𝑇1 −𝑡) ℚ 𝑆𝑇1 < 𝑆̂𝑇1 | 𝑆𝑡 | ( ) 1 2 | (𝑟−𝐷− 𝜎 )(𝑇1 −𝑡)+𝜎𝑊𝑇ℚ −𝑡 −𝑟(𝑇1 −𝑡) 2 | ̂ 1 = 𝐾1 𝑒 ℚ 𝑆𝑡 𝑒 < 𝑆𝑇1 | 𝑆𝑡 | ( ) ̂ log(𝑆𝑇1 ∕𝑆𝑡 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇1 − 𝑡) || ℚ −𝑟(𝑇1 −𝑡) |𝑆 ℚ 𝑊𝑇 −𝑡 < = 𝐾1 𝑒 | 𝑡 1 𝜎 | | = 𝐾1 𝑒−𝑟(𝑇1 −𝑡) Φ(−𝛼− )

log(𝑆𝑡 ∕𝑆̂𝑇1 ) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇1 − 𝑡) . where 𝛼− = √ 𝜎 𝑇1 − 𝑡

598

6.2.2 Path-Dependent Options

As for the case 𝐵2 = 𝐾2 𝑒−𝑟(𝑇2 −𝑡) ( ̂ 𝑆𝑇

×

1

∫0

Φ

− log(𝑆𝑇1 ∕𝐾2 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇2 − 𝑇1 ) √ 𝜎 𝑇2 − 𝑇1

) 𝑓 (𝑆𝑇1 |𝑆𝑡 ) 𝑑𝑆𝑇1

log(𝑆𝑇1 ∕𝑆𝑡 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇1 − 𝑡) we have and by setting 𝑥 = √ 𝜎 𝑇1 − 𝑡 𝐵2 = 𝐾2 𝑒−𝑟(𝑇2 −𝑡) ( −𝛼−

×

∫−∞

Φ

√ ) log(𝐾2 ∕𝑆𝑡 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇2 − 𝑡) − 𝑥𝜎 𝑇1 − 𝑡 1 − 1 𝑥2 √ √ 𝑒 2 𝑑𝑥. 𝜎 𝑇2 − 𝑇1 2𝜋

By simplifying the integrands and from Problem 1.2.2.16 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus we can write 𝐵2 = 𝐾2 𝑒−𝑟(𝑇2 −𝑡)

(

−𝛼−

∫−∞

Φ

−𝛽− − 𝜌𝑥 √ 1 − 𝜌2

) 1 − 1 𝑥2 √ 𝑒 2 𝑑𝑥 2𝜋

= 𝐾2 𝑒−𝑟(𝑇2 −𝑡) 𝚽(−𝛼− , −𝛽− , 𝜌) log(𝑆𝑡 ∕𝐾2 ) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇2 − 𝑡) and 𝜌 = where 𝛽− = √ 𝜎 𝑇2 − 𝑡 Finally, for the case

√

𝑇1 − 𝑡 . 𝑇2 − 𝑡

𝐵3 = 𝑒−𝑟(𝑇1 −𝑡)−𝐷(𝑇2 −𝑇1 ) ( ) 𝑆̂𝑇 log(𝐾2 ∕𝑆𝑇1 ) − (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇2 − 𝑇1 ) 1 × 𝑆𝑇1 Φ 𝑓 (𝑆𝑇1 |𝑆𝑡 ) 𝑑𝑆𝑇1 √ ∫0 𝜎 𝑇2 − 𝑇1 and by setting 𝑥 =

log(𝑆𝑇1 ∕𝑆𝑡 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇1 − 𝑡) we have √ 𝜎 𝑇1 − 𝑡 1 2

𝐵3 = 𝑆𝑡 𝑒−𝑟(𝑇1 −𝑡)−𝐷(𝑇2 −𝑇1 )+(𝑟−𝐷− 2 𝜎 )(𝑇1 −𝑡) ( ) √ 𝛼 √ 𝑚 + 𝑥𝜎 𝑇1 − 𝑡 1 − 1 (𝑥2 −2𝑥𝜎 𝑇2 −𝑡) × Φ 𝑑𝑥 √ √ 𝑒 2 ∫−∞ 𝜎 𝑇 −𝑇 2𝜋 2

1

or 𝐵3 = 𝑆𝑡 𝑒

−𝐷(𝑇2 −𝑡)

(

𝛼

∫−∞

Φ

) √ √ 𝑚 − 𝑥𝜎 𝑇1 − 𝑡 1 − 1 (𝑥−𝜎 𝑇1 −𝑡)2 𝑑𝑥 √ √ 𝑒 2 𝜎 𝑇2 − 𝑇1 2𝜋

6.2.2 Path-Dependent Options

599

where log(𝑆̃𝑇1 ∕𝑆𝑡 ) − (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇1 − 𝑡) √ 𝜎 𝑇1 − 𝑡 1 1 𝑚 = log(𝐾2 ∕𝑆𝑡 ) − (𝑟 − 𝐷 + 𝜎 2 )(𝑇2 − 𝑇1 ) − (𝑟 − 𝐷 − 𝜎 2 )(𝑇1 − 𝑡). 2 2 √ By setting 𝑦 = 𝑥 − 𝜎 𝑇1 − 𝑡 we have ( ) √ −𝛼+ 𝑚 − 𝜎 2 (𝑇1 − 𝑡) − 𝑦𝜎 𝑇1 − 𝑡 1 − 1 𝑦2 −𝐷(𝑇2 −𝑡) Φ 𝐵3 = 𝑆𝑡 𝑒 √ √ 𝑒 2 𝑑𝑦 ∫−∞ 𝜎 𝑇 2 − 𝑇1 2𝜋 ( ) −𝛼+ 1 −𝛽+ − 𝜌𝑦 1 − 𝑦2 = 𝑆𝑡 𝑒−𝐷(𝑇2 −𝑡) Φ √ √ 𝑒 2 𝑑𝑦 ∫−∞ 2 1−𝜌 2𝜋 𝛼=

= 𝑆𝑡 𝑒−𝐷(𝑇2 −𝑡) 𝚽(−𝛼+ , −𝛽+ , 𝜌). Thus, the put-on-a-put option price at time 𝑡 ≤ 𝑇1 ≤ 𝑇2 is 𝑃𝑝 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇1 , 𝐾2 , 𝑇2 ) = 𝐾1 𝑒−𝑟(𝑇1 −𝑡) Φ(−𝛼− ) − 𝐾2 𝑒−𝑟(𝑇2 −𝑡) 𝚽(−𝛼− , −𝛽− , 𝜌) +𝑆𝑡 𝑒−𝐷(𝑇2 −𝑡) 𝚽(−𝛼+ , −𝛽+ , 𝜌). For the case of a call-on-a-put option, the payoff at time 𝑇1 is 𝐶𝑝 (𝑆𝑇1 , 𝑇1 ; 𝐾1 , 𝑇1 , 𝐾2 , 𝑇2 ) = max{𝑃𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) − 𝐾1 , 0} whilst the payoff at time 𝑇1 for a put-on-a-put option is 𝑃𝑝 (𝑆𝑇1 , 𝑇1 ; 𝐾1 , 𝑇1 , 𝐾2 , 𝑇2 ) = max{𝐾1 − 𝑃𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ), 0}. Hence, 𝐶𝑝 (𝑆𝑇1 , 𝑇1 ; 𝐾1 , 𝑇1 , 𝐾2 , 𝑇2 ) − 𝑃𝑝 (𝑆𝑇1 , 𝑇1 ; 𝐾1 , 𝑇1 , 𝐾2 , 𝑇2 ) ⎧ 𝑃 (𝑆 , 𝑇 ; 𝐾 , 𝑇 ) − 𝐾 1 ⎪ 𝑏𝑠 𝑇1 1 2 2 =⎨ ⎪ 𝑃𝑏𝑠 (𝑆𝑇 , 𝑇1 ; 𝐾2 , 𝑇2 ) − 𝐾1 1 ⎩

if 𝑃𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) > 𝐾1 if 𝑃𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) ≤ 𝐾1

= 𝑃𝑏𝑠 (𝑆𝑇1 , 𝑇1 ; 𝐾2 , 𝑇2 ) − 𝐾1 . By discounting the payoff back to time 𝑡 under the risk-neutral measure ℚ, we have 𝐶𝑝 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇1 , 𝐾2 , 𝑇2 ) − 𝑃𝑝 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇1 , 𝐾2 , 𝑇2 ) = 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇2 ) − 𝐾1 𝑒−𝑟(𝑇1 −𝑡) or 𝐶𝑝 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇1 , 𝐾2 , 𝑇2 ) = 𝑃𝑝 (𝑆𝑡 , 𝑡; 𝐾1 , 𝑇1 , 𝐾2 , 𝑇2 ) + 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾2 , 𝑇2 ) − 𝐾1 𝑒−𝑟(𝑇1 −𝑡) .

600

6.2.2 Path-Dependent Options

5. Simple Chooser Option. Let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. In addition, we let 𝑟 be the risk-free interest rate. A simple chooser option is an option contract whereby it gives the holder of the option at a fixed time 𝜏, 𝑡 ≤ 𝜏 ≤ 𝑇 the right but not the obligation to decide whether the contract is a European call or put option with the following payoff ( ) Ψ 𝑆𝜏 = max{𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ), 𝑃𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 )} where 𝑆𝜏 is the stock price at time 𝜏, 𝑇 is the chooser option expiry time, 𝐾 is the strike price, 𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) = 𝑆𝜏 𝑒−𝐷(𝑇 −𝜏) Φ(𝑑+ ) − 𝐾𝑒−𝑟(𝑇 −𝜏) Φ(𝑑− ) and 𝑃𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) = 𝐾𝑒−𝑟(𝑇 −𝜏) Φ(−𝑑− ) − 𝑆𝜏 𝑒−𝐷(𝑇 −𝜏) Φ(−𝑑+ ) are the European call and put options at time 𝜏, respectively with log(𝑆𝜏 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝜏) 𝑑± = √ 𝜎 𝑇 −𝜏 and Φ(⋅) is the cdf of a standard normal. Show that 𝑒−𝑟𝑡 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is a martingale under the risk-neutral measure ℚ, where 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is the price of a European option at time 𝑡 given as ⎧ 𝐶 (𝑆 , 𝑡; 𝐾, 𝑇 ) ⎪ 𝑏𝑠 𝑡 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ ⎪ 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎩

if payoff Ψ(𝑆𝑇 ) = max{𝑆𝑇 − 𝐾, 0} if payoff Ψ(𝑆𝑇 ) = max{𝐾 − 𝑆𝑇 , 0}.

Given the simple chooser option is valued at time 𝑡, 𝑡 ≤ 𝜏 show using the put–call parity that the payoff at intermediate time 𝜏 can be expressed as ̃ − 𝑆𝜏 , 0} Ψ(𝑆𝜏 ) = 𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) + 𝑒−𝐷(𝑇 −𝜏) max{𝐾 ̃ = 𝐾𝑒−(𝑟−𝐷)(𝑇 −𝜏) . where 𝐾 Hence, show that the simple chooser option price at time 𝑡 ≤ 𝜏 ≤ 𝑇 is ̃ 𝜏, 𝐾, 𝑇 ) = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑒−𝐷(𝑇 −𝜏) 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, ̃ 𝜏). 𝑉𝑠𝑐 (𝑆𝑡 , 𝑡; 𝐾,

6.2.2 Path-Dependent Options

601

Solution: Under the risk-neutral measure ℚ, 𝑆𝑡 follows

where 𝑊𝑡ℚ = 𝑊𝑡 +

(𝜇 − 𝑟)

𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡

𝑡 is a ℚ-standard Wiener process. To show that 𝜎 𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is a ℚ-martingale, see Problem 2.2.2.8 (page 109). From the simple chooser payoff at time 𝜏 ≤ 𝑇

𝑒−𝑟𝑡 𝑉

Ψ(𝑆𝜏 ) = max{𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ), 𝑃𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 )} and knowing the put–call parity at time 𝜏 is 𝑃𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) = 𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) + 𝐾𝑒−𝑟(𝑇 −𝜏) − 𝑆𝜏 𝑒−𝐷(𝑇 −𝜏) we can rewrite the payoff as Ψ(𝑆𝜏 ) = max{𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ), 𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) + 𝐾𝑒−𝑟(𝑇 −𝜏) − 𝑆𝜏 𝑒−𝐷(𝑇 −𝜏) } = 𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) + max{0, 𝐾𝑒−𝑟(𝑇 −𝜏) − 𝑆𝜏 𝑒−𝐷(𝑇 −𝜏) } = 𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) + 𝑒−𝐷(𝑇 −𝜏) max{𝐾𝑒−(𝑟−𝐷)(𝑇 −𝜏) − 𝑆𝜏 , 0} ̃ − 𝑆𝜏 , 0} = 𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) + 𝑒−𝐷(𝑇 −𝜏) max{𝐾 ̃ = 𝐾𝑒−(𝑟−𝐷)(𝑇 −𝜏) . where 𝐾 Therefore, the price of a simple chooser option at time 𝑡 ≤ 𝜏 ≤ 𝑇 is [ ] ̃ 𝜏, 𝐾, 𝑇 ) = 𝑒−𝑟(𝜏−𝑡) 𝔼ℚ Ψ(𝑆𝜏 )|ℱ𝑡 𝑉𝑠𝑐 (𝑆𝑡 , 𝑡; 𝐾, = 𝑒−𝑟(𝜏−𝑡) [ ] ̃ − 𝑆𝜏 , 0}|| ℱ𝑡 ×𝔼ℚ 𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 ) + 𝑒−𝐷(𝑇 −𝜏) max{𝐾 | −𝑟(𝜏−𝑡) 𝑟(𝜏−𝑡) =𝑒 ⋅𝑒 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) [ ] −𝐷(𝑇 −𝜏) −𝑟(𝜏−𝑡) ℚ ̃ − 𝑆𝜏 , 0}|| ℱ𝑡 ⋅𝑒 𝔼 max{𝐾 +𝑒 | ̃ 𝜏) = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑒−𝐷(𝑇 −𝜏) 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, [ ] since 𝔼ℚ 𝑒−𝑟𝜏 𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇 )|| ℱ𝑡 = 𝑒−𝑟𝑡 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is a ℚ-martingale. 6. Complex Chooser Option. Let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. In addition, we let 𝑟 be the risk-free interest rate.

602

6.2.2 Path-Dependent Options

A complex chooser option is an option contract whereby it gives the holder of the option at a fixed time 𝜏, 𝑡 ≤ 𝜏 ≤ min{𝑇𝑐 , 𝑇𝑝 } the right to decide whether the contract is a European call or put option of different time to expiry and strike prices with the following payoff ( ) Ψ 𝑆𝜏 = max{𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾𝑐 , 𝑇𝑐 ), 𝑃𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾𝑝 , 𝑇𝑝 )} where 𝑆𝜏 is the stock price at time 𝜏, 𝑇𝑐 is the call option expiry time, 𝑇𝑝 is the put option expiry time, 𝐾𝑐 is the call option strike, 𝐾𝑝 is the put option strike so that 𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾𝑐 , 𝑇𝑐 ) = 𝑆𝜏 𝑒−𝐷(𝑇𝑐 −𝜏) Φ(𝑑+𝑐 ) − 𝐾𝑐 𝑒−𝑟(𝑇𝑐 −𝜏) Φ(𝑑−𝑐 ) and 𝑃𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾, 𝑇𝑝 ) = 𝐾𝑝 𝑒−𝑟(𝑇𝑝 −𝜏) Φ(−𝑑−𝑝 ) − 𝑆𝜏 𝑒−𝐷(𝑇𝑝 −𝜏) Φ(−𝑑+𝑝 ) are the European call and put options at time 𝜏, respectively with log(𝑆𝜏 ∕𝐾𝑐 ) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇𝑐 − 𝜏) log(𝑆𝜏 ∕𝐾𝑝 ) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇𝑝 − 𝜏) 𝑝 𝑐 𝑑± = , 𝑑± = √ √ 𝜎 𝑇𝑝 − 𝜏 𝜎 𝑇𝑐 − 𝜏 and Φ(⋅) is the cdf of a standard normal. Show that under the risk-neutral measure ℚ, the complex chooser option price at time 𝑡 ≤ 𝜏 is 𝑉𝑐𝑐 (𝑆𝑡 , 𝑡; 𝜏, 𝐾𝑐 , 𝑇𝑐 , 𝐾𝑝 , 𝑇𝑝 ) = 𝑆𝑡 𝑒−𝐷(𝑇𝑐 −𝑡) 𝚽(𝛼+ , 𝛽+ , 𝜌𝑐 ) −𝐾𝑐 𝑒−𝑟(𝑇𝑐 −𝑡) 𝚽(𝛼− , 𝛽− , 𝜌𝑐 ) +𝐾𝑝 𝑒−𝑟(𝑇𝑝 −𝑡) 𝚽(−𝛼− , −𝛾− , 𝜌𝑝 ) −𝑆𝑡 𝑒−𝐷(𝑇𝑝 −𝑡) 𝚽(−𝛼+ , −𝛾+ , 𝜌𝑝 ) where

log(𝑆𝑡 ∕𝑋) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝜏 − 𝑡) log(𝑆𝑡 ∕𝐾𝑐 ) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇𝑐 − 𝑡) , 𝛽± = 𝛼± = , √ √ 𝜎 𝑇𝑐 − 𝑡 𝜎 𝜏 −𝑡 √ √ log(𝑆𝑡 ∕𝐾𝑝 ) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇𝑝 − 𝑡) 𝜏 −𝑡 𝜏 −𝑡 𝛾± = , 𝜌𝑝 = , 𝑋 solves the , 𝜌𝑐 = √ 𝑇𝑐 − 𝑡 𝑇𝑝 − 𝑡 𝜎 𝑇𝑝 − 𝑡 equation 𝐶𝑏𝑠 (𝑋, 𝜏; 𝐾𝑐 , 𝑇𝑐 ) = 𝑃𝑏𝑠 (𝑋, 𝜏; 𝐾𝑝 , 𝑇𝑝 ) and 𝚽(𝑢, 𝑣, 𝜌𝑢𝑣 ) is the cdf of a standard bivariate normal with correlation coefficient 𝜌𝑢𝑣 ∈ (−1, 1). Solution: Under the risk-neutral measure ℚ, the asset price 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡

6.2.2 Path-Dependent Options

603

(𝜇 − 𝑟) where 𝑊𝑡ℚ = 𝑊𝑡 + 𝑡 is a ℚ-standard Wiener process. From It¯o’s lemma we can 𝜎 easily show for 𝑇 > 𝑡 ( log

𝑆𝑇 𝑆𝑡

) ∼

[(

) ] 1 𝑟 − 𝐷 − 𝜎 2 (𝑇 − 𝑡), 𝜎 2 (𝑇 − 𝑡) 2

with density function (

𝑓 (𝑆𝑇 |𝑆𝑡 ) =

√

− 21

1

𝑆𝑇 𝜎 2𝜋(𝑇 − 𝑡)

𝑒

log(𝑆𝑇 ∕𝑆𝑡 )−(𝑟−𝐷− 1 𝜎 2 )(𝑇 −𝑡) 2 √ 𝜎 𝑇 −𝑡

)2

.

By definition, the complex chooser option price at time 𝑡, 𝑡 ≤ 𝜏 ≤ min{𝑇𝑐 , 𝑇𝑝 } is 𝑉𝑐𝑐 (𝑆𝑡 , 𝑡; 𝜏, 𝐾𝑐 , 𝑇𝑐 , 𝐾𝑝 , 𝑇𝑝 ) [ ] | = 𝑒−𝑟(𝜏−𝑡) 𝔼ℚ max{𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾𝑐 , 𝑇𝑐 ), 𝑃𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾𝑝 , 𝑇𝑝 )}| ℱ𝑡 | = 𝑒−𝑟(𝜏−𝑡)

∞

∫0

max{𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾𝑐 , 𝑇𝑐 ), 𝑃𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾𝑝 , 𝑇𝑝 )}𝑓 (𝑆𝜏 |𝑆𝑡 ) 𝑑𝑆𝜏 .

Let 𝑋 solve the nonlinear equation 𝐶𝑏𝑠 (𝑋, 𝜏; 𝐾𝑐 , 𝑇𝑐 ) = 𝑃𝑏𝑠 (𝑋, 𝜏; 𝐾𝑝 , 𝑇𝑝 ), then we can rewrite 𝑉𝑐𝑐 (𝑆𝑡 , 𝑡; 𝜏, 𝐾𝑐 , 𝑇𝑐 , 𝐾𝑝 , 𝑇𝑝 ) = 𝑒−𝑟(𝜏−𝑡)

𝑋

𝑃𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾𝑝 , 𝑇𝑝 )𝑓 (𝑆𝜏 |𝑆𝑡 ) 𝑑𝑆𝜏

∫0

∞

+𝑒−𝑟(𝜏−𝑡)

∫𝑋

𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾𝑐 , 𝑇𝑐 )𝑓 (𝑆𝜏 |𝑆𝑡 ) 𝑑𝑆𝜏 .

From Problem 6.2.2.3 (page 591) we can deduce that 𝑒−𝑟(𝜏−𝑡)

∞

∫𝑋

= 𝑒−𝑟(𝜏−𝑡)

( ) 𝑆𝜏 𝑒−𝐷(𝑇𝑐 −𝜏) Φ 𝑑+𝑐 𝑓 (𝑆𝜏 |𝑆𝑡 ) 𝑑𝑆𝜏

∞

∫𝑋

−𝑒−𝑟(𝜏−𝑡) = 𝑆𝑡 𝑒

𝐶𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾𝑐 , 𝑇𝑐 )𝑓 (𝑆𝜏 |𝑆𝑡 ) 𝑑𝑆𝜏

∞

∫𝑋 −𝐷(𝑇𝑐 −𝑡)

( ) 𝐾𝑐 𝑒−𝑟(𝑇𝑐 −𝜏) Φ 𝑑−𝑐 𝑓 (𝑆𝜏 |𝑆𝑡 ) 𝑑𝑆𝜏

𝚽(𝛼+ , 𝛽+ , 𝜌𝑐 ) − 𝐾𝑐 𝑒−𝑟(𝑇𝑐 −𝑡) 𝚽(𝛼− , 𝛽− , 𝜌𝑐 )

log(𝑆𝑡 ∕𝑋) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝜏 − 𝑡) log(𝑆𝑡 ∕𝐾𝑐 ) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇𝑐 − 𝑡) , 𝛽 = √ √ ± 𝜎 𝑇𝑐 − 𝑡 𝜎 𝜏 −𝑡 √ 𝜏 −𝑡 . and 𝜌𝑐 = 𝑇𝑐 − 𝑡 where 𝛼± =

604

6.2.2 Path-Dependent Options

Furthermore, from Problem 6.2.2.4 (page 596) we can also deduce that 𝑒−𝑟(𝜏−𝑡)

𝑋

𝑃𝑏𝑠 (𝑆𝜏 , 𝜏; 𝐾𝑝 , 𝑇𝑝 )𝑓 (𝑆𝜏 |𝑆𝑡 ) 𝑑𝑆𝜏

∫0

= 𝑒−𝑟(𝜏−𝑡)

𝑋

∫0

𝑋

−𝑒−𝑟(𝜏−𝑡) = 𝐾𝑝 𝑒

) ( 𝐾𝑝 𝑒−𝑟(𝑇𝑝 −𝜏) Φ −𝑑−𝑝 𝑓 (𝑆𝜏 |𝑆𝑡 ) 𝑑𝑆𝜏

∫0 −𝑟(𝑇𝑝 −𝑡)

) ( 𝑆𝜏 𝑒−𝐷(𝑇𝑝 −𝜏) Φ −𝑑+𝑝 𝑓 (𝑆𝜏 |𝑆𝑡 ) 𝑑𝑆𝜏

𝚽(−𝛼− , −𝛾− , 𝜌𝑝 ) − 𝑆𝑡 𝑒−𝐷(𝑇𝑝 −𝑡) 𝚽(−𝛼+ , −𝛾+ , 𝜌𝑝 )

√ log(𝑆𝑡 ∕𝐾𝑝 ) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇𝑝 − 𝑡) 𝜏 −𝑡 where 𝛾± = . and 𝜌𝑝 = √ 𝑇𝑝 − 𝑡 𝜎 𝑇𝑝 − 𝑡 Hence, the price of a complex chooser option at time 𝑡 ≤ 𝜏 is 𝑉𝑐𝑐 (𝑆𝑡 , 𝑡; 𝜏, 𝐾𝑐 , 𝑇𝑐 , 𝐾𝑝 , 𝑇𝑝 ) = 𝑆𝑡 𝑒−𝐷(𝑇𝑐 −𝑡) 𝚽(𝛼+ , 𝛽+ , 𝜌𝑐 ) −𝐾𝑐 𝑒−𝑟(𝑇𝑐 −𝑡) 𝚽(𝛼− , 𝛽− , 𝜌𝑐 ) +𝐾𝑝 𝑒−𝑟(𝑇𝑝 −𝑡) 𝚽(−𝛼− , −𝛾− , 𝜌𝑝 ) −𝑆𝑡 𝑒−𝐷(𝑇𝑝 −𝑡) 𝚽(−𝛼+ , −𝛾+ , 𝜌𝑝 ).

7. Black–Scholes Equation for Lookback Option I. Let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the stock price 𝑆𝑡 follow a GBM 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. In addition, we let 𝑟 be the risk-free interest rate. Consider a European-style lookback option with terminal payoff 𝑉 (𝑆𝑇 , 𝑀𝑇 , 𝑇 ) = Ψ(𝑆𝑇 , 𝑀𝑇 ) which depends on the maximum of the stock price 𝑀𝑇 reached within the lookback time period [𝑡0 , 𝑇 ], 𝑡0 ≥ 0 where 𝑇 , 𝑇 > 𝑡 is the option expiry time. By defining [ 𝑀𝑡(𝑛)

=

𝑡

∫𝑡0

]1 𝑆𝑢𝑛 𝑑𝑢

𝑛

and

for 𝑛 ∈ ℕ, show that lim 𝑀𝑡(𝑛) = 𝑀𝑡 .

𝑛→∞

𝑀𝑡 = max 𝑆𝑢 𝑡0 ≤𝑢≤𝑡

6.2.2 Path-Dependent Options

605

Explain why, when the current stock price equals its current maximum, we would have the following result 𝜕𝑉 || = 0. 𝜕𝑀𝑡 ||𝑆𝑡 =𝑀𝑡 By considering a hedging portfolio involving both lookback option 𝑉 (𝑆𝑡 , 𝑀𝑡(𝑛) , 𝑡) and stock price 𝑆𝑡 , show that for 0 < 𝑆𝑡 < 𝑀𝑡(𝑛) , 𝑉 (𝑆𝑡 , 𝑀𝑡(𝑛) , 𝑡) satisfies the following PDE 𝑆𝑡𝑛 1 2 2 𝜕2𝑉 𝜕𝑉 𝜕𝑉 + + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑀𝑡(𝑛) , 𝑡) = 0. 𝜎 𝑆𝑡 ]𝑛−1 (𝑛) 2 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 𝜕𝑀𝑡 𝑀 (𝑛)

𝜕𝑉 1 + [ 𝜕𝑡 𝑛

𝑡

Finally, by taking 𝑛 → ∞, show that for 0 < 𝑆𝑡 < 𝑀𝑡 , 𝑉 (𝑆𝑡 , 𝑀𝑡 , 𝑡) satisfies 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 − 𝑟𝑉 (𝑆𝑡 , 𝑀𝑡 , 𝑡) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 subject to the boundary conditions 𝜕𝑉 || = 0. 𝜕𝑀𝑡 ||𝑆𝑡 =𝑀𝑡

𝑉 (𝑆𝑇 , 𝑀𝑇 , 𝑇 ) = Ψ(𝑆𝑇 , 𝑀𝑇 ) and Solution: By definition, for 𝑛 ∈ ℕ [ 𝑀𝑡(𝑛)

=

]1

𝑡

𝑛

𝑆𝑢𝑛 𝑑𝑢

∫𝑡0

and

𝑀𝑡 = max 𝑆𝑢 𝑡0 ≤𝑢≤𝑡

and because 𝑆𝑡 is continuous within the interval [𝑡0 , 𝑡], therefore the integral exists as well as the maximum 𝑀𝑡(𝑛) and 𝑀𝑡 . Since 𝑆𝑢 ≤ 𝑀𝑡 , 𝑡0 ≤ 𝑢 ≤ 𝑡, therefore for any 𝑛 ∈ ℕ 0
0 and define 𝜖 (𝑡) to be the set of 𝑢 ∈ [𝑡0 , 𝑡] for which 𝑆𝑢 ≥ 𝑀𝑡 − 𝜖,

𝑢 ∈ [𝑡0 , 𝑡]

and let 𝜆𝜖 (𝑡) =

∫𝜖 (𝑡)

𝑑𝑢

be the total length of the interval in which 𝑆𝑢 ≥ 𝑀𝑡 − 𝜖. Since 𝑆𝑢 is continuous for 𝑢 ∈ [𝑡0 , 𝑡], 𝜆𝜖 (𝑡) > 0 and 𝜆𝜖 (𝑡) ≤ 𝑡 − 𝑡0 therefore 𝑡

∫𝑡0

𝑆𝑢𝑛 𝑑𝑢 ≥

∫𝜖 (𝑡)

𝑆𝑢𝑛 𝑑𝑢 ≥

∫𝜖 (𝑡)

(𝑀𝑡 − 𝜖)𝑛 𝑑𝑢 = (𝑀𝑡 − 𝜖)𝑛 𝜆𝜖 (𝑡)

and hence [

𝑡

∫𝑡0

]1 𝑆𝑢𝑛 𝑑𝑢

𝑛

1

≥ 𝜆𝜖 (𝑡) 𝑛 (𝑀𝑡 − 𝜖).

Taking limits 𝑛 → ∞ lim 𝑀𝑡(𝑛) ≥ 𝑀𝑡 − 𝜖

𝑛→∞

and thus we have 𝑀𝑡 − 𝜖 ≤ lim 𝑀𝑡(𝑛) ≤ 𝑀𝑡 𝑛→∞

and because 𝜖 > 0 is a small number we can deduce that lim 𝑀𝑡(𝑛) = 𝑀𝑡 .

𝑛→∞

To show that

𝜕𝑉 = 0 on 𝑆𝑡 = 𝑀𝑡 , we note that if 𝑆𝑡 = 𝑀𝑡 then 𝜕𝑀𝑡 {

𝑀𝑇 = max 𝑆𝑢 = max 𝑡0 ≤𝑢≤𝑇

} max 𝑆𝑢 , max 𝑆𝑢

𝑡0 ≤𝑢≤𝑡

𝑡≤𝑢≤𝑇

≥ max 𝑆𝑢 = 𝑀𝑡 𝑡0 ≤𝑢≤𝑡

and hence ℙ(𝑀𝑇 = 𝑀𝑡 ) = 0, which shows that 𝑀𝑡 cannot be the final maximum at option expiry time 𝑇 .

6.2.2 Path-Dependent Options

607

Since 𝑀𝑇 ≠ 𝑀𝑡 , this implies that the lookback option 𝑉 (𝑆𝑡 , 𝑀𝑡 , 𝑡) at time 𝑡 is insensitive to small changes in 𝑀𝑡 . Therefore, 𝜕𝑉 || = 0. 𝜕𝑀𝑡 ||𝑆𝑡 =𝑀𝑡 To find the PDE satisfied by 𝑉 (𝑆𝑡 , 𝑀𝑡(𝑛) , 𝑡) we first construct a Δ-hedged portfolio having one option 𝑉 (𝑆𝑡 , 𝑀𝑡(𝑛) , 𝑡) and −Δ number of shares 𝑆𝑡 . Thus, the hedged portfolio Π𝑡 at time 𝑡 is Π𝑡 = 𝑉 (𝑆𝑡 , 𝑀𝑡(𝑛) , 𝑡) − Δ𝑆𝑡 . From 𝑡 to 𝑡 + 𝑑𝑡, and because the holder receives 𝐷𝑆𝑡 𝑑𝑡 for every asset held, the portfolio value changes by an amount 𝑑Π𝑡 = 𝑑𝑉 − Δ(𝑑𝑆𝑡 + 𝐷𝑆𝑡 𝑑𝑡) where 𝑑𝑉 =

𝜕𝑉 1 𝜕2𝑉 1 𝜕2𝑉 𝜕𝑉 𝜕𝑉 𝑑𝑆𝑡 + 𝑑𝑀𝑡(𝑛) + (𝑑𝑆𝑡 )2 + (𝑑𝑀𝑡(𝑛) )2 + … 𝑑𝑡 + (𝑛) 2 𝜕𝑡 𝜕𝑆𝑡 2 𝜕𝑆𝑡 2 𝜕(𝑀 (𝑛) )2 𝜕𝑀𝑡 𝑡

and 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 . [ Since

𝑀𝑡(𝑛)

=

𝑡

∫𝑡0

]1 𝑆𝑢𝑛 𝑑𝑢

𝑑𝑀𝑡(𝑛)

𝑛

we have

] 1 −1 [ 𝑛 𝑡 𝑆𝑡𝑛 1 1 𝑛 = 𝑆𝑢 𝑑𝑢 𝑆𝑡𝑛 𝑑𝑡 = 𝑑𝑡. 𝑛 ∫𝑡0 𝑛 (𝑀 (𝑛) )𝑛−1 𝑡

From It¯o’s lemma 𝑑𝑉 =

𝑆𝑡𝑛 ] 1 𝜕𝑉 𝜕𝑉 [ 𝜕𝑉 (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 + 𝑑𝑡 𝑑𝑡 + 𝜕𝑡 𝜕𝑆𝑡 𝑛 (𝑀 (𝑛) )𝑛−1 𝜕𝑀 (𝑛) 𝑡 𝑡

1 𝜕2𝑉 + 𝜎 2 𝑆𝑡2 2 𝑑𝑡 2 𝜕𝑆𝑡 [ ] 𝑆𝑡𝑛 𝜕𝑉 1 2 2 𝜕2𝑉 𝜕𝑉 𝜕𝑉 1 = + 𝜎 𝑆𝑡 + (𝜇 − 𝐷)𝑆𝑡 + 𝑑𝑡 𝜕𝑡 𝑛 (𝑀 (𝑛) )𝑛−1 𝜕𝑀 (𝑛) 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 𝑡

𝜕𝑉 𝑑𝑊𝑡 +𝜎𝑆𝑡 𝜕𝑆𝑡

𝑡

608

6.2.2 Path-Dependent Options

and by substituting the above equation back into 𝑑Π𝑡 and rearranging terms, we have [

] 𝑆𝑡𝑛 𝜕𝑉 1 2 2 𝜕2𝑉 𝜕𝑉 𝜕𝑉 1 + 𝜎 𝑆𝑡 + (𝜇 − 𝐷)𝑆𝑡 − 𝜇Δ𝑆𝑡 𝑑𝑡 + 𝑑Π𝑡 = 2 𝜕𝑡 𝑛 (𝑀 (𝑛) )𝑛−1 𝜕𝑀 (𝑛) 2 𝜕𝑆𝑡 𝜕𝑆 𝑡 𝑡 𝑡 ) ( 𝜕𝑉 − Δ 𝑑𝑊𝑡 . +𝜎𝑆𝑡 𝜕𝑆𝑡 To eliminate the random term we set Δ=

𝜕𝑉 𝜕𝑆𝑡

and hence [ 𝑑Π𝑡 =

] 𝑆𝑡𝑛 𝜕𝑉 1 2 2 𝜕2𝑉 𝜕𝑉 𝜕𝑉 1 + 𝜎 𝑆𝑡 − 𝐷𝑆𝑡 + 𝑑𝑡. 𝜕𝑡 𝑛 (𝑀 (𝑛) )𝑛−1 𝜕𝑀 (𝑛) 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 𝑡

𝑡

Under the no-arbitrage condition, the return on the amount Π𝑡 invested in a risk-free interest rate would see a growth of 𝑑Π𝑡 = 𝑟Π𝑡 𝑑𝑡 and therefore [

𝑛

𝑆𝑡 𝜕𝑉 1 2 2 𝜕2𝑉 𝜕𝑉 1 + + 𝜎 𝑆𝑡 𝜕𝑡 𝑛 (𝑀 (𝑛) )𝑛−1 𝜕𝑀 (𝑛) 2 𝜕𝑆𝑡2 𝑡 ] 𝑡 𝜕𝑉 −𝐷𝑆𝑡 𝑑𝑡 𝜕𝑆𝑡 [ ) ( 𝑆𝑡𝑛 𝜕𝑉 1 𝜕𝑉 𝜕2𝑉 1 (𝑛) + 𝜎 2 𝑆𝑡2 2 𝑟 𝑉 (𝑆𝑡 , 𝑀𝑡 , 𝑡) − Δ𝑆𝑡 𝑑𝑡 = + (𝑛) (𝑛) 𝜕𝑡 𝑛 (𝑀 )𝑛−1 𝜕𝑀 2 𝜕𝑆𝑡 𝑡 ] 𝑡 𝜕𝑉 −𝐷𝑆𝑡 𝑑𝑡 𝜕𝑆𝑡 [ ( ) 𝑆𝑡𝑛 1 𝜕𝑉 𝜕𝑉 𝜕2𝑉 1 𝜕𝑉 (𝑛) 𝑟 𝑉 (𝑆𝑡 , 𝑀𝑡 , 𝑡) − 𝑆𝑡 + 𝜎 2 𝑆𝑡2 2 + 𝑑𝑡 = (𝑛) (𝑛) 𝜕𝑆𝑡 𝜕𝑡 𝑛 (𝑀 )𝑛−1 𝜕𝑀 2 𝜕𝑆𝑡 𝑡 ] 𝑡 𝜕𝑉 −𝐷𝑆𝑡 𝑑𝑡. 𝜕𝑆𝑡 𝑟Π𝑡 𝑑𝑡 =

By removing 𝑑𝑡 and rearranging terms, we finally have 𝑛

𝑆𝑡 1 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑀𝑡(𝑛) , 𝑡) = 0. + (𝑛) (𝑛) 𝜕𝑡 𝑛 (𝑀 )𝑛−1 𝜕𝑀 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝑡

𝑡

6.2.2 Path-Dependent Options

609

If we take the limit 𝑛 → ∞ then lim 𝑀𝑡(𝑛) = 𝑀𝑡 and since 𝑆𝑡 ≤ max 𝑆𝑢 = 𝑀𝑡 we have 𝑛→∞

𝑡0 ≤𝑢≤𝑡

𝑛

𝑆𝑡 1 = 0. 𝑛→∞ 𝑛 (𝑀 (𝑛) )𝑛−1 lim

𝑡

Thus, in the limit 𝑉 (𝑆𝑡 , 𝑀𝑡 , 𝑡) satisfies 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 − 𝑟𝑉 (𝑆𝑡 , 𝑀𝑡 , 𝑡) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 with boundary conditions 𝑉 (𝑆𝑇 , 𝑀𝑇 , 𝑇 ) = Ψ(𝑆𝑇 , 𝑀𝑇 ) and

𝜕𝑉 || = 0. 𝜕𝑀𝑡 ||𝑆𝑡 =𝑀𝑡

8. Stop-Loss Option. At time 𝑡, let the asset price 𝑆𝑡 follow a GBM 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝑊𝑡 is a standard Wiener process on the probability space (Ω, ℱ, ℙ), 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. In addition, let 𝑟 be the risk-free interest rate. Consider a stop-loss option 𝑉 (𝑆𝑡 , 𝑀𝑡 ), which is a perpetual barrier lookback option with a rebate 𝜆𝑀𝑡 where 𝜆 ∈ (0, 1) is a fixed proportion of the maximum realised asset price at time 𝑡, 𝑀𝑡 = max 𝑆𝑢 where 𝑡0 ≥ 0. By setting a time-dependent barrier 𝐵𝑡 = 𝜆𝑀𝑡 𝑡0 ≤𝑢≤𝑡

such that if 𝑆𝑡 ≤ 𝐵𝑡 then the option pays the holder 𝐵𝑡 , and because the option is time independent, the option is not triggered until the barrier is hit. From the above information show that for 𝐵𝑡 < 𝑆𝑡 < 𝑀𝑡 , the stop-loss option price 𝑉 (𝑆𝑡 , 𝑀𝑡 ) satisfies 1 2 2 𝜕2𝑉 𝜕𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑀𝑡 ) = 0 𝜎 𝑆𝑡 2 2 𝜕𝑆𝑡 𝜕𝑆𝑡 with boundary conditions 𝑉 (𝐵𝑡 , 𝑀𝑡 ) = 𝐵𝑡

and

𝜕𝑉 || = 0. 𝜕𝑀𝑡 ||𝑆𝑡 =𝑀𝑡

By considering the change of variables 𝑉 (𝑆𝑡 , 𝑀𝑡 ) = 𝑀𝑡 𝜙(𝜉),

𝜉 = 𝑆𝑡 ∕𝑀𝑡

610

6.2.2 Path-Dependent Options

show that for 𝜆 < 𝜉 < 1, 𝜙(𝜉) satisfies 𝑑𝜙 1 2 2 𝑑2𝜙 + (𝑟 − 𝐷)𝜉 𝜎 𝜉 − 𝑟𝜙(𝜉) = 0 2 2 𝑑𝜉 𝑑𝜉 with boundary conditions 𝜙(𝜆) = 𝜆 and

𝜙(1) =

𝑑𝜙 || . 𝑑𝜉 ||𝜉=1

By setting 𝜙(𝜉) = 𝐶𝜉 𝑚 where 𝐶 and 𝑚 are constants, show that the stop-loss option price is [ 𝑉 (𝑆𝑡 , 𝑀𝑡 ) = 𝜆𝑀𝑡

)𝛼 )𝛽 ] ( ( (1 − 𝛽) 𝑆𝑡 ∕𝑀𝑡 − (1 − 𝛼) 𝑆𝑡 ∕𝑀𝑡 (1 − 𝛽)𝜆𝛼 − (1 − 𝛼)𝜆𝛽

where

𝛼=

−(𝑟 − 𝐷 − 12 𝜎 2 ) +

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟 𝜎2

and

𝛽=

−(𝑟 − 𝐷 − 12 𝜎 2 ) −

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟 𝜎2

.

What is the option price if 𝐷 = 0? Solution: Following the steps given in Problem 6.2.1.7 (page 547), we first define [ 𝑀𝑡(𝑛)

=

𝑡

∫𝑡0

]1 𝑆𝑢𝑛 𝑑𝑢

𝑛

.

To find the PDE satisfied by 𝑉 (𝑆𝑡 , 𝑀𝑡(𝑛) ) we construct a Δ-hedged portfolio Π𝑡 = 𝑉 (𝑆𝑡 , 𝑀𝑡(𝑛) ) − Δ𝑆𝑡 having one option 𝑉 (𝑆𝑡 , 𝑀𝑡(𝑛) ) and short Δ number of shares 𝑆𝑡 . Using similar steps as described in Problem 6.2.1.7 (page 547), we can easily show that 𝑛

𝑆𝑡 1 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑀𝑡(𝑛) ) = 0. (𝑛) (𝑛) 𝑛 (𝑀 )𝑛−1 𝜕𝑀 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝑡

𝑡

6.2.2 Path-Dependent Options

611

By taking lim𝑛→∞ 𝑀𝑡(𝑛) = 𝑀𝑡 then 𝑉 (𝑆𝑡 , 𝑀𝑡 ) will satisfy 𝜕𝑉 1 2 2 𝜕2 𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑀𝑡 , 𝑡) = 0 𝜎 𝑆𝑡 2 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 for 𝜆𝑀𝑡 < 𝑆𝑡 < 𝑀𝑡 with boundary conditions 𝑉 (𝜆𝑀𝑡 , 𝑀𝑡 ) = 𝜆𝑀𝑡

and

𝜕𝑉 || = 0. 𝜕𝑀𝑡 ||𝑆𝑡 =𝑀𝑡

Setting 𝑉 (𝑆𝑡 , 𝑀𝑡 ) = 𝑀𝑡 𝜙(𝜉) with 𝜉 = 𝑆𝑡 ∕𝑀𝑡 we have 𝑑𝜙 𝑑𝜙 𝑑𝜉 𝜕𝑉 = 𝑀𝑡 = , 𝜕𝑆𝑡 𝑑𝜉 𝑑𝑆𝑡 𝑑𝜉

𝑑 2 𝜙 𝑑𝜉 1 𝑑2𝜙 𝜕2 𝑉 = = 𝑀𝑡 𝑑𝜉 2 𝑑𝜉 2 𝑑𝑆𝑡 𝜕𝑆𝑡2

𝑑𝜙 𝑑𝜉 𝑑𝜙 𝜕𝑉 = 𝜙(𝜉) + 𝑀𝑡 = 𝜙(𝜉) + 𝑀𝑡 𝜕𝑀𝑡 𝑑𝜉 𝑑𝑀𝑡 𝑑𝜉 ( 𝑉 (𝜆𝑀𝑡 , 𝑀𝑡 ) = 𝜆𝑀𝑡 ⟹ 𝑀𝑡 𝜙

𝜆𝑀𝑡 𝑀𝑡

( −

𝑆𝑡

) = 𝜙(𝜉) − 𝜉

𝑀𝑡2

𝑑𝜙 𝑑𝜉

) = 𝜆𝑀𝑡

or

𝜙(𝜆) = 𝜆

and 𝑑𝜙 || 𝜕𝑉 || = 𝜙(1) − = 0. | 𝜕𝑀𝑡 |𝑆𝑡 =𝑀𝑡 𝑑𝜉 ||𝜉=1 Substituting the above results into the PDE and boundary conditions, and because 𝜉 = 𝑆𝑡 ∕𝑀𝑡 , we will eventually arrive at a second-order ODE 𝑑𝜙 1 2 2 𝑑2𝜙 + (𝑟 − 𝐷)𝜉 𝜎 𝜉 − 𝑟𝜙(𝜉) = 0 2 2 𝑑𝜉 𝑑𝜉 for 𝜆 < 𝜉 < 1 with boundary conditions 𝜙(𝜆) = 𝜆 and

𝜙(1) =

To solve the ODE we let 𝜙(𝜉) = 𝐶𝜉 𝑚

𝑑𝜙 || . 𝑑𝜉 ||𝜉=1

612

6.2.2 Path-Dependent Options

where 𝐶 and 𝑚 are constants. Substituting 𝜙(𝜉) = 𝐶𝜉 𝑚 ,

𝑑𝜙 = 𝑚𝐶𝜉 𝑚−1 , 𝑑𝜉

𝑑2𝜙 = 𝑚(𝑚 − 1)𝐶𝜉 𝑚−2 𝑑𝜉 2

into the ODE we have 1 2 𝜎 𝑚(𝑚 − 1) + (𝑟 − 𝐷)𝑚 − 𝑟 = 0 2 or ) ( 1 2 2 1 𝜎 𝑚 + 𝑟 − 𝐷 − 𝜎 2 𝑚 − 𝑟 = 0. 2 2 Therefore,

𝑚=

−(𝑟 − 𝐷 − 12 𝜎 2 ) ±

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟 𝜎2

.

Since √

1 1 (𝑟 − 𝐷 − 𝜎 2 )2 + 2𝜎 2 𝑟 > (𝑟 − 𝐷 − 𝜎 2 ) 2 2

the solution of the ODE must be of the form 𝜙(𝜉) = 𝐴𝜉 𝛼 + 𝐵𝜉 𝛽 where 𝐴 and 𝐵 are unknown constants,

𝛼=

−(𝑟 − 𝐷 − 12 𝜎 2 ) +

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟 𝜎2

>0

and

𝛽=

−(𝑟 − 𝐷 − 12 𝜎 2 ) −

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟 𝜎2

< 0.

Substituting 𝜙(𝜉) = 𝐴𝜉 𝛼 + 𝐵𝜉 𝛽 into the boundary conditions 𝜙(𝜆) = 𝜆

and

𝜙(1) = 𝜙′ (1)

we have 𝐴𝜆𝛼 + 𝐵𝜆𝛽 = 𝜆

and

𝐴(𝛼 − 1) + 𝐵(𝛼 − 1) = 0.

6.2.2 Path-Dependent Options

613

Solving the two equations simultaneously, we have 𝐴=

(1 − 𝛽)𝜆 (1 − 𝛽)𝜆𝛼 + (𝛼 − 1)𝜆𝛽

and

𝐵=

−(1 − 𝛼)𝜆 (1 − 𝛽)𝜆𝛼 + (𝛼 − 1)𝜆𝛽

and hence 𝜙(𝜉) = 𝐴𝜉 𝛼 + 𝐵𝜉 𝛽 =

(1 − 𝛽)𝜆𝜉 𝛼 − (1 − 𝛼)𝜆𝜉 𝛽 . (1 − 𝛽)𝜆𝛼 + (𝛼 − 1)𝜆𝛽

Because 𝑉 (𝑆𝑡 , 𝑀𝑡 ) = 𝑀𝑡 𝜙(𝜉) and 𝜉 = 𝑆𝑡 ∕𝑀𝑡 , the stop-loss option price at time 𝑡 is [ 𝑉 (𝑆𝑡 , 𝑀𝑡 ) = 𝜆𝑀𝑡

)𝛼 )𝛽 ] ( ( (1 − 𝛽) 𝑆𝑡 ∕𝑀𝑡 − (1 − 𝛼) 𝑆𝑡 ∕𝑀𝑡 (1 − 𝛽)𝜆𝛼 − (1 − 𝛼)𝜆𝛽

.

Finally, if 𝐷 = 0 then 𝛼 = 1 and 𝛽 = 0, and therefore 𝐴 = 1 and 𝐵 = 0. Thus, 𝜙(𝜉) = 𝜉 is independent of 𝜆 and hence the stop-loss option price is 𝑉 (𝑆𝑡 , 𝑀𝑡 ) = 𝑀𝑡 𝜙(𝜉) = 𝑀𝑡 𝜉 = 𝑆𝑡 which is equivalent to the underlying stock price 𝑆𝑡 . 9. Perpetual American Fixed Strike Lookback Option. At time 𝑡 let the asset price 𝑆𝑡 follow a GBM 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝑊𝑡 is a standard Wiener process on the probability space (Ω, ℱ, ℙ), 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. In addition, let 𝑟 be the risk-free interest rate. Consider a perpetual American fixed strike lookback option 𝑉 (𝑆𝑡 , 𝑀𝑡 ; 𝐾) which gives the holder the right to buy at the specified fixed strike 𝐾 > 0 with the intrinsic payoff max{𝑀𝑡 − 𝐾, 0}, 𝑀𝑡 = max 𝑆𝑢 , 𝑡0 ≥ 0 up to the date chosen by the option holder. Let 𝑡0 ≤𝑢≤𝑡

𝑆 ∞ < 𝑀𝑡 where 𝑆 ∞ > 𝐾 is the unknown optimal exercise boundary such that for 𝑆𝑡 ≥ 𝑆 ∞ the option should be exercised whilst for 𝑆𝑡 < 𝑆 ∞ the option should be held. Show that for 0 < 𝑆𝑡 < 𝑆 ∞ < 𝑀𝑡 , the option price 𝑉 (𝑆𝑡 , 𝑀𝑡 ; 𝐾) satisfies 1 2 2 𝜕2𝑉 𝜕𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑀𝑡 ; 𝐾) = 0 𝜎 𝑆𝑡 2 2 𝜕𝑆𝑡 𝜕𝑆𝑡 with boundary conditions 𝑉 (𝑆 ∞ , 𝑀𝑡 ; 𝐾) = 𝑀𝑡 − 𝐾

and

𝜕𝑉 || = 0. 𝜕𝑀𝑡 ||𝑆𝑡 =𝑀𝑡

614

6.2.2 Path-Dependent Options

By considering the change of variables 𝑉 (𝑆𝑡 , 𝑀𝑡 ; 𝐾) = 𝑀𝑡 𝜙(𝜉),

𝜉 = 𝑆𝑡 ∕𝑀𝑡

show that for 0 < 𝜉 < 𝜉 ∞ < 1 where 𝜉 ∞ = 𝑆 ∞ ∕𝑀𝑡 , then 𝜙(𝜉) satisfies 𝑑𝜙 1 2 2 𝑑2𝜙 + (𝑟 − 𝐷)𝜉 𝜎 𝜉 − 𝑟𝜙(𝜉) = 0 2 2 𝑑𝜉 𝑑𝜉 with boundary conditions 𝜙(𝜉 ∞ ) =

𝑀𝑡 − 𝐾 , 𝑀𝑡

𝑑𝜙 || 𝐾 =− ∞ 𝑑𝜉 ||𝜉=𝜉 ∞ 𝑆

and

𝜙(1) =

𝑑𝜙 || . 𝑑𝜉 ||𝜉=1

By setting 𝜙(𝜉) = 𝐶𝜉 𝑚 where 𝐶 and 𝑚 are constants, show that the option price is 𝑉 (𝑆𝑡 , 𝑀𝑡 ; 𝐾) {[ ] ( )𝛽 ⎧ (𝑀𝑡 − 𝐾)𝛼 + 𝐾 𝜉 ⎪ 1 ⎪𝛼−𝛽 𝑀𝑡 𝜉∞ ⎪ if 𝑆𝑡 < 𝑆 ∞ ⎪ = ⎨ [ (𝑀𝑡 − 𝐾)𝛽 + 𝐾 ] ( 𝜉 )𝛼 } ⎪− 𝑀𝑡 𝜉∞ ⎪ ⎪ ⎪ 𝑀𝑡 − 𝐾 if 𝑆𝑡 ≥ 𝑆 ∞ ⎩ where 𝛼= 𝛽=

−(𝑟 − 𝐷 − 12 𝜎 2 ) +

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟

1 2 𝜎 )− 2

𝜎2 √ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟

−(𝑟 − 𝐷 −

𝜎2

and { 𝜉∞ =

1 ] } 𝛽−𝛼 [ (𝛽 − 1) (𝑀𝑡 − 𝐾)𝛼 + 𝐾 . [ ] (𝛼 − 1) (𝑀𝑡 − 𝐾)𝛽 + 𝐾

What is the option price if 𝐷 = 0? Solution: To show that 𝑉 (𝑆𝑡 , 𝑀𝑡 ; 𝐾) satisfies 1 2 2 𝜕2𝑉 𝜕𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑀𝑡 ; 𝐾) = 0 𝜎 𝑆𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 with boundary condition

𝜕𝑉 || = 0, see Problem 6.2.2.8 (page 609). 𝜕𝑀𝑡 ||𝑆𝑡 =𝑀𝑡

6.2.2 Path-Dependent Options

615

Let 𝐾 < 𝑆 ∞ < 𝑀𝑡 be the optimal exercise boundary such that 𝑆𝑡 ≥ 𝑆 ∞ , then the perpetual American fixed strike lookback option price is equal to its intrinsic value 𝑉 (𝑆 ∞ , 𝑀𝑡 ; 𝐾) = 𝑀𝑡 − 𝐾. In contrast, for 𝑆𝑡 < 𝑆 ∞ the option should be held. Thus, for 0 < 𝑆𝑡 < 𝑆 ∞ < 𝑀𝑡 the option problem can be expressed as 1 2 2 𝜕2𝑉 𝜕2𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑀𝑡 ; 𝐾) = 0 𝜎 𝑆𝑡 2 2 𝜕𝑆𝑡 𝜕𝑆𝑡 with boundary conditions 𝑉 (𝑆 ∞ , 𝑀𝑡 ; 𝐾) = 𝑀𝑡 − 𝐾

and

𝜕𝑉 || = 0. 𝜕𝑀𝑡 ||𝑆𝑡 =𝑀𝑡

Using the change of variables 𝑉 (𝑆𝑡 , 𝑀𝑡 ; 𝐾) = 𝑀𝑡 𝜙(𝜉),

𝜉 = 𝑆𝑡 ∕𝑀𝑡

and

𝜉 ∞ = 𝑆 ∞ ∕𝑀𝑡

and following the steps given in Problem 6.2.2.8 (page 609), we can easily show that 𝜙(𝜉) satisfies 𝑑𝜙 1 2 2 𝑑2𝜙 + (𝑟 − 𝐷)𝜉 𝜎 𝜉 − 𝑟𝜙(𝜉) = 0 2 2 𝑑𝜉 𝑑𝜉 for 0 < 𝜉 < 𝜉 ∞ < 1. As for the boundary conditions, we note that for 𝑉 (𝑆 ∞ , 𝑀𝑡 ; 𝐾) = 𝑀𝑡 − 𝐾 𝑀𝑡 𝜙(𝜉 ∞ ) = 𝑀𝑡 − 𝐾

or

𝜙(𝜉 ∞ ) =

𝑀𝑡 − 𝐾 . 𝑀𝑡

In addition, because 𝑑𝜙 𝑑𝜙 𝑑𝜉 𝜕𝑉 = 𝜙(𝜉) + 𝑀𝑡 = 𝜙(𝜉) − 𝜉 𝜕𝑀𝑡 𝑑𝜉 𝑑𝑀𝑡 𝑑𝜉 therefore

𝜕𝑉 || = 1 becomes 𝜕𝑀𝑡 ||𝑆𝑡 =𝑆 ∞ 𝜙(𝜉 ∞ ) − 𝜉 ∞

𝑑𝜙 || =1 𝑑𝜉 ||𝜉=𝜉 ∞

or

𝑑𝜙 || 𝐾 =− ∞ 𝑑𝜉 ||𝜉=𝜉 ∞ 𝑆

𝑀𝑡 − 𝐾 . 𝑀𝑡 𝜕𝑉 || Finally, = 0 becomes 𝜕𝑀𝑡 ||𝑆𝑡 =𝑀𝑡

since 𝜙(𝜉 ∞ ) =

𝜙(1) −

𝑑𝜙 || = 0. 𝑑𝜉 ||𝜉=1

616

6.2.2 Path-Dependent Options

Therefore, for 0 < 𝜉 < 𝜉 ∞ < 1, 𝜙(𝜉) satisfies 𝑑𝜙 1 2 2 𝑑2𝜙 + (𝑟 − 𝐷)𝜉 𝜎 𝜉 − 𝑟𝜙(𝜉) = 0 2 2 𝑑𝜉 𝑑𝜉 with boundary conditions 𝜙(𝜉 ∞ ) =

𝑀𝑡 − 𝐾 , 𝑀𝑡

𝑑𝜙 || 𝐾 =− ∞ 𝑑𝜉 ||𝜉=𝜉 ∞ 𝑆

and

𝜙(1) =

𝑑𝜙 || . 𝑑𝜉 ||𝜉=1

By setting 𝜙(𝜉) = 𝐶𝜉 𝑚 where 𝐶 and 𝑚 are constants, then following Problem 6.2.2.8 (page 609) the solution of the Cauchy–Euler equation is 𝜙(𝜉) = 𝐴𝜉 𝛼 + 𝐵𝜉 𝛽 where 𝐴 and 𝐵 are constants to be determined with

𝛼=

−(𝑟 − 𝐷 − 12 𝜎 2 ) +

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟 𝜎2

and −(𝑟 − 𝐷 − 12 𝜎 2 ) −

𝛽=

√ (𝑟 − 𝐷 − 12 𝜎 2 )2 + 2𝜎 2 𝑟 𝜎2

.

Substituting 𝜙(𝜉) = 𝐴𝜉 𝛼 + 𝐵𝜉 𝛽 and 𝜙′ (𝜉) = 𝛼𝐴𝜉 𝛼−1 + 𝛽𝐵𝜉 𝛽−1 into the boundary conditions, we have 𝑀𝑡 − 𝐾 𝑀𝑡 𝐾 𝛼𝐴(𝜉 ∞ )𝛼−1 + 𝛽𝐵(𝜉 ∞ )𝛽−1 = − ∞ 𝑆 (𝛼 − 1)𝐴 + (𝛽 − 1)𝐵 = 0. 𝐴(𝜉 ∞ )𝛼 + 𝐵(𝜉 ∞ )𝛽 =

By solving 𝐴(𝜉 ∞ )𝛼 + 𝐵(𝜉 ∞ )𝛽 =

𝑀𝑡 − 𝐾 𝑀𝑡

and

𝛼𝐴(𝜉 ∞ )𝛼−1 + 𝛽𝐵(𝜉 ∞ )𝛽−1 = −

simultaneously we have 𝐴=

(𝑀𝑡 − 𝐾)𝛽 + 𝐾 𝑀𝑡 (𝛽 − 𝛼)(𝜉 ∞ )𝛼

and

𝐵=

(𝑀𝑡 − 𝐾)𝛼 + 𝐾 . 𝑀𝑡 (𝛼 − 𝛽)(𝜉 ∞ )𝛽

𝐾 𝑆∞

6.2.2 Path-Dependent Options

617

By substituting the expressions for 𝐴 and 𝐵 into 𝛼𝐴(𝜉 ∞ )𝛼−1 + 𝛽𝐵(𝜉 ∞ )𝛽−1 = 0 we have the identity ] [ ] [ (𝛼 − 1) (𝑀𝑡 − 𝐾)𝛽 + 𝐾 (𝜉 ∞ )𝛽 = (𝛽 − 1) (𝑀𝑡 − 𝐾)𝛼 + 𝐾 (𝜉 ∞ )𝛼 and hence { 𝜉∞ =

1 [ ] } 𝛽−𝛼 (𝛽 − 1) (𝑀𝑡 − 𝐾)𝛼 + 𝐾 . [ ] (𝛼 − 1) (𝑀𝑡 − 𝐾)𝛽 + 𝐾

Therefore, 𝜙(𝜉) = 𝐴𝜉 𝛼 + 𝐵𝜉 𝛽 1 = 𝑀𝑡 (𝛼 − 𝛽)

{[

(𝑀𝑡 − 𝐾)𝛼 + 𝐾 𝑀𝑡

](

𝜉 𝜉∞

)𝛽

[

(𝑀𝑡 − 𝐾)𝛽 + 𝐾 − 𝑀𝑡

](

𝜉 𝜉∞

)𝛼 }

and hence the option price for 0 < 𝑆𝑡 < 𝑆 ∞ < 𝑀𝑡 is 𝑉 (𝑆𝑡 , 𝑀𝑡 ; 𝐾) {[ ] ( )𝛽 [ ] ( )𝛼 } (𝑀𝑡 − 𝐾)𝛽 + 𝐾 (𝑀𝑡 − 𝐾)𝛼 + 𝐾 𝜉 𝜉 1 = − . 𝛼−𝛽 𝑀𝑡 𝜉∞ 𝑀𝑡 𝜉∞ Collectively, we can therefore write 𝑉 (𝑆𝑡 , 𝑀𝑡 ; 𝐾) {[ ] ( )𝛽 ⎧ (𝑀𝑡 − 𝐾)𝛼 + 𝐾 𝜉 ⎪ 1 ⎪𝛼−𝛽 𝑀𝑡 𝜉∞ ⎪ if 𝑆𝑡 < 𝑆 ∞ ⎪ = ⎨ [ (𝑀𝑡 − 𝐾)𝛽 + 𝐾 ] ( 𝜉 )𝛼 } ⎪− 𝑀𝑡 𝜉∞ ⎪ ⎪ ⎪ 𝑀𝑡 − 𝐾 if 𝑆𝑡 ≥ 𝑆 ∞ . ⎩ 2𝑟 . Thus, 𝜉 ∞ is unde𝜎2 fined, which implies that the option price problem does not have a solution. This shows that it is never optimal to hold such an option unless 𝑆𝑡 pays a continuous stream of dividends. N.B. For the case when 𝐾 = 0, the option is known as a Russian option, which pays out the maximum realised asset price 𝑀𝑡 up to the date chosen by the holder. When the continuous dividend yield 𝐷 = 0 then 𝛼 = 1 and 𝛽 = −

618

6.2.2 Path-Dependent Options

10. Black–Scholes Equation for Lookback Option II. Let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the stock price 𝑆𝑡 follow a GBM 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. Consider a European-style lookback option with terminal payoff 𝑉 (𝑆𝑇 , 𝑚𝑇 , 𝑇 ) = Ψ(𝑆𝑇 , 𝑚𝑇 ) which depends on the minimum of the stock price 𝑚𝑇 reached within the lookback time period [𝑡0 , 𝑇 ], 𝑡0 ≥ 0 where 𝑇 > 𝑡 is the option expiry time. By defining [ 𝑚(𝑛) 𝑡 =

𝑡

∫𝑡0

]− 1 𝑆𝑢−𝑛 𝑑𝑢

𝑛

and

𝑚𝑡 = min 𝑆𝑢 𝑡0 ≤𝑢≤𝑡

for 𝑛 ∈ ℕ, show that lim 𝑚(𝑛) 𝑡 = 𝑚𝑡 .

𝑛→∞

Explain why, when the current stock price equals its current minimum, we would have the following result 𝜕𝑉 || = 0. 𝜕𝑚𝑡 ||𝑆𝑡 =𝑚𝑡 By considering a hedging portfolio involving both lookback option 𝑉 (𝑆𝑡 , 𝑚(𝑛) 𝑡 , 𝑡) and stock (𝑛) (𝑛) 𝑆𝑡 , show that for 𝑆𝑡 > 𝑚𝑡 > 0, 𝑉 (𝑆𝑡 , 𝑚𝑡 , 𝑡) satisfies the following PDE [ 1 𝜕𝑉 − 𝜕𝑡 𝑛

𝑚(𝑛) 𝑡

]𝑛+1

𝑆𝑡𝑛

𝜕𝑉 𝜕𝑚(𝑛) 𝑡

1 𝜕2𝑉 𝜕𝑉 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑀𝑡(𝑛) , 𝑡) = 0. 2 𝜕𝑆 𝜕𝑆𝑡 𝑡

Finally, by taking 𝑛 → ∞, show that for 𝑆𝑡 > 𝑚𝑡 > 0, 𝑉 (𝑆𝑡 , 𝑚𝑡 , 𝑡) satisfies 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 − 𝑟𝑉 (𝑆𝑡 , 𝑚𝑡 , 𝑡) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 subject to the boundary conditions 𝑉 (𝑆𝑇 , 𝑚𝑇 , 𝑇 ) = Ψ(𝑆𝑇 , 𝑚𝑇 )

and

𝜕𝑉 || = 0. 𝜕𝑚𝑡 ||𝑆𝑡 =𝑚𝑡

6.2.2 Path-Dependent Options

619

Solution: By definition, for 𝑛 ∈ ℕ [ 𝑚(𝑛) 𝑡

=

𝑡

∫𝑡0

]1 𝑛

𝑆𝑢𝑛 𝑑𝑢

𝑚𝑡 = min 𝑆𝑢

and

𝑡0 ≤𝑢≤𝑡

and because 𝑆𝑡 is continuous within the interval [𝑡0 , 𝑡], therefore the integral exists as well as the minimum 𝑚(𝑛) 𝑡 and 𝑚𝑡 . Since we can write [ 𝑚𝑡 =

𝑡

∫𝑡0

𝑆𝑢−𝑛 𝑑𝑢

]−1

max 𝑆 −1 𝑡 ≤𝑢≤𝑡 𝑢 0

therefore −1 𝑚−1 𝑡 = max 𝑆𝑢 𝑡0 ≤𝑢≤𝑡

and for any 𝑛 ∈ ℕ 𝑡

∫𝑡0

𝑚−𝑛 𝑡 𝑑𝑢 ≥

𝑡

∫𝑡0

𝑆𝑢−𝑛 𝑑𝑢

or 𝑚−𝑛 𝑡 (𝑡 − 𝑡0 ) ≥

𝑡

∫0

𝑆𝑢−𝑛 𝑑𝑢

and hence 𝑚𝑡 (𝑡 − 𝑡0 )

− 1𝑛

[ ≤

𝑡

∫0

]− 1 𝑆𝑢−𝑛 𝑑𝑢

𝑛

.

Taking limits 𝑛 → ∞ lim 𝑚𝑡 (𝑡 − 𝑡0 )

𝑛→∞

− 1𝑛

[ ≤ lim

𝑛→∞

𝑡

∫0

]− 1 𝑆𝑢−𝑛 𝑑𝑢

𝑛

we therefore have 𝑚𝑡 ≤ lim 𝑚(𝑛) 𝑡 . 𝑛→∞

We next choose a small 𝜖 > 0 and define 𝜖 (𝑡) to be the set of 𝑢 ∈ [𝑡0 , 𝑡] for which 𝑆𝑢 ≤ 𝑚𝑡 + 𝜖,

𝑢 ∈ [𝑡0 , 𝑡]

620

6.2.2 Path-Dependent Options

and let 𝜆𝜖 (𝑡) =

∫𝜖 (𝑡)

𝑑𝑢

be the total length of the interval in which 𝑆𝑢 ≤ 𝑚𝑡 + 𝜖 (or 𝑆𝑢−1 ≥ (𝑚𝑡 + 𝜖)−1 ). Since 𝑆𝑢 is continuous for 𝑢 ∈ [𝑡0 , 𝑡], 𝜆𝜖 (𝑡) > 0 and 𝜆𝜖 (𝑡) ≤ 𝑡 − 𝑡0 therefore 𝑡

∫𝑡0

𝑆𝑢−𝑛 𝑑𝑢 ≥

∫𝜖 (𝑡)

𝑆𝑢−𝑛 𝑑𝑢 ≥

∫𝜖 (𝑡)

(𝑚𝑡 + 𝜖)−𝑛 𝑑𝑢 = (𝑚𝑡 + 𝜖)−𝑛 𝜆𝜖 (𝑡)

and hence [

𝑡

∫𝑡0

]− 1 𝑛

𝑆𝑢−𝑛 𝑑𝑢

1

≤ 𝜆𝜖 (𝑡)− 𝑛 (𝑚𝑡 + 𝜖).

Taking limits 𝑛 → ∞ lim 𝑚(𝑛) 𝑡 ≤ 𝑚𝑡 + 𝜖

𝑛→∞

and thus we have 𝑚𝑡 ≤ lim 𝑀𝑡(𝑛) ≤ 𝑚𝑡 + 𝜖 𝑛→∞

and because 𝜖 > 0 is a small number we can deduce that lim 𝑚(𝑛) 𝑡 = 𝑚𝑡 .

𝑛→∞

To show that

𝜕𝑉 = 0 on 𝑆𝑡 = 𝑚𝑡 we note that if 𝑆𝑡 = 𝑚𝑡 then 𝜕𝑚𝑡 { 𝑚𝑇 = min 𝑆𝑢 = min 𝑡0 ≤𝑢≤𝑇

} min 𝑆𝑢 , min 𝑆𝑢

𝑡0 ≤𝑢≤𝑡

𝑡≤𝑢≤𝑇

≤ min 𝑆𝑢 = 𝑚𝑡 0≤𝑢≤𝑡

and hence ℙ(𝑚𝑇 = 𝑚𝑡 ) = 0, which shows that 𝑚𝑡 cannot be the final maximum at option expiry time 𝑇 . Since 𝑚𝑇 ≠ 𝑚𝑡 , this implies that the lookback option 𝑉 (𝑆𝑡 , 𝑚𝑡 , 𝑡) at time 𝑡 is insensitive to small changes in 𝑚𝑡 . Therefore, 𝜕𝑉 || = 0. 𝜕𝑚𝑡 ||𝑆𝑡 =𝑚𝑡 To find the PDE satisfied by 𝑉 (𝑆𝑡 , 𝑚(𝑛) 𝑡 , 𝑡) we first construct a Δ-hedged portfolio having (𝑛) one option 𝑉 (𝑆𝑡 , 𝑚𝑡 , 𝑡) and shorting Δ number of shares 𝑆𝑡 . Thus, the hedged portfolio Π𝑡 at time 𝑡 is Π𝑡 = 𝑉 (𝑆𝑡 , 𝑚(𝑛) 𝑡 , 𝑡) − Δ𝑆𝑡 .

6.2.2 Path-Dependent Options

621

From 𝑡 to 𝑡 + 𝑑𝑡, and because the holder receives 𝐷𝑆𝑡 𝑑𝑡 for every asset held, the portfolio value changes by an amount 𝑑Π𝑡 = 𝑑𝑉 − Δ(𝑑𝑆𝑡 + 𝐷𝑆𝑡 𝑑𝑡) where 𝜕𝑉 𝜕𝑉 1 𝜕2𝑉 1 𝜕2𝑉 𝜕𝑉 2 𝑑𝑆𝑡 + 𝑑𝑚(𝑛) + (𝑑𝑆𝑡 )2 + (𝑑𝑚(𝑛) 𝑑𝑡 + 𝑡 𝑡 ) +… (𝑛) 2 𝜕𝑡 𝜕𝑆𝑡 2 𝜕𝑆𝑡 2 𝜕(𝑚(𝑛) )2 𝜕𝑚𝑡 𝑡

𝑑𝑉 = and

𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 . [ Since 𝑚(𝑛) 𝑡 =

𝑡

∫𝑡0

]− 1 𝑆𝑢−𝑛 𝑑𝑢

𝑑𝑚(𝑛) 𝑡

𝑛

we have

]− 1 −1 [ 𝑡 (𝑛) 𝑛+1 𝑛 1 1 (𝑚𝑡 ) −𝑛 −𝑛 =− 𝑆 𝑑𝑢 𝑆𝑡 𝑑𝑡 = − 𝑑𝑡. 𝑛 ∫0 𝑢 𝑛 𝑆𝑡𝑛

From It¯o’s lemma 𝑑𝑉 =

(𝑛) ] 1 (𝑚𝑡 )𝑛+1 𝜕𝑉 𝜕𝑉 𝜕𝑉 [ (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 − 𝑑𝑡 𝑑𝑡 + 𝜕𝑡 𝜕𝑆𝑡 𝑛 𝑆𝑡𝑛 𝜕𝑚(𝑛) 𝑡

1 𝜕2𝑉 + 𝜎 2 𝑆𝑡2 2 𝑑𝑡 2 𝜕𝑆𝑡 ] [ (𝑛) 𝑛+1 1 2 2 𝜕2𝑉 𝜕𝑉 𝜕𝑉 1 (𝑚𝑡 ) 𝜕𝑉 + 𝜎 𝑆𝑡 + (𝜇 − 𝐷)𝑆𝑡 − 𝑑𝑡 = 𝜕𝑡 𝑛 𝑆𝑡𝑛 𝜕𝑆𝑡 𝜕𝑆𝑡2 𝜕𝑚(𝑛) 2 𝜕𝑉 𝑑𝑊𝑡 +𝜎𝑆𝑡 𝜕𝑆𝑡

𝑡

and by substituting 𝑑𝑉 into 𝑑Π𝑡 and rearranging terms, we have [ ] (𝑛) 𝑛+1 𝜕𝑉 1 2 2 𝜕2 𝑉 𝜕𝑉 𝜕𝑉 1 (𝑚𝑡 ) + 𝜎 𝑆𝑡 + (𝜇 − 𝐷)𝑆𝑡 − 𝜇Δ𝑆𝑡 𝑑𝑡 − 𝑑Π𝑡 = 𝜕𝑡 𝑛 𝑆𝑡𝑛 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 𝜕𝑚(𝑛) 𝑡 ) ( 𝜕𝑉 − Δ 𝑑𝑊𝑡 . +𝜎𝑆𝑡 𝜕𝑆𝑡 To eliminate the random term we set Δ=

𝜕𝑉 𝜕𝑆𝑡

and hence [ 𝑑Π𝑡 =

] (𝑛) 𝑛+1 𝜕𝑉 1 2 2 𝜕2𝑉 𝜕𝑉 𝜕𝑉 1 (𝑚𝑡 ) + 𝜎 𝑆𝑡 − 𝐷𝑆𝑡 − 𝑑𝑡. 𝜕𝑡 𝑛 𝑆𝑡𝑛 𝜕𝑆𝑡 𝜕𝑆𝑡2 𝜕𝑚(𝑛) 2 𝑡

622

6.2.2 Path-Dependent Options

Under the no-arbitrage condition, the return on the amount Π𝑡 invested in a risk-free interest rate would see a growth of 𝑑Π𝑡 = 𝑟Π𝑡 𝑑𝑡 and therefore [

(𝑛) 𝑛+1 𝜕𝑉 1 𝜕𝑉 𝜕2𝑉 1 (𝑚𝑡 ) + 𝜎 2 𝑆𝑡2 2 − 𝑛 (𝑛) 𝜕𝑡 𝑛 𝑆𝑡 2 𝜕𝑆𝑡 𝜕𝑚𝑡 ] 𝜕𝑉 −𝐷𝑆𝑡 𝑑𝑡 𝜕𝑆𝑡 [ (𝑛) 𝑛+1 ) ( 𝜕𝑉 1 𝜕𝑉 𝜕2𝑉 1 (𝑚𝑡 ) (𝑛) + 𝜎 2 𝑆𝑡2 2 − 𝑟 𝑉 (𝑆𝑡 , 𝑚𝑡 , 𝑡) − Δ𝑆𝑡 𝑑𝑡 = 𝑛 (𝑛) 𝜕𝑡 𝑛 𝑆𝑡 2 𝜕𝑆𝑡 𝜕𝑀𝑡 ] 𝜕𝑉 −𝐷𝑆𝑡 𝑑𝑡 𝜕𝑆𝑡 [ ) ( (𝑛) 𝑛+1 1 𝜕𝑉 𝜕𝑉 𝜕2𝑉 1 (𝑚𝑡 ) 𝜕𝑉 (𝑛) + 𝜎 2 𝑆𝑡2 2 − 𝑑𝑡 = 𝑟 𝑉 (𝑆𝑡 , 𝑚𝑡 , 𝑡) − 𝑆𝑡 𝑛 (𝑛) 𝜕𝑆𝑡 𝜕𝑡 𝑛 𝑆𝑡 2 𝜕𝑆𝑡 𝜕𝑚𝑡 ] 𝜕𝑉 −𝐷𝑆𝑡 𝑑𝑡. 𝜕𝑆𝑡

𝑟Π𝑡 𝑑𝑡 =

By removing 𝑑𝑡 and rearranging terms, we finally have (𝑛) 𝑛+1 1 2 2 𝜕2𝑉 𝜕𝑉 𝜕𝑉 1 (𝑚𝑡 ) 𝜕𝑉 + + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑚(𝑛) − 𝜎 𝑆𝑡 𝑛 𝑡 , 𝑡) = 0. (𝑛) 2 𝜕𝑡 𝑛 𝑆𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑚 𝑡

If we take the limit 𝑛 → ∞ then lim 𝑚(𝑛) 𝑡 = 𝑚𝑡 , and since 𝑚𝑡 = min 𝑆𝑢 ≤= 𝑆𝑡 , we have 𝑛→∞

(𝑛) 𝑛+1 1 (𝑚𝑡 ) 1 = lim lim 𝑛→∞ 𝑛 𝑛→∞ 𝑛 𝑆𝑡𝑛

𝑡0 ≤𝑢≤𝑡

(

𝑚(𝑛) 𝑡 𝑆𝑡

)𝑛 𝑚(𝑛) 𝑡 = 0.

Thus, in the limit, 𝑉 (𝑆𝑡 , 𝑚𝑡 , 𝑡) satisfies 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 − 𝑟𝑉 (𝑆𝑡 , 𝑚𝑡 , 𝑡) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 with boundary conditions 𝑉 (𝑆𝑇 , 𝑚𝑇 , 𝑇 ) = Ψ(𝑆𝑇 , 𝑚𝑇 )

and

𝜕𝑉 || = 0. 𝜕𝑚𝑡 ||𝑆𝑡 =𝑚𝑡

6.2.2 Path-Dependent Options

623

11. Perpetual American Floating Strike Lookback Option. At time 𝑡 let the asset price 𝑆𝑡 follow a GBM 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝑊𝑡 is a standard Wiener process on the probability space (Ω, ℱ, ℙ), 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. In addition, we let 𝑟 be the risk-free interest rate. Consider a perpetual American floating strike lookback option 𝑉 (𝑆𝑡 , 𝑚𝑡 ) which gives the holder the right but not the obligation to buy at the lowest realised asset price with the intrinsic payoff max{𝑆𝑡 − 𝑚𝑡 , 0}, 𝑚𝑡 = min 𝑆𝑢 , 𝑡0 ≥ 0 up to the date chosen by the 𝑡0 ≤𝑢≤𝑡

option holder. Let 𝑆 ∗ > 𝑚𝑡 be the unknown optimal exercise boundary such that for 𝑆𝑡 ≥ 𝑆 ∗ the option should be exercised whilst for 𝑆𝑡 < 𝑆 ∗ the option should be held. Show that for 𝑚𝑡 < 𝑆𝑡 < 𝑆 ∗ , the option price 𝑉 (𝑆𝑡 , 𝑚𝑡 ) satisfies 𝜕𝑉 1 2 2 𝜕2𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑚𝑡 ) = 0 𝜎 𝑆𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡2 with boundary conditions 𝑉 (𝑆 ∗ , 𝑚𝑡 ) = 𝑆 ∗ − 𝑚𝑡

and

𝜕𝑉 || = 0. 𝜕𝑚𝑡 ||𝑆𝑡 =𝑚𝑡

By considering the change of variables 𝑉 (𝑆𝑡 , 𝑚𝑡 ) = 𝑆𝑡 𝜑(𝜁 ), 𝜁 = 𝑚𝑡 ∕𝑆𝑡 show that for 0 < 𝜁 ∗ < 𝜁 < 1, where 𝜁 ∗ = 𝑚𝑡 ∕𝑆 ∗ , then 𝜑(𝜁 ) satisfies 𝑑𝜑 1 2 2 𝑑2𝜑 + (𝐷 − 𝑟)𝜁 𝜎 𝜁 − 𝐷𝜑(𝜁 ) = 0 2 𝑑𝜁 𝑑𝜁 2 with boundary conditions 𝜑(𝜁 ∗ ) = 1 − 𝜁 ∗ ,

𝑑𝜑 || = −1 and 𝑑𝜁 ||𝜁 =𝜁 ∗

𝜑(1) =

𝑑𝜑 || . 𝑑𝜁 ||𝜁 =1

By setting 𝜑(𝜁 ) = 𝐶𝜁 𝑚 where 𝐶 and 𝑚 are constants, show that the option price 𝑉 (𝑆𝑡 , 𝑚𝑡 ) is 𝑉 (𝑆𝑡 , 𝑚𝑡 )

{ ( ) ( ) } ⎧ [ ] 𝜁 𝛽 [ ] 𝜁 𝛼 ⎪ 1 ∗ ∗ (1 − 𝛼)𝜁 + 𝛼 − (1 − 𝛽)𝜁 + 𝛽 if 𝑆𝑡 < 𝑆 ∗ ∗ ∗ ⎪𝛼−𝛽 𝜁 𝜁 =⎨ ⎪ ⎪ 𝑆 ∗ − 𝑚𝑡 if 𝑆𝑡 ≥ 𝑆 ∗ ⎩

624

6.2.2 Path-Dependent Options

where 𝛼=

𝛽=

−(𝐷 − 𝑟 − 12 𝜎 2 ) +

√ (𝐷 − 𝑟 − 12 𝜎 2 )2 + 2𝜎 2 𝐷

𝜎2 √ −(𝐷 − 𝑟 − 12 𝜎 2 ) − (𝐷 − 𝑟 − 12 𝜎 2 )2 + 2𝜎 2 𝐷 𝜎2

and 0 < 𝜁 ∗ < 1 satisfies 𝑓 (𝜁 ∗ ) = 0 where 𝑓 (𝜁 ) = 𝛽(1 − 𝛼)𝜁 𝛼−𝛽+1 + 𝛼𝛽𝜁 𝛼−𝛽 − 𝛼(1 − 𝛽)𝜁 − 𝛼𝛽. Show that if 𝐷 > 0 then 𝑓 (𝜁 ) = 0 has a unique solution in 𝜁 ∈ (0, 1). What is the option price when 𝐷 = 0? Solution: To show that 𝑉 (𝑆𝑡 , 𝑚𝑡 ) satisfies 𝜕𝑉 1 2 2 𝜕2𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑚𝑡 ) = 0 𝜎 𝑆𝑡 2 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 with boundary condition

𝜕𝑉 || = 0, see Problem 6.2.2.10 (page 618) and Problem 𝜕𝑚𝑡 ||𝑆𝑡 =𝑚𝑡

6.2.2.8 (page 609). Given that 𝑆 ∗ is the optimal exercise boundary then for 𝑆𝑡 ≥ 𝑆 ∗ , the option price is equal to its intrinsic value 𝑉 (𝑆 ∗ , 𝑚𝑡 ) = 𝑆 ∗ − 𝑚𝑡 . Therefore, for 𝑚𝑡 < 𝑆𝑡 < 𝑆 ∗ , 𝑉 (𝑆𝑡 , 𝑚𝑡 ) satisfies 𝜕𝑉 1 2 2 𝜕2𝑉 + (𝑟 − 𝐷)𝑆𝑡 − 𝑟𝑉 (𝑆𝑡 , 𝑚𝑡 ) = 0 𝜎 𝑆𝑡 2 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 with boundary conditions 𝑉 (𝑆 ∗ , 𝑚𝑡 ) = 𝑆 ∗ − 𝑚𝑡

and

𝜕𝑉 || = 0. 𝜕𝑚𝑡 ||𝑆𝑡 =𝑚𝑡

By the change of variables 𝑉 (𝑆𝑡 , 𝑚𝑡 ) = 𝑆𝑡 𝜑(𝜁 ),

𝜁 = 𝑚𝑡 ∕𝑆𝑡

and

𝜁 ∗ = 𝑚𝑡 ∕𝑆 ∗

we have 𝑑𝜑 𝑑𝜑 𝑑𝜁 𝜕𝑉 = 𝜑(𝜁 ) + 𝑆𝑡 = 𝜑(𝜁 ) − 𝜁 𝜕𝑆𝑡 𝑑𝜁 𝑑𝑆𝑡 𝑑𝜁 𝑑𝜑 𝑑𝜁 𝑑𝜁 𝑑𝜑 𝜁 2 𝑑2𝜑 𝑑 2 𝜑 𝑑𝜁 𝜕2𝑉 = − = − 𝜁 𝑑𝜁 𝑑𝑆𝑡 𝑑𝑆𝑡 𝑑𝜁 𝑆𝑡 𝑑𝜁 2 𝑑𝜁 2 𝑑𝑆𝑡 𝜕𝑆𝑡2 𝑑𝜑 𝑑𝜑 𝑑𝜁 𝜕𝑉 = 𝑆𝑡 = 𝜕𝑚𝑡 𝑑𝜁 𝑑𝑚𝑡 𝑑𝜁 𝑑𝜑 || 𝜕𝑉 || = | 𝜕𝑚𝑡 |𝑆𝑡 =𝑚𝑡 𝑑𝜁 ||𝜁 =1

and

𝑑𝜑 || 𝜕𝑉 || = . | 𝜕𝑚𝑡 |𝑆𝑡 =𝑆 ∗ 𝑑𝜁 ||𝜁 =𝜁 ∗

6.2.2 Path-Dependent Options

625

Substituting the above results into the PDE and because 𝜁 = 𝑚𝑡 ∕𝑆𝑡 , we eventually arrive at a second-order ODE of the form 𝑑𝜑 1 2 2 𝑑2𝜑 + (𝐷 − 𝑟)𝜁 𝜎 𝜉 − 𝐷𝜑(𝜁 ) = 0 2 2 𝑑𝜁 𝑑𝜁 for 0 < 𝜁 ∗ < 𝜁 < 1. As for the boundary conditions, we note that for 𝑉 (𝑆 ∗ , 𝑚𝑡 ) = 𝑆 ∗ − 𝑚𝑡 𝑆 ∗ 𝜑(𝜁 ∗ ) = 𝑆 ∗ − 𝑚𝑡 In addition, because

or

𝜑(𝜁 ∗ ) = 1 − 𝜁 ∗ .

𝑑𝜑 || 𝜕𝑉 || 𝜕𝑉 || = − 1 we have = − 1 and =0 𝜕𝑚𝑡 ||𝑆𝑡 =𝑆 ∗ 𝑑𝜁 ||𝜁 =𝜁 ∗ 𝜕𝑚𝑡 ||𝑆𝑡 =𝑚𝑡

𝑑𝜑 || = 0. 𝑑𝜁 ||𝜁 =1 Therefore, for 0 < 𝜁 ∗ < 𝜁 < 1, 𝜑(𝜁 ) satisfies

becomes

𝑑𝜑 1 2 2 𝑑2𝜑 + (𝐷 − 𝑟)𝜁 𝜎 𝜉 − 𝐷𝜑(𝜁 ) = 0 2 2 𝑑𝜁 𝑑𝜁 with boundary conditions 𝑑𝜑 || = −1 𝑑𝜁 ||𝜁 =𝜁 ∗

𝜑(𝜁 ∗ ) = 1 − 𝜁 ∗ ,

and

𝑑𝜑 || = 0. 𝑑𝜁 ||𝜁 =1

By setting 𝜑(𝜁 ) = 𝐶𝜁 𝑛 where 𝐶 and 𝑛 are constants, then following Problem 6.2.2.8 (page 609) the solution of the ODE is 𝜑(𝜁 ) = 𝐴𝜁 𝛼 + 𝐵𝜁 𝛽 where 𝐴 and 𝐵 are constants to be determined with

𝛼=

−(𝐷 − 𝑟 − 12 𝜎 2 ) +

√ (𝐷 − 𝑟 − 12 𝜎 2 )2 + 2𝜎 2 𝐷 𝜎2

and

𝛽=

−(𝐷 − 𝑟 − 12 𝜎 2 ) −

√ (𝐷 − 𝑟 − 12 𝜎 2 )2 + 2𝜎 2 𝐷 𝜎2

.

Substituting 𝜑(𝜁 ) = 𝐴𝜁 𝛼 + 𝐵𝜁 𝛽 and 𝜑′ (𝜉) = 𝛼𝐴𝜁 𝛼−1 + 𝛽𝐵𝜁 𝛽−1 into the boundary conditions we have 𝐴(𝜁 ∗ )𝛼 + 𝐵(𝜁 ∗ )𝛽 = 1 − 𝜁 ∗ ∗ 𝛼−1

𝛼𝐴(𝜁 )

+ 𝛽𝐵(𝜁 ∗ )𝛽−1 = −1 𝛼𝐴 + 𝛽𝐵 = 0.

626

6.2.2 Path-Dependent Options

By solving 𝐴(𝜁 ∗ )𝛼 + 𝐵(𝜁 ∗ )𝛽 = 1 − 𝜁 ∗

and

𝛼𝐴(𝜁 ∗ )𝛼−1 + 𝛽𝐵(𝜁 ∗ )𝛽−1 = −1

simultaneously we have 𝐴=

(1 − 𝛽)𝜁 ∗ + 𝛽 (𝛽 − 𝛼)(𝜁 ∗ )𝛼

and

𝐵=

(1 − 𝛼)𝜁 ∗ + 𝛼 . (𝛼 − 𝛽)(𝜁 ∗ )𝛽

By substituting the expressions for 𝐴 and 𝐵 into 𝛼𝐴 + 𝛽𝐵 = 0 we have 𝛼(1 − 𝛽)𝜁 ∗ + 𝛼𝛽 𝛽(1 − 𝛼)𝜁 ∗ + 𝛼𝛽 = (𝛼 − 𝛽)(𝜁 ∗ )𝛼 (𝛼 − 𝛽)(𝜁 ∗ )𝛽 or 𝛽(1 − 𝛼)(𝜁 ∗ )𝛼−𝛽+1 + 𝛼𝛽(𝜁 ∗ )𝛼−𝛽 − 𝛼(1 − 𝛽)𝜁 ∗ − 𝛼𝛽 = 0. Thus, 0 < 𝜁 ∗ < 1 satisfies 𝑓 (𝜁 ) = 0 where 𝑓 (𝜁 ) = 𝛽(1 − 𝛼)𝜁 𝛼−𝛽+1 + 𝛼𝛽𝜁 𝛼−𝛽 − 𝛼(1 − 𝛽)𝜁 − 𝛼𝛽. To show that 𝜁 ∗ ∈ (0, 1) is unique for 𝐷 > 0, we note that √ √ 1 22 1 (𝐷 − 𝑟 − 𝜎 ) + 2𝜎 2 𝐷 = (𝐷 − 𝑟)2 − (𝐷 − 𝑟)𝜎 2 + 𝜎 4 + 2𝜎 2 𝐷 2 4 √ 1 = (𝐷 − 𝑟)2 + (𝐷 + 𝑟)𝜎 2 + 𝜎 4 4 √ 1 > (𝐷 − 𝑟)2 + (𝐷 − 𝑟)𝜎 2 + 𝜎 4 4 1 2 = 𝐷−𝑟+ 𝜎 2 and hence 𝛼=

−(𝐷 − 𝑟 − 12 𝜎 2 ) +

√ (𝐷 − 𝑟 + 12 𝜎 2 )2 + 2𝜎 2 𝐷 𝜎2

>1

and 𝛽=

−(𝐷 − 𝑟 − 12 𝜎 2 ) −

√ (𝐷 − 𝑟 + 12 𝜎 2 )2 + 2𝜎 2 𝐷 𝜎2

< 0.

Since 𝑓 (0) = −𝛼𝛽 > 0 and

𝑓 (1) = 𝛽 − 𝛼 < 0

then from the intermediate value theorem, 𝑓 must have at least one root in the interval (0,1).

6.2.2 Path-Dependent Options

627

For 𝜁 ∈ (0, 1) and because 𝛽(1 − 𝛼)(𝛼 − 𝛽 + 1) > 0,

𝛼𝛽(𝛼 − 𝛽) > 0

and

𝛼−𝛽 >1

therefore 𝑓 ′ (𝜁 ) = 𝛽(1 − 𝛼)(𝛼 − 𝛽 + 1)𝜁 𝛼−𝛽 + 𝛼𝛽(𝛼 − 𝛽)𝜁 𝛼−𝛽−1 − 𝛼(1 − 𝛽) < 𝛽(1 − 𝛼)(𝛼 − 𝛽 + 1) + 𝛼𝛽(𝛼 − 𝛽) − 𝛼(1 − 𝛽) = (𝛼 − 𝛽)(𝛽 − 1) < 0. Thus, 𝑓 is a monotonically decreasing function in 𝜁 ∈ (0, 1) which implies that 𝑓 (𝜁 ) = 0 has a unique solution in (0, 1). Finally, when 𝐷 = 0 then 2𝑟 + 1 and 𝛽 = 0 𝜎2 ( ) 2𝑟 and hence 𝜁 ∗ = 0 since 𝑓 (𝜁 ) = − 2 + 1 𝜁 , which implies that the option problem does 𝜎 not have a solution. This shows that it is never optimal to hold such an option in the absence of an asset paying continuous dividend yields. 𝛼=

{ } 12. European Fixed Strike Lookback Option I. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. Consider a European-style fixed strike lookback call option with terminal payoff { Ψ(𝑆𝑇 ) = max

} max 𝑆𝑢 − 𝐾, 0

𝑡0 ≤𝑢≤𝑇

which depends on the maximum of the asset price reached within the lookback period [𝑡0 , 𝑇 ], 𝑡0 ≥ 0 where 𝑇 is the expiry time and 𝐾 > 0 is the strike price. By definition, under the risk-neutral measure ℚ, the value of a fixed strike lookback call option at time 𝑡, 𝑡0 ≤ 𝑡 ≤ 𝑇 is [ 𝐶𝐹 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒

−𝑟(𝑇 −𝑡) ℚ

𝔼

{ max

where 𝑟 is the risk-free interest rate.

{ max

} max 𝑆𝑢 , max 𝑆𝑢

𝑡0 ≤𝑢≤𝑡

𝑡≤𝑢≤𝑇

}| ] | − 𝐾, 0 | ℱ𝑡 | |

628

6.2.2 Path-Dependent Options

By considering two cases max 𝑆𝑢 ≤ 𝐾 and max 𝑆𝑢 > 𝐾, show that 𝐶𝐹 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) 𝑡0 ≤𝑢≤𝑡

𝑡0 ≤𝑢≤𝑡

can be written as

{

𝐶𝐹 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) max

} ̂ 𝑇) max 𝑆𝑢 − 𝐾, 0 + 𝐶̂𝐹 (𝑆𝑡 , 𝑡; 𝐾,

𝑡0 ≤𝑢≤𝑡

[ ̂ 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max where 𝐶̂𝐹 (𝑆𝑡 , 𝑡; 𝐾, } { ̂ 𝐾 = max max 𝑆𝑢 , 𝐾 .

{

}| ] ̂ 0 || ℱ𝑡 such that max 𝑆𝑢 − 𝐾, | 𝑡≤𝑢≤𝑇 |

𝑡0 ≤𝑢≤𝑡

Using the identity 1

𝑈 𝑎𝑤− 1 ( √𝑤 )2 2 𝜏

𝑒 √ 2𝜋𝜏 ∫𝐿

[ ( 𝑑𝑤 = 𝑒

1 2 𝑎 𝜏 2

Φ

𝑈 − 𝑎𝜏 √ 𝜏

)

( −Φ

𝐿 − 𝑎𝜏 √ 𝜏

)]

show that ̂ 𝑇 ) = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, ̂ 𝑇) 𝐶̂𝐹 (𝑆𝑡 , 𝑡; 𝐾, ( )−𝛼 ] ( )[ 𝑆𝑡 𝑆𝑡 −𝐷(𝑇 −𝑡) −𝑟(𝑇 −𝑡) ̂ ̂ Φ(𝑑+ ) − 𝑒 Φ(𝑑− ) + 𝑒 𝛼 ̂ 𝐾 ̂ 𝑇 ) is the vanilla (or European) call option price defined as where 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, ̂ 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑒 ̂ −𝑟(𝑇 −𝑡) Φ(𝑑− ) 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, with ̂ + (𝑟 − 𝐷 ± 1 𝜎 2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝐾) 2 𝑑± = √ 𝜎 𝑇 −𝑡

𝑑̂± = 𝛼=

𝑟−𝐷 1 2 𝜎 2

𝑥

and Φ(𝑥) =

∫−∞

̂ ± (𝑟 − 𝐷 ± 1 𝜎 2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝐾) 2 √ 𝜎 𝑇 −𝑡 1 − 1 𝑢2 √ 𝑒 2 𝑑𝑢 is the cdf of a standard normal. 2𝜋

Solution: Under the risk-neutral measure ℚ, 𝑆𝑡 follows

where 𝑊𝑡ℚ = 𝑊𝑡 + By writing

(𝜇 − 𝑟) 𝜎

𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡 𝑡 is a ℚ-standard Wiener process.

max max 𝑆𝑢 = 𝑆𝑡−𝑡 ,

𝑡0 ≤𝑢≤𝑡

0

max 𝑆𝑢 = 𝑆𝑇max −𝑡

𝑡≤𝑢≤𝑇

6.2.2 Path-Dependent Options

629

hence [

}| ] | 𝔼 max max max 𝑆𝑢 , max 𝑆𝑢 − 𝐾, 0 | ℱ𝑡 𝐶𝐹 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒 | 𝑡0 ≤𝑢≤𝑡 𝑡≤𝑢≤𝑇 | [ ] } }| { { max | = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max max 𝑆𝑡−𝑡 , 𝑆𝑇max −𝑡 − 𝐾, 0 | ℱ𝑡 . 0 | −𝑟(𝑇 −𝑡) ℚ

{

{

}

max ≤ 𝐾 then If 𝑆𝑡−𝑡 0

{ { } } { } max max max 𝑆𝑡−𝑡 − 𝐾, 0 = max 𝑆𝑇max , 𝑆𝑇max −𝑡 −𝑡 − 𝐾, 0 0

max > 𝐾 then and if 𝑆𝑡−𝑡 0

{ { } } { } max max max max max max 𝑆𝑡−𝑡 , 𝑆𝑇max −𝑡 − 𝐾, 0 = 𝑆𝑡−𝑡 − 𝐾 + max 𝑆𝑇 −𝑡 − 𝑆𝑡−𝑡 , 0 . 0

0

0

Thus, we can write } } { } { { max max − 𝐾, 0 = max 𝑆 , 𝑆𝑇max − 𝐾, 0 max max 𝑆𝑡−𝑡 𝑡−𝑡0 −𝑡 0 } { max + max 𝑆𝑇max −𝑡 − max{𝑆𝑡−𝑡 , 𝐾}, 0 . 0

By substituting this back into the fixed strike lookback option price, we have 𝐶𝐹 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) [ } { }| ] { max max | ℱ𝑡 = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝑆𝑡−𝑡 − 𝐾, 0 + max 𝑆𝑇max − max{𝑆 , 𝐾}, 0 | 𝑡−𝑡0 −𝑡 0 | { } −𝑟(𝑇 −𝑡) max =𝑒 max 𝑆𝑡−𝑡 − 𝐾, 0 0 [ }| ] { max |ℱ − max{𝑆 , 𝐾}, 0 +𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝑆𝑇max | 𝑡 𝑡−𝑡0 −𝑡 | { } max ̂ 𝑇) − 𝐾, 0 + 𝐶̂𝐹 (𝑆𝑡 , 𝑡; 𝐾, = 𝑒−𝑟(𝑇 −𝑡) max 𝑆𝑡−𝑡 0

[ }| ] } { { max −𝑟(𝑇 −𝑡) ℚ ̂ ̂ ̂ ̂ = max 𝑆 max , 𝐾 . 𝔼 max 𝑆𝑇 −𝑡 − 𝐾, 0 || ℱ𝑡 , 𝐾 where 𝐶𝐹 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒 𝑡−𝑡0 | From It¯o’s lemma we can write 𝑆𝑇max −𝑡 = max 𝑆𝑢 𝑡≤𝑢≤𝑇

1 2 ℚ )(𝑢−𝑡)+𝜎𝑊𝑢−𝑡

= max 𝑆𝑡 𝑒(𝑟−𝐷− 2 𝜎 𝑡≤𝑢≤𝑇

= 𝑆𝑡 𝑒𝑀𝑇 −𝑡

630

6.2.2 Path-Dependent Options ℚ where 𝑀𝑇 −𝑡 = max 𝜈(𝑢 − 𝑡) + 𝜎𝑊𝑢−𝑡 such that 𝜈 = 𝑟 − 𝐷 − 12 𝜎 2 . From Problem 𝑡≤𝑢≤𝑇

4.2.2.15 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus, (

(

)

ℚ 𝑀𝑇 −𝑡 ≤ 𝑥 = Φ

𝑥 − 𝜈(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

(

) −𝑒

2𝜈𝑥 𝜎2

Φ

−𝑥 − 𝜈(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

) ,

𝑥≥0

and hence [ }| ] { ̂ 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝑆𝑡 𝑒𝑀𝑇 −𝑡 − 𝐾, ̂ 0 | ℱ𝑡 𝐶̂𝐹 (𝑆𝑡 , 𝑡; 𝐾, | | = 𝑒−𝑟(𝑇 −𝑡)

∞

∫𝐾̂

( ) ℚ 𝑆𝑡 𝑒𝑀𝑇 −𝑡 ≥ 𝑥 𝑑𝑥

( ) 𝑥 ℚ 𝑒𝑀𝑇 −𝑡 ≥ 𝑑𝑥 ∫𝐾̂ 𝑆𝑡 ( ( )) ∞ 𝑥 ℚ 𝑀𝑇 −𝑡 ≥ log 𝑑𝑥. = 𝑒−𝑟(𝑇 −𝑡) ∫𝐾̂ 𝑆𝑡 ∞

= 𝑒−𝑟(𝑇 −𝑡)

) ( By substituting 𝑦 = log 𝑥∕𝑆𝑡 we have ̂ 𝑇) 𝐶̂𝐹 (𝑆𝑡 , 𝑡; 𝐾, = 𝑒−𝑟(𝑇 −𝑡) = 𝑆𝑡 𝑒

∞

∫log(𝐾∕𝑆 ̂ 𝑡)

( ) 𝑆𝑡 𝑒𝑦 ℚ 𝑀𝑇 −𝑡 ≥ 𝑦 𝑑𝑦

∞ −𝑟(𝑇 −𝑡)

∫log(𝐾∕𝑆 ̂ 𝑡)

[ ( 𝑒

𝑦

Φ

−𝑦 + 𝜈(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

)

( +𝑒

2𝜈𝑦 𝜎2

Φ

−𝑦 − 𝜈(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

= 𝐴 1 + 𝐴2 where 𝐴1 = 𝑆𝑡 𝑒−𝑟(𝑇 −𝑡)

(

∞

∫log(𝐾∕𝑆 ̂ 𝑡)

𝑒𝑦 Φ

−𝑦 + 𝜈(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

) 𝑑𝑦

and 𝐴2 = 𝑆𝑡 𝑒

∞ −𝑟(𝑇 −𝑡)

∫log(𝐾∕𝑆 ̂ 𝑡)

𝑒

) 1+ 2𝜈2 𝑦

(

(

𝜎

Φ

−𝑦 − 𝜈(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

) 𝑑𝑦.

)] 𝑑𝑦

6.2.2 Path-Dependent Options

For the case 𝐴1 = 𝑆𝑡

631

𝑒−𝑟(𝑇 −𝑡)

(

∞

∫log(𝐾∕𝑆 ̂ 𝑡)

−𝑦 + 𝜈(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

𝑦

𝑒 Φ

parts with

) 𝑑𝑦 and using integration by

𝑑𝑈 = 𝑒𝑦 ⇒ 𝑈 = 𝑒 𝑦 𝑑𝑦 ( 𝑉 =Φ

−𝑦 + 𝜈(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

)

( 1

− 𝑑𝑉 1 𝑒 2 ⇒ =− √ 𝑑𝑦 𝜎 2𝜋(𝑇 − 𝑡)

−𝑦+𝜈(𝑇 −𝑡) √ 𝜎 𝑇 −𝑡

)2

we can write )]∞ ( ⎧[ ⎪ −𝑦 + 𝜈(𝑇 − 𝑡) 𝐴1 = 𝑆𝑡 𝑒−𝑟(𝑇 −𝑡) ⎨ 𝑒𝑦 Φ √ 𝜎 𝑇 −𝑡 ⎪ ̂ 𝑡) log(𝐾∕𝑆 ⎩ ( ∞

+

(

∫log

̂ 𝑡 𝐾∕𝑆

)

1

𝑒 √ 𝜎 2𝜋(𝑇 − 𝑡) (

= −𝐾𝑒−𝑟(𝑇 −𝑡) Φ

+𝑆𝑡 𝑒

𝑦− 21

−𝑦+𝜈(𝑇 −𝑡) √ 𝜎 𝑇 −𝑡

̂ + 𝜈(𝑇 − 𝑡) log(𝑆𝑡 ∕𝐾) √ 𝜎 𝑇 −𝑡

) ( 2 − 𝑟+ 𝜈 2 (𝑇 −𝑡) 2𝜎

∞ (

∫log

̂ 𝑡 𝐾∕𝑆

)

√

)

)2

⎫ ⎪ 𝑑𝑦⎬ ⎪ ⎭

1

𝜎 2𝜋(𝑇 − 𝑡)

( ( ) 𝑦 √ 1+ 𝜈2 𝑦− 21

𝑒

𝜎

)2

𝜎 𝑇 −𝑡

𝑑𝑦.

From the identity √

1 2𝜋𝜏 ∫𝐿

𝑈 𝑎𝑤− 1 ( √𝑤 )2 2 𝜏

we have 𝐴1 = −𝐾𝑒

𝑒

( −𝑟(𝑇 −𝑡)

Φ

[ ( 𝑑𝑤 = 𝑒

1 2 𝑎 𝜏 2

̂ + 𝜈(𝑇 − 𝑡) log(𝑆𝑡 ∕𝐾) √ 𝜎 𝑇 −𝑡

Φ

𝑈 − 𝑎𝜏 √ 𝜏

)

( −Φ

𝐿 − 𝑎𝜏 √ 𝜏

)]

)

) ( ⎛ ̂ 𝑡 ) − 1 + 𝜈2 𝜎 2 (𝑇 − 𝑡) ⎞⎤ log( 𝐾∕𝑆 ⎜ ⎟⎥ 𝜎 2𝜎 𝜎 +𝑆𝑡 𝑒 √ ⎢1 − Φ ⎜ ⎟⎥ 𝜎 𝑇 −𝑡 ⎢ ⎜ ⎟⎥ ⎣ ⎝ ⎠⎦ ) ( ̂ + 𝜈(𝑇 − 𝑡) log(𝑆𝑡 ∕𝐾) ̂ −𝑟(𝑇 −𝑡) Φ = −𝐾𝑒 √ 𝜎 𝑇 −𝑡 ) ( ⎛ ⎞ 𝜈 ( ) ̂ log(𝑆 𝜎 2 (𝑇 − 𝑡) ⎟ ∕ 𝐾) + 1 + 𝑡 𝜎2 𝜈+ 21 𝜎 2 −𝑟 (𝑇 −𝑡) ⎜ +𝑆𝑡 𝑒 Φ⎜ √ ⎟. 𝜎 𝑇 −𝑡 ⎜ ⎟ ⎝ ⎠ ( ) ( )2 ⎡ 2 − 𝑟+ 𝜈 2 (𝑇 −𝑡)+ 12 1+ 𝜈2 𝜎 2 (𝑇 −𝑡) ⎢

632

6.2.2 Path-Dependent Options

By substituting 𝜈 = 𝑟 − 𝐷 − 12 𝜎 2 we obtain ̂ −𝑟(𝑇 −𝑡) Φ(𝑑− ) 𝐴1 = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑒 ̂ + (𝑟 − 𝐷 ± 1 𝜎 2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝐾) 2 such that 𝑑± = . √ 𝜎 𝑇 −𝑡 Finally, for the case 𝐴2 = 𝑆𝑡 𝑒−𝑟(𝑇 −𝑡)

∞

)𝑒 ( ̂ 𝑡 𝐾∕𝑆

∫log

( ) 1+ 2𝜈2 𝑦 𝜎

( Φ

−𝑦 + 𝜈(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

) 𝑑𝑦 and

using integration by parts with 𝑑𝑈 =𝑒 𝑑𝑦 ( 𝑉 =Φ

(

) 1+ 2𝜈2 𝑦

(

2𝜈 ⇒𝑈 = 1+ 2 𝜎

𝜎

−𝑦 − 𝜈(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

)

)−1

𝑒

( ) 1+ 2𝜈2 𝑦 𝜎

(

⇒

− 21

𝑑𝑉 1 𝑒 =− √ 𝑑𝑦 𝜎 2𝜋(𝑇 − 𝑡)

−𝑦−𝜈(𝑇 −𝑡) √ 𝜎 𝑇 −𝑡

)2

we can write )]∞ ( ⎧[ ( ) 𝑆𝑡 𝑒−𝑟(𝑇 −𝑡) ⎪ −𝑦 − 𝜈(𝑇 − 𝑡) 1+ 2𝜈2 𝑦 𝐴2 = ( )⎨ 𝑒 𝜎 Φ √ 𝜎 𝑇 −𝑡 1 + 𝜎2𝜈2 ⎪ ̂ 𝑡) log(𝐾∕𝑆 ⎩ ( )2 ) −𝑡) √ 1+ 2𝜈2 𝑦− 12 −𝑦−𝜈(𝑇 𝜎 𝜎 𝑇 −𝑡

⎫ ⎪ 𝑒 + 𝑑𝑦⎬ ) √ ( ∫log 𝐾∕𝑆 ̂ 𝑡 𝜎 2𝜋(𝑇 − 𝑡) ⎪ ⎭ ) ( ( ) 2𝜈 ( ) − 1+ ̂ − 𝜈(𝑇 − 𝑡) 𝜎2 𝑆𝑡 𝑒−𝑟(𝑇 −𝑡) 𝑆𝑡 log(𝑆𝑡 ∕𝐾) = −( Φ ) √ ̂ 𝐾 𝜎 𝑇 −𝑡 1 + 2𝜈2 (

∞

+

𝑆𝑡 𝑒

1

𝜎 ) ( 2 − 𝑟+ 𝜈 2 (𝑇 −𝑡) 2𝜎

( 1+

2𝜈 𝜎2

)

∞ (

∫log

̂ 𝑡 𝐾∕𝑆

)

1

( ( ) 𝑦 √ 1+ 𝜈2 𝑦− 12

𝑒 √ 𝜎 2𝜋(𝑇 − 𝑡)

𝜎

)2

𝜎 𝑇 −𝑡

𝑑𝑦.

Using the identity 1

𝑈 𝑎𝑤− 1 ( √𝑤 )2 2 𝜏

𝑒 √ 2𝜋𝜏 ∫𝐿

[ ( 𝑑𝑤 = 𝑒

1 2 𝑎 𝜏 2

Φ

𝑈 − 𝑎𝜏 √ 𝜏

)

( −Φ

𝐿 − 𝑎𝜏 √ 𝜏

)]

6.2.2 Path-Dependent Options

633

again we have 𝑆 𝑒−𝑟(𝑇 −𝑡) 𝐴2 = − (𝑡 ) 1 + 𝜎2𝜈2

(

(

𝑆𝑡 ̂ 𝐾

)−

1+ 2𝜈2 𝜎

)

( Φ

̂ − 𝜈(𝑇 − 𝑡) log(𝑆𝑡 ∕𝐾) √ 𝜎 𝑇 −𝑡

)

) ( ( )2 2 − 𝑟+ 𝜈 2 (𝑇 −𝑡)+ 12 1+ 𝜈2 𝜎 2 (𝑇 −𝑡) 2𝜎 𝜎

) ( ⎡ ⎛ ̂ 𝑡 ) − 1 + 𝜈2 𝜎 2 (𝑇 − 𝑡) ⎞⎤ log( 𝐾∕𝑆 𝑆𝑒 ⎢ ⎜ ⎟⎥ 𝜎 1 − Φ⎜ + 𝑡 ( ) √ ⎢ ⎟⎥ 𝜎 𝑇 −𝑡 ⎢ ⎜ ⎟⎥ 1 + 𝜎2𝜈2 ⎣ ⎝ ⎠⎦ ) ( ( ) 2𝜈 ( ) ̂ − 𝜈(𝑇 − 𝑡) log(𝑆𝑡 ∕𝐾) 𝑆 𝑒−𝑟(𝑇 −𝑡) 𝑆𝑡 − 1+ 𝜎 2 Φ = − (𝑡 ) √ ̂ 𝐾 𝜎 𝑇 −𝑡 1 + 𝜎2𝜈2 ) ( ⎛ ̂ + 1 + 𝜈2 𝜎 2 (𝑇 − 𝑡) ⎞ (𝜈 + 12 𝜎 2 − 𝑟)(𝑇 − 𝑡) ⎜ log(𝑆𝑡 ∕𝐾) ⎟ 𝜎 Φ⎜ + ( ) √ ⎟. 𝜎 𝑇 −𝑡 ⎜ ⎟ 1 + 𝜎2𝜈2 ⎝ ⎠

By substituting 𝜈 = 𝑟 − 𝐷 − 12 𝜎 2 and setting 𝛼 =

𝑟−𝐷 1 2 𝜎 2

𝑆 𝑒−𝐷(𝑇 −𝑡) 𝑆 𝑒−𝑟(𝑇 −𝑡) 𝐴2 = 𝑡 Φ(𝑑̂+ ) − 𝑡 𝛼 𝛼 where 𝑑̂± = Hence,

we obtain

(

𝑆𝑡 ̂ 𝐾

)−𝛼

Φ(𝑑̂− )

̂ ± (𝑟 − 𝐷 ± 1 𝜎 2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝐾) 2 . √ 𝜎 𝑇 −𝑡

̂ 𝑇 ) = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, ̂ 𝑇) 𝐶̂𝐹 (𝑆𝑡 , 𝑡; 𝐾, ( )−𝛼 ] ( )[ 𝑆𝑡 𝑆𝑡 −𝐷(𝑇 −𝑡) −𝑟(𝑇 −𝑡) ̂ ̂ Φ(𝑑+ ) − 𝑒 Φ(𝑑− ) . + 𝑒 𝛼 ̂ 𝐾 By gathering all the results, we finally have { } max ̂ 𝑇) − 𝐾, 0 + 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝐶𝐹 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) max 𝑆𝑡−𝑡 0 ( )[ ( )−𝛼 ] 𝑆𝑡 𝑆𝑡 −𝐷(𝑇 −𝑡) −𝑟(𝑇 −𝑡) ̂ ̂ + Φ(𝑑+ ) − 𝑒 Φ(𝑑− ) 𝑒 𝛼 ̂ 𝐾 { } ̂ = max 𝑆 max , 𝐾 . where 𝐾 𝑡−𝑡 0

634

6.2.2 Path-Dependent Options

{ } 13. European Fixed Strike Lookback Option II. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. We consider a European-style fixed strike lookback put option with terminal payoff { Ψ(𝑆𝑇 ) = max

} 𝐾 − min 𝑆𝑢 , 0 𝑡0 ≤𝑢≤𝑇

which depends on the minimum of the asset price reached within the lookback period [𝑡0 , 𝑇 ], 𝑡0 ≥ 0 where 𝑇 is the expiry time and 𝐾 > 0 is the strike price. By definition, under the risk-neutral measure ℚ the value of a fixed strike lookback put option at time 𝑡, 𝑡0 ≤ 𝑡 ≤ 𝑇 is [ 𝑃𝐹 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒

−𝑟(𝑇 −𝑡) ℚ

𝔼

{

{ 𝐾 − min

max

} min 𝑆𝑢 , min 𝑆𝑢

𝑡0 ≤𝑢≤𝑡

𝑡≤𝑢≤𝑇

}| ] | , 0 | ℱ𝑡 | |

where 𝑟 is the risk-free interest rate. By considering two cases min 𝑆𝑢 ≤ 𝐾 and min 𝑆𝑢 > 𝐾, show that 𝑃𝐹 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) 𝑡0 ≤𝑢≤𝑡

𝑡0 ≤𝑢≤𝑡

can be written as 𝑃𝐹 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) max

{ } ̂ 𝑇) 𝐾 − min 𝑆𝑢 , 0 + 𝑃̂𝐹 (𝑆𝑡 , 𝑡; 𝐾,

[ ̂ 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max where 𝑃̂𝐹 (𝑆𝑡 , 𝑡; 𝐾, } { ̂ = min min 𝑆𝑢 , 𝐾 . 𝐾

𝑡0 ≤𝑢≤𝑡

}| ] | ̂ 𝐾 − min 𝑆𝑢 , 0 | ℱ𝑡 such that | 𝑡≤𝑢≤𝑇 |

{

𝑡0 ≤𝑢≤𝑡

Using the identity 1

𝑈 𝑎𝑤− 1 ( √𝑤 )2 2 𝜏

𝑒 √ 2𝜋𝜏 ∫𝐿

[ ( 𝑑𝑤 = 𝑒

1 2 𝑎 𝜏 2

Φ

𝑈 − 𝑎𝜏 √ 𝜏

)

( −Φ

𝐿 − 𝑎𝜏 √ 𝜏

)]

show that ̂ 𝑇 ) = 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, ̂ 𝑇) 𝑃̂𝐹 (𝑆𝑡 , 𝑡; 𝐾, ] ( ) [( )−𝛼 𝑆𝑡 𝑆𝑡 −𝑟(𝑇 −𝑡) −𝐷(𝑇 −𝑡) ̂ ̂ 𝑒 Φ(−𝑑− ) − 𝑒 Φ(−𝑑+ ) + 𝛼 ̂ 𝐾

6.2.2 Path-Dependent Options

635

̂ 𝑇 ) is the vanilla (or European) put option price defined as where 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, ̂ 𝑇 ) = 𝐾𝑒 ̂ −𝑟(𝑇 −𝑡) Φ(−𝑑− ) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑+ ) 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, with ̂ + (𝑟 − 𝐷 ± 1 𝜎 2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝐾) 2 𝑑± = √ 𝜎 𝑇 −𝑡 ̂ ± (𝑟 − 𝐷 ± 1 𝜎 2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝐾) 2 √ 𝜎 𝑇 −𝑡

𝑑̂± =

𝛼=

𝑟−𝐷 1 2 𝜎 2

𝑥

and Φ(𝑥) =

∫−∞

1 − 1 𝑢2 √ 𝑒 2 𝑑𝑢 is the cdf of a standard normal. 2𝜋

Solution: From Girsanov’s theorem, under the risk-neutral measure ℚ, 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡 where 𝑊𝑡ℚ = 𝑊𝑡 + By writing

(𝜇 − 𝑟) 𝜎

𝑡 is a ℚ-standard Wiener process.

min min 𝑆𝑢 = 𝑆𝑡−𝑡 ,

𝑡0 ≤𝑢≤𝑡

0

min 𝑆𝑢 = 𝑆𝑇min −𝑡

𝑡≤𝑢≤𝑇

hence [

}| ] | 𝔼 max 𝐾 − min min 𝑆𝑢 , min 𝑆𝑢 , 0 | ℱ𝑡 𝑃𝐹 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒 | 𝑡0 ≤𝑢≤𝑡 𝑡≤𝑢≤𝑇 | [ } }| ] { { min | = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝐾 − min 𝑆𝑡−𝑡 , 𝑆𝑇min −𝑡 , 0 | ℱ𝑡 . 0 | {

−𝑟(𝑇 −𝑡) ℚ

{

}

min ≤ 𝐾 then If 𝑆𝑡−𝑡 0

} } { { { } min min min min , 0 = 𝐾 − 𝑆 max 𝐾 − min 𝑆𝑡−𝑡 , 𝑆𝑇min + max 𝑆 − 𝑆 , 0 𝑡−𝑡 𝑡−𝑡 −𝑡 𝑇 −𝑡 0

0

0

min > 𝐾 then and if 𝑆𝑡−𝑡 0

} } { { { } min min max 𝐾 − min 𝑆𝑡−𝑡 , 𝑆𝑇min −𝑡 , 0 = max 𝐾 − 𝑆𝑇 −𝑡 , 0 . 0

636

6.2.2 Path-Dependent Options

Thus, we can write } } { } { { min min , 0 = max 𝐾 − 𝑆 , 𝑆𝑇min , 0 max 𝐾 − min 𝑆𝑡−𝑡 𝑡−𝑡0 −𝑡 0 } { min , 𝐾} − 𝑆𝑇min , 0 . + max min{𝑆𝑡−𝑡 −𝑡 0

By substituting this back into the fixed strike lookback option price, we have 𝑃𝐹 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) [ } { }| ] { min min |ℱ = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝐾 − 𝑆𝑡−𝑡 , 0 + max min{𝑆𝑡−𝑡 , 𝐾} − 𝑆𝑇min , 0 | 𝑡 −𝑡 0 0 | { } −𝑟(𝑇 −𝑡) min max 𝐾 − 𝑆𝑡−𝑡 , 0 =𝑒 0 [ }| ] { min | ℱ𝑡 , 𝐾} − 𝑆𝑇min , 0 +𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max min{𝑆𝑡−𝑡 | −𝑡 0 | { } min ̂ 𝑇) = 𝑒−𝑟(𝑇 −𝑡) max 𝐾 − 𝑆𝑡−𝑡 , 0 + 𝑃̂𝐹 (𝑆𝑡 , 𝑡; 𝐾, 0

[ }| ] } { { ̂ 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝐾 ̂ = min 𝑆 min , 𝐾 . ̂ − 𝑆 min , 0 | ℱ𝑡 , 𝐾 where 𝑃̂𝐹 (𝑆𝑡 , 𝑡; 𝐾, | 𝑡−𝑡0 𝑇 −𝑡 | From It¯o’s lemma we can write 𝑆𝑇max −𝑡 = min 𝑆𝑢 𝑡≤𝑢≤𝑇

1 2 ℚ )(𝑢−𝑡)+𝜎𝑊𝑢−𝑡

= min 𝑆𝑡 𝑒(𝑟−𝐷− 2 𝜎 𝑡≤𝑢≤𝑇

= 𝑆𝑡 𝑒𝑚𝑇 −𝑡 ℚ where 𝑚𝑇 −𝑡 = min 𝜈(𝑢 − 𝑡) + 𝜎𝑊𝑢−𝑡 such that 𝜈 = 𝑟 − 𝐷 − 12 𝜎 2 . From Problem 4.2.2.15 𝑡≤𝑢≤𝑇

of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus ( ( ) ) 2𝜈𝑥 ) ( 𝑥 − 𝜈(𝑇 − 𝑡) 𝑥 + 𝜈(𝑇 − 𝑡) + 𝑒 𝜎2 Φ , 𝑥 ≤ 0. ℚ 𝑚𝑇 −𝑡 ≤ 𝑥 = Φ √ √ 𝜎 𝑇 −𝑡 𝜎 𝑇 −𝑡 Hence, [ }| ] { −𝑟(𝑇 −𝑡) ℚ 𝑚𝑇 −𝑡 ̂ ̂ ̂ 𝑃𝐹 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒 𝔼 max 𝐾 − 𝑆𝑡 𝑒 , 0 || ℱ𝑡 | = 𝑒−𝑟(𝑇 −𝑡) =𝑒

−𝑟(𝑇 −𝑡)

=𝑒

−𝑟(𝑇 −𝑡)

̂ 𝐾

∫−∞ ̂ 𝐾

∫−∞ ̂ 𝐾

∫−∞

( ) ℚ 𝑆𝑡 𝑒𝑚𝑇 −𝑡 ≤ 𝑥 𝑑𝑥 ( ℚ 𝑒 (

𝑚𝑇 −𝑡

𝑥 ≤ 𝑆𝑡

)

ℚ 𝑚𝑇 −𝑡 ≤ log

𝑑𝑥 (

𝑥 𝑆𝑡

)) 𝑑𝑥.

6.2.2 Path-Dependent Options

637

) ( By substituting 𝑦 = log 𝑥∕𝑆𝑡 we have ̂ 𝑡) log(𝐾∕𝑆

̂ 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝑃̂𝐹 (𝑆𝑡 , 𝑡; 𝐾, = 𝑆𝑡 𝑒

∫−∞ −𝑟(𝑇 −𝑡) ̂ 𝑡) log(𝐾∕𝑆

×

∫−∞

( ) 𝑆𝑡 𝑒𝑦 ℚ 𝑚𝑇 −𝑡 ≤ 𝑦 𝑑𝑦

[ ( 𝑒

𝑦

Φ

= 𝐵1 + 𝐵2

𝑦 − 𝜈(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

)

( +𝑒

2𝜈𝑦 𝜎2

Φ

𝑦 + 𝜈(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

)] 𝑑𝑦

where 𝐵1 = 𝑆𝑡 𝑒

−𝑟(𝑇 −𝑡)

(

̂ 𝑡) log(𝐾∕𝑆

𝑦

𝑒 Φ

∫−∞

𝑦 − 𝜈(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

) 𝑑𝑦

and ) 𝑦 + 𝜈(𝑇 − 𝑡) 𝑑𝑦. 𝑒 𝜎 Φ 𝐵2 = 𝑆𝑡 𝑒 √ ∫−∞ 𝜎 𝑇 −𝑡 ( ) ̂ 𝑡) log(𝐾∕𝑆 𝑦 − 𝜈(𝑇 − 𝑡) For the case 𝐵1 = 𝑆𝑡 𝑒−𝑟(𝑇 −𝑡) 𝑒𝑦 Φ 𝑑𝑦 and using integration by √ ∫−∞ 𝜎 𝑇 −𝑡 parts with ̂ 𝑡 ) ( 2𝜈 ) log(𝐾∕𝑆 1+ 2 𝑦

−𝑟(𝑇 −𝑡)

(

𝑑𝑈 = 𝑒𝑦 ⇒ 𝑈 = 𝑒 𝑦 𝑑𝑦 ( 𝑉 =Φ

𝑦 − 𝜈(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

)

( 1

− 𝑑𝑉 1 𝑒 2 ⇒ = √ 𝑑𝑦 𝜎 2𝜋(𝑇 − 𝑡)

𝑦−𝜈(𝑇 −𝑡) √ 𝜎 𝑇 −𝑡

)2

we can write ⎧[

𝐵1 = 𝑆𝑡 𝑒

𝑦

⎨ 𝑒 Φ ⎪ ⎩

̂ 𝑡) log(𝐾∕𝑆

−

(

−𝑟(𝑇 −𝑡) ⎪

∫−∞

= 𝐾𝑒

−𝑆𝑡 𝑒

Φ

𝑦− 21

1

𝑒 √ 𝜎 2𝜋(𝑇 − 𝑡)

−∞ 𝑦−𝜈(𝑇 −𝑡) √ 𝜎 𝑇 −𝑡

̂ 𝑡 ) − 𝜈(𝑇 − 𝑡) log(𝐾∕𝑆 √ 𝜎 𝑇 −𝑡

) ( 2 − 𝑟+ 𝜈 2 (𝑇 −𝑡) 2𝜎

)]log(𝐾∕𝑆 ̂ 𝑡)

(

( −𝑟(𝑇 −𝑡)

𝑦 − 𝜈(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

̂ 𝑡) log(𝐾∕𝑆

∫−∞

)

)2

⎫ ⎪ 𝑑𝑦⎬ ⎪ ⎭

(

1 𝑒 √ 𝜎 2𝜋(𝑇 − 𝑡)

( )2 ) 𝑦 √ 1+ 𝜈2 𝑦− 12 𝜎 𝜎 𝑇 −𝑡

𝑑𝑦.

638

6.2.2 Path-Dependent Options

From the identity

1

𝑈 𝑎𝑤− 1 ( √𝑤 )2 2 𝜏

𝑒 √ 2𝜋𝜏 ∫𝐿

[ ( 𝑑𝑤 = 𝑒

1 2 𝑎 𝜏 2

Φ

𝑈 − 𝑎𝜏 √ 𝜏

)

( −Φ

𝐿 − 𝑎𝜏 √ 𝜏

)]

we have ( 𝐵1 = 𝐾𝑒

−𝑟(𝑇 −𝑡)

Φ

̂ 𝑡 ) − 𝜈(𝑇 − 𝑡) log(𝐾∕𝑆 √ 𝜎 𝑇 −𝑡

)

) ( ⎛ ̂ 𝑡 ) − 1 + 𝜈2 𝜎 2 (𝑇 − 𝑡) ⎞ log( 𝐾∕𝑆 ⎜ ⎟ 𝜎 2𝜎 𝜎 −𝑆𝑡 𝑒 Φ⎜ √ ⎟ 𝜎 𝑇 −𝑡 ⎜ ⎟ ⎝ ⎠ ) ( ̂ 𝑡 ) − 𝜈(𝑇 − 𝑡) log(𝐾∕𝑆 ̂ −𝑟(𝑇 −𝑡) Φ = 𝐾𝑒 √ 𝜎 𝑇 −𝑡 ) ( ⎛ ( ) ̂ 𝑡 ) − 1 + 𝜈2 𝜎 2 (𝑇 − 𝑡) ⎞ log( 𝐾∕𝑆 1 2 ⎟ 𝜎 𝜈+ 𝜎 −𝑟 (𝑇 −𝑡) ⎜ −𝑆𝑡 𝑒 2 Φ⎜ √ ⎟. 𝜎 𝑇 −𝑡 ⎜ ⎟ ⎝ ⎠ ) ( ( )2 2 − 𝑟+ 𝜈 2 (𝑇 −𝑡)+ 12 1+ 𝜈2 𝜎 2 (𝑇 −𝑡)

By substituting 𝜈 = 𝑟 − 𝐷 − 12 𝜎 2 we have ) ) ( ( ̂ −𝑟(𝑇 −𝑡) Φ −𝑑− − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ −𝑑+ 𝐵1 = 𝐾𝑒 ̂ + (𝑟 − 𝐷 ± 1 𝜎 2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝐾) 2 . such that 𝑑± = √ 𝜎 𝑇 −𝑡 Finally, for the case 𝐵2 = 𝑆𝑡

𝑒−𝑟(𝑇 −𝑡)

̂ 𝑡) ( log(𝐾∕𝑆

𝑒

∫−∞

) 1+ 2𝜈2 𝑦 𝜎

( Φ

using integration by parts with 𝑑𝑈 =𝑒 𝑑𝑦 ( 𝑉 =Φ

(

) 1+ 2𝜈2 𝑦 𝜎

𝑦 + 𝜈(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

)

(

2𝜈 ⇒𝑈 = 1+ 2 𝜎

)−1

𝑒

𝑦 + 𝜈(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

( ) 1+ 2𝜈2 𝑦 𝜎

( 1

− 𝑑𝑉 1 𝑒 2 ⇒ = √ 𝑑𝑦 𝜎 2𝜋(𝑇 − 𝑡)

𝑦+𝜈(𝑇 −𝑡) √ 𝜎 𝑇 −𝑡

)2

) 𝑑𝑦 and

6.2.2 Path-Dependent Options

639

we can write )]log(𝐾∕𝑆 ( ̂ 𝑡) ⎧[ ( ) 𝑆𝑡 𝑒−𝑟(𝑇 −𝑡) ⎪ 𝑦 + 𝜈(𝑇 − 𝑡) 1+ 2𝜈2 𝑦 𝜎 Φ 𝐵2 = ( )⎨ 𝑒 √ 𝜎 𝑇 −𝑡 1 + 𝜎2𝜈2 ⎪ −∞ ⎩ −

−

1

𝑒 √ 𝜎 2𝜋(𝑇 − 𝑡)

∫−∞

𝑆 𝑒−𝑟(𝑇 −𝑡) = (𝑡 ) 1 + 𝜎2𝜈2 𝑆𝑡 𝑒

(

(

𝑆𝑡 ̂ 𝐾

)−

) ( 2 − 𝑟+ 𝜈 2 (𝑇 −𝑡) 2𝜎

( 1+

( )2 ) −𝑡) √ 1+ 2𝜈2 𝑦− 12 𝑦+𝜈(𝑇 𝜎 𝜎 𝑇 −𝑡

(

̂ 𝑡) log(𝐾∕𝑆

2𝜈 𝜎2

)

1+ 2𝜈2

)

𝜎

( Φ

̂ 𝑡) log(𝐾∕𝑆

∫−∞

⎫ ⎪ 𝑑𝑦⎬ ⎪ ⎭

̂ 𝑡 ) + 𝜈(𝑇 − 𝑡) log(𝐾∕𝑆 √ 𝜎 𝑇 −𝑡

)

( ( ) 𝑦 √ 1+ 𝜈2 𝑦− 12

1 𝑒 √ 𝜎 2𝜋(𝑇 − 𝑡)

𝜎

)2

𝜎 𝑇 −𝑡

𝑑𝑦.

Using the identity

√

1 2𝜋𝜏 ∫𝐿

𝑈 𝑎𝑤− 1 ( √𝑤 )2 2 𝜏

𝑒

[ ( 𝑑𝑤 = 𝑒

1 2 𝑎 𝜏 2

Φ

𝑈 − 𝑎𝜏 √ 𝜏

)

( −Φ

𝐿 − 𝑎𝜏 √ 𝜏

)]

once again we have

𝑆 𝑒−𝑟(𝑇 −𝑡) 𝐵2 = (𝑡 ) 1 + 𝜎2𝜈2

(

(

𝑆𝑡 ̂ 𝐾

)−

1+ 2𝜈2 𝜎

)

( Φ

̂ 𝑡 ) + 𝜈(𝑇 − 𝑡) log(𝐾∕𝑆 √ 𝜎 𝑇 −𝑡

) ( ( )2 2 − 𝑟+ 𝜈 2 (𝑇 −𝑡)+ 21 1+ 𝜈2 𝜎 2 (𝑇 −𝑡) 2𝜎 𝜎

)

) ( ⎛ ̂ 𝑡 ) − 1 + 𝜈2 𝜎 2 (𝑇 − 𝑡) ⎞ log( 𝐾∕𝑆 ⎜ ⎟ 𝜎 Φ⎜ − ( ) √ ⎟ 2𝜈 𝜎 𝑇 −𝑡 ⎜ ⎟ 1 + 𝜎2 ⎝ ⎠ ) ( ( ) 2𝜈 ( ) ̂ 𝑡 ) + 𝜈(𝑇 − 𝑡) log(𝐾∕𝑆 𝑆 𝑒−𝑟(𝑇 −𝑡) 𝑆𝑡 − 1+ 𝜎 2 Φ = (𝑡 ) √ ̂ 𝐾 𝜎 𝑇 −𝑡 1 + 𝜎2𝜈2 ) ( ⎛ ̂ 𝑡 ) − 1 + 𝜈2 𝜎 2 (𝑇 − 𝑡) ⎞ (𝜈 + 12 𝜎 2 − 𝑟)(𝑇 − 𝑡) ⎜ log(𝐾∕𝑆 ⎟ 𝜎 Φ⎜ − ( ) √ ⎟. 2𝜈 𝜎 𝑇 −𝑡 ⎜ ⎟ 1 + 𝜎2 ⎝ ⎠ 𝑆𝑡 𝑒

640

6.2.2 Path-Dependent Options

By substituting 𝜈 = 𝑟 − 𝐷 − 12 𝜎 2 and setting 𝛼 = 𝑆 𝑒−𝑟(𝑇 −𝑡) 𝐵2 = 𝑡 𝛼 where 𝑑̂± = Hence,

(

𝑆𝑡 ̂ 𝐾

)−𝛼

𝑟−𝐷 1 2 𝜎 2

Φ(−𝑑̂− ) −

we obtain

𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑̂+ ) 𝛼

̂ ± (𝑟 − 𝐷 ± 1 𝜎 2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝐾) 2 . √ 𝜎 𝑇 −𝑡

̂ 𝑇 ) = 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, ̂ 𝑇) 𝑃̂𝐹 (𝑆𝑡 , 𝑡; 𝐾, ] ( ) [( )−𝛼 𝑆𝑡 𝑆𝑡 𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑̂− ) − 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑̂+ ) . + 𝛼 ̂ 𝐾 By gathering all the results, we finally have { } min ̂ 𝑇) , 0 + 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑃𝐹 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) max 𝐾 − 𝑆𝑡−𝑡 0 ( ) [( )−𝛼 ] 𝑆𝑡 𝑆𝑡 + 𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑̂− ) − 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑̂+ ) 𝛼 ̂ 𝐾 } { ̂ = min 𝑆 min , 𝐾 . where 𝐾 𝑡−𝑡 0

14. European Floating Strike Lookback Option I. Let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. We consider a European-style floating strike lookback call option with terminal payoff Ψ(𝑆𝑇 ) = max{𝑆𝑇 − 𝐾min , 0} { where the floating strike price 𝐾min is defined as 𝐾min = min

} min 𝑆𝑢 , min 𝑆𝑢 ,

𝑡0 ≤𝑢≤𝑡

𝑡≤𝑢≤𝑇

which depends on the minimum of the asset price reached within the lookback period [𝑡0 , 𝑇 ], 𝑡0 ≥ 0 where 𝑇 is the option expiry time. By definition, under the risk-neutral measure ℚ the value of a floating strike lookback call option at time 𝑡, 𝑡0 ≤ 𝑡 ≤ 𝑇 is [ ] 𝐶𝑓 (𝑆𝑡 , 𝑡; 𝐾min , 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑆𝑇 − 𝐾min , 0}|| ℱ𝑡

6.2.2 Path-Dependent Options

641

min = min 𝑆 , show that where 𝑟 is the risk-free interest rate. By setting 𝑆𝑡−𝑡 𝑢 𝑡0 ≤𝑢≤𝑡

0

min min 𝐶𝑓 (𝑆𝑡 , 𝑡; 𝐾min , 𝑇 ) = 𝑆𝑡 − 𝑒−𝑟(𝑇 −𝑡) 𝑆𝑡−𝑡 + 𝑃̂𝐹 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 ,𝑇) 0

0

where min min , 𝑇 ) = 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 ,𝑇) 𝑃̂𝐹 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 0 0 ] [( )−𝛼 ( ) 𝑆𝑡 𝑆𝑡 −𝑟(𝑇 −𝑡) −𝐷(𝑇 −𝑡) + 𝑒 Φ(−𝑑̂− ) − 𝑒 Φ(−𝑑̂+ ) min 𝛼 𝑆𝑡−𝑡 0

min , 𝑇 ) is the European put option price defined as such that 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 0

) ) ( ( min min −𝑟(𝑇 −𝑡) , 𝑇 ) = 𝑆𝑡−𝑡 𝑒 Φ −𝑑− − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ −𝑑+ 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 0

0

with min ) + (𝑟 − 𝐷 ± 1 𝜎 2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝑆𝑡−𝑡 2 0 𝑑± = √ 𝜎 𝑇 −𝑡

𝑑̂± =

𝛼=

𝑟−𝐷 1 2 𝜎 2

𝑥

and Φ(𝑥) =

∫−∞

min ) ± (𝑟 − 𝐷 ± 1 𝜎 2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝑆𝑡−𝑡 2 0 √ 𝜎 𝑇 −𝑡

1 − 1 𝑢2 √ 𝑒 2 𝑑𝑢 is the cdf of a standard normal. 2𝜋

min = min 𝑆 and 𝑆 min = min 𝑆 , we can write the payoff as Solution: By setting 𝑆𝑡−𝑡 𝑢 𝑢 𝑇 −𝑡 0

𝑡0 ≤𝑢≤𝑡

𝑡≤𝑢≤𝑇

{ } Ψ(𝑆𝑇 ) = max 𝑆𝑇 − 𝐾min , 0 } } { { min ,0 , 𝑆𝑇min = max 𝑆𝑇 − min 𝑆𝑡−𝑡 −𝑡 0 } { min ≤ 𝑆 min ⎧ max 𝑆 − 𝑆 min , 0 if 𝑆𝑡−𝑡 𝑇 𝑡−𝑡0 𝑇 −𝑡 0 ⎪ =⎨ ⎪ max {𝑆 − 𝑆 min , 0} if 𝑆 min > 𝑆 min 𝑇 𝑡−𝑡 ⎩ 𝑇 −𝑡 𝑇 −𝑡 0

⎧ 𝑆 − 𝑆 min if 𝑆 min ≤ 𝑆 min 𝑡−𝑡0 𝑡−𝑡0 𝑇 −𝑡 ⎪ 𝑇 =⎨ min > 𝑆 min ⎪ 𝑆𝑇 − 𝑆 min if 𝑆𝑡−𝑡 𝑇 −𝑡 𝑇 −𝑡 0 ⎩ } { min = 𝑆𝑇 − min 𝑆𝑡−𝑡 , 𝑆𝑇min −𝑡 0

{ } min min + max 𝑆𝑡−𝑡 − 𝑆𝑇min = 𝑆𝑇 − 𝑆𝑡−𝑡 −𝑡 , 0 . 0

0

642

6.2.2 Path-Dependent Options

Under the risk-neutral measure ℚ, 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡 where 𝑊𝑡ℚ = 𝑊𝑡 + time 𝑡, 𝑡0 ≤ 𝑡 ≤ 𝑇 is

(𝜇 − 𝑟) 𝜎

𝑡 is a ℚ-standard Wiener process. Thus, the option price at

𝐶𝑓 (𝑆𝑡 , 𝑡; 𝐾min , 𝑇 ) [ { }| ] −𝑟(𝑇 −𝑡) ℚ min min min =𝑒 𝔼 𝑆𝑇 − 𝑆𝑡−𝑡 + max 𝑆𝑡−𝑡 − 𝑆𝑇 −𝑡 , 0 || ℱ𝑡 0 0 | [ [ ] }| ] { −𝑟(𝑇 −𝑡) ℚ min | −𝑟(𝑇 −𝑡) ℚ min min 𝔼 𝑆𝑇 − 𝑆𝑡−𝑡 | ℱ𝑡 + 𝑒 𝔼 max 𝑆𝑡−𝑡 − 𝑆𝑇 −𝑡 , 0 || ℱ𝑡 =𝑒 0| 0 | [ ( ) }| ] { min min | + 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝑆𝑡−𝑡 − 𝑆𝑇min = 𝑒−𝑟(𝑇 −𝑡) 𝑆𝑡 𝑒𝑟(𝑇 −𝑡) − 𝑆𝑡−𝑡 −𝑡 , 0 | ℱ𝑡 0 0 | [ }| ] { min min | + 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝑆𝑡−𝑡 − 𝑆𝑇min = 𝑆𝑡 − 𝑒−𝑟(𝑇 −𝑡) 𝑆𝑡−𝑡 −𝑡 , 0 | ℱ𝑡 . 0 0 | From Problem 6.2.2.13 (page 634) we can write [ }| ] { min −𝑟(𝑇 −𝑡) ℚ min min ̂ 𝔼 max 𝑆𝑡−𝑡 − 𝑆𝑇 −𝑡 , 0 || ℱ𝑡 𝑃𝐹 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 , 𝑇 ) = 𝑒 0 0 | where min min 𝑃̂𝐹 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 , 𝑇 ) = 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 ,𝑇) 0 0 ] )−𝛼 ( ) [( 𝑆𝑡 𝑆𝑡 −𝑟(𝑇 −𝑡) −𝐷(𝑇 −𝑡) ̂ ̂ + 𝑒 Φ(−𝑑− ) − 𝑒 Φ(−𝑑+ ) min 𝛼 𝑆𝑡−𝑡 0

min , 𝑇 ) is the European put option price defined as such that 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 0

min min −𝑟(𝑇 −𝑡) , 𝑇 ) = 𝑆𝑡−𝑡 𝑒 Φ(−𝑑− ) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑+ ) 𝑃𝑏𝑠 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 0

0

with min ) + (𝑟 − 𝐷 ± 1 𝜎 2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝑆𝑡−𝑡 2 0 𝑑± = √ 𝜎 𝑇 −𝑡 min ) ± (𝑟 − 𝐷 ± 1 𝜎 2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝑆𝑡−𝑡 2 0 𝑑̂± = √ 𝜎 𝑇 −𝑡

and 𝛼 =

𝑟−𝐷 1 2 𝜎 2

.

6.2.2 Path-Dependent Options

643

Hence, min min + 𝑃̂𝐹 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 , 𝑇 ). 𝐶𝑓 (𝑆𝑡 , 𝑡; 𝐾min , 𝑇 ) = 𝑆𝑡 − 𝑒−𝑟(𝑇 −𝑡) 𝑆𝑡−𝑡 0

0

15. European Floating Strike Lookback Option II. Let {𝑊𝑡 : 𝑡 ≥ 0} be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow a GBM with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield and 𝜎 is the volatility parameter. We consider a European-style floating strike lookback put option with terminal payoff Ψ(𝑆𝑇 ) = max{𝐾max − 𝑆𝑇 , 0} { where the floating strike price 𝐾max is defined as 𝐾max = max

} max 𝑆𝑢 , max 𝑆𝑢

𝑡0 ≤𝑢≤𝑡

𝑡≤𝑢≤𝑇

which depends on the maximum of the asset price reached within the lookback period [𝑡0 , 𝑇 ], 𝑡0 ≥ 0 where 𝑇 is the option expiry time. By definition, under the risk-neutral measure ℚ the value of a floating strike lookback put option at time 𝑡, 𝑡0 ≤ 𝑡 ≤ 𝑇 is [ ] 𝑃𝑓 (𝑆𝑡 , 𝑡; 𝐾max , 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝐾max − 𝑆𝑇 , 0}|| ℱ𝑡 max = max 𝑆 , show that where 𝑟 is the risk-free interest rate. By setting 𝑆𝑡−𝑡 𝑢 0

𝑡0 ≤𝑢≤𝑡

max max 𝑃𝑓 (𝑆𝑡 , 𝑡; 𝐾max , 𝑇 ) = 𝐶̂𝐹 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 , 𝑇 ) + 𝑒−𝑟(𝑇 −𝑡) 𝑆𝑡−𝑡 − 𝑆𝑡 0

0

where max max 𝐶̂𝐹 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 , 𝑇 ) = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 ,𝑇) 0 0 ( ] [ )−𝛼 ( ) 𝑆𝑡 𝑆𝑡 −𝐷(𝑇 −𝑡) −𝑟(𝑇 −𝑡) + Φ(𝑑̂+ ) − 𝑒 Φ(𝑑̂− ) 𝑒 max 𝛼 𝑆𝑡−𝑡 0

max , 𝑇 ) is the European call option price defined as such that 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 0

( ) ( ) max max −𝑟(𝑇 −𝑡) , 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ 𝑑+ − 𝑆𝑡−𝑡 𝑒 Φ 𝑑− 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 0

0

with max ) + (𝑟 − 𝐷 ± 1 𝜎 2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝑆𝑡−𝑡 2 0 𝑑± = √ 𝜎 𝑇 −𝑡

644

6.2.2 Path-Dependent Options

𝑑̂± =

𝛼=

𝑟−𝐷 1 2 𝜎 2

max ) ± (𝑟 − 𝐷 ± 1 𝜎 2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝑆𝑡−𝑡 2 0 √ 𝜎 𝑇 −𝑡

𝑥

and Φ(𝑥) =

∫−∞

1 − 1 𝑢2 √ 𝑒 2 𝑑𝑢 is the cdf of a standard normal. 2𝜋

max = max 𝑆 and 𝑆 max = max 𝑆 , we can write the payoff as Solution: By setting 𝑆𝑡−𝑡 𝑢 𝑢 𝑇 −𝑡 0

𝑡0 ≤𝑢≤𝑡

𝑡≤𝑢≤𝑇

{ } Ψ(𝑆𝑇 ) = max 𝐾max − 𝑆𝑇 , 0 } } { { max − 𝑆 , 𝑆𝑇max , 0 = max max 𝑆𝑡−𝑡 𝑇 −𝑡 0 } { max ⎧ max 𝑆 max − 𝑆 , 0 if 𝑆𝑇max ≤ 𝑆𝑡−𝑡 𝑇 𝑡−𝑡0 −𝑡 0 ⎪ =⎨ ⎪ max {𝑆 max − 𝑆 , 0} if 𝑆 max > 𝑆 max 𝑇 𝑡−𝑡 ⎩ 𝑇 −𝑡 𝑇 −𝑡 0

max ⎧ 𝑆 max − 𝑆 if 𝑆𝑇max ≤ 𝑆𝑡−𝑡 𝑇 −𝑡 0 ⎪ 𝑡−𝑡0 =⎨ max ⎪ 𝑆𝑇max − 𝑆𝑇 if 𝑆𝑇max > 𝑆𝑡−𝑡 −𝑡 0 ⎩ −𝑡 } { max = max 𝑆𝑡−𝑡 , 𝑆𝑇max −𝑡 − 𝑆𝑇 0

{ } max max = max 𝑆𝑇max − 𝑆 , 0 + 𝑆𝑡−𝑡 − 𝑆𝑇 . 𝑡−𝑡 −𝑡 0

0

Under the risk-neutral measure ℚ, 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡ℚ 𝑆𝑡 where 𝑊𝑡ℚ = 𝑊𝑡 + time 𝑡, 𝑡0 ≤ 𝑡 ≤ 𝑇 is

(𝜇 − 𝑟) 𝜎

𝑡 is a ℚ-standard Wiener process. Hence, the option price at

𝑃𝑓 (𝑆𝑡 , 𝑡; 𝐾max , 𝑇 ) [ ] } { | max max |ℱ = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝑆𝑇max − 𝑆 , 0 + 𝑆 − 𝑆 𝑇| 𝑡 𝑡−𝑡0 𝑡−𝑡0 −𝑡 | [ ] [ ] }| { | max −𝑟(𝑇 −𝑡) ℚ max | 𝔼 𝑆𝑡−𝑡 − 𝑆𝑇 | ℱ𝑡 = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝑆𝑇max −𝑡 − 𝑆𝑡−𝑡0 , 0 | ℱ𝑡 + 𝑒 0 | | [ ( ) }| ] { max | ℱ + 𝑒−𝑟(𝑇 −𝑡) 𝑆 max − 𝑆 𝑒𝑟(𝑇 −𝑡) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝑆𝑇max − 𝑆 , 0 𝑡 𝑡 | 𝑡−𝑡0 𝑡−𝑡0 −𝑡 | [ }| ] { max −𝑟(𝑇 −𝑡) max | = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝑆𝑇max 𝑆𝑡−𝑡 − 𝑆𝑡 . −𝑡 − 𝑆𝑡−𝑡0 , 0 | ℱ𝑡 + 𝑒 0 |

6.2.2 Path-Dependent Options

645

From Problem 6.2.2.12 (page 627) we can write [ }| ] { max max |ℱ , 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max 𝑆𝑇max − 𝑆 , 0 𝐶̂𝐹 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 | 𝑡 𝑡−𝑡0 −𝑡 0 | where max max , 𝑇 ) = 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 ,𝑇) 𝐶̂𝐹 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 0 0 ( ] [ )−𝛼 ( ) 𝑆𝑡 𝑆𝑡 −𝐷(𝑇 −𝑡) −𝑟(𝑇 −𝑡) + Φ(𝑑̂+ ) − 𝑒 Φ(𝑑̂− ) 𝑒 max 𝛼 𝑆𝑡−𝑡 0

max , 𝑇 ) is the European call option price defined as such that 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 0

( ) ( ) max max −𝑟(𝑇 −𝑡) , 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ 𝑑+ − 𝑆𝑡−𝑡 𝑒 Φ 𝑑− 𝐶𝑏𝑠 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 0

0

with max ) + (𝑟 − 𝐷 ± 1 𝜎 2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝑆𝑡−𝑡 2 0 𝑑± = √ 𝜎 𝑇 −𝑡

𝑑̂± = and 𝛼 = Thus,

𝑟−𝐷 1 2 𝜎 2

max ) ± (𝑟 − 𝐷 ± 1 𝜎 2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝑆𝑡−𝑡 2 0 √ 𝜎 𝑇 −𝑡

.

max max 𝑃𝑓 (𝑆𝑡 , 𝑡; 𝐾max , 𝑇 ) = 𝐶̂𝐹 (𝑆𝑡 , 𝑡; 𝑆𝑡−𝑡 , 𝑇 ) + 𝑒−𝑟(𝑇 −𝑡) 𝑆𝑡−𝑡 − 𝑆𝑡 . 0

0

7 Volatility Models Within the framework of the Black–Scholes model, one of the incorrect assumptions of the model is that the volatility of the underlying asset or stock price is constant. Empirical studies have found that by equating the theoretical Black–Scholes formula with market-quoted European option prices, different volatility values are obtained for different strikes and option expiries. However, this is due to the inherent property of the geometric Brownian motion model, which tries to fit the log returns of asset prices that do not conform to the normality assumption as observed in the market. In this chapter we will discuss further developments beyond the Black–Scholes framework where the volatility takes centre stage.

7.1 INTRODUCTION In this section we consider different types of volatility which are used by practitioners as a form of risk measurement.

Historical Volatility Historical volatility reflects the past price movements of the underlying asset observed in the market and is also referred to as the asset’s actual, realised or statistical volatility. In general, historical volatility can also be considered as a standard deviation, which is a measure of the amount of variation or dispersion for a specified period of asset prices. Thus, a higher dispersion value implies a higher risk that future asset prices will be further away from the current price. To estimate the volatility parameter of a diffusion model, the most systematic way is to use the maximum-likelihood method. Suppose we have a sequence of random variables 𝑋1 , 𝑋2 , … , 𝑋𝑛 having a joint density 𝑓𝑋1 ,𝑋2 ,…,𝑋𝑛 ( 𝑥1 , 𝑥2 , … , 𝑥𝑛 || 𝜃) which depends on an unknown parameter 𝜃. Here, 𝜃 may be a vector. Given the observed values 𝑋𝑖 = 𝑥𝑖 where 𝑖 = 1, 2, … , 𝑛, the likelihood of 𝜃 as a function of 𝑥1 , 𝑥2 , … , 𝑥𝑛 is defined as 𝓁(𝜃) = 𝑓𝑋1 ,𝑋2 ,…,𝑋𝑛 ( 𝑥1 , 𝑥2 , … , 𝑥𝑛 || 𝜃). The maximum-likelihood estimate (mle) of 𝜃 is that the value of 𝜃 is chosen via optimisation to maximise the joint density 𝑓𝑋1 ,𝑋2 ,…,𝑋𝑛 ( 𝑥1 , 𝑥2 , … , 𝑥𝑛 || 𝜃). If the random variables 𝑋1 , 𝑋2 , … , 𝑋𝑛 are assumed to be independent and identically distributed, their joint density is the product of the marginal densities, and the likelihood becomes 𝓁(𝜃) = 𝑓𝑋1 ( 𝑥1 || 𝜃)𝑓𝑋2 ( 𝑥2 || 𝜃) … 𝑓𝑋𝑛 ( 𝑥𝑛 || 𝜃).

648

7.1 Introduction

In contrast, if 𝑋1 , 𝑋2 , … , 𝑋𝑛 are assumed to follow a Markov process, the joint density can be represented as a product of conditional densities 𝓁(𝜃) = 𝑓𝑋1 ( 𝑥1 || 𝜃)𝑓𝑋2 |𝑋1 ( 𝑥2 || 𝑥1 , 𝜃) … 𝑓𝑋𝑛 |𝑋𝑛−1 ( 𝑥𝑛 || 𝑥𝑛−1 , 𝜃). For certain probability density functions, rather than maximising the likelihood function, it is more efficient to maximise the natural logarithm (since the logarithm is a monotone function). Hence, for an independent and identically distributed sample, the log-likelihood in terms of marginal densities is log 𝓁(𝜃) = log 𝑓𝑋1 ( 𝑥1 || 𝜃) + log 𝑓𝑋2 ( 𝑥2 || 𝜃) + … + log 𝑓𝑋𝑛 ( 𝑥𝑛 || 𝜃) whilst for the conditional densities log 𝓁(𝜃) = log 𝑓𝑋1 ( 𝑥1 || 𝜃) + log 𝑓𝑋2 |𝑋1 ( 𝑥2 || 𝑥1 , 𝜃) + … + log 𝑓𝑋𝑛 |𝑋𝑛−1 ( 𝑥𝑛 || 𝑥𝑛−1 , 𝜃). ̂ and is In statistics, the maximum-likelihood estimate for a parameter 𝜃 is denoted by 𝜃, obtained by solving the first-order necessary condition of a local maximiser 𝜕 log 𝓁(𝜃) || 𝜕𝓁(𝜃) || = 0 or | | ̂ = 0. 𝜕𝜃 |𝜃=𝜃̂ 𝜕𝜃 |𝜃=𝜃 Implied Volatility Recall that the Black–Scholes formula for a European option price 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) at time 𝑡, written on asset price 𝑆𝑡 with strike price 𝐾 and expiry 𝑇 > 𝑡, is ⎧ 𝑆 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑 ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑 ) + − ⎪ 𝑡 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ ⎪ 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑− ) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑+ ) ⎩

for call option for put option

with 𝑑± =

𝑥 log(𝑆𝑡 ∕𝐾) + (𝑟 + 𝐷 ± 12 𝜎 2 (𝑇 − 𝑡)) 1 − 1 𝑢2 , Φ(𝑥) = √ 𝑒 2 𝑑𝑢 √ ∫−∞ 2𝜋 𝜎 𝑇 −𝑡

where 𝑟 is the risk-free interest rate, 𝐷 is the continuous dividend yield on the underlying asset and 𝜎 is the asset price volatility which is the only parameter that is not directly observed. In the simplest terms, the volatility parameter can be estimated from historical price data on the asset price 𝑆𝑡 , and with the calibrated value it can then be used in the Black–Scholes formula to value any contingent claim written on 𝑆𝑡 . In addition, all other options can also be priced with the same estimated volatility irrespective of the strike price 𝐾 and expiry time 𝑇 . Although this approach is feasible and easy to implement, there is no single calculation for historical volatility. For example, the choice of the number of historical days for the volatility estimation will have a direct impact on the calculation and hence, the estimated volatility might be unreliable.

7.1 Introduction

649

Instead of relying on historical price movements of the asset price, an alternate volatility estimation can be calculated involving the Black–Scholes theoretical formula. As the volatility 𝜎 is the only unobservable parameter, and given that the vega of the Black–Scholes formula of either a call or put option is positive, 𝜕𝑉𝑏𝑠 >0 𝜕𝜎 for all 𝜎, therefore if we know the market price of the European option 𝑉mkt written on asset price 𝑆𝑡 and traded at time 𝑡 with strike 𝐾 and expiry 𝑇 , then there exists a unique 𝜎 = 𝜎imp known as the implied volatility such that 𝑉𝑏𝑠 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎imp ) = 𝑉mkt provided 𝑉mkt lies between

𝑉mkt

⎧𝑆 ⎪ 𝑡 ≤⎨ ⎪𝐾 ⎩

for call option for put option

and

𝑉mkt

⎧ max {𝑆 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) , 0} for call option 𝑡 ⎪ ≥⎨ } { ⎪ max 𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) , 0 for put option. ⎩

According to the Black–Scholes assumptions, the implied volatility calculated at time 𝑡 should be a constant value or time dependent across all strike prices 𝐾 and option expiries 𝑇 . However, in reality, it shows variations with both strike (known as volatility smile or volatility skew) and expiry 𝑇 (known as term structure of volatility). Given that the implied volatility 𝜎imp (𝐾, 𝑇 ) for a certain strike 𝐾 and expiry 𝑇 has a fixed value, the relation 𝜎imp (𝐾, 𝑇 ) is called an implied volatility surface. Local Volatility As the calculated implied volatility is not a constant value by varying the strike and option expiry, the most straightforward strategy is to modify the Black–Scholes model by assuming the asset price volatility 𝜎(𝑆𝑡 , 𝑡) is a function of both the underlying asset price and time. By doing so, not only will it be able to accommodate the implied volatility for discrete strike prices and expiry times, but it will also allow us to price options consistently at any moment in time. In a local volatility model, the underlying asset price 𝑆𝑡 follows a modified SDE of the form 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎(𝑆𝑡 , 𝑡)𝑑𝑊𝑡 𝑆𝑡

650

7.1 Introduction

where 𝑊𝑡 is the standard Wiener process, 𝜇 is the drift, 𝐷 is the continuous dividend yield and 𝜎(𝑆𝑡 , 𝑡) is the local volatility function depending on the asset price 𝑆𝑡 and time 𝑡. By substituting the constant or time-dependent volatility with 𝜎(𝑆𝑡 , 𝑡), the Black–Scholes PDE satisfied by a European-style option 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) with strike 𝐾, expiry 𝑇 > 𝑡 and having a payoff Ψ(𝑆𝑇 ), 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎(𝑆𝑡 , 𝑡)2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 𝑉 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = Ψ(𝑆𝑇 ) still remains valid and the option price at time 𝑡, [ ] 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ Ψ(𝑆𝑇 )|| ℱ𝑡 as the present value of the expected payoff under the risk-neutral measure ℚ remains the same. However, given that the volatility is now a function of the underlying asset price and time, the Black–Scholes analytical solution is not extended for this model. Given the current asset price 𝑆𝑡 , the local volatility function 𝜎(𝑆𝑡 , 𝑡) can be obtained uniquely from market quotes of European call option prices 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) (or put option prices 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 )) across all strikes 𝐾 and expiry times 𝑇 through the Dupire formula √( ) ⎧√ 𝜕𝐶 𝜕𝐶 ⎪√ + (𝑟 − 𝐷)𝐾 + 𝐷𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) √ 𝜕𝐾 ⎪√ √ 𝜕𝑇 ⎪√ 𝜕2𝐶 𝐾2 ⎪ 𝜕𝐾 2 ⎪ 𝜎(𝐾, 𝑇 ) = ⎨ √ ( ) ⎪√ 𝜕𝑃 𝜕𝑃 + (𝑟 − 𝐷)𝐾 + 𝐷𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ⎪√ √ 𝜕𝐾 ⎪√ √ 𝜕𝑇 ⎪√ 𝜕2𝑃 𝐾2 ⎪ 𝜕𝐾 2 ⎩

for European call option

for European put option.

If we have a sequence of market-quoted European call (or put) option prices 𝐶mkt (𝑆𝑡 , 𝑡; 𝐾𝑖 , 𝑇𝑗 ) (or 𝑃mkt (𝑆𝑡 , 𝑡; 𝐾𝑖 , 𝑇𝑗 )) of different strikes 𝐾𝑖 , 𝑖 = 1, 2, … , 𝑁 and expiries 𝑇𝑗 , 𝑗 = 1, 2, … , 𝑀, we can first interpolate and extrapolate these prices to produce a smooth surface. Assuming that the surface is twice continuously differentiable in 𝐾 and 𝑇 , we can then subsequently utilise the Dupire formula to determine the local volatility for any arbitrary strikes and expiries. In addition, the local volatility function can also be expressed in terms of the implied volatility 𝜎imp , given as √ ( ) √ 𝜕𝜎imp 𝜕𝜎imp 1 𝜎imp √ √ 2 + (𝑟 − 𝐷)𝐾 + √ 𝜕𝑇 𝜕𝐾 2𝑇 −𝑡 √ 𝜎(𝐾, 𝑇 ) = √ ) √ imp imp imp ( 2 2 𝜕𝜎imp 2 𝜕 𝜎imp 𝜕𝜎imp 𝐾 𝑑+ 𝑑− √ 2𝐾𝑑+ 1 √ 𝐾2 + + + √ √ 𝜕𝐾 𝜎 𝜕𝐾 𝜕𝐾 2 imp 𝜎imp 𝑇 − 𝑡 𝜎imp 𝑇 − 𝑡

7.1 Introduction

651

such that

imp

𝑑±

=

2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎imp √ 𝜎imp 𝑇 − 𝑡

where, analogous to the Dupire formula, an implied volatility surface is first produced using the market-quoted European call (or put) option prices for a range of strikes and expiries before utilising the above formula to calculate the local volatility function for any arbitrary strikes and expiries.

Stochastic Volatility As observed in the market, the distribution of the returns of the asset prices shows that it is highly peaked and fat-tailed, which deviates from the assumption of normality. To resolve this shortcoming of the Black–Scholes model, we can either choose a jump-diffusion model or we can assume that the volatility of the underlying asset price is a stochastic process (i.e., a stochastic volatility model). When modelling the asset price as a jump-diffusion model, we assume that the prices do not move continuously in time and we can then let the model make discrete jumps based on a Poisson process. In contrast, when modelling the asset price volatility as a stochastic process we assume that the volatility is a continuous random variable in order to describe the fat-tailed distribution of the asset price returns. In a general stochastic volatility model, the underlying asset price return and its instantaneous volatility have the following diffusion processes 𝑑𝑆𝑡 = 𝜇(𝑆𝑡 , 𝑌𝑡 , 𝑡)𝑑𝑡 + 𝜎(𝑌𝑡 , 𝑡)𝑑𝑊𝑡𝑆 𝑆𝑡

𝑑𝑌𝑡 = 𝛼(𝑆𝑡 , 𝑌𝑡 , 𝑡)𝑑𝑡 + 𝛽(𝑆𝑡 , 𝑌𝑡 , 𝑡)𝑑𝑊𝑡𝑌

𝑑𝑊𝑡𝑆 ⋅ 𝑑𝑊𝑡𝑌 = 𝜌𝑑𝑡

where {𝑊𝑡𝑆 : 𝑡 ≥ 0} and {𝑊𝑡𝑌 : 𝑡 ≥ 0} are two correlated standard Wiener processes under the physical measure ℙ with correlation value 𝜌 ∈ (−1, 1), 𝑆𝑡 is the underlying asset price, 𝜇(𝑆𝑡 , 𝑌𝑡 , 𝑡) is the asset price drift and 𝜎(𝑌𝑡 , 𝑡) is the asset instantaneous volatility, which is assumed to be a stochastic process having drift 𝛼(𝑆𝑡 , 𝑌𝑡 , 𝑡) and volatility (known as vol-of-vol) 𝛽(𝑆𝑡 , 𝑌𝑡 , 𝑡). Like the jump-diffusion model, the market we have described above is incomplete since there is only one traded asset and two sources of uncertainty (Wiener processes 𝑊𝑡𝑆 and 𝑊𝑡𝑌 ). In addition, the modelling of the volatility as a stochastic process is more challenging, since it is a hidden process and is not directly observable. Thus, the calibration of the model parameters against existing market data is harder as it contains more parameters to estimate than a simple Black–Scholes model.

652

7.2.1 Historical and Implied Volatility

7.2 PROBLEMS AND SOLUTIONS 7.2.1

Historical and Implied Volatility

1. Estimation of Geometric Brownian Motion Parameters. Let 𝑆𝑡 , 𝑆𝑡+Δ𝑡 , 𝑆𝑡+2Δ𝑡 , …, 𝑆𝑡+𝑁Δ𝑡 with Δ𝑡 = (𝑇 − 𝑡)∕𝑁 be a sequence of discrete values observed at regular time intervals Δ𝑡 > 0 which are assumed to follow a GBM process 𝑑𝑆𝑡 = 𝜇𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 where the drift 𝜇 and volatility 𝜎 are constant parameters and 𝑊𝑡 is a standard Wiener process on the probability space (Ω, ℱ, ℙ). Using maximum-likelihood estimation or otherwise, calculate the historical estimates of 𝜇 and 𝜎. Solution: From It¯o’s formula ( ) 𝑑𝑆𝑡 1 𝑑𝑆𝑡 2 − +… 𝑆𝑡 2 𝑆𝑡 ) ( 1 = 𝜇 − 𝜎 2 𝑑𝑡 + 𝜎𝑑𝑊𝑡 2

𝑑 log 𝑆𝑡 =

and taking integrals ) 𝑡+Δ𝑡 ( 1 𝜇 − 𝜎 2 𝑑𝑢 + 𝑑 log 𝑆𝑢 = ∫𝑡 ∫ ∫𝑡 2 ( ) (𝑡 ) 𝑆𝑡+Δ𝑡 1 log = 𝜇 − 𝜎 2 Δ𝑡 + 𝜎𝑊Δ𝑡 𝑆𝑡 2 𝑡+Δ𝑡

where 𝑊Δ𝑡 ∼  (0, Δ𝑡). Therefore, log

(

𝑆𝑡+Δ𝑡 𝑆𝑡

) ∼

[(

𝑡+Δ𝑡

𝜎𝑑𝑊𝑢

) ] 1 𝜇 − 𝜎 2 Δ𝑡, 𝜎 2 Δ𝑡 . 2

Maximum-Likelihood Estimation Method Using method, function is the joint density of ) ( ) ( the likelihood ) ( the maximum-likelihood 𝑆𝑡+2Δ𝑡 𝑆𝑡+𝑁Δ𝑡 𝑆𝑡+Δ𝑡 , log , … , log which is a product of their marginal log 𝑆𝑡 𝑆𝑡+Δ𝑡 𝑆𝑡+(𝑁−1)Δ𝑡 densities ( ) ( 2 ) ⎡ 𝑆𝑡+𝑖Δ𝑡 ⎛ ⎞ ⎤ 1 2 ⎢ Δ𝑡 log 𝜎 − 𝜇 − 𝑁 ⎟ ⎥⎥ 2 ∏ 𝑆𝑡+(𝑖−1)Δ𝑡 ⎢ 1 ⎜⎜ 1 ⎟ ⎥ exp ⎢− 𝓁(𝜇, 𝜎) = √ √ ⎜ ⎟ ⎥ 2 𝜎 Δ𝑡 𝜎 2𝜋Δ𝑡 ⎢ 𝑖=1 ⎜ ⎟ ⎥ ⎢ ⎝ ⎠ ⎦ ⎣ 2 ) ( ⎞ ⎤ ⎡ ⎛ 1 2 𝑁 𝑅 Δ𝑡 − 𝜇 − 𝜎 ∏ ⎟ ⎥ ⎢ 1⎜ 𝑖 2 1 = exp ⎢− ⎜ √ √ ⎟ ⎥ 𝜎 Δ𝑡 ⎟ ⎥ ⎢ 2⎜ 𝑖=1 𝜎 2𝜋Δ𝑡 ⎣ ⎠ ⎦ ⎝

7.2.1 Historical and Implied Volatility

( where 𝑅𝑖 = log

𝑆𝑡+𝑖Δ𝑡

653

) , 𝑖 = 1, 2, … , 𝑁.

𝑆𝑡+(𝑖−1)Δ𝑡 Taking the log-likelihood

log 𝓁(𝜇, 𝜎) = −𝑁 log 𝜎 −

𝑁 ( ) )2 ( 1 1 ∑ 𝑁 𝑅𝑖 − 𝜇 − 𝜎 2 Δ𝑡 log(2𝜋Δ𝑡) − 2 2 2 2𝜎 Δ𝑡 𝑖=1

and partial differentials with respect to 𝜇 and 𝜎, 𝑁 ( ) ) ( 𝜕 log 𝓁 1 1 ∑ 𝑅𝑖 − 𝜇 − 𝜎 2 Δ𝑡 = 2 𝜕𝜇 2 𝜎 𝑖=1 𝑁 ( 𝑁 ( ) )2 ) ) ( ( 𝜕 log 𝓁 1 1∑ 1 𝑁 1 ∑ 𝑅𝑖 − 𝜇 − 𝜎 2 Δ𝑡 − 𝑅𝑖 − 𝜇 − 𝜎 2 Δ𝑡 . =− + 3 𝜕𝜎 𝜎 2 𝜎 𝑖=1 2 𝜎 Δ𝑡 𝑖=1

By setting satisfy

𝜕 log 𝓁 𝜕 log 𝓁 = 0 and = 0, the maximum-likelihood estimates 𝜇̂ and 𝜎̂ 𝜕𝜇 𝜕𝜎 (

) 1 𝜇̂ − 𝜎̂ 2 Δ𝑡 = 𝑅̄ 2

and

−

𝑁 𝑁 ( )2 1 ∑ ) 𝑁 1 ∑( 𝑅𝑖 − 𝑅̄ − 𝑅𝑖 − 𝑅̄ = 0 + 3 𝜎̂ 𝜎 ̂ 𝜎̂ Δ𝑡 𝑖=1 𝑖=1

𝑁

1 ∑ 𝑅. 𝑁 𝑖=1 𝑖 Thus, we have

where 𝑅̄ =

( 𝜇̂ =

1 Δ𝑡

𝑁 1 ∑ ̄ 2 (𝑅 − 𝑅) 𝑅̄ + 2𝑁 𝑖=1 𝑖

) and

√ √ 𝑁 √ 1 ∑ ( )2 𝑅𝑖 − 𝑅̄ . 𝜎̂ = √ 𝑁Δ𝑡 𝑖=1

Moment-Matching Estimation Method Since [

)| ] ( ) 1 | log 𝔼 | ℱ𝑡 = 𝜇 − 𝜎 2 Δ𝑡 | 2 | [ ( )| ] 𝑆𝑡+Δ𝑡 | Varℚ log | ℱ𝑡 = 𝜎 2 Δ𝑡 | 𝑆𝑡 | ℚ

(

𝑆𝑡+Δ𝑡 𝑆𝑡

654

7.2.1 Historical and Implied Volatility

and by setting the logarithm return ( 𝑅𝑖 = log

𝑆𝑡+𝑖Δ𝑡

)

𝑆𝑡+(𝑖−1)Δ𝑡

where 𝑅𝑖 = 1, 2, … , 𝑁 are independent and identically distributed, the estimates of 𝜇̂ and 𝜎̂ are given by (

) 1 𝜇̂ − 𝜎̂ 2 Δ𝑡 = 𝑅̄ 2

and 𝑁

𝜎̂ 2 Δ𝑡 =

∑( )2 1 𝑅𝑖 − 𝑅̄ (𝑁 − 1)Δ𝑡 𝑖=1

𝑁 1 ∑ ̄ where 𝑅 = 𝑅 . Thus, we have 𝑁 𝑖=1 𝑖

( 𝜇̂ =

1 Δ𝑡

𝑅̄ +

1 2(𝑁 − 1)

𝑁 ∑ 𝑖=1

)

̄ 2 (𝑅𝑖 − 𝑅)

and

√ √ √ 𝜎̂ = √

𝑁 ∑ ( )2 1 𝑅𝑖 − 𝑅̄ . (𝑁 − 1)Δ𝑡 𝑖=1

2. Estimation of Ornstein–Uhlenbeck Process Parameters. Let 𝑆𝑡 , 𝑆𝑡+Δ𝑡 , 𝑆𝑡+2Δ𝑡 , …, 𝑆𝑡+𝑁Δ𝑡 with Δ𝑡 = (𝑇 − 𝑡)∕𝑁 be a sequence of discrete values observed at regular time intervals Δ𝑡 > 0 which are assumed to follow an Ornstein–Uhlenbeck process 𝑑𝑆𝑡 = 𝜅(𝜃 − 𝑆𝑡 )𝑑𝑡 + 𝜎𝑑𝑊𝑡 where the mean-reversion rate 𝜅, long-term mean 𝜃 and volatility 𝜎 are constant parameters and 𝑊𝑡 is a standard Wiener process on the probability space (Ω, ℱ, ℙ). Using maximum-likelihood estimation or otherwise, calculate the historical estimates of 𝜅, 𝜃 and 𝜎. Solution: From the Ornstein–Uhlenbeck process 𝑑𝑆𝑡 = 𝜅(𝜃 − 𝑆𝑡 )𝑑𝑡 + 𝜎𝑑𝑊𝑡 and by applying It¯o’s formula on 𝑒𝜅𝑡 𝑆𝑡 we have 1 𝑑(𝑒𝜅𝑡 𝑆𝑡 ) = 𝜅 𝑒𝜅𝑡 𝑆𝑡 𝑑𝑡 + 𝑒𝜅𝑡 𝑑𝑆𝑡 + 𝜅 2 𝑒𝜅𝑡 𝑆𝑡 (𝑑𝑡)2 + … 2 ( ) = 𝜅 𝑒𝜅𝑡 𝑆𝑡 𝑑𝑡 + 𝑒𝜅𝑡 𝜅(𝜃 − 𝑆𝑡 )𝑑𝑡 + 𝜎𝑑𝑊𝑡 = 𝜅𝜃𝑒𝜅𝑡 𝑑𝑡 + 𝜎𝑒𝜅𝑡 𝑑𝑊𝑡 .

7.2.1 Historical and Implied Volatility

655

Integrating the above expression, 𝑡+Δ𝑡

∫𝑡

𝑡+Δ𝑡

𝑑(𝑒𝜅𝑢 𝑆𝑢 ) =

∫𝑡

𝜅𝜃𝑒𝜅𝑢 𝑑𝑢 +

𝑡+Δ𝑡

∫𝑡

𝜎𝑒𝜅𝑢 𝑑𝑊𝑢

) ( 𝑆𝑡+Δ𝑡 = 𝑆𝑡 𝑒−𝜅Δ𝑡 + 𝜃 1 − 𝑒−𝜅Δ𝑡 +

𝑡+Δ𝑡

∫𝑡

𝜎𝑒−𝜅(𝑡+Δ𝑡−𝑢) 𝑑𝑊𝑢 .

Given that both {𝑊𝑡 : 𝑡 ≥ 0} and {𝑊𝑡2 − 𝑡 : 𝑡 ≥ 0} are ℙ-martingales, we have [ 𝔼

𝑡+Δ𝑡

∫𝑡

| | 𝜎𝑒−𝜅(𝑡+Δ𝑡−𝑢) 𝑑𝑊𝑢 | ℱ𝑡 | |

] =0

and [( 𝔼

𝑡+Δ𝑡

∫𝑡

𝜎𝑒

−𝜅(𝑡+Δ𝑡−𝑢)

[ ] )2 | ] 𝑡+Δ𝑡 | | 2 −2𝜅(𝑡+Δ𝑡−𝑢) | | 𝑑𝑊𝑢 | ℱ𝑡 = 𝔼 𝜎 𝑒 𝑑𝑢| ℱ𝑡 | ∫𝑡 | | | 2 ( ) 𝜎 1 − 𝑒−2𝜅Δ𝑡 = 2𝜅

and following the arguments given in Problem 2.2.2.31 (page 161) we can deduce 𝑆𝑡+Δ𝑡 || 𝑆𝑡 ∼ 

( 𝑆𝑡 𝑒

−𝜅Δ𝑡

) ) 𝜎2 ( ) ( −𝜅Δ𝑡 −2𝜅Δ𝑡 1−𝑒 , +𝜃 1−𝑒 . 2𝜅

To calculate the estimates of 𝜅, 𝜃 and 𝜎, we consider two approaches. Maximum-Likelihood Estimation Method ) 𝜎2 ( 1 − 𝑒−2𝜅Δ𝑡 , the likelihood function is the joint density of 𝑆𝑡 , 𝑆𝑡+Δ𝑡 , By setting 𝑠2 = 2𝜅 …, 𝑆𝑡+𝑁Δ𝑡 such that

𝓁(𝜅, 𝜃, 𝑠) =

𝑁 ∏ 𝑖=1

⎡ 1 1 √ exp ⎢− ⎢ 2 𝑠 2𝜋 ⎣

(

𝑆𝑡+𝑖Δ𝑡 − 𝑆𝑡+(𝑖−1)Δ𝑡 𝑒−𝜅Δ𝑡 − 𝜃(1 − 𝑒−𝜅Δ𝑡 )

)2

𝑠

and taking the log-likelihood log 𝓁(𝜅, 𝜃, 𝑠) = −𝑁 log 𝑠 − −

𝑁 log(2𝜋) 2

𝑁 )]2 ( 1 ∑[ 𝑆𝑡+𝑖Δ𝑡 − 𝑆𝑡+(𝑖−1)Δ𝑡 𝑒−𝜅Δ𝑡 − 𝜃 1 − 𝑒−𝜅Δ𝑡 . 2 2𝑠 𝑖=1

⎤ ⎥ ⎥ ⎦

656

7.2.1 Historical and Implied Volatility

Taking partial differentials with respect to 𝜅, 𝜃 and 𝑠, 𝑁 )] [ ( ] 𝜕 log 𝓁 1 ∑[ 𝑆𝑡+𝑖Δ𝑡 − 𝑆𝑡+(𝑖−1)Δ𝑡 𝑒−𝜅Δ𝑡 − 𝜃 1 − 𝑒−𝜅Δ𝑡 𝑆𝑡+(𝑖−1)Δ𝑡 − 𝜃 Δ𝑡𝑒−𝜅Δ𝑡 =− 2 𝜕𝜅 𝑠 𝑖=1

=−

𝑁 [ )( ) ( )2 −𝜅Δ𝑡 ] Δ𝑡𝑒−𝜅Δ𝑡 ∑ ( 𝑆 − 𝜃 𝑆 − 𝜃 − 𝑆 − 𝜃 𝑒 𝑡+𝑖Δ𝑡 𝑡+(𝑖−1)Δ𝑡 𝑡+(𝑖−1)Δ𝑡 𝑠2 𝑖=1

𝑁 )] ( 𝜕 log 𝓁 1 − 𝑒−𝜅Δ𝑡 ∑ [ 𝑆𝑡+𝑖Δ𝑡 − 𝑆𝑡+(𝑖−1)Δ𝑡 𝑒−𝜅Δ𝑡 − 𝜃 1 − 𝑒−𝜅Δ𝑡 = 2 𝜕𝜃 𝑠 𝑖=1 𝑁 )]2 ( 𝜕 log 𝓁 𝑁 1 ∑[ 𝑆𝑡+𝑖Δ𝑡 − 𝑆𝑡+(𝑖−1)Δ𝑡 𝑒−𝜅Δ𝑡 − 𝜃 1 − 𝑒−𝜅Δ𝑡 . =− + 3 𝜕𝑠 𝑠 𝑠 𝑖=1

𝜕 log 𝓁 𝜕 log 𝓁 𝜕 log 𝓁 = 0, = 0 and = 0, the maximum-likelihood estimates 𝜕𝜅 𝜕𝜃 𝜕𝑠 𝜅, ̂ 𝜃̂ and 𝑠̂ are By setting

[ ∑𝑁 ] ̂ ̂ 1 𝑖=1 (𝑆𝑡+𝑖Δ𝑡 − 𝜃)(𝑆𝑡+(𝑖−1)Δ𝑡 − 𝜃) 𝜅̂ = − log ∑𝑁 Δ𝑡 ̂2 𝑖=1 (𝑆𝑡+(𝑖−1)Δ𝑡 − 𝜃) ∑𝑁 −𝜅Δ𝑡 ̂ ) 𝑖=1 (𝑆𝑡+𝑖Δ𝑡 − 𝑆𝑡+(𝑖−1)Δ𝑡 𝑒 𝜃̂ = ̂ ) 𝑁(1 − 𝑒−𝜅Δ𝑡 𝑠̂2 =

𝑁 ]2 1 ∑[ ̂ ̂ −𝜅Δ𝑡 𝑆𝑡+𝑖Δ𝑡 − 𝜃̂ − (𝑆𝑡+(𝑖−1)Δ𝑡 − 𝜃)𝑒 . 𝑁 𝑖=1

Letting 𝑆𝑥 =

𝑆𝑥𝑥 =

𝑁 ∑ 𝑖=1

2 𝑆𝑡+(𝑖−1)Δ𝑡 ,

𝑁 ∑ 𝑖=1

𝑆𝑡+(𝑖−1)Δ𝑡 , 𝑆𝑦 =

𝑆𝑦𝑦 =

𝑁 ∑ 𝑖=1

2 𝑆𝑡+𝑖Δ𝑡 ,

𝑁 ∑ 𝑖=1

𝑆𝑡+𝑖Δ𝑡

𝑆𝑥𝑦 =

𝑁 ∑ 𝑖=1

𝑆𝑡+(𝑖−1)Δ𝑡 𝑆𝑡+𝑖Δ𝑡

we have ] [ ̂ 𝑥 + 𝑆𝑦 ) + 𝑁 𝜃̂ 2 𝑆𝑥𝑦 − 𝜃(𝑆 1 𝜅̂ = − log ̂ 𝑥 + 𝑁 𝜃̂ 2 Δ𝑡 𝑆𝑥𝑥 − 2𝜃𝑆 𝜃̂ =

̂ 𝑆𝑦 − 𝑆𝑥 𝑒−𝜅Δ𝑡

̂ ) 𝑁(1 − 𝑒−𝜅Δ𝑡 [ 1 ̂ ̂ ̂ ̂ ̂ 𝑦 − 𝑆𝑥 𝑒−𝜅Δ𝑡 𝑆 − 2𝑆𝑥𝑦 𝑒−𝜅Δ𝑡 + 𝑆𝑥𝑥 𝑒−2𝜅Δ𝑡 − 2𝜃(𝑆 )(1 − 𝑒−𝜅Δ𝑡 ) 𝑠̂2 = 𝑁 𝑦𝑦 ] ̂ )2 . +𝑁 𝜃̂ 2 (1 − 𝑒−𝜅Δ𝑡

7.2.1 Historical and Implied Volatility

657

̂ Substituting 𝜅̂ into 𝜃, 𝑁 𝜃̂ = =

̂ 𝑥 − 𝜃𝑆 ̂ 𝑦 + 𝑁 𝜃̂ 2 ) 𝑆𝑦 (𝑆𝑥𝑥 − 2𝜃̂ + 𝑁 𝜃̂ 2 ) − 𝑆𝑥 (𝑆𝑥𝑦 − 𝜃𝑆 ̂ 𝑦 − 𝑆𝑥 ) 𝑆𝑥𝑥 − 𝑆𝑥𝑦 + 𝜃(𝑆 ̂ 2 − 𝑆𝑥 𝑆𝑦 ) + 𝑁 𝜃̂ 2 (𝑆𝑦 − 𝑆𝑥 ) 𝑆𝑦 𝑆𝑥𝑥 − 𝑆𝑥 𝑆𝑥𝑦 + 𝜃(𝑆 𝑥

̂ 𝑦 − 𝑆𝑥 ) 𝑆𝑥𝑥 − 𝑆𝑥𝑦 + 𝜃(𝑆

or ̂ 2 − 𝑆𝑥 𝑆𝑦 ) + 𝑁 𝜃̂ 2 (𝑆𝑦 − 𝑆𝑥 ) ̂ 𝑥𝑥 − 𝑆𝑥𝑦 ) + 𝑁 𝜃̂ 2 (𝑆𝑦 − 𝑆𝑥 ) = 𝑆𝑦 𝑆𝑥𝑥 − 𝑆𝑥 𝑆𝑥𝑦 + 𝜃(𝑆 𝑁 𝜃(𝑆 𝑥 and hence 𝜃̂ =

𝑆𝑦 𝑆𝑥𝑥 − 𝑆𝑥 𝑆𝑥𝑦 𝑁(𝑆𝑥𝑥 − 𝑆𝑥𝑦 ) − 𝑆𝑥2 + 𝑆𝑥 𝑆𝑦

with the maximum-likelihood estimates of mean-reversion rate 𝜅̂ and volatility 𝜎̂ being [ ] ̂ 𝑥 + 𝑆𝑦 ) + 𝑁 𝜃̂ 2 𝑆𝑥𝑦 − 𝜃(𝑆 1 𝜅̂ = − log ̂ 𝑥 + 𝑁 𝜃̂ 2 Δ𝑡 𝑆𝑥𝑥 − 2𝜃𝑆

√ and

𝜎̂ =

2𝜅̂ 𝑠̂ ̂ 1 − 𝑒−2𝜅Δ𝑡

respectively. Ordinary Least-Squares Method We let the relationship between consecutive 𝑆𝑡 , 𝑆𝑡+Δ𝑡 , … , 𝑆𝑡+𝑁Δ𝑡 be 𝑆𝑡+Δ𝑡 = 𝑚𝑆𝑡 + 𝑐 + 𝜖𝑡 , 𝜖𝑡 ∼  (0, 𝜎𝜖2 ) where 𝑚 and 𝑐 are the regression parameters, 𝜖𝑡 is normally distributed and is independent and identically distributed. By comparing the relationship between the linear fit and the solution of the Ornstein– Uhlenbeck process model ) ( 𝑆𝑡+Δ𝑡 = 𝑆𝑡 𝑒−𝜅Δ𝑡 + 𝜃 1 − 𝑒−𝜅Δ𝑡 + 𝑡+Δ𝑡

where

( 𝜎𝑒−𝜅(𝑡+Δ𝑡−𝑢) 𝑑𝑊𝑢 ∼ 

∫𝑡 parameters can be equated as log 𝑚 , 𝜅=− Δ𝑡

0,

𝑐 𝜃= 1−𝑚

𝑡+Δ𝑡

∫𝑡

𝜎𝑒−𝜅(𝑡+Δ𝑡−𝑢) 𝑑𝑊𝑢

) ) 𝜎2 ( 1 − 𝑒−2𝜅Δ𝑡 , the Ornstein–Uhlenbeck 2𝜅 √ and

𝜎 = 𝜎𝜖

2𝜅 . 1 − 𝑒−2𝜅Δ𝑡

658

7.2.1 Historical and Implied Volatility

As for estimating the pair (𝑚, 𝑐), we can compute it by solving the following ordinary least-squares problem { OLS

By setting 𝑓 (𝑚, 𝑐) = respect to 𝑚 and 𝑐,

𝑁 ∑ ( )2 𝑆𝑡+𝑖Δ𝑡 − 𝑚𝑆𝑡+(𝑖−1)Δ𝑡 − 𝑐 . minimise 𝑚,𝑐∈ℝ

𝑖=1

𝑁 ∑ ( 𝑖=1

𝑆𝑡+𝑖Δ𝑡 − 𝑚𝑆𝑡+(𝑖−1)Δ𝑡 − 𝑐

)2

and taking partial derivatives with

𝑁

∑( ) 𝜕𝑓 𝑆𝑡+𝑖Δ𝑡 − 𝑚𝑆𝑡+(𝑖−1)Δ𝑡 − 𝑐 𝑆𝑡+(𝑖−1)Δ𝑡 = −2(𝑆𝑥𝑦 − 𝑚𝑆𝑥𝑥 − 𝑐𝑆𝑥 ) = −2 𝜕𝑚 𝑖=1 𝑁

∑( ) 𝜕𝑓 𝑆𝑡+𝑖Δ𝑡 − 𝑚𝑆𝑡+(𝑖−1)Δ𝑡 − 𝑐 = −2(𝑆𝑦 − 𝑚𝑆𝑥 − 𝑐𝑁) = −2 𝜕𝑐 𝑖=1 where 𝑆𝑥 =

𝑆𝑥𝑥 =

𝑁 ∑ 𝑖=1

2 𝑆𝑡+(𝑖−1)Δ𝑡 ,

𝑁 ∑ 𝑖=1

𝑆𝑡+(𝑖−1)Δ𝑡 ,

𝑆𝑦𝑦 =

𝑁 ∑ 𝑖=1

𝑆𝑦 =

2 𝑆𝑡+𝑖Δ𝑡 ,

𝑁 ∑ 𝑖=1

𝑆𝑡+𝑖Δ𝑡

𝑆𝑥𝑦 =

𝑁 ∑ 𝑖=1

𝑆𝑡+(𝑖−1)Δ𝑡 𝑆𝑡+𝑖Δ𝑡 .

By setting the partial derivatives to zero and solving the linear equations simultaneously, the estimates 𝑚̂ and 𝑐̂ that minimise 𝑂𝐿𝑆 are given as 𝑚̂ =

𝑁𝑆𝑥𝑦 − 𝑆𝑥 𝑆𝑦 𝑁𝑆𝑥𝑥 − 𝑆𝑥2

,

𝑐̂ =

̂ 𝑥 𝑆𝑦 − 𝑚𝑆 𝑁

with the unbiased estimated standard error 𝜎̂ 𝜖 given as

𝜎̂ 𝜖 =

√ √ 𝑁 √∑ ( )2 √ √ 𝑆𝑡+𝑖Δ𝑡 − 𝑚𝑆 ̂ 𝑡+(𝑖−1)Δ𝑡 − 𝑐̂ √ √ 𝑖=1 𝑁 −2

.

Take note that the divisor 𝑁 − 2 is used instead of 𝑁 because two parameters 𝑚 and 𝑐 have been estimated, thus giving 𝑁 − 2 degrees of freedom. 3. Estimation of Geometric Mean-Reverting Process Parameters. Let 𝑆𝑡 , 𝑆𝑡+Δ𝑡 , 𝑆𝑡+2Δ𝑡 , …, 𝑆𝑡+𝑁Δ𝑡 with Δ𝑡 = (𝑇 − 𝑡)∕𝑁 be a sequence of discrete values observed at regular time intervals Δ𝑡 > 0 which are assumed to follow a geometric mean-reverting process with

7.2.1 Historical and Implied Volatility

659

the following SDE 𝑑𝑆𝑡 = 𝜅(𝜃 − log 𝑆𝑡 )𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡 ,

𝑆0 > 0

where the mean-reversion rate 𝜅, long-term mean 𝜃 and volatility 𝜎 are constant parameters and 𝑊𝑡 is a standard Wiener process on the probability space (Ω, ℱ, ℙ). Using maximum-likelihood estimation or otherwise, calculate the historical estimates of 𝜅, 𝜃 and 𝜎. Solution: By expanding log 𝑆𝑡 using Taylor’s formula and subsequently applying It¯o’s formula we have )2 1 1 ( 𝑑𝑆𝑡 + … 𝑑𝑆 − 𝑆𝑡 𝑡 2𝑆𝑡2 ) ( 1 = 𝜅 𝜃 − log 𝑆𝑡 𝑑𝑡 + 𝜎𝑑𝑊𝑡 − 𝜎 2 𝑑𝑡 2 ( ( ) 1 2) = 𝜅 𝜃 − log 𝑆𝑡 − 𝜎 𝑑𝑡 + 𝜎𝑑𝑊𝑡 2

𝑑(log 𝑆𝑡 ) =

and setting 𝑋𝑡 = log 𝑆𝑡 we can rewrite the SDE as ) ( 1 𝑑𝑋𝑡 = 𝜅(𝜃 − 𝑋𝑡 ) − 𝜎 2 𝑑𝑡 + 𝜎𝑑𝑊𝑡 2 = 𝜅(𝜗 − 𝑋𝑡 )𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝜎2 where 𝜗 = 𝜃 − . 2𝜅 By letting

𝑆𝑥 = 𝑆𝑦𝑦 =

𝑁 ∑ 𝑖=1 𝑁 ∑ 𝑖=1

log 𝑆𝑡+(𝑖−1)Δ𝑡 , (

log 𝑆𝑡+𝑖Δ𝑡

)2

𝑆𝑦 =

𝑆𝑥𝑦 =

𝑁 ∑ 𝑖=1

log 𝑆𝑡+𝑖Δ𝑡 𝑆𝑥𝑥 =

𝑁 ∑ ( 𝑖=1

log 𝑆𝑡+(𝑖−1)Δ𝑡

)2

,

𝑁 ∑ ( )( ) log 𝑆𝑡+(𝑖−1)Δ𝑡 log 𝑆𝑡+𝑖Δ𝑡 𝑖=1

then from analogy with the Ornstein–Uhlenbeck process estimation of parameters (see Problem 7.2.1.2, page 654), the maximum-likelihood estimates of 𝜗, 𝜅 and 𝜎 are 𝜗̂ =

𝑆𝑦 𝑆𝑥𝑥 − 𝑆𝑥 𝑆𝑥𝑦

𝑁(𝑆𝑥𝑥 − 𝑆𝑥𝑦 ) − 𝑆𝑥2 + 𝑆𝑥 𝑆𝑦 [ ] ̂ 𝑥 + 𝑆𝑦 ) + 𝑁 𝜗̂ 2 𝑆𝑥𝑦 − 𝜗(𝑆 1 𝜅̂ = − log ̂ 𝑥 + 𝑁 𝜗̂ 2 Δ𝑡 𝑆𝑥𝑥 − 2𝜃𝑆 √ 2𝜅̂ 𝑠̂ 𝜎̂ = ̂ 1 − 𝑒−2𝜅Δ𝑡

660

7.2.1 Historical and Implied Volatility

where 𝑠̂2 =

1 [ ̂ ̂ ̂ ̂ ̂ 𝑦 − 𝑆𝑥 𝑒−𝜅Δ𝑡 𝑆 − 2𝑆𝑥𝑦 𝑒−𝜅Δ𝑡 + 𝑆𝑥𝑥 𝑒−2𝜅Δ𝑡 − 2𝜗(𝑆 )(1 − 𝑒−𝜅Δ𝑡 ) 𝑁 𝑦𝑦 ] ̂ )2 . +𝑁 𝜗̂ 2 (1 − 𝑒−𝜅Δ𝑡

Alternatively, using ordinary least squares, the estimates of 𝜗, 𝜅 and 𝜎 are log 𝑚̂ 𝜅̂ = − Δ𝑡

𝑐̂ 𝜗̂ = , 1 − 𝑚̂

√ and

𝜎̂ = 𝜎̂ 𝜖

2𝜅̂ ̂ 1 − 𝑒−2𝜅Δ𝑡

where 𝑁𝑆𝑥𝑦 − 𝑆𝑥 𝑆𝑦

𝑚̂ = 𝑐̂ =

𝜎̂ 𝜖 =

𝑁𝑆𝑥𝑥 − 𝑆𝑥2 ̂ 𝑥 𝑆𝑦 − 𝑚𝑆 𝑁 √ √ 𝑁 √∑ ( )2 √ √ log 𝑆𝑡+𝑖Δ𝑡 − 𝑚̂ log 𝑆𝑡+(𝑖−1)Δ𝑡 − 𝑐̂ √ √ 𝑖=1 𝑁 −2

From the relationship 𝜗 = 𝜃 −

.

𝜎2 , the estimate 𝜃̂ is recovered from 2𝜅 𝜎̂ 2 . 𝜃̂ = 𝜗̂ + 2𝜅̂

4. Generalised Historical Volatility. Let 𝑆𝑡 , 𝑆𝑡+Δ𝑡 , 𝑆𝑡+2Δ𝑡 , … , 𝑆𝑡+𝑁Δ𝑡 with Δ𝑡 = (𝑇 − 𝑡)∕𝑁 be a sequence of discrete asset prices observed at regular time intervals Δ𝑡 > 0. Assume that the asset prices follow a lognormal model with daily closing prices (i.e., Δ𝑡 = 1 trading day) and let 𝜎sd be the standard deviation of logarithm returns of asset prices. Assuming 252 trading days per year, find the estimated annualised, monthly and weekly volatilities. Solution: Following Problem 7.2.1.1 (page 651) using the maximum-likelihood method, the biased standard deviation 𝜎sd of the logarithm returns of asset prices can be estimated as √ √ 𝑁 √1 ∑ ( )2 𝑅𝑖 − 𝑅̄ 𝜎sd = √ 𝑁 𝑖=1 ( where 𝑅𝑖 = log

𝑆𝑡+𝑖Δ𝑡 𝑆𝑡+(𝑖−1)Δ𝑡

)

, 𝑖 = 1, 2, … , 𝑁 and 𝑅̄ =

𝑁 1 ∑ 𝑅. 𝑁 𝑖=1 𝑖

7.2.1 Historical and Implied Volatility

661

For the case of annualised volatility, we set Δ𝑡 =

1 252

and have

√ 𝜎 𝜎 𝜎annualised = √sd = √ sd = 𝜎sd 252. 1 Δ𝑡 252

For the case of monthly volatility, we set Δ𝑡 = 𝜎monthly

12 252

and therefore

𝜎 𝜎 = √sd = √ sd = 𝜎sd 12 Δ𝑡

√

252 . 12

252

Finally, for the case of weekly volatility, we set Δ𝑡 = 𝜎weekly

52 252

𝜎 𝜎 = √sd = √ sd = 𝜎sd 52 Δ𝑡

and hence

√

252 . 52

252

5. Brenner–Subrahmanyam Approximation. Show that the cdf of a standard normal 𝑥

1 2 1 𝑒− 2 𝑢 𝑑𝑢 Φ(𝑥) = √ 2𝜋 ∫−∞

can be expressed as Φ(𝑥) =

( ) 1 1 1 1 1 𝑥− 1 +√ 𝑥3 + 2 𝑥5 − 3 𝑥7 + … . 2 2 ⋅ 1! ⋅ 3 2 ⋅ 2! ⋅ 5 2 ⋅ 3! ⋅ 7 2𝜋

Let the Black–Scholes formula for the value of a European option 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be ⎧ 𝑆 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑 ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑 ) for call option + − ⎪ 𝑡 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ ⎪ 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑− ) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑+ ) for put option ⎩ where log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝑡) 𝑑± = √ 𝜎 𝑇 −𝑡 such that Φ(⋅) is the cumulative distribution of a standard normal, 𝑆𝑡 is the spot price at time 𝑡 < 𝑇 , 𝑇 is the option expiry time, 𝐾 is the strike price, 𝑟 is the risk-free interest rate, 𝐷 is the continuous dividend yield and 𝜎 is the volatility.

662

7.2.1 Historical and Implied Volatility

By defining ATM as 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) = 𝐾𝑒−𝑟(𝑇 −𝑡) (i.e., continuously paid dividend stock price equal to discounted strike price) and taking a linear approximation of the cumuatm of an ATM lative distribution of a standard normal, show that the implied volatility 𝜎imp option in a forward sense (𝐹 (𝑡, 𝑇 ) = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) = 𝐾) satisfying atm 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎imp ) = 𝑉mkt

where 𝑉mkt is the market-observed ATM European option price can be approximated as atm 𝜎imp

≈

√

𝑉mkt 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡)

2𝜋 . 𝑇 −𝑡

1 2

Solution: Given that 𝑒− 2 𝑥 is an even function 𝑥

1 2 1 𝑒− 2 𝑢 𝑑𝑢 Φ(𝑥) = √ 2𝜋 ∫−∞ 𝑥 1 2 1 1 𝑒− 2 𝑢 𝑑𝑢 = +√ ∫ 2 2𝜋 0 1 2

and using Taylor’s expansion of 𝑓 (𝑢) = 𝑒− 2 𝑢 about 𝑢 = 0 so that 𝑓 (𝑢) = 𝑓 (0) + 𝑓 ′ (0)𝑢 +

1 ′′ 1 𝑓 (0)𝑢2 + 𝑓 ′′′ (0)𝑢3 + … 2! 3!

we have 𝑓 ′ (𝑢) = −𝑢𝑓 (𝑢),

𝑓 ′′ (𝑢) = (𝑢2 − 1)𝑓 (𝑢),

𝑓 (4) (𝑢) = (𝑢4 − 6𝑢2 + 3)𝑓 (𝑢), 𝑓 (6) (𝑢) = (𝑢6 − 15𝑢4 + 45𝑢2 + 15)𝑓 (𝑢),

𝑓 ′′′ (𝑢) = (−𝑢3 + 3𝑢)𝑓 (𝑢)

𝑓 (5) (𝑢) = (−𝑢5 + 10𝑢3 − 15𝑢)𝑓 (𝑢) 𝑓 (7) (𝑢) = (−𝑢7 + 21𝑢5 − 105𝑢3 + 75𝑢)𝑓 (𝑢) ⋮

Hence, 𝑥

∫0

1 2

𝑥(

) 1 2 3 4 15 6 𝑢 + 𝑢 − 𝑢 + … 𝑑𝑢 ∫0 2! 4! 6! ) 𝑥( 1 1 1 1 − 𝑢2 + 2 = 𝑢4 − 3 𝑢6 + … 𝑑𝑢 ∫0 2! 2 ⋅ 2! 2 ⋅ 3! 1 3 1 1 5 = 𝑥− 𝑥 + 2 𝑥 − 3 𝑥7 + … 2! ⋅ 3 2 ⋅ 2! ⋅ 5 2 ⋅ 3! ⋅ 7

𝑒− 2 𝑢 𝑑𝑢 =

1−

7.2.1 Historical and Implied Volatility

663

and therefore Φ(𝑥) =

( ) 1 1 1 1 1 𝑥− 1 +√ 𝑥3 + 2 𝑥5 − 3 𝑥7 + … . 2 2 ⋅ 1! ⋅ 3 2 ⋅ 2! ⋅ 5 2 ⋅ 3! ⋅ 7 2𝜋

Setting 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) = 𝐾𝑒−𝑟(𝑇 −𝑡) so that 1 √ 𝑑± = ± 𝜎 𝑇 − 𝑡 2 and taking a linear approximation of Φ(𝑑± ), 1 1 Φ(𝑑± ) ≈ ± 𝜎 2 2

√

𝑇 −𝑡 . 2𝜋

By equating the Black–Scholes theoretical price with the European option market price 𝑉mkt = 𝛿𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝛿𝑑+ ) − 𝛿𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝛿𝑑− ) 1 1 atm where 𝛿 ∈ {−1, 1} and substituting Φ(𝑑± ) ≈ ± 𝜎imp 2 2 (

𝑉mkt

atm √

𝛿 𝛿𝜎imp ≈ 𝛿𝑆𝑡 𝑒 + 2 2 √ 𝑇 −𝑡 atm 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) = 𝜎imp 2𝜋 −𝐷(𝑇 −𝑡)

𝑇 −𝑡 2𝜋

√

𝑇 −𝑡 , we have 2𝜋 (

) − 𝛿𝑆𝑡 𝑒

−𝐷(𝑇 −𝑡)

atm 𝛿 𝛿𝜎imp − 2 2

√

since 𝛿 2 = 1. Therefore, the implied volatility can approximated by atm ≈ 𝜎imp

𝑉mkt

√

𝑆𝑡 𝑒−𝐷(𝑇 −𝑡)

2𝜋 . 𝑇 −𝑡

6. Li ATM Volatility Approximation. Prove the following trigonometry identity cos 3𝜃 = 4 cos3 𝜃 − 3 cos 𝜃. Consider the depressed cubic equation 𝑥3 + 𝑝𝑥 + 𝑞 = 0

𝑇 −𝑡 2𝜋

)

664

7.2.1 Historical and Implied Volatility

√ where 𝑝, 𝑞 ∈ ℝ. By setting 𝑥 = 2𝑦 as

|𝑝| , show that the cubic equation can be expressed 3

4𝑦3 + 3 sgn(𝑝)𝑦 = 𝐶 √ 3𝑞 3 . Hence, if 𝑝 < 0 and |𝐶| < 1 then show that the roots of the 2|𝑝| |𝑝| depressed cubic equation are where 𝐶 = −

√ 𝑥=2

√ ) ) ( ( |𝑝| |𝑝| 1 2𝜋 1 −1 cos cos 𝐶 , 2 cos ± cos−1 𝐶 . 3 3 3 3 3

Let the Black–Scholes theoretical price for the value of a European option 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be ⎧ 𝑆 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑 ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑 ) + − ⎪ 𝑡 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = ⎨ ⎪ 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑− ) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑+ ) ⎩

for call option for put option

where 𝑑± =

log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

such that Φ(⋅) is the cdf of a standard normal, 𝑆𝑡 is the spot price at time 𝑡 < 𝑇 , 𝑇 is the option expiry time, 𝐾 is the strike price, 𝑟 is the risk-free interest rate, 𝐷 is the continuous dividend yield and 𝜎 is the volatility. By taking the third-order approximation of Φ(𝑥) =

( ) 1 1 1 1 1 𝑥− 1 +√ 𝑥3 + 2 𝑥5 − 3 𝑥7 + … 2 2 ⋅ 1! ⋅ 3 2 ⋅ 2! ⋅ 5 2 ⋅ 3! ⋅ 7 2𝜋

and by considering an ATM option in a forward sense such that 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) = 𝐾𝑒−𝑟(𝑇 −𝑡) , show that by equating atm ) = 𝑉mkt 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎imp

where 𝑉mkt is the market-observed ATM European option price, the corresponding ATM atm can be approximated as implied volatility 𝜎imp √ atm 𝜎imp

≈4

) ( 2 2𝜋 1 cos − cos−1 𝐶 𝑇 −𝑡 3 3

7.2.1 Historical and Implied Volatility

where 𝐶 = −

665

√ 3 𝜋𝑉mkt

. 4𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) √ Solution: For 𝑖 = −1, from De Moivre’s formula (cos 𝜃 + 𝑖 sin 𝜃)3 = cos 3𝜃 + 𝑖 sin 3𝜃 cos3 𝜃 + 3𝑖 cos2 𝜃 sin 𝜃 − 3 cos 𝜃 sin2 𝜃 − 𝑖 sin3 𝜃 = cos 3𝜃 + 𝑖 sin 3𝜃 ( 3 ) ( ) cos 𝜃 − 3 cos 𝜃 sin2 𝜃 + 𝑖 3 cos2 𝜃 sin 𝜃 − sin3 𝜃 = cos 3𝜃 + 𝑖 sin 3𝜃. By equating the real terms and because cos2 𝜃 + sin2 𝜃 = 1, we have cos 3𝜃 = cos3 𝜃 − 3 cos 𝜃 sin2 𝜃 ( ) = cos3 𝜃 − 3 cos 𝜃 1 − cos2 𝜃 or cos 3𝜃 = 4 cos3 𝜃 − 3 cos 𝜃. √ By substituting 𝑥 = 2

|𝑝| 𝑦 into the depressed cubic equation 3

8|𝑝| 3

√

√ |𝑝| 3 |𝑝| 𝑦 + 2𝑝 𝑦+𝑞 = 0 3 3 √ 𝑝 3𝑞 3 3 4𝑦 + 3 𝑦 = − |𝑝| 2|𝑝| |𝑝| √ 3𝑞 3 4𝑦3 + 3 sgn(𝑝)𝑦 = − 2|𝑝| |𝑝|

or 4𝑦3 + 3 sgn(𝑝)𝑦 = 𝐶 √ 3𝑞 3 where 𝐶 = − . 2|𝑝| |𝑝| For 𝑝 < 0 and |𝐶| ≤ 1 we let 𝑦 = cos 𝜃 so that the domain of 𝑦 ∈ [−1, 1]. Therefore, 4 cos3 𝜃 − 3 cos 𝜃 = 𝐶 or cos 3𝜃 = 𝐶. Hence, 3𝜃 = cos−1 𝐶, 2𝜋 + cos−1 𝐶, 2𝜋 − cos−1 𝐶

666

7.2.1 Historical and Implied Volatility

or 𝜃=

1 2𝜋 1 2𝜋 1 cos−1 𝐶, + cos−1 𝐶, − cos−1 𝐶. 3 3 3 3 3

Thus, the roots of the depressed cubic equation are √ 𝑥=2

) √ 𝑝 ) ( ( 𝑝 1 2𝜋 1 −1 − cos cos 𝐶 , 2 − cos ± cos−1 𝐶 . 3 3 3 3 3

By setting 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) = 𝐾𝑒−𝑟(𝑇 −𝑡) so that 1 √ 𝑑± = ± 𝜎 𝑇 − 𝑡 2 and taking the cubic approximation of Φ(𝑑± ) we have Φ(𝑑± ) ≈

1 1 1 ± √ 𝜉 ∓ √ 𝜉3 2 2𝜋 6 2𝜋

√ where 𝜁 = 12 𝜎 𝑇 − 𝑡. Equating the Black–Scholes theoretical price with the European ATM option market price 𝑉mkt = 𝛿𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝛿𝑑+ ) − 𝛿𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝛿𝑑− ) where 𝛿 ∈ {−1, 1} and substituting 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) = 𝐾𝑒−𝑟(𝑇 −𝑡) , Φ(𝑑± ) ≈ 1 atm √ 1 3 √ 𝜉 , 𝜉 = 𝜎imp 𝑇 − 𝑡 and because 𝛿 2 = 1 we have 2 6 2𝜋 (

𝑉mkt

) 1 1 1 3 ≈ 𝛿 𝑆𝑡 𝑒 +√ 𝜉− √ 𝜉 2 2𝜋 6 2𝜋 ( ) 1 1 1 −𝛿 2 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − √ 𝜉 + √ 𝜉3 2 2𝜋 6 2𝜋 ) ( −𝐷(𝑇 −𝑡) 𝑆𝑒 1 2𝜉 − 𝜉 3 = 𝑡√ 3 2𝜋 2

−𝐷(𝑇 −𝑡)

or 𝜉 3 − 6𝜉 +

√ 3 2𝜋𝑉mkt 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡)

≈ 0.

1 1 ±√ 𝜉∓ 2 2𝜋

7.2.1 Historical and Implied Volatility

By setting 𝑝 = −6 and 𝑞 =

667

√ 3 2𝜋𝑉mkt 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡)

in the depressed cubic equation we have

𝐶=−

√ 3 𝜋𝑉mkt 4𝑆𝑡 𝑒−𝐷(𝑇 −𝑡)

and hence the roots of the cubic equation are ( ) √ ( ) √ 1 2𝜋 1 𝜉 ≈ 2 2 cos cos−1 𝐶 , 2 2 cos ± cos−1 𝐶 . 3 3 3 atm ∈ (0, 1), 𝑉 −𝐷(𝑇 −𝑡) > 0 we only consider the case when −1 < Since 𝜎imp mkt > 0 and 𝑆𝑡 𝑒 𝐶 < 0, which implies

−1 < −

√ 3 𝜋𝑉mkt 4𝑆𝑡 𝑒−𝐷(𝑇 −𝑡)

0. By equating 𝑉mkt = 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎imp ) and taking a linear approximation of Φ(𝛿𝑑± ) such that

𝑑± =

̄ log(𝑆̄𝑡 ∕𝐾) 1 √ ± 𝜎 𝑇 −𝑡 √ 𝜎imp 𝑇 − 𝑡 2

we have ̄ ) 𝑉mkt = 𝛿 𝑆̄𝑡 Φ(𝛿𝑑+ ) − 𝛿 𝐾Φ(𝛿𝑑 ( ) − ( ) 1 1 𝛿 𝛿 ̄ ̄ − 𝛿𝐾 +√ +√ ≈ 𝛿 𝑆𝑡 2 2 2𝜋𝑑+ 2𝜋𝑑− √ ( ̄ ) ̄ ̄ ̄ ̄ ̄ 𝑆𝑡 − 𝐾 (𝑆 − 𝐾) log(𝑆𝑡 ∕𝐾) 1 𝑇 − 𝑡 ̄ + =𝛿 + 𝑡 √ (𝑆̄𝑡 + 𝐾) 2 2 2𝜋 𝜎imp 2𝜋(𝑇 − 𝑡) since 𝛿 2 = 1. Thus, ( ( ̄ )) √ 𝑆𝑡 − 𝐾̄ 2 ̄ ̄ (𝑆𝑡 + 𝐾)(𝑇 − 𝑡)𝜎imp − 2 2𝜋(𝑇 − 𝑡) 𝑉mkt − 𝛿 𝜎imp 2 ̄ log(𝑆̄𝑡 ∕𝐾) ̄ ≈0 +2(𝑆̄𝑡 − 𝐾)

676

7.2.1 Historical and Implied Volatility

and taking the largest root, ( ̄ ) 𝑆𝑡 − 𝐾̄ ⎞ ⎛ 𝑉 − 𝛿 ⎟ ⎜ mkt 2 2𝜋 ⎜ ⎟ ≈ ⎟ 𝑇 −𝑡⎜ 𝑆̄𝑡 + 𝐾̄ ⎟ ⎜ ⎠ ⎝ √ √ ( ̄ ) 2 √ 𝑆𝑡 − 𝐾̄ ⎞ ⎛ √ 𝑉 − 𝛿 ) ( ̄ √ ⎟ √ 2𝜋 ⎜ mkt 2 𝑆𝑡 − 𝐾̄ 2 √ ⎟ ⎜ ̄ +√ − log(𝑆̄𝑡 ∕𝐾). ⎟ 𝑇 − 𝑡 𝑆̄𝑡 + 𝐾̄ √𝑇 − 𝑡 ⎜ 𝑆̄𝑡 + 𝐾̄ ⎟ ⎜ ⎠ ⎝ √

𝜎imp

̄ ≈ 2(𝑆̄𝑡 − 𝐾)∕( ̄ 𝑆̄𝑡 + 𝐾), ̄ Taking a linear approximation of log(𝑆̄𝑡 ∕𝐾)

𝜎imp

( ̄ ) 𝑆𝑡 − 𝐾̄ ⎞ ⎛ √ ⎟ ⎜ 𝑉mkt − 𝛿 2 2𝜋 ⎜ ⎟ ≈ ⎟ 𝑇 −𝑡⎜ 𝑆̄𝑡 + 𝐾̄ ⎟ ⎜ ⎠ ⎝ √ √ ( ̄ ) 2 √ 𝑆𝑡 − 𝐾̄ ⎞ ⎛ √ 𝑉 − 𝛿 )2 ( ̄ √ ⎟ √ 2𝜋 ⎜ mkt 2 𝑆𝑡 − 𝐾̄ 4 √ ⎟ ⎜ +√ − . ⎟ 𝑇 − 𝑡 𝑆̄𝑡 + 𝐾̄ √𝑇 − 𝑡 ⎜ 𝑆̄𝑡 + 𝐾̄ ⎟ ⎜ ⎠ ⎝

9. We consider the Black–Scholes formula for the value of a European option 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎) such that ⎧ 𝑆 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑 ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑 ) for call option + − ⎪ 𝑡 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎) = ⎨ ⎪ 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑− ) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑+ ) for put option ⎩ with 𝑑± =

log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝑡) √ 𝜎 𝑇 −𝑡

such that Φ(⋅) is the cdf of a standard normal, 𝑆𝑡 is the spot price at time 𝑡 < 𝑇 , 𝑇 is the option expiry time, 𝐾 is the strike price, 𝑟 is the risk-free interest rate, 𝐷 is the continuous dividend yield and 𝜎 is the spot volatility. Show that 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎) is a monotonically increasing function in 𝜎 over (0, ∞).

7.2.1 Historical and Implied Volatility

677

∞ as the implied volatility such that We define 𝜎imp ∞ 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎imp ) = 𝑉mkt

that is we equate the Black–Scholes theoretical price with the observed European option price obtained in the market, 𝑉mkt . ∞ ? Under what conditions does there exist a unique solution 𝜎imp Solution: From Problem 2.2.4.6 (page 224), the vega of a European option is 𝜕𝑉 = 𝜕𝜎

√

1 2 𝑇 −𝑡 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑒− 2 𝑑+ . 2𝜋

𝜕𝑉 Since 𝑆𝑡 > 0, therefore > 0 for all 𝜎 ∈ (0, ∞). Thus, 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) is a monotonically 𝜕𝜎 increasing function in 𝜎 ∈ (0, ∞). From Problems 2.2.1.3 (page 74), 2.2.1.4 (page 75), 2.2.1.5 (page 75) and 2.2.1.6 (page 76), 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) satisfies ⎧ max{𝑆 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) , 0} 𝑡 ⎪ 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎) ≥ ⎨ ⎪ max{𝐾𝑒−𝑟(𝑇 −𝑡) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) , 0} ⎩

for call option for put option

and ⎧𝑆 ⎪ 𝑡 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎) ≤ ⎨ ⎪𝐾 ⎩

for call option for put option.

If the market price 𝑉mkt lies between the bounded region as described above for either a call or put option, and from the monotonicity and continuity of 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎) in 𝜎 ∈ (0, ∞), ∞ exists and is a unique solution of 𝑉 (𝑆 , 𝑡; 𝐾, 𝑇 , 𝜎 ∞ ) = 𝑉 then 𝜎imp 𝑡 mkt . imp 10. Manaster–Koehler Method. Consider the Black–Scholes formula for the value of a European option 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎) such that ⎧ 𝑆 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑 ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑 ) + − ⎪ 𝑡 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎) = ⎨ ⎪ 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(−𝑑− ) − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(−𝑑+ ) ⎩ with log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝑡) 𝑑± = √ 𝜎 𝑇 −𝑡

for call option for put option

678

7.2.1 Historical and Implied Volatility

such that Φ(⋅) is the cdf of a standard normal, 𝑆𝑡 is the spot price at time 𝑡 < 𝑇 , 𝑇 is the option expiry time, 𝐾 is the strike price, 𝑟 is the risk-free interest rate, 𝐷 is the continuous dividend yield and 𝜎 is the spot volatility. 𝜕𝑉 is maximised at Show that 𝜕𝜎 𝜎max

√ | 2 ( )| log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷)(𝑇 − 𝑡) || = || |𝑇 − 𝑡 |

and prove that 𝜕2 𝑉 𝜕𝑉 = 𝜕𝜎 𝜕𝜎 2

(

) ) 𝑇 −𝑡 ( 4 𝜎max − 𝜎 4 3 4𝜎

for all 𝛼 ∈ (0, ∞). ∞ as the implied volatility such that We define 𝜎imp ∞ ) = 𝑉mkt 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎imp

where 𝑉mkt is the option price obtained in the market. Deduce that 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎) is ∞ 𝜎max . Finally, if the sequence {𝜎𝑛 } is generated by the Newton–Raphson method

𝜎𝑛+1

⎫ ⎧ ⎪ ⎪ ⎪ 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎𝑛 ) − 𝑉mkt ⎪ = 𝜎𝑛 − ⎨ ⎬,𝑛 ≥ 0 𝜕𝑉 || ⎪ ⎪ ⎪ ⎪ 𝜕𝜎 ||𝜎=𝜎𝑛 ⎭ ⎩

∞ )=𝑉 with initial iterate 𝜎0 = 𝜎max and if 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎imp mkt has a solution, show that ∞ with a quadratic rate of conver{𝜎𝑛 } converges monotonically to the unique solution 𝜎imp gence, i.e. ∞ ∞ 2 |𝜎𝑛+1 − 𝜎imp | = 𝑂(|𝜎𝑛 − 𝜎imp | ).

Solution: From Problem 2.2.4.6 (page 224), the vega of a European option is 𝜕𝑉 = 𝜕𝜎 To find the local extrema of

√

1 2 𝑇 −𝑡 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑒− 2 𝑑+ . 2𝜋

𝜕𝑉 𝜕𝑉 , we first differentiate with respect to 𝜎 𝜕𝜎 𝜕𝜎

√ 𝜕𝑑+ − 1 𝑑 2 𝑇 −𝑡 𝜕2𝑉 =− 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑑+ 𝑒 2 + 2 2𝜋 𝜕𝜎 𝜕𝜎

7.2.1 Historical and Implied Volatility

679

and since ( )√ 𝑇 −𝑡 𝜎 2 (𝑇 − 𝑡)3∕2 − log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 + 12 𝜎 2 )

𝜕𝑑+ = 𝜕𝜎 = −𝑑−

𝜎 2 (𝑇 − 𝑡)

therefore √ 1 𝑇 −𝑡 𝜕2𝑉 −𝐷(𝑇 −𝑡) − 2 𝑑+2 = 𝑑 𝑑 𝑒 𝑒 𝑆 + − 𝑡 2𝜋 𝜕𝜎 2 ( ) 𝑑+ 𝑑− 𝜕𝑉 = . 𝜎 𝜕𝜎 By setting

𝜕2𝑉 = 0, we have either 𝑑+ = 0 or 𝑑− = 0, which implies 𝜕𝜎 2 𝜎2 = −

) 2 ( log(𝑆𝑡 ∕𝐾) + 𝑟 − 𝐷 𝑇 −𝑡

or 𝜎2 = Taking second derivatives of (

𝜕𝑉 𝜕𝜎

( ) ( ) 𝑑+ 𝑑− 𝜕 2 𝑉 𝜕𝑑− 𝑑+ 𝜕𝑉 𝜕𝑉 + + 𝜕𝜎 𝜕𝜎 𝜎 𝜕𝜎 𝜎 𝜕𝜎 2 ( ) ( ) 𝑑+2 𝜕𝑉 𝑑 2 𝜕𝑉 𝑑+ 𝑑− 𝜕𝑉 𝑑+ 𝑑− 𝜕 2 𝑉 =− − − − + 𝜎 𝜕𝜎 𝜎 𝜕𝜎 𝜎 𝜕𝜎 𝜎 𝜕𝜎 2

𝜕3𝑉 =− 𝜕𝜎 3

𝑑+ 𝑑− 𝜎

)

) 2 ( log(𝑆𝑡 ∕𝐾) + 𝑟 − 𝐷 . 𝑇 −𝑡

𝜕𝑑+ 𝜕𝑉 + 𝜕𝜎 𝜕𝜎

(

𝑑− 𝜎

)

) ) 2 ( 2 ( log(𝑆𝑡 ∕𝐾) + 𝑟 − 𝐷 or 𝜎 2 = log(𝑆𝑡 ∕𝐾) + 𝑟 − 𝐷 and substituting 𝜎 2 = − 𝑇 −𝑡 𝑇 −𝑡 gives 𝜕3 𝑉 < 0. 𝜕𝜎 3 Hence, 𝜎max is the maximum point of

√ | 2 ( )| log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷)(𝑇 − 𝑡) || = || 𝑇 − 𝑡 | | 𝜕𝑉 (since 𝜎max is the only extremum point). 𝜕𝜎

680

7.2.1 Historical and Implied Volatility

From 𝜎max

√ | 2 ( )| log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷)(𝑇 − 𝑡) || = || |𝑇 − 𝑡 |

we can write 4 = 𝜎max

( )2 4 log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷)(𝑇 − 𝑡) 2 (𝑇 − 𝑡)

or ( )2 𝜎 4 (𝑇 − 𝑡)2 log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷)(𝑇 − 𝑡) = max . 4 Expanding 𝑑+ 𝑑− , ( 𝑑+ 𝑑− = ( =

log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 + 12 𝜎 2 )(𝑇 − 𝑡) )2

log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷)(𝑇 − 𝑡)

)( ) log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 − 12 𝜎 2 )(𝑇 − 𝑡)

𝜎 2 (𝑇 − 𝑡) − 14 𝜎 4 (𝑇 − 𝑡)2

𝜎 2 (𝑇 − 𝑡) ) 𝑇 −𝑡( 4 𝜎max − 𝜎 4 . = 2 4𝜎

Therefore, 𝜕2 𝑉 𝜕𝑉 = 2 𝜕𝜎 𝜕𝜎

(

) ) 𝑇 −𝑡 ( 4 𝜎max − 𝜎 4 3 4𝜎

for all 𝜎 ∈ (0, ∞).

𝜕2𝑉 𝜕𝑉 > 0, which implies is a monotoni2 𝜕𝜎 𝜕𝜎 ∞ cally increasing function in 𝜎 ∈ [𝜎imp , 𝜎max ], and hence 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎) is strictly convex ∞ ,𝜎 in 𝜎 ∈ [𝜎imp max ]. If the implied volatility 𝜎 ∞ < 𝜎max we have

𝜕2 𝑉 𝜕𝑉 < 0, which implies is a monotonically 2 𝜕𝜎 𝜕𝜎 ∞ decreasing function in 𝜎 ∈ [𝜎imp , 𝜎max ], and hence 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎) is strictly concave in ∞ ,𝜎 𝜎 ∈ [𝜎imp max ]. To show that the sequence {𝜎𝑛 } is monotonic and bounded, we note that from the Newton–Raphson formula In contrast, if 𝜎 ∞ > 𝜎max we have

𝜎𝑛+1

⎫ ⎧ ⎪ ⎪ ⎪ 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎𝑛 ) − 𝑉mkt ⎪ = 𝜎𝑛 − ⎨ ⎬, 𝜕𝑉 || ⎪ ⎪ ⎪ ⎪ 𝜕𝜎 ||𝜎=𝜎𝑛 ⎭ ⎩

𝑛≥0

7.2.1 Historical and Implied Volatility

681

∞ up to the first order and by expanding 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎𝑛 ) at 𝜎imp

| | ( | | ∞ | ∞ )) | | | 𝜎𝑛+1 − 𝜎imp 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎𝑛 ) − 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 , 𝜎imp 1 | | | | 1 − = | | | | ∞ | | | 𝜎𝑛 − 𝜎 ∞ | | 𝜎𝑛 − 𝜎imp 𝜕𝑉 | imp | | | | | | 𝜕𝜎 |𝜎=𝜎𝑛 || | ( ) /( )| | | | | 𝜕𝑉 𝜕𝑉 || | | | = |1 − | | ∗ | 𝜕𝜎 𝜕𝜎 | |𝜎=𝜎𝑛 |𝜎=𝜎𝑛 | | | ∞ . where 𝜎𝑛∗ lies between 𝜎𝑛 and 𝜎imp For 𝑛 = 0

( ) /( )| ∞ | | | 𝜎1 − 𝜎imp | 𝜕𝑉 || 𝜕𝑉 || | || | | | = 1− | | | ∗ | | 𝜎0 − 𝜎 ∞ | || 𝜕𝜎 𝜕𝜎 |𝜎=𝜎0 |𝜎=𝜎0 | imp | | | | and because 𝜎0 = 𝜎max maximises 0
0 for all 𝜎 ∈ (0, ∞), thus 𝜕𝜎 𝜕𝜎 𝜕𝑉 || 𝜕𝑉 || < | 𝜕𝜎 |𝜎=𝜎 ∗ 𝜕𝜎 ||𝜎=𝜎0 0

and we obtain ∞ | | 𝜎1 − 𝜎imp | | | < 1. | | 𝜎0 − 𝜎 ∞ | imp | |

Assume the result ∞ | | 𝜎𝑘+1 − 𝜎imp | | | | 0 for all 𝜎 ∈ (0, ∞), thus 𝜕𝜎 0
0. Let 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) be the transition pdf for the asset price where the asset price is 𝑆𝑡 at time 𝑡 given that the asset price is 𝑆𝑇 at time 𝑇 > 𝑡. From the Chapman–Kolmogorov equation for Δ𝑡 > 0,

𝑝(𝑆𝑡 , 𝑡 − Δ𝑡; 𝑆𝑇 , 𝑇 ) =

∞

∫0

𝑝(𝑆𝑡 , 𝑡 − Δ𝑡; 𝑧, 𝑡)𝑝(𝑧, 𝑡; 𝑆𝑇 , 𝑇 ) 𝑑𝑧

show that by expanding 𝑝(𝑧, 𝑡; 𝑆𝑇 , 𝑇 ) using Taylor series centred on 𝑆𝑡 up to second order and taking limits Δ𝑡 → 0, 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) satisfies the backward Kolmogorov equation 𝜕 1 𝜕2 𝜕 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) = 0 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) + 𝜎(𝑆𝑡 , 𝑡)2 𝑆𝑡2 2 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡 with boundary condition 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑡) = 𝛿(𝑆𝑡 − 𝑆𝑇 ),

∀𝑡.

Solution: From Taylor’s theorem ) )( ( ) 1 𝑎+𝑏 𝑎+𝑏 + 𝑓′ 𝑥 − (𝑎 + 𝑏) 2 2 2 (( )( ( )2 )3 ) 1 1 1 ′′ 𝑎 + 𝑏 𝑥 − (𝑎 + 𝑏) 𝑥 − (𝑎 + 𝑏) + 𝑂 + 𝑓 2 2 2 2

𝑓 (𝑥) = 𝑓

(

7.2.2 Local Volatility

689

and taking integrals 𝑏

∫𝑎

) )( ( ) 𝑏 1 𝑎+𝑏 𝑎+𝑏 𝑑𝑢 + 𝑢 − (𝑎 + 𝑏) 𝑑𝑢 𝑓′ ∫𝑎 ∫𝑎 2 2 2 )( ( )2 𝑏 1 1 ′′ 𝑎 + 𝑏 𝑢 − (𝑎 + 𝑏) 𝑑𝑢 𝑓 + ∫𝑎 2 2 2 (( )3 ) 𝑏 1 𝑢 − (𝑎 + 𝑏) 𝑂 + 𝑑𝑢 ∫𝑎 2 ) )( ( )2 |𝑏 ( 1 1 𝑎+𝑏 𝑎+𝑏 | (𝑏 − 𝑎) + 𝑓 ′ 𝑢 − (𝑎 + 𝑏) | =𝑓 | 2 2 2 2 |𝑎 ( 𝑏 ( )( ( )3 | )4 )|𝑏 1 1 1 ′ 𝑎+𝑏 | | 𝑢 − (𝑎 + 𝑏) 𝑢 − (𝑎 + 𝑏) | + 𝑂 + 𝑓 | | | 3! 2 2 2 |𝑎 |𝑎 ) ( ( ) 𝑎+𝑏 (𝑏 − 𝑎) + 𝑂 (𝑏 − 𝑎)3 . =𝑓 2

𝑓 (𝑢) 𝑑𝑢 =

𝑏

𝑓

(

From Girsanov’s theorem, under the risk-neutral measure ℚ, 𝑆𝑡 follows 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎(𝑆𝑡 , 𝑡)𝑑𝑊𝑡ℚ 𝑆𝑡 𝑡

𝜇−𝑟 𝑑𝑢 is a ℚ-standard Wiener process. ∫0 𝜎(𝑆𝑢 , 𝑢) By using the risk-neutral dynamics

where 𝑊𝑡ℚ = 𝑊𝑡 +

𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑆𝑡 + 𝜎(𝑆𝑡 , 𝑡)𝑆𝑡 𝑑𝑊𝑡ℚ and taking integrals we have 𝑡+Δ𝑡

∫𝑡

𝑑𝑆𝑢 =

𝑡+Δ𝑡

∫𝑡

(𝑟 − 𝐷)𝑆𝑢 𝑑𝑢 +

𝑡+Δ𝑡

∫𝑡

𝜎(𝑆𝑢 , 𝑢)𝑆𝑢 𝑑𝑊𝑢ℚ

or 𝑆𝑡+Δ𝑡 − 𝑆𝑡 =

𝑡+Δ𝑡

∫𝑡

(𝑟 − 𝐷)𝑆𝑢 𝑑𝑢 +

𝑡+Δ𝑡

∫𝑡

𝜎(𝑆𝑢 , 𝑢)𝑆𝑢 𝑑𝑊𝑢ℚ .

Thus, by taking expectations and using the approximation integration formula as well as the It¯o calculus property, 𝔼

ℚ

[

𝑡+Δ𝑡

]

𝑆𝑡+Δ𝑡 − 𝑆𝑡 || ℱ𝑡 =

∫𝑡 𝑡+Δ𝑡

=

∫𝑡

[ (𝑟 − 𝐷)𝑆𝑢 𝑑𝑢 + 𝔼 (𝑟 − 𝐷)𝑆𝑢 𝑑𝑢

ℚ

𝑡+Δ𝑡

∫𝑡

) ( = (𝑟 − 𝐷)𝑆𝑡+ 1 Δ𝑡 Δ𝑡 + 𝑂 (Δ𝑡)3 2

|

| 𝜎(𝑆𝑢 , 𝑢)𝑆𝑢 𝑑𝑊𝑢ℚ | ℱ𝑡 | |

]

690

7.2.2 Local Volatility

[( )2 | ] 𝑆𝑡+Δ𝑡 − 𝑆𝑡 | ℱ𝑡 | )2 ( 𝑡+Δ𝑡 = (𝑟 − 𝐷)2 𝑆𝑢 𝑑𝑢 ∫𝑡 [( ( 𝑡+Δ𝑡 ) )| ] 𝑡+Δ𝑡 | +2(𝑟 − 𝐷) 𝑆𝑢 𝑑𝑢 𝔼ℚ 𝜎(𝑆𝑢 , 𝑢)𝑆𝑢 𝑑𝑊𝑢ℚ | ℱ𝑡 | ∫𝑡 ∫𝑡 | [( )2 | ] 𝑡+Δ𝑡 | +𝔼ℚ 𝜎(𝑆𝑢 , 𝑢)𝑆𝑢 𝑑𝑊𝑢ℚ || ℱ𝑡 ∫𝑡 | | )2 ( 𝑡+Δ𝑡 𝑆𝑢 𝑑𝑢 = (𝑟 − 𝐷)2 ∫𝑡 [( ( 𝑡+Δ𝑡 ) )| ] 𝑡+Δ𝑡 | +2(𝑟 − 𝐷) 𝑆𝑢 𝑑𝑢 𝔼ℚ 𝜎(𝑆𝑢 , 𝑢)𝑆𝑢 𝑑𝑊𝑢ℚ | ℱ𝑡 | ∫𝑡 ∫𝑡 |

𝔼ℚ

𝑡+Δ𝑡

𝜎(𝑆𝑢 , 𝑢)2 𝑆𝑢2 𝑑𝑢 ∫𝑡 )2 ( ) ( 1 = 𝜎 𝑆𝑡+ 1 Δ𝑡 , 𝑡 + Δ𝑡 𝑆 2 1 Δ𝑡 + 𝑂 (Δ𝑡)2 Δ𝑡 𝑡+ 2 2 2 +

and [( )3 | ] 𝑆𝑡+Δ𝑡 − 𝑆𝑡 | ℱ𝑡 | )3 ( 𝑡+Δ𝑡 3 = (𝑟 − 𝐷) 𝑆𝑢 𝑑𝑢 ∫𝑡 [( )2 ( 𝑡+Δ𝑡 )| ] 𝑡+Δ𝑡 2 ℚ ℚ | +2(𝑟 − 𝐷) 𝑆𝑢 𝑑𝑢 𝔼 𝜎(𝑆𝑢 , 𝑢)𝑆𝑢 𝑑𝑊𝑢 | ℱ𝑡 | ∫𝑡 ∫𝑡 | ) ( 𝑡+Δ𝑡 ) ( 𝑡+Δ𝑡 𝑆𝑢 𝑑𝑢 𝜎(𝑆𝑢 , 𝑢)2 𝑆𝑢2 𝑑𝑢 +(𝑟 − 𝐷) ∫𝑡 ∫𝑡 [( )2 ( 𝑡+Δ𝑡 )| ] 𝑡+Δ𝑡 2 ℚ ℚ | +(𝑟 − 𝐷) 𝑆𝑢 𝑑𝑢 𝔼 𝜎(𝑆𝑢 , 𝑢)𝑆𝑢 𝑑𝑊𝑢 | ℱ𝑡 | ∫𝑡 ∫𝑡 | ) ( 𝑡+Δ𝑡 ) ( 𝑡+Δ𝑡 𝑆𝑢 𝑑𝑢 𝜎(𝑆𝑢 , 𝑢)2 𝑆𝑢2 𝑑𝑢 +2(𝑟 − 𝐷) ∫𝑡 ∫𝑡 [( ( 𝑡+Δ𝑡 ) )| ] 𝑡+Δ𝑡 2 2 ℚ ℚ | + 𝜎(𝑆𝑢 , 𝑢) 𝑆𝑢 𝑑𝑢 𝔼 𝜎(𝑆𝑢 , 𝑢)𝑆𝑢 𝑑𝑊𝑢 | ℱ𝑡 | ∫𝑡 ∫𝑡 | ) ( 2 = 𝑂 (Δ𝑡) . 𝔼ℚ

7.2.2 Local Volatility

691

From the Chapman–Kolmogorov equation we have 𝑝(𝑆𝑡 , 𝑡 − Δ𝑡; 𝑆𝑇 , 𝑇 ) =

∞

∫0

𝑝(𝑆𝑡 , 𝑡 − Δ𝑡; 𝑧, 𝑡)𝑝(𝑧, 𝑡; 𝑆𝑇 , 𝑇 ) 𝑑𝑧

and expanding 𝑝(𝑧, 𝑡; 𝑆𝑇 , 𝑇 ) using Taylor series centred on 𝑆𝑡 up to second order yields 𝑝(𝑧, 𝑡; 𝑆𝑇 , 𝑇 ) = 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) + (𝑧 − 𝑆𝑡 )

𝜕 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) 𝜕𝑆𝑡

1 𝜕2 + (𝑧 − 𝑆𝑡 )2 2 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) + 𝑂((𝑧 − 𝑆𝑡 )3 ). 2 𝜕𝑆𝑡 By substituting the Taylor series into the Chapman–Kolmogorov equation we have 𝑝(𝑆𝑡 , 𝑡 − Δ𝑡; 𝑆𝑇 , 𝑇 ) = 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 )

∞

∫0

𝑝(𝑆𝑡 , 𝑡 − Δ𝑡; 𝑧, 𝑡) 𝑑𝑧 ∞

+

𝜕 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) (𝑧 − 𝑆𝑡 )𝑝(𝑆𝑡 , 𝑡 − Δ𝑡; 𝑧, 𝑡) 𝑑𝑧 ∫0 𝜕𝑆𝑡 ∞

1 𝜕2 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) (𝑧 − 𝑆𝑡 )2 𝑝(𝑆𝑡 , 𝑡 − Δ𝑡; 𝑧, 𝑡) 𝑑𝑧 ∫0 2 𝜕𝑆𝑡2 ( ∞ ) +𝑂 (𝑧 − 𝑆𝑡 )3 𝑝(𝑆𝑡 , 𝑡 − Δ𝑡; 𝑧, 𝑡) 𝑑𝑧 . ∫0 +

Since ∞

∫0 ∞

∫0 ∞

∫0 ∞

∫0

𝑝(𝑆𝑡 , 𝑡 − Δ𝑡; 𝑧, 𝑡) 𝑑𝑧 = 1

(𝑧 − 𝑆𝑡 )𝑝(𝑆𝑡 , 𝑡 − Δ𝑡; 𝑧, 𝑡) 𝑑𝑧 = (𝑟 − 𝐷)𝑆𝑡+ 1 Δ𝑡 Δ𝑡 + 𝑂((Δ𝑡)2 ) (

2

)2 1 (𝑧 − 𝑆𝑡 )2 𝑝(𝑆𝑡 , 𝑡 − Δ𝑡; 𝑧, 𝑡) 𝑑𝑧 = 𝜎 𝑆𝑡+ 1 Δ𝑡 , 𝑡 + Δ𝑡 𝑆 2 1 Δ𝑡 + 𝑂((Δ𝑡)2 ) 𝑡+ 2 Δ𝑡 2 2 (𝑧 − 𝑆𝑡 )3 𝑝(𝑆𝑡 , 𝑡 − Δ𝑡; 𝑧, 𝑡) 𝑑𝑧 = 𝑂((Δ𝑡)2 )

then by taking limits Δ𝑡 → 0, 𝑝(𝑆𝑡 , 𝑡 − Δ𝑡; 𝑆𝑇 , 𝑇 ) − 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) Δ𝑡→0 Δ𝑡 𝜕 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) = (𝑟 − 𝐷)𝑆𝑡 𝜕𝑆𝑡 ( )2 1 𝜕2 1 + lim 𝜎 𝑆𝑡+ 1 Δ𝑡 , 𝑡 + Δ𝑡 𝑆𝑡2 2 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) + lim 𝑂((Δ𝑡)2 ) Δ𝑡→0 2 Δ𝑡→0 2 2 𝜕𝑆𝑡 lim

692

7.2.2 Local Volatility

we have −

1 𝜕 𝜕2 𝜕 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) + 𝜎(𝑆𝑡 , 𝑡)2 𝑆𝑡2 2 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) = (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 𝜕𝑆𝑡 2 𝜕𝑆𝑡

or 1 𝜕2 𝜕 𝜕 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) = 0 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) + 𝜎(𝑆𝑡 , 𝑡)2 𝑆𝑡2 2 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 with boundary condition 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑡) = 𝛿(𝑆𝑡 − 𝑆𝑇 ) for all 𝑡. 3. Black–Scholes Equation – Local Volatility Model. Under the risk-neutral measure ℚ, let ̃𝑡 : 𝑡 ≥ 0} be a ℚ-standard Wiener process on the probability space (Ω, ℱ, ℚ) and let {𝑊 the asset price 𝑆𝑡 follow a local volatility model with the following SDE 𝑑𝑆𝑡 ̃𝑡 = (𝑟 − 𝐷) 𝑑𝑡 + 𝜎(𝑆𝑡 , 𝑡)𝑑 𝑊 𝑆𝑡 where 𝑟 is the risk-free interest-rate parameter from the money-market account, 𝐷 is the continuous dividend yield and 𝜎(𝑆𝑡 , 𝑡) is the local volatility function. Let 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) be the transition pdf of the asset price where the asset price is 𝑆𝑡 at time 𝑡 given that the asset price is 𝑆𝑇 at time 𝑇 > 𝑡. We consider a European option written on 𝑆𝑡 with strike price 𝐾 > 0 expiring at time 𝑇 > 𝑡 with payoff Ψ(𝑆𝑇 ), where the option price at time 𝑡 under the risk-neutral measure ℚ is [ ] 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ Ψ(𝑆𝑇 )|| ℱ𝑡 = 𝑒−𝑟(𝑇 −𝑡)

∞

∫0

Ψ(𝑧)𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧.

By using the backward Kolmogorov equation, show that 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) satisfies the Black– Scholes equation 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 + 𝜎(𝑆𝑡 , 𝑡)2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 with boundary condition 𝑉 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = Ψ(𝑆𝑇 ). From the above PDE and boundary condition, can we price a European option analytically?

7.2.2 Local Volatility

693

Solution: Given 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡)

∞

∫0

Ψ(𝑧)𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧

by taking first and second-order differentials we have 𝜕𝑉 = 𝑟𝑒−𝑟(𝑇 −𝑡) ∫0 𝜕𝑡

∞

Ψ(𝑧)𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 + 𝑒−𝑟(𝑇 −𝑡)

= 𝑟𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑒−𝑟(𝑇 −𝑡) 𝜕𝑉 = 𝑒−𝑟(𝑇 −𝑡) ∫0 𝜕𝑆𝑡 𝜕2𝑉 = 𝑒−𝑟(𝑇 −𝑡) ∫0 𝜕𝑆𝑡2

∞

∞

∫0

Ψ(𝑧)

∫0

Ψ(𝑧)

𝜕 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 𝜕𝑡

𝜕 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 𝜕𝑡

Ψ(𝑧)

𝜕 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 𝜕𝑆𝑡

Ψ(𝑧)

𝜕2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 𝜕𝑆𝑡2

∞

∞

and from Problem 7.2.2.2 (page 687), the backward Kolmogorov equation satisfied by 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) is 1 𝜕2 𝜕 𝜕 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) = 0 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) + 𝜎(𝑆𝑡 , 𝑡)2 𝑆𝑡2 2 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 with boundary condition 𝑝(𝑆𝑡 , 𝑇 ; 𝑆𝑇 , 𝑇 ) = 𝛿(𝑆𝑡 − 𝑆𝑇 ) for all 𝑡. By multiplying the backward Kolmogorov equation with Ψ(𝑆𝑇 ) and taking integrals we have ∞

∫0

Ψ(𝑧)

𝜕 1 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 + 𝜎(𝑆𝑡 , 𝑡)2 𝑆𝑡2 ∫0 𝜕𝑡 2

+(𝑟 − 𝐷)𝑆𝑡

∞

∫0

Ψ(𝑧)

[ 𝑒

Ψ(𝑧)

𝜕2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 𝜕𝑆𝑡2

𝜕 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 = 0. 𝜕𝑆𝑡

Writing the above equation in terms of

−𝑟(𝑇 −𝑡)

∞

𝜕2𝑉 𝜕𝑉 𝜕𝑉 and , , 𝜕𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡2

1 𝜕2𝑉 𝜕𝑉 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝜎(𝑆𝑡 , 𝑡)2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡

]

or 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 − 𝑟𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0. + 𝜎(𝑆𝑡 , 𝑡)2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆𝑡 𝜕𝑆𝑡

=0

694

7.2.2 Local Volatility

Finally, for the boundary condition we note that 𝑉 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) =

∞

∫0

Ψ(𝑧)𝑝(𝑆𝑇 , 𝑇 ; 𝑧, 𝑇 ) 𝑑𝑧

and from the definition of the Dirac delta function 𝑉 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) =

∞

∫0

Ψ(𝑧)𝑝(𝑆𝑇 , 𝑇 ; 𝑧, 𝑇 ) 𝑑𝑧

∞

Ψ(𝑧)𝛿(𝑆𝑇 − 𝑧) 𝑑𝑧 ∫0 = Ψ(𝑆𝑇 ). =

Given that the solution of 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) depends on the unknown local volatility function, we cannot rely on the analytical Black–Scholes formula to price a European option. In general, this PDE has to be solved numerically. { } 4. Forward Kolmogorov Equation – Local Volatility Model. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ). We consider an asset price 𝑆𝑡 following a local volatility model SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎(𝑆𝑡 , 𝑡)𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎(𝑆𝑡 , 𝑡) is the local volatility function and let 𝑟 be the risk-free interest-rate parameter. Using Girsanov’s theorem, show that under the risk-neutral measure ℚ, the above SDE is 𝑑𝑆𝑡 ̃𝑡 = (𝑟 − 𝐷) 𝑑𝑡 + 𝜎(𝑆𝑡 , 𝑡)𝑑 𝑊 𝑆𝑡 ̃𝑡 is a ℚ-standard Wiener process. where 𝑊 Let 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) be the transition pdf of the asset price where the asset price is 𝑆𝑡 at time 𝑡 given that the asset price is 𝑆𝑇 at time 𝑇 > 𝑡. From the Chapman–Kolmogorov equation for Δ𝑇 > 0, 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 + Δ𝑇 ) =

∞

∫0

𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 )𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 + Δ𝑇 ) 𝑑𝑧

show that in the limit Δ𝑇 → 0, 𝜕 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) = − ∫0 𝜕𝑇

∞

𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 )

𝜕 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) 𝑑𝑧. 𝜕𝑇

7.2.2 Local Volatility

695

Finally, using the backward Kolmogorov equation, show that 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) satisfies the forward Kolmogorov equation ] 1 𝜕2 [ 𝜕 𝜎(𝑆𝑇 , 𝑇 )2 𝑆𝑇2 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) − 2 𝜕𝑇 2 𝜕𝑆 𝑇 +(𝑟 − 𝐷)

] 𝜕 [ 𝑆𝑇 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) = 0 𝜕𝑆𝑇

with boundary condition 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑡) = 𝛿(𝑆𝑡 − 𝑆𝑇 ),

∀𝑡.

Solution: At time 𝑡, the portfolio Π𝑡 is valued as Π𝑡 = 𝜙𝑡 𝑆𝑡 + 𝜓𝑡 𝐵𝑡 where 𝐵𝑡 is the risk-free asset having the following diffusion 𝑑𝐵𝑡 = 𝑟𝐵𝑡 𝑑𝑡. Since the holder of the portfolio will receive 𝐷𝑆𝑡 𝑑𝑡 for every stock held, then in differential form ( ) 𝑑Π𝑡 = 𝜙𝑡 𝑑𝑆𝑡 + 𝐷𝑆𝑡 𝑑𝑡 + 𝜓𝑡 𝑑𝐵𝑡 ( ) = 𝜙𝑡 𝜇𝑆𝑡 𝑑𝑡 + 𝜎(𝑆𝑡 , 𝑡)𝑆𝑡 𝑑𝑊𝑡 + 𝜓𝑡 𝑟𝐵𝑡 𝑑𝑡 = 𝑟Π𝑡 𝑑𝑡 + 𝜙𝑡 (𝜇 − 𝑟)𝑆𝑡 𝑑𝑡 + 𝜙𝑡 𝜎(𝑆𝑡 , 𝑡)𝑆𝑡 𝑑𝑊𝑡 ( ) = 𝑟Π𝑡 𝑑𝑡 + 𝜙𝑡 𝜎(𝑆𝑡 , 𝑡)𝑆𝑡 𝜆𝑡 𝑑𝑡 + 𝑑𝑊𝑡 𝜇−𝑟 . 𝜎(𝑆𝑡 , 𝑡) From the discounted portfolio,

where 𝜆𝑡 =

𝑑(𝑒−𝑟𝑡 Π𝑡 ) = −𝑟𝑒−𝑟𝑡 Π𝑡 𝑑𝑡 + 𝑒−𝑟𝑡 𝑑Π𝑡 ( ) = 𝑒−𝑟𝑡 𝜙𝑡 𝜎(𝑆𝑡 , 𝑡)𝑆𝑡 𝜆𝑡 𝑑𝑡 + 𝑑𝑊𝑡 ̃𝑡 = 𝜙𝑡 𝜎(𝑆𝑡 , 𝑡)𝑒−𝑟𝑡 𝑑 𝑊 where ̃𝑡 = 𝑊𝑡 + 𝑊

𝑡

𝜇−𝑟 𝑑𝑢. ∫0 𝜎(𝑆𝑢 , 𝑢)

From Girsanov’s theorem, there exists an equivalent martingale measure or risk-neutral measure ℚ on the filtration ℱ𝑠 , 0 ≤ 𝑠 ≤ 𝑡 defined by the Radon–Nikod´ym derivative 𝑡 2

1

𝑡

𝑍𝑠 = 𝑒− ∫0 𝜆𝑢 𝑑𝑢− 2 ∫0 𝜆𝑢 𝑑𝑊𝑢 ̃𝑡 is a ℚ-standard Wiener process. Given that under the risk-neutral measure ℚ, so that 𝑊 the discounted portfolio has no 𝑑𝑡 term, therefore 𝑒−𝑟𝑡 Π𝑡 is a ℚ-martingale.

696

7.2.2 Local Volatility

By substituting ̃𝑡 − 𝜇 − 𝑟 𝑑𝑡 𝑑𝑊𝑡 = 𝑑 𝑊 𝜎(𝑆𝑡 , 𝑡) into 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎(𝑆𝑡 , 𝑡)𝑑𝑊𝑡 𝑆𝑡 the asset price diffusion process under ℚ becomes 𝑑𝑆𝑡 ̃𝑡 . = (𝑟 − 𝐷) 𝑑𝑡 + 𝜎(𝑆𝑡 , 𝑡)𝑑 𝑊 𝑆𝑡 From the Chapman–Kolmogorov equation for Δ𝑇 > 0, 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 + Δ𝑇 ) =

∞

∫0

𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 )𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 + Δ𝑇 ) 𝑑𝑧

we can write 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 + Δ𝑇 ) − 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) ∞

=

∫0 ∞

=

∫0

𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 )𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 + Δ𝑇 ) 𝑑𝑧 − 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 )𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 + Δ𝑇 ) 𝑑𝑧 ∞

−

∫0 ∞

=

∫0

𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 )𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 + Δ𝑇 ) 𝑑𝑧 ∞

−

∫0

𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 )𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) 𝑑𝑧

𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 )𝛿(𝑧 − 𝑆𝑇 ) 𝑑𝑧.

By dividing the expression with Δ𝑇 and taking limits Δ𝑇 → 0, 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 + Δ𝑇 ) − 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) Δ𝑇 →0 Δ𝑇 ∞ 𝑝(𝑆 , 𝑡; 𝑧, 𝑇 )𝑝(𝑧, 𝑇 ; 𝑆 , 𝑇 + Δ𝑇 ) 𝑡 𝑇 𝑑𝑧 = lim Δ𝑡→0 ∫0 Δ𝑇 ∞ 𝑝(𝑆 , 𝑡; 𝑧, 𝑇 )𝛿(𝑧 − 𝑆 ) 𝑡 𝑇 𝑑𝑧 − lim Δ𝑇 →0 ∫0 Δ𝑇 [ ] ∞ 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 + Δ𝑇 ) − 𝛿(𝑧 − 𝑆𝑇 ) = lim 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 Δ𝑇 →0 ∫0 Δ𝑇 [ ] ∞ 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 + Δ𝑇 ) − 𝑝(𝑧, 𝑇 + Δ𝑇 ; 𝑆𝑇 , 𝑇 + Δ𝑇 ) 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) = lim 𝑑𝑧. Δ𝑇 →0 ∫0 Δ𝑇 lim

7.2.2 Local Volatility

697

Since 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 + Δ𝑇 ) − 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) 𝜕 = 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) Δ𝑇 →0 Δ𝑇 𝜕𝑇 lim

and

lim

Δ𝑇 →0

𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 + Δ𝑇 ) − 𝑝(𝑧, 𝑇 + Δ𝑇 ; 𝑆𝑇 , 𝑇 + Δ𝑇 ) 𝜕 = − 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) Δ𝑇 𝜕𝑇

thus 𝜕 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) = − ∫0 𝜕𝑇

∞

𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 )

𝜕 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) 𝑑𝑧. 𝜕𝑇

From the backward Kolmogorov equation on 𝑝(𝑧, 𝑡; 𝑆𝑇 , 𝑇 ) for the local volatility model 1 𝜕 𝜕2 𝜕 𝑝(𝑧, 𝑡; 𝑆𝑇 , 𝑇 ) + 𝜎(𝑧, 𝑡)2 𝑧2 2 𝑝(𝑧, 𝑡; 𝑆𝑇 , 𝑇 ) + (𝑟 − 𝐷)𝑧 𝑝(𝑧, 𝑡; 𝑆𝑇 , 𝑇 ) = 0 𝜕𝑡 2 𝜕𝑧 𝜕𝑧 we have 𝜕 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) 𝜕𝑇 [ ] ∞ 𝜕 1 𝜕2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) (𝑟 − 𝐷)𝑧 𝑧 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) + 𝜎(𝑧, 𝑇 )2 𝑧2 2 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) 𝑑𝑧 = ∫0 𝜕 2 𝜕𝑧 ∞

=

∫0

∞

+

∫0

𝜕 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) 𝑑𝑧 𝜕𝑧 𝜕2 1 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 2 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) 𝑑𝑧. 2 𝜕𝑧

(𝑟 − 𝐷)𝑧 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 )

∞

Integrating by parts for

∫0

(𝑟 − 𝐷)𝑧 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 )

𝑢 = (𝑟 − 𝐷)𝑧 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 )

and

𝜕 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) 𝑑𝑧, we let 𝜕𝑧 𝑑𝑣 𝜕 = 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) 𝑑𝑧 𝜕𝑧

so that ] 𝑑𝑢 𝜕 [ (𝑟 − 𝐷)𝑧 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) = 𝑑𝑧 𝜕𝑧

and

𝑣 = 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ).

698

7.2.2 Local Volatility

We then have ∞

𝜕 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 )𝑑𝑧 𝜕𝑧 ∞ = (𝑟 − 𝐷)𝑧 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 )𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 )||0 ∞ ] 𝜕 [ (𝑟 − 𝐷)𝑧 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 − 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) ∫0 𝜕𝑧 ∞ ] 𝜕 [ (𝑟 − 𝐷)𝑧 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧. 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) =− ∫0 𝜕𝑧 ∫0

(𝑟 − 𝐷)𝑧 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 )

∞

Taking integration by parts for

𝑢=

∫0

𝜕2 1 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 2 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) 𝑑𝑧, we let 2 𝜕𝑧

1 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 2

𝑑𝑣 𝜕2 = 2 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) 𝑑𝑧 𝜕𝑧

and

so that [ ] 𝜕 1 𝑑𝑢 = 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 𝜕𝑧 2

𝑣=

and

𝜕 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) 𝜕𝑧

and hence ∞

𝜕2 1 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 2 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) 𝑑𝑧 ∫0 2 𝜕𝑧 |∞ 𝜕 1 = 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 )|| 2 𝜕𝑧 |0 [ ] ∞ 𝜕 1 𝜕 − 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 ∫0 𝜕𝑧 𝜕𝑧 2 [ ] ∞ 𝜕 1 𝜕 =− 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧. ∫0 𝜕𝑧 𝜕𝑧 2 Using integration by parts again, we let 𝑢=

[ ] 𝜕 1 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝜕𝑧 2

and

𝑑𝑣 𝜕 = 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) 𝑑𝑧 𝜕𝑧

so that [ ] 𝜕2 1 𝑑𝑢 = 2 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 𝜕𝑧 2

and

𝑣 = 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 )

7.2.2 Local Volatility

699

and therefore ∞

𝜕2 1 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 2 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) 𝑑𝑧 ∫0 2 𝜕𝑧 [ ] |∞ 𝜕 1 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 )|| =− 𝜕𝑧 2 |0 [ ] ∞ 2 1 𝜕 + 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) 2 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 ∫0 𝜕𝑧 2 [ ] ∞ 2 1 𝜕 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) 2 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧. = ∫0 𝜕𝑧 2 Thus, 𝜕 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) 𝜕𝑇

] 𝜕 [ (𝑟 − 𝐷)𝑧 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 ∫0 𝜕𝑧 [ ] ∞ 𝜕2 1 𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 ) 2 + 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 ∫0 𝜕𝑧 2 } ] ∞{ 2 [ ] 1 𝜕 [ 𝜕 2 2 (𝑟 − 𝐷)𝑧 𝑝(𝑆 = 𝑧 𝑝(𝑆 , 𝑡; 𝑧, 𝑇 ) − , 𝑡; 𝑧, 𝑇 ) 𝜎(𝑧, 𝑇 ) 𝛿(𝑧 − 𝑆𝑇 ) 𝑑𝑧 𝑡 𝑡 ∫0 𝜕𝑧 𝜕𝑧2 2 [ ] ] 𝜕 [ 𝜕2 1 2 2 (𝑟 − 𝐷)𝑆𝑇 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) , 𝑇 ) 𝑆 𝑝(𝑆 , 𝑡; 𝑆 , 𝑇 ) − 𝜎(𝑆 = 𝑇 𝑡 𝑇 𝑇 2 𝜕𝑆𝑇 𝜕𝑆 2 ∞

=−

𝑝(𝑧, 𝑇 ; 𝑆𝑇 , 𝑇 )

𝑇

and rearranging terms we have the forward Kolmogorov equation ] 1 𝜕2 [ 𝜕 2 2 𝜎(𝑆 , 𝑇 ) 𝑆 𝑝(𝑆 , 𝑡; 𝑆 , 𝑇 ) 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) − 𝑇 𝑡 𝑇 𝑇 𝜕𝑇 2 𝜕𝑆 2 𝑇 ] 𝜕 [ 𝑆 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) = 0 +(𝑟 − 𝐷) 𝜕𝑆𝑇 𝑇 with boundary condition 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑡) = 𝛿(𝑆𝑡 − 𝑆𝑇 ), ∀𝑡. { } 5. Dupire Equation. Let 𝑊𝑡 : 𝑡 ≥ 0 be a ℙ-standard Wiener process on the probability space (Ω, ℱ, ℙ) and let the stock price 𝑆𝑡 follow a local volatility model with the following SDE 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎(𝑆𝑡 , 𝑡)𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎(𝑆𝑡 , 𝑡) is the local volatility function and let 𝑟 be the risk-free interest rate.

700

7.2.2 Local Volatility

Let 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) be the transition pdf of the stock price where the stock is worth 𝑆𝑡 at time 𝑡 given that the stock price is 𝑆𝑇 at time 𝑇 > 𝑡. We consider a European call option written on 𝑆𝑡 with strike price 𝐾 > 0 expiring at time 𝑇 > 𝑡 with payoff Ψ(𝑆𝑇 ) = max{𝑆𝑇 − 𝐾, 0} where, under the risk-neutral measure ℚ, the option price at time 𝑡 is [ ] 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑆𝑇 − 𝐾, 0}|| ℱ𝑡 = 𝑒−𝑟(𝑇 −𝑡)

∞

∫0

max{𝑧 − 𝐾, 0}𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧.

Show that the following identities are true

𝑧

𝜕 (𝑧 − 𝐾) = 𝜕𝐾 𝜕 (𝑧 − 𝐾) = 𝜕𝑧 𝜕2 (𝑧 − 𝐾) = 𝜕𝑧2

𝜕 (𝑧 − 𝐾) − (𝑧 − 𝐾) 𝜕𝐾 𝜕 − (𝑧 − 𝐾) 𝜕𝐾 𝜕2 (𝑧 − 𝐾). 𝜕𝐾 2 𝐾

By using the forward Kolmogorov equation and the above identities or otherwise, show that 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) satisfies the Dupire equation 𝜕𝐶 𝜕2𝐶 𝜕𝐶 1 + (𝑟 − 𝐷)𝐾 − 𝜎(𝐾, 𝑇 )2 𝐾 2 + 𝐷𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 𝜕𝑇 2 𝜕𝐾 𝜕𝐾 2 with boundary condition 𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0}. Explain the significance of this equation. Finally, deduce that a European put option price 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) written on 𝑆𝑡 at time 𝑡, with payoff Ψ(𝑆𝑇 ) = max{𝐾 − 𝑆𝑇 , 0} expiring at time 𝑇 > 𝑡, satisfies 𝜕𝑃 𝜕𝑃 𝜕2 𝑃 1 + (𝑟 − 𝐷)𝐾 − 𝜎(𝐾, 𝑇 )2 𝐾 2 + 𝐷𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 2 𝜕𝑇 2 𝜕𝐾 𝜕𝐾 with boundary condition 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝐾 − 𝑆𝑇 , 0}.

7.2.2 Local Volatility

701

Solution: To show the first identity, we can write 𝑧

𝜕 𝜕 (𝑧 − 𝐾) = (𝑧 − 𝐾 + 𝐾) (𝑧 − 𝐾) 𝜕𝐾 𝜕𝐾 𝜕 𝜕 (𝑧 − 𝐾) + 𝐾 (𝑧 − 𝐾) = (𝑧 − 𝐾) 𝜕𝐾 𝜕𝐾 𝜕 (𝑧 − 𝐾) = −(𝑧 − 𝐾) + 𝐾 𝜕𝐾 𝜕 (𝑧 − 𝐾) − (𝑧 − 𝐾). =𝐾 𝜕𝐾

As for the second identity, we note that 𝜕 𝜕 (𝑧 − 𝐾) = 1 = − (𝑧 − 𝐾). 𝜕𝑧 𝜕𝐾 Finally, [ ] 𝜕2 𝜕 𝜕 (𝑧 − 𝐾) = (𝑧 − 𝐾) 𝜕𝑧 𝜕𝑧 𝜕𝑧2 [ ] 𝜕 𝜕 (𝑧 − 𝐾) =− 𝜕𝑧 𝜕𝐾 [ ] 𝜕 𝜕 (𝑧 − 𝐾) =− 𝜕𝐾 𝜕𝑧 𝜕2 (𝑧 − 𝐾). = 𝜕𝐾 2 By definition [ ] 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑆𝑇 − 𝐾, 0}|| ℱ𝑡 = 𝑒−𝑟(𝑇 −𝑡) = 𝑒−𝑟(𝑇 −𝑡)

∞

∫0 ∞

∫𝐾

max{𝑧 − 𝐾, 0}𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 (𝑧 − 𝐾)𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧.

Differentiating 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) with respect to 𝑇 and from the forward Kolmogorov equation ] 𝜕 1 𝜕2 [ 𝜎(𝑆𝑇 , 𝑇 )2 𝑆𝑇2 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) − 𝜕𝑇 2 𝜕𝑆 2 𝑇 ] 𝜕 [ 𝑆𝑇 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) = 0 +(𝑟 − 𝐷) 𝜕𝑆𝑇

702

7.2.2 Local Volatility

we have ∞

𝜕𝐶 (𝑧 − 𝐾)𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 = −𝑟𝑒−𝑟(𝑇 −𝑡) ∫𝐾 𝜕𝑇 ∞ 𝜕 (𝑧 − 𝐾) 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 +𝑒−𝑟(𝑇 −𝑡) ∫𝐾 𝜕𝑇 ∞ ] 1 𝜕2 [ (𝑧 − 𝐾) 2 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 = −𝑟𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑒−𝑟(𝑇 −𝑡) ∫𝐾 2 𝜕𝑧 ∞ ] 𝜕 [ −𝑟(𝑇 −𝑡) 𝑧 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧. (𝑧 − 𝐾) −(𝑟 − 𝐷)𝑒 ∫𝐾 𝜕𝑧 ∞

Integrate by parts for

∫𝐾

] 𝜕 [ 𝑧 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 and let 𝜕𝑧

(𝑧 − 𝐾)

] 𝜕 [ 𝑑𝑣 𝑧 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) = 𝑑𝑧 𝜕𝑧

𝑢 = (𝑧 − 𝐾), so that

𝑑𝑢 𝜕 𝜕 = (𝑧 − 𝐾) = − (𝑧 − 𝐾) and 𝑑𝑧 𝜕𝑧 𝜕𝐾

𝑣 = 𝑧 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ).

Therefore, ∞

∫𝐾

(𝑧 − 𝐾)

] 𝜕 [ 𝑧 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 𝜕𝑧

∞ = (𝑧 − 𝐾)𝑧 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 )||𝐾 + ∞

∞

∫𝐾

𝑧 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 )

𝜕 (𝑧 − 𝐾) 𝑑𝑧 𝜕𝐾

𝜕 (𝑧 − 𝐾) 𝑑𝑧 𝜕𝐾 ] ∞[ 𝜕 𝐾 (𝑧 − 𝐾) − (𝑧 − 𝐾) 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 = ∫𝐾 𝜕𝐾 ∞ ∞ ] 𝜕 [ (𝑧 − 𝐾)𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 − (𝑧 − 𝐾)𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 =𝐾 ∫𝐾 ∫𝐾 𝜕𝐾

=

∫𝐾

𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 )𝑧

and hence 𝑒−𝑟(𝑇 −𝑡)

∞

∫𝐾

= 𝐾𝑒−𝑟(𝑇 −𝑡)

∞

∫𝐾

∞

−𝑒−𝑟(𝑇 −𝑡) =𝐾

] 𝜕 [ 𝑧 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 𝜕𝑧 ] 𝜕 [ (𝑧 − 𝐾)𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 𝜕𝐾

(𝑧 − 𝐾)

∫𝐾

(𝑧 − 𝐾)𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧

𝜕𝐶 − 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). 𝜕𝐾

7.2.2 Local Volatility ∞

For

∫𝐾

703

(𝑧 − 𝐾)

] 𝜕2 [ 2 2 𝜎(𝑧, 𝑇 ) 𝑧 𝑝(𝑆 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 we let 𝑡 𝜕𝑧2

𝑢 = (𝑧 − 𝐾) and

] 𝑑𝑣 𝜕2 [ = 2 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 𝜕𝑧

so that 𝜕 𝑑𝑢 = (𝑧 − 𝐾) and 𝑑𝑧 𝜕𝑧

𝑣=

] 𝜕 [ 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝜕𝑧

and therefore ] 𝜕2 [ 2 2 𝜎(𝑧, 𝑇 ) 𝑧 𝑝(𝑆 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 𝑡 ∫𝐾 𝜕𝑧2 ]|∞ 𝜕 [ 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) || = (𝑧 − 𝐾) 𝜕𝑧 |𝐾 ∞ ] 𝜕 𝜕 [ 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 − (𝑧 − 𝐾) ∫𝐾 𝜕𝑧 𝜕𝑧 ∞ ] 𝜕 𝜕 [ 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧. =− (𝑧 − 𝐾) ∫𝐾 𝜕𝑧 𝜕𝑧 ∞

∞

To integrate −

∫𝐾 𝑢=

(𝑧 − 𝐾)

] 𝜕 𝜕 [ 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 we set (𝑧 − 𝐾) 𝜕𝑧 𝜕𝑧

𝜕 (𝑧 − 𝐾) and 𝜕𝑧

] 𝑑𝑣 𝜕 [ 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) = 𝑑𝑧 𝜕𝑧

so that 𝜕2 𝑑𝑢 = 2 (𝑧 − 𝐾) and 𝑑𝑧 𝜕𝑧

𝑣 = 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 )

and hence ] 𝜕2 [ 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 2 𝜕𝑧 |∞ 𝜕 = − 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) (𝑧 − 𝐾)|| 𝜕𝑧 |𝐾 ∞ 𝜕2 2 2 + 𝜎(𝑧, 𝑇 ) 𝑧 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 2 (𝑧 − 𝐾) 𝑑𝑧 ∫𝐾 𝜕𝑧 ∞

−

∫𝐾

(𝑧 − 𝐾)

= 𝜎(𝐾, 𝑇 )2 𝐾 2 𝑝(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) +

∞

∫𝐾

𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 )

= 𝜎(𝐾, 𝑇 ) 𝐾 𝑝(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) 2

since

2

𝜕2 𝜕2 𝜕 (𝑧 − 𝐾) = 0. (𝑧 − 𝐾) = 1 and 2 (𝑧 − 𝐾) = 𝜕𝑧 𝜕𝑧 𝜕𝐾 2

𝜕2 (𝑧 − 𝐾) 𝑑𝑧 𝜕𝐾 2

704

7.2.2 Local Volatility

Given that ( ) 𝜕𝐶 𝜕2𝐶 𝜕 = 𝜕𝐾 𝜕𝐾 𝜕𝐾 2 ] [ ∞ ] 𝜕 𝜕 [ −𝑟(𝑇 −𝑡) (𝑧 − 𝐾)𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 = 𝑒 ∫𝐾 𝜕𝐾 𝜕𝐾 ] [ ∞ 𝜕 𝜕2 −𝑟(𝑇 −𝑡) =𝑒 (𝑧 − 𝐾)𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 (𝑧 − 𝐾) + −𝑝(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) ∫𝐾 𝜕𝐾 2 𝜕𝐾 = 𝑒−𝑟(𝑇 −𝑡) 𝑝(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) therefore ∞ ] 𝜕2 [ 1 𝜕2𝐶 1 −𝑟(𝑇 −𝑡) (𝑧 − 𝐾) 2 𝜎(𝑧, 𝑇 )2 𝑧2 𝑝(𝑆𝑡 , 𝑡; 𝑧, 𝑇 ) 𝑑𝑧 = 𝜎(𝐾, 𝑇 )2 𝐾 2 . 𝑒 ∫𝐾 2 2 𝜕𝑧 𝜕𝐾 2

Thus, [ ] 1 𝜕𝐶 𝜕𝐶 𝜕2𝐶 − (𝑟 − 𝐷) 𝐾 , 𝑡; 𝐾, 𝑇 ) = −𝑟𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝜎(𝐾, 𝑇 )2 𝐾 2 − 𝐶(𝑆 𝑡 𝜕𝑇 2 𝜕𝐾 𝜕𝐾 2 or 𝜕𝐶 𝜕2𝐶 𝜕𝐶 1 + (𝑟 − 𝐷)𝐾 − 𝜎(𝐾, 𝑇 )2 𝐾 2 + 𝐷𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 𝜕𝑇 2 𝜕𝐾 𝜕𝐾 2 with boundary condition 𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0}. From the equation, we can write ( 2 𝜎(𝐾, 𝑇 )2 =

𝜕𝐶 𝜕𝐶 + (𝑟 − 𝐷)𝐾 + 𝐷𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) 𝜕𝑇 𝜕𝐾 𝜕2𝐶 𝐾2 𝜕𝐾 2

)

such that given the current spot price 𝑆𝑡 , the local volatility function 𝜎(𝑆𝑡 , 𝑡) can be obtained from market quotes of call options of arbitrary strikes 𝐾 and expiry times 𝑇 . From the put–call parity 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡)

7.2.2 Local Volatility

705

we have 𝜕𝐶 𝜕𝑃 = − 𝐷𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) + 𝑟𝐾𝑒−𝑟(𝑇 −𝑡) 𝜕𝑇 𝜕𝑇 𝜕𝐶 𝜕𝑃 = − 𝑒−𝑟(𝑇 −𝑡) 𝜕𝐾 𝜕𝐾 𝜕2 𝑃 𝜕2𝐶 = . 𝜕𝐾 2 𝜕𝐾 2 Substituting the above information into the Dupire equation for 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ), we have ] [ 1 𝜕𝑃 𝜕2 𝑃 𝜕𝑃 −𝑟(𝑇 −𝑡) + (𝑟 − 𝐷)𝐾 − 𝐷𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) + 𝑟𝐾𝑒−𝑟(𝑇 −𝑡) − 𝜎(𝐾, 𝑇 )2 − 𝑒 𝜕𝑇 2 𝜕𝐾 𝜕𝐾 2 ] [ +𝐷 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) = 0 or 𝜕𝑃 𝜕𝑃 𝜕2𝑃 1 + (𝑟 − 𝐷)𝐾 − 𝜎(𝐾, 𝑇 )2 𝐾 2 + 𝐷𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 𝜕𝑇 2 𝜕𝐾 𝜕𝐾 2 with boundary condition 𝑃 (𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝐾 − 𝑆𝑇 , 0}.

6. Time-Dependent Volatility. Consider a time-dependent (or term-structure) volatility function 𝜎(𝑡) and the implied volatility function 𝜎imp (𝑡, 𝑇 ), which is a function of 𝑡 and 𝑇 , 𝑇 > 𝑡 and assume their relationship is defined by 2 (𝑡, 𝑇 ) = 𝜎imp

1 𝑇 − 𝑡 ∫𝑡

𝑇

𝜎 2 (𝑢) 𝑑𝑢.

For two given expiry times 𝑇1 < 𝑇2 , show that 𝑇2

∫𝑇1

2 2 𝜎 2 (𝑢) 𝑑𝑢 = (𝑇2 − 𝑡)𝜎imp (𝑡, 𝑇2 ) − (𝑇1 − 𝑡)𝜎imp (𝑡, 𝑇1 ).

If the volatility is a piecewise constant over the time interval [𝑇1 , 𝑇2 ], find 𝜎(𝑢), 𝑇1 < 𝑢 < 𝑇2 . Solution: By definition 𝑇2

∫𝑇1

𝜎 2 (𝑢) 𝑑𝑢 =

𝑇2

∫𝑡

𝜎 2 (𝑢) 𝑑𝑢 −

𝑇1

∫𝑡

𝜎 2 (𝑢) 𝑑𝑢

2 2 (𝑡, 𝑇2 ) − (𝑇1 − 𝑡)𝜎imp (𝑡, 𝑇1 ). = (𝑇2 − 𝑡)𝜎imp

706

7.2.2 Local Volatility

Assuming 𝑇2

∫ 𝑇1

𝜎 2 (𝑢) 𝑑𝑢 = 𝜎 2 (𝑢)(𝑇2 − 𝑇1 ), 𝑇1 < 𝑢 < 𝑇2

and using the above result, we have 2 2 (𝑡, 𝑇2 ) − (𝑇1 − 𝑡)𝜎imp (𝑡, 𝑇1 ) 𝜎 2 (𝑢)(𝑇2 − 𝑇1 ) = (𝑇2 − 𝑡)𝜎imp

or √ √ √ (𝑇2 − 𝑡)𝜎 2 (𝑡, 𝑇2 ) − (𝑇1 − 𝑡)𝜎 2 (𝑡, 𝑇1 ) √ imp imp , 𝑇 1 < 𝑢 < 𝑇2 . 𝜎(𝑢) = 𝑇2 − 𝑇1

7. Relationship Between Local Volatility and Implied Volatility. Let the relationship between the market-quoted call prices 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) and their corresponding implied volatilities 𝜎imp (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) be given by the Black–Scholes formula imp

𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑+ ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑−imp ) 2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎imp 𝑑± = √ 𝜎imp 𝑇 − 𝑡 𝑥

1 2 1 where Φ(𝑥) = √ 𝑒− 2 𝑥 𝑑𝑥 is the cdf of a standard normal and 𝜎imp ≡ ∫ 2𝜋 −∞ 𝜎imp (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ). From the Dupire equation, the local volatility 𝜎(𝐾, 𝑇 ) is extracted from

( 2 𝜎(𝐾, 𝑇 )2 =

) 𝜕𝐶 𝜕𝐶 + (𝑟 − 𝐷)𝐾 + 𝐷𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) 𝜕𝑇 𝜕𝐾 . 𝜕2 𝐶 𝐾2 𝜕𝐾 2

By writing the Black–Scholes formula in the form 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑆𝑡 𝑁1 − 𝐾𝑁2 imp

where 𝑁1 = 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑+ ), 𝑁2 = 𝑒−𝑟(𝑇 −𝑡) Φ(𝑑−imp ) and using the property imp

𝑑+ show that

√ = 𝑑−imp + 𝜎imp (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) 𝑇 − 𝑡

7.2.2 Local Volatility

(a) 𝑆𝑡

𝜕𝑁1

imp 𝜕𝑑+ imp 𝜕𝑑+

707

−𝐾

𝜕𝑁2 imp

𝜕𝑑−

=0

√ 𝜕𝜎imp 𝜎imp + 𝑇 −𝑡 = √ 𝜕𝑇 𝜕𝑇 𝜕𝑇 2 𝑇 −𝑡 imp 𝜕𝜎imp 𝜕𝑑 imp √ 𝜕𝑑+ − − = 𝑇 −𝑡 (c) 𝜕𝐾 𝜕𝐾 𝜕𝐾 ) ( 𝜕𝜎imp 𝜕𝑑−imp 𝑑−imp √ 1 + 𝑇 −𝑡 (d) − =− . √ 𝜕𝐾 𝜎imp 𝜕𝐾 𝐾𝜎imp 𝑇 − 𝑡 Hence, using the above properties or otherwise, show that the local volatility function in terms of implied volatility is (b)

−

𝜕𝑑−imp

(

𝜎(𝐾, 𝑇 )2 = 𝐾2

𝜕 2 𝜎imp 𝜕𝐾 2

) 1 𝜎imp 2 + (𝑟 − 𝐷)𝐾 + 𝜕𝑇 𝜕𝐾 2𝑇 −𝑡 . ) ( imp imp 𝜕𝜎imp 𝐾 2 𝑑+ 𝑑−imp 𝜕𝜎imp 2 2𝐾𝑑+ 1 + + + √ √ 𝜎imp 𝜕𝐾 𝜎imp 𝑇 − 𝑡 𝜕𝐾 𝜎imp 𝑇 − 𝑡 𝜕𝜎imp

𝜕𝜎imp

Solution: √ imp (a) Using the identity 𝑑+ = 𝑑−imp − 𝜎imp 𝑇 − 𝑡, from Problem 2.2.4.1 (page 218), we can easily show that 1

imp 2 )

𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑒− 2 (𝑑+

1

imp )2

= 𝐾𝑒−𝑟(𝑇 −𝑡) 𝑒− 2 (𝑑−

or 1 imp 2 1 imp 2 1 1 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) √ 𝑒− 2 (𝑑+ ) = 𝐾𝑒−𝑟(𝑇 −𝑡) √ 𝑒− 2 (𝑑− ) 2𝜋 2𝜋

which implies 𝑆𝑡

𝜕𝑁1 imp 𝜕𝑑+

=𝐾

𝜕𝑁2 imp

𝜕𝑑−

and hence 𝑆𝑡

𝜕𝑁1 imp 𝜕𝑑+

−𝐾

𝜕𝑁2 imp

𝜕𝑑−

= 0.

√ imp (b) From 𝑑+ − 𝑑−imp = 𝜎imp 𝑇 − 𝑡 and taking partial derivatives with respect to 𝑇 , we have imp

𝜕𝑑+

𝜕𝑇

−

𝜕𝑑−imp 𝜕𝑇

√ 𝜕𝜎imp 𝜎imp + 𝑇 −𝑡 = √ . 𝜕𝑇 2 𝑇 −𝑡

708

7.2.2 Local Volatility

√ imp (c) From 𝑑+ − 𝑑−imp = 𝜎imp 𝑇 − 𝑡 and taking partial derivatives with respect to 𝐾, we have imp

𝜕𝑑+

𝜕𝐾

−

𝜕𝑑−imp 𝜕𝐾

=

√ 𝜕𝜎imp 𝑇 −𝑡 . 𝜕𝐾

2 )(𝑇 − 𝑡) log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 − 12 𝜎imp = and taking partial derivatives with (d) From √ 𝜎imp 𝑇 − 𝑡 respect to 𝐾, we have

𝑑−imp

𝜕𝑑−imp 𝜕𝐾

=

( ) √ 𝜕𝜎imp 1 −𝜎imp 𝑇 − 𝑡 + 𝜎imp (𝑇 − 𝑡) 𝐾 𝜕𝐾 2 (𝑇 − 𝑡) 𝜎imp

√

𝑇 −𝑡

−

𝜕𝜎imp ( 𝜕𝐾

) ) ( 1 2 log(𝑆𝑡 ∕𝐾) + 𝑟 − 𝐷 − 𝜎imp (𝑇 − 𝑡) 2 2 (𝑇 − 𝑡) 𝜎imp

√ 𝜕𝜎imp 𝑑−imp 𝜕𝜎imp 1 𝑇 − 𝑡 − − √ 𝜕𝐾 𝜎imp 𝜕𝐾 𝐾𝜎imp 𝑇 − 𝑡 ( ) 𝜕𝜎imp 𝑑−imp √ 1 =− − + 𝑇 −𝑡 . √ 𝜎imp 𝜕𝐾 𝐾𝜎imp 𝑇 − 𝑡 =−

Using the above information, imp

𝜕𝐶 imp imp 𝜕𝑑+ = −𝐷𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝑑+ ) + 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ′ (𝑑+ ) 𝜕𝑇 𝜕𝑇 +𝑟𝐾𝑒−𝑟(𝑇 −𝑡) Φ(𝑑−imp ) − 𝐾𝑒−𝑟(𝑇 −𝑡) Φ′ (𝑑−imp ) imp

𝜕𝑑−imp 𝜕𝑇

𝜕𝑁2 𝜕𝑑−imp 𝜕𝑁1 𝜕𝑑+ − 𝐾 imp 𝜕𝑇 imp 𝜕𝑇 𝜕𝑑+ 𝜕𝑑− ) ( imp 𝜕𝑑−imp 𝜕𝑑+ 𝜕𝑁2 = −𝐷𝑆𝑡 𝑁1 + 𝑟𝐾𝑁2 + 𝐾 imp − 𝜕𝑇 𝜕𝑇 𝜕𝑑− ( ) √ 𝜎imp 𝜕𝜎imp 𝜕𝑁2 = −𝐷𝑆𝑡 𝑁1 + 𝑟𝐾𝑁2 + 𝐾 imp + 𝑇 −𝑡 √ 𝜕𝑇 𝜕𝑑− 2 𝑇 −𝑡

= −𝐷𝑆𝑡 𝑁1 + 𝑟𝐾𝑁2 + 𝑆𝑡

7.2.2 Local Volatility

709

[ imp 𝜕𝑑+ 𝜕𝐶 −𝐷(𝑇 −𝑡) ′ imp (𝑟 − 𝐷)𝐾 Φ (𝑑+ ) = (𝑟 − 𝐷)𝐾 𝑆𝑡 𝑒 − 𝑒−𝑟(𝑇 −𝑡) Φ(𝑑−imp ) 𝜕𝐾 𝜕𝐾 ] 𝜕𝑑−imp −𝑟(𝑇 −𝑡) ′ imp −𝐾𝑒 Φ (𝑑− ) 𝜕𝐾 [ ( imp ) 𝜕𝑑+ 𝜕𝑑−imp −𝑟(𝑇 −𝑡) ′ imp = (𝑟 − 𝐷)𝐾 𝐾𝑒 Φ (𝑑− ) − 𝜕𝐾 𝜕𝐾 ] −𝑒−𝑟(𝑇 −𝑡) Φ(𝑑−imp ) = (𝑟 − 𝐷)𝐾 2

𝜕𝜎imp 𝜕𝑁2 √ 𝑇 − 𝑡 − (𝑟 − 𝐷)𝐾𝑁2 imp 𝜕𝐾 𝜕𝑑−

𝐷𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝐷𝑆𝑡 𝑁1 − 𝐷𝐾𝑁2 and )] 𝜕𝜎imp 𝜕𝑁2 √ =𝐾 𝐾 imp 𝑇 − 𝑡 − 𝑁2 𝐾 𝜕𝐾 𝜕𝐾 2 𝜕𝑑− [ 𝜕𝜎imp 𝜕𝜎imp 𝜕𝑁2 √ 𝜕𝑁2 𝜕𝑑−imp √ = 𝐾2 𝑇 −𝑡 𝑇 −𝑡 − 𝐾𝑑−imp imp 𝜕𝑑− 𝜕𝐾 𝜕𝐾 𝜕𝑑− 𝜕𝐾 ] 𝜕 2 𝜎imp 𝜕𝑁2 𝜕𝑑−imp 𝜕𝑁2 √ 𝑇 −𝑡 − +𝐾 imp imp 𝜕𝐾 2 𝜕𝑑− 𝜕𝑑− 𝜕𝐾 [ √ 𝜕𝜎imp 𝜕𝑑 imp 𝜕𝜎imp 𝜕𝑁2 √ 𝑇 −𝑡 = 𝐾 2 imp − 𝐾 𝑇 − 𝑡𝑑−imp − 𝜕𝐾 𝜕𝐾 𝜕𝐾 𝜕𝑑− ] √ 𝜕 2 𝜎imp 𝜕𝑑−imp − +𝐾 𝑇 − 𝑡 𝜕𝐾 𝜕𝐾 2 [ √ 𝜕𝜎imp 𝜕 2 𝜎imp 𝜕𝑁2 √ 𝑇 −𝑡 = 𝐾 2 imp +𝐾 𝑇 −𝑡 𝜕𝐾 𝜕𝐾 2 𝜕𝑑− ) ( ( )⎛ √ 𝜕𝜎imp ⎞⎤ 𝜕𝜎imp 𝑑−imp √ 1 imp ⎟⎥ + 𝐾𝑑− 𝑇 −𝑡 + 𝑇 − 𝑡 + +1 ⎜ √ ⎜ 𝐾𝜎 𝜕𝐾 𝜎imp 𝜕𝐾 ⎟⎥ 𝑇 − 𝑡 ⎝ imp ⎠⎦ ) [ ( 2𝜎 imp √ √ 𝜕𝜎 𝜕 𝑑 𝜕𝑁2 imp imp − = 𝐾 2 imp + 𝑇 −𝑡 +𝐾 𝑇 −𝑡 2 2 𝜎 𝜕𝐾 𝜕𝐾 imp 𝜕𝑑− ( )( ) ⎤ √ 𝜕𝜎imp 2 𝑑−imp √ 1 imp ⎥ +𝐾𝑑− 𝑇 −𝑡 + 𝑇 −𝑡 + √ 𝜎imp 𝜕𝐾 𝐾𝜎imp 𝑇 − 𝑡 ⎥⎦ 2𝜕

2𝐶

[

2

𝜕 𝜕𝐾

(

710

7.2.3 Stochastic Volatility

=𝐾

2

𝜕𝑁2

[

imp

2𝑑+

√ 𝜕 2 𝜎imp +𝐾 𝑇 −𝑡 𝜕𝐾 𝜕𝐾 2

𝜕𝜎imp

imp 𝜎imp 𝜕𝑑− ( imp ) ( ) ⎤ 𝜕𝜎imp 2 𝑑+ 𝑑−imp √ 1 ⎥ +𝐾 𝑇 −𝑡 + √ 𝜎imp 𝜕𝐾 𝐾𝜎imp 𝑇 − 𝑡 ⎥⎦ imp ⎡ 𝜕2 𝜎 𝜕𝜎imp 2𝐾𝑑+ 𝜕𝑁2 √ imp 𝑇 − 𝑡 ⎢𝐾 2 + √ 2 ⎢ 𝜕𝑑− 𝜕𝐾 𝜎imp 𝑇 − 𝑡 𝜕𝐾 ⎣ ) ( imp ⎤ 𝐾 2 𝑑+ 𝑑−imp 𝜕𝜎imp 2 1 ⎥. + + √ 𝜎imp 𝜕𝐾 𝜎imp 𝑇 − 𝑡 ⎥⎦

=𝐾

Gathering all this information, we have ( 2 𝜎(𝐾, 𝑇 )2 = 𝐾

7.2.3

𝜕 2 𝜎imp 2 𝜕𝐾 2

+

𝜕𝜎imp 𝜕𝑇

imp 2𝐾𝑑+

+ (𝑟 − 𝐷)𝐾

𝜕𝜎imp

+ √ 𝜎imp 𝑇 − 𝑡 𝜕𝐾

𝜕𝜎imp 𝜕𝐾

+

imp 𝐾 2 𝑑+ 𝑑−imp

𝜎imp

) 1 𝜎imp 2𝑇 −𝑡 ) ( 𝜕𝜎imp 2 𝜕𝐾

. +

1 √

𝜎imp 𝑇 − 𝑡

Stochastic Volatility

1. Generalised Stochastic Volatility Model. Let {𝑊𝑡𝑆 : 𝑡 ≥ 0} and {𝑊𝑡𝑌 : 𝑡 ≥ 0} be two correlated ℙ-standard Wiener processes on the probability space (Ω, ℱ, ℙ). Suppose we have a stochastic volatility model having the following diffusion processes 𝑑𝑆𝑡 = 𝜇(𝑆𝑡 , 𝑌𝑡 , 𝑡)𝑆𝑡 𝑑𝑡 + 𝜎(𝑌𝑡 , 𝑡)𝑆𝑡 𝑑𝑊𝑡𝑆 𝑑𝑌𝑡 = 𝛼(𝑆𝑡 , 𝑌𝑡 , 𝑡)𝑑𝑡 + 𝛽(𝑆𝑡 , 𝑌𝑡 , 𝑡)𝑑𝑊𝑡𝑌

𝑑𝑊𝑡𝑆 ⋅ 𝑑𝑊𝑡𝑌 = 𝜌𝑑𝑡

where 𝑆𝑡 is the asset price which pays no dividends, 𝜎(𝑌𝑡 , 𝑡) is the volatility process, 𝜇(𝑆𝑡 , 𝑌𝑡 , 𝑡), 𝛼(𝑆𝑡 , 𝑌𝑡 , 𝑡) and 𝛽(𝑆𝑡 , 𝑌𝑡 , 𝑡) are continuous functions and 𝜌 ∈ (−1, 1) is the correlation. In addition, let 𝐵𝑡 be a risk-free asset having the following differential equation 𝑑𝐵𝑡 = 𝑟 𝐵𝑡 𝑑𝑡 where 𝑟 is a constant risk-free interest rate. ⟂ 𝑊𝑡𝑌 , show that we By defining {𝑊𝑡 : 𝑡 ≥ 0} as a standard Wiener process where 𝑊𝑡 ⟂ can write √ 𝑊𝑡𝑆 = 𝜌𝑊𝑡𝑌 + 1 − 𝜌2 𝑊𝑡 .

7.2.3 Stochastic Volatility

711

Using the two-dimensional Girsanov’s theorem, show that under the risk-neutral measure ℚ, ( ) √ ̃ 𝑌 + 1 − 𝜌2 𝑑 𝑊 ̃𝑡 𝑑𝑆𝑡 = 𝑟𝑆𝑡 𝑑𝑡 + 𝜎(𝑌𝑡 , 𝑡)𝑆𝑡 𝜌𝑑 𝑊 𝑡 [ ] ̃𝑌 𝑑𝑌𝑡 = 𝛼(𝑆𝑡 , 𝑌𝑡 , 𝑡) − 𝛾𝑡 𝛽(𝑆𝑡 , 𝑌𝑡 , 𝑡) 𝑑𝑡 + 𝛽(𝑆𝑡 , 𝑌𝑡 , 𝑡)𝑑 𝑊 𝑡

̃𝑡 and 𝑊 ̃ 𝑌 are ℚ-standard Wiener processes and 𝛾𝑡 is the market price of volatility where 𝑊 𝑡 risk. √ Solution: For the first part of the solution, from 𝑊𝑡𝑆 = 𝜌𝑊𝑡𝑌 + 1 − 𝜌2 𝑊𝑡 we have 𝔼(𝑊𝑡𝑆 ) = 𝔼(𝜌𝑊𝑡𝑌 +

√ √ 1 − 𝜌2 𝑊𝑡 ) = 𝜌𝔼(𝑊𝑡𝑌 ) + 1 − 𝜌2 𝔼(𝑊𝑡 ) = 0

and Var(𝑊𝑡𝑆 ) = Var(𝜌𝑊𝑡𝑌 +

√

1 − 𝜌2 𝑊𝑡 ) = 𝜌2 Var(𝑊𝑡𝑌 ) + (1 − 𝜌2 )Var(𝑊𝑡 ) = 𝑡.

Given both 𝑊𝑡𝑌 ∼  (0, 𝑡), 𝑊𝑡 ∼  (0, 𝑡) and 𝑊𝑡𝑌 ⟂ ⟂ 𝑊𝑡 therefore 𝜌𝑊𝑡𝑌 +

√ 1 − 𝜌2 𝑊𝑡 ∼  (0, 𝑡).

In addition, using It¯o’s formula and taking note that 𝑊𝑡𝑌 ⟂ ⟂ 𝑊𝑡 , √ 1 − 𝜌2 𝑊𝑡 ) ⋅ 𝑑𝑊𝑡𝑌 √ = (𝜌𝑑𝑊𝑡𝑌 + 1 − 𝜌2 𝑑𝑊𝑡 ) ⋅ 𝑑𝑊𝑡𝑌 √ = 𝜌(𝑑𝑊𝑡𝑌 )2 + 1 − 𝜌2 𝑑𝑊𝑡 ⋅ 𝑑𝑊𝑡𝑌

𝑑𝑊𝑡𝑆 ⋅ 𝑑𝑊𝑡𝑌 = 𝑑(𝜌𝑊𝑡𝑌 +

= 𝜌𝑑𝑡. Thus, we can write 𝑊𝑡𝑆 = 𝜌𝑊𝑡𝑌 + Under the ℙ-measure we have

√ 1 − 𝜌2 𝑊𝑡 .

𝑑𝑆𝑡 = 𝜇(𝑆𝑡 , 𝑌𝑡 , 𝑡)𝑆𝑡 𝑑𝑡 + 𝜎(𝑌𝑡 , 𝑡)𝑆𝑡

(√ ) 1 − 𝜌2 𝑑𝑊𝑡 + 𝜌𝑑𝑊𝑡𝑌

𝑑𝑌𝑡 = 𝛼(𝑆𝑡 , 𝑌𝑡 , 𝑡)𝑑𝑡 + 𝛽(𝑆𝑡 , 𝑌𝑡 , 𝑡)𝑑𝑊𝑡𝑌 and by defining ̃𝑌 = 𝑊 𝑌 + 𝑊 𝑡 𝑡 ̃𝑡 = 𝑊𝑡 + 𝑊

𝑡

∫0 𝑡

∫0

𝛾𝑢 𝑑𝑢

𝜆𝑢 𝑑𝑢

where 𝛾𝑡 is the market price of volatility risk and 𝜆𝑡 is the market price of asset price risk, ̃𝑡 ⟂ ̃𝑌 . and since 𝑊𝑡 ⟂ ⟂ 𝑊𝑡𝑌 , therefore 𝑊 ⟂𝑊 𝑡

712

7.2.3 Stochastic Volatility

From the two-dimensional Girsanov’s theorem, there exists a risk-neutral measure ℚ on the filtration ℱ𝑠 , 0 ≤ 𝑠 ≤ 𝑡 defined by the Radon–Nikod´ym process 𝑡 𝑡 1 𝑡 2 1 𝑡 2 𝑌 𝑑ℚ = 𝑍𝑡 = 𝑒− 2 ∫0 𝜆𝑢 𝑑𝑢−∫0 𝜆𝑢 𝑑𝑊𝑢 ⋅ 𝑒− 2 ∫0 𝛾𝑢 𝑑𝑢−∫0 𝛾𝑢 𝑑𝑊𝑢 𝑑ℙ

̃𝑡 and 𝑊 ̃ 𝑌 are ℚ-standard Wiener processes and 𝑊 ̃𝑡 ⟂ ̃𝑌 . so that 𝑊 ⟂𝑊 𝑡 𝑡 −𝑟𝑡 Let 𝑋𝑡 = 𝑒 𝑆𝑡 be the discounted asset price, and from the application of It¯o’s formula 2 2 𝜕𝑋𝑡 𝜕𝑋𝑡 1 𝜕 𝑋𝑡 1 𝜕 𝑋𝑡 2 𝑑𝑆𝑡 + (𝑑𝑡) + (𝑑𝑆𝑡 )2 + … 𝑑𝑡 + 𝜕𝑡 𝜕𝑆𝑡 2 𝜕𝑡2 2 𝜕𝑆𝑡2 ( √ = −𝑟𝑒−𝑟𝑡 𝑆𝑡 𝑑𝑡 + 𝑒−𝑟𝑡 𝜇(𝑆𝑡 , 𝑌𝑡 , 𝑡)𝑆𝑡 𝑑𝑡 + 1 − 𝜌2 𝜎(𝑌𝑡 , 𝑡)𝑆𝑡 𝑑𝑊𝑡 ) +𝜌𝜎(𝑌𝑡 , 𝑡)𝑆𝑡 𝑑𝑊𝑡𝑌 √ = −𝑟𝑋𝑡 𝑑𝑡 + 𝜇(𝑆𝑡 , 𝑌𝑡 , 𝑡)𝑋𝑡 𝑑𝑡 + 1 − 𝜌2 𝜎(𝑌𝑡 , 𝑡)𝑋𝑡 𝑑𝑊𝑡 + 𝜌𝜎(𝑌𝑡 , 𝑡)𝑋𝑡 𝑑𝑊𝑡𝑌 ) [( ] √ 𝜇(𝑆𝑡 , 𝑌𝑡 , 𝑡) − 𝑟 = 𝜎(𝑌𝑡 , 𝑡)𝑋𝑡 𝑑𝑡 + 1 − 𝜌2 𝑑𝑊𝑡 + 𝜌𝑑𝑊𝑡𝑌 𝜎(𝑌𝑡 , 𝑡) ) [( ( ) √ 𝜇(𝑆𝑡 , 𝑌𝑡 , 𝑡) − 𝑟 ̃𝑡 − 𝜆𝑡 𝑑𝑡 = 𝜎(𝑌𝑡 , 𝑡)𝑋𝑡 𝑑𝑡 + 1 − 𝜌2 𝑑 𝑊 𝜎(𝑌𝑡 , 𝑡) )] ( ̃ 𝑌 − 𝛾𝑡 𝑑𝑡 +𝜌 𝑑 𝑊 𝑡 ) √ [{( } 𝜇(𝑆𝑡 , 𝑌𝑡 , 𝑡) − 𝑟 2 − 1 − 𝜌 𝜆𝑡 − 𝜌𝛾𝑡 𝑑𝑡 = 𝜎(𝑌𝑡 , 𝑡)𝑋𝑡 𝜎(𝑌𝑡 , 𝑡) ] √ ̃𝑡 + 𝜌𝑑 𝑊 ̃𝑌 . + 1 − 𝜌2 𝑑 𝑊 𝑡

𝑑𝑋𝑡 =

To ensure that 𝑒−𝑟𝑡 𝑆𝑡 is a ℚ-martingale, we therefore set √ 𝜇(𝑆𝑡 , 𝑌𝑡 , 𝑡) − 𝑟 1 − 𝜌2 𝜆𝑡 + 𝜌𝛾𝑡 = . 𝜎(𝑌𝑡 , 𝑡) Using the above relationship and substituting ̃𝑡 − 𝜆𝑡 𝑑𝑡 𝑑𝑊𝑡 = 𝑑 𝑊 ̃ 𝑌 − 𝛾𝑡 𝑑𝑡 𝑑𝑊 𝑌 = 𝑑 𝑊 𝑡

𝑡

into 𝑑𝑆𝑡 = 𝜇(𝑆𝑡 , 𝑌𝑡 , 𝑡)𝑆𝑡 𝑑𝑡 + 𝜎(𝑌𝑡 , 𝑡)𝑆𝑡

(√

𝑑𝑌𝑡 = 𝛼(𝑆𝑡 , 𝑌𝑡 , 𝑡)𝑑𝑡 + 𝛽(𝑆𝑡 , 𝑌𝑡 , 𝑡)𝑑𝑊𝑡𝑌

1 − 𝜌2 𝑑𝑊𝑡 + 𝜌𝑑𝑊𝑡𝑌

)

7.2.3 Stochastic Volatility

713

the asset price under the risk-neutral measure ℚ becomes ( ) √ ̃𝑡 − 𝜆𝑡 𝑑𝑡 𝑑𝑆𝑡 = 𝜇(𝑆𝑡 , 𝑌𝑡 , 𝑡)𝑆𝑡 𝑑𝑡 + 1 − 𝜌2 𝜎(𝑌𝑡 , 𝑡)𝑆𝑡 𝑑 𝑊 ( ) ̃ 𝑌 − 𝛾𝑡 𝑑𝑡 +𝜌𝜎(𝑌𝑡 , 𝑡)𝑆𝑡 𝑑 𝑊 𝑡 √ ̃𝑡 + 𝜌𝜎(𝑌𝑡 , 𝑡)𝑆𝑡 𝑑 𝑊 ̃𝑌 = 𝑟 𝑆𝑡 𝑑𝑡 + 1 − 𝜌2 𝜎(𝑌𝑡 , 𝑡)𝑆𝑡 𝑑 𝑊 𝑡 and the stochastic volatility under the risk-neutral measure ℚ is [ ] ̃𝑌 . 𝑑𝑌𝑡 = 𝛼(𝑆𝑡 , 𝑌𝑡 , 𝑡) − 𝛾𝑡 𝛽(𝑆𝑡 , 𝑌𝑡 , 𝑡) 𝑑𝑡 + 𝛽(𝑆𝑡 , 𝑌𝑡 , 𝑡)𝑑 𝑊 𝑡

2. Backward Kolmogorov Equation – Stochastic Volatility Model. Let {𝑊𝑡𝑆 : 𝑡 ≥ 0} and {𝑊𝑡𝜎 : 𝑡 ≥ 0} be two correlated ℙ-standard Wiener processes on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow a stochastic volatility model with the following SDEs 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡𝑆 𝑆𝑡 𝑑𝜎𝑡 = 𝛼(𝜎𝑡 , 𝑡)𝑑𝑡 + 𝛽(𝜎𝑡 , 𝑡)𝑑𝑊𝑡𝜎

𝑑𝑊𝑡𝑆 ⋅ 𝑑𝑊𝑡𝜎 = 𝜌𝑑𝑡

where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎𝑡 is the volatility process, 𝛼(𝜎𝑡 , 𝑡) and 𝛽(𝜎𝑡 , 𝑡) are continuous functions, 𝜌 ∈ (−1, 1) is the correlation parameter, and let 𝑟 be the risk-free interest-rate parameter from the money-market account. By introducing {𝑍𝑡 : 𝑡 ≥ 0} as a standard Wiener process, independent of 𝑊𝑡𝜎 , show that we can write 𝑊𝑡𝑆 = 𝜌𝑊𝑡𝜎 +

√ 1 − 𝜌2 𝑍𝑡 .

Using the two-dimensional Girsanov’s theorem, show that under the risk-neutral measure ℚ, ̃𝑆 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑 𝑊 𝑡 ̃𝜎 𝑑𝜎𝑡 = 𝛼 ̃(𝜎𝑡 , 𝑡)𝑑𝑡 + 𝛽(𝜎𝑡 , 𝑡)𝑑 𝑊 𝑡

√ ̃ 𝑆 = 𝜌𝑊 ̃ 𝜎 + 1 − 𝜌2 𝑍 ̃ 𝜎 and 𝑍 ̃𝑡 are ℚ-standard Wiener processes, 𝑊 ̃𝜎 ⟂ ̃𝑡 , 𝑊 where 𝑊 𝑡 𝑡 𝑡 𝑡 ⟂ ̃ ̃(𝜎𝑡 , 𝑡) = 𝛼(𝜎𝑡 , 𝑡) − 𝛾𝑡 𝛽(𝜎𝑡 , 𝑡) such that 𝛾𝑡 is the market price of volatility risk. 𝑍𝑡 and 𝛼 Using Taylor’s series, show that for a definite integral of a smooth function 𝑓 (𝑥), 𝑏

∫𝑎

𝑓 (𝑢) 𝑑𝑢 = 𝑓

(

) ( ) 𝑎+𝑏 (𝑏 − 𝑎) + 𝑂 (𝑏 − 𝑎)3 2

714

7.2.3 Stochastic Volatility

and hence show that under the risk-neutral measure ℚ, [ ] ) ( 𝔼ℚ 𝑆𝑡+Δ𝑡 − 𝑆𝑡 || ℱ𝑡 = (𝑟 − 𝐷)𝑆𝑡+ 1 Δ𝑡 Δ𝑡 + 𝑂 (Δ𝑡)3 2 ( ) [ ] ) ( 1 ℚ | ̃ 𝜎𝑡+ 1 Δ𝑡 , 𝑡 + Δ𝑡 Δ𝑡 + 𝑂 (Δ𝑡)2 𝔼 𝜎𝑡+Δ𝑡 − 𝜎𝑡 | ℱ𝑡 = 𝛼 2 2 [( )2 | ] ) ( ℚ 2 2 𝑆𝑡+Δ𝑡 − 𝑆𝑡 | ℱ𝑡 = 𝜎 1 𝑆 1 Δ𝑡 + 𝑂 (Δ𝑡)2 𝔼 | 𝑡+ 2 Δ𝑡 𝑡+ 2 Δ𝑡 [( ( )2 )2 | ] ) ( 1 𝔼ℚ 𝜎𝑡+Δ𝑡 − 𝜎𝑡 | ℱ𝑡 = 𝛽 𝜎𝑡+ 1 Δ𝑡 , 𝑡 + Δ𝑡 Δ𝑡 + 𝑂 (Δ𝑡)2 | 2 2 [( ( ) )( )| ] 1 ℚ 𝑆𝑡+Δ𝑡 − 𝑆𝑡 𝜎𝑡+Δ𝑡 − 𝜎𝑡 | ℱ𝑡 = 𝜌𝛽 𝜎𝑡+ 1 Δ𝑡 , 𝑡 + Δ𝑡 𝜎𝑡+ 1 Δ𝑡 𝑆𝑡+ 1 Δ𝑡 Δ𝑡 𝔼 | 2 2 2 2 ) ( +𝑂 (Δ𝑡)2 [( )3 | ] ) ( 𝔼ℚ 𝑆𝑡+Δ𝑡 − 𝑆𝑡 | ℱ𝑡 = 𝑂 (Δ𝑡)2 | [( )2 ( )| ] ) ( 𝜎𝑡+Δ𝑡 − 𝜎𝑡 | ℱ𝑡 = 𝑂 (Δ𝑡)2 𝔼ℚ 𝑆𝑡+Δ𝑡 − 𝑆𝑡 | [( )( )2 | ] ) ( 𝔼ℚ 𝑆𝑡+Δ𝑡 − 𝑆𝑡 𝜎𝑡+Δ𝑡 − 𝜎𝑡 | ℱ𝑡 = 𝑂 (Δ𝑡)2 | [( )3 | ] ) ( ℚ 𝜎𝑡+Δ𝑡 − 𝜎𝑡 | ℱ𝑡 = 𝑂 (Δ𝑡)2 𝔼 | for Δ𝑡 > 0. Let 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) be the transition pdf of the asset price, where the asset price and volatility are 𝑆𝑡 and 𝜎𝑡 at time 𝑡, respectively given that the asset price and volatility are 𝑆𝑇 and 𝜎𝑇 at time 𝑇 > 𝑡, respectively. From the Chapman–Kolmogorov equation for Δ𝑡 > 0, 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) =

∞

∫0

∞

∫0

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡)𝑝(𝑥, 𝑦, 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑥𝑑𝑦

show that by expanding 𝑝(𝑥, 𝑦, 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) using Taylor series centred on 𝑆𝑡 and 𝜎𝑡 up to second order and taking limits Δ𝑡 → 0, 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) satisfies the backward Kolmogorov equation 1 𝜕2 𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) + 𝜎𝑡2 𝑆𝑡2 2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝑡 2 𝜕𝑆𝑡 1 𝜕2 𝜕2 + 𝛽(𝜎𝑡 , 𝑡)2 2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) + 𝜌𝛽(𝜎𝑡 , 𝑡)𝜎𝑡 𝑆𝑡 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 2 𝜕𝜎𝑡 𝜕𝑆𝑡 𝜕𝜎𝑡 +(𝑟 − 𝐷)𝑆𝑡

𝜕 𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) + 𝛼 ̃(𝜎𝑡 , 𝑡) 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) = 0 𝜕𝑆𝑡 𝜕𝜎𝑡

with boundary condition 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑡) = 𝛿(𝑆𝑡 − 𝑆𝑇 )𝛿(𝜎𝑡 − 𝜎𝑇 ),

∀𝑡.

7.2.3 Stochastic Volatility

715

Note that if {𝑊𝑡 : 𝑡 ≥ 0} and {𝐵𝑡 : 𝑡 ≥ 0} are two correlated standard Wiener processes on the probability space (Ω, ℱ, ℙ) with correlation 𝜌 ∈ (−1, 1), and if 𝑓 and 𝑔 are simple processes, then the covariance between the It¯o integrals 𝐼𝑡 =

𝑡

∫0

𝑓 (𝑊𝑠 , 𝑠) 𝑑𝑊𝑠 𝑎𝑛𝑑 𝐽𝑡 =

𝑡

∫0

𝑔(𝐵𝑠 , 𝑠) 𝑑𝐵𝑠

is [( 𝔼

𝑡

∫0 (

=𝔼 = 𝜌𝔼

)( 𝑓 (𝑊𝑠 , 𝑠) 𝑑𝑊𝑠 𝑡

∫0 ( ∫0

𝑡

∫0

)] 𝑔(𝐵𝑠 , 𝑠) 𝑑𝐵𝑠 )

𝑓 (𝑊𝑠 , 𝑠)𝑔(𝐵𝑠 , 𝑠) 𝑑⟨𝑊 , 𝐵⟩𝑠 𝑡

) 𝑓 (𝑊𝑠 , 𝑠)𝑔(𝐵𝑠 , 𝑠) 𝑑𝑠

where ⟨𝑊 , 𝐵⟩𝑡 = lim

𝑛→∞

𝑛−1 ∑ (𝑊𝑡𝑖+1 − 𝑊𝑡𝑖 )(𝐵𝑡𝑖+1 − 𝐵𝑡𝑖 ) 𝑖=0

such that 𝑡𝑖 = 𝑖𝑡∕𝑛, 0 = 𝑡0 < 𝑡1 < 𝑡2 < ⋯ < 𝑡𝑛−1 < 𝑡𝑛 = 𝑡, 𝑛 ∈ ℕ. √ ⟂ 𝑊𝑡𝜎 , see Problem Solution: To show that we can set 𝑊𝑡𝑆 = 𝜌𝑊𝑡𝜎 + 1 − 𝜌2 𝑍𝑡 , 𝑍𝑡 ⟂ 7.2.2.1 (page 685). We define ̃𝑡 = 𝑍𝑡 + 𝑍

𝑡

∫0

̃𝜎 = 𝑊 𝜎 + 𝑊 𝑡 𝑡

∫0

𝜆𝑢 𝑑𝑢 𝑡

𝛾𝑢 𝑑𝑢

where 𝜆𝑡 is the market price of asset risk and 𝛾𝑡 is the market price of volatility risk. Since ̃𝑡 ⟂ ̃𝜎. 𝑍𝑡 ⟂ ⟂ 𝑊𝑡𝜎 , we can easily deduce that 𝑍 ⟂𝑊 𝑡 Let the portfolio Π𝑡 be defined as Π𝑡 = 𝜙𝑡 𝑆𝑡 + 𝜓𝑡 𝐵𝑡 where 𝜙𝑡 units are invested in risky asset 𝑆𝑡 and 𝜓𝑡 units are invested in risk-free asset 𝐵𝑡 . Given that the holder of the portfolio will receive 𝐷𝑆𝑡 𝑑𝑡 for every risky asset held, thus ( ) 𝑑Π𝑡 = 𝜙𝑡 𝑑𝑆𝑡 + 𝐷𝑆𝑡 𝑑𝑡 + 𝜓𝑡 𝑑𝐵𝑡 ( ) ( ) = 𝜙𝑡 𝜇𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑𝑊𝑡𝑆 + 𝜓𝑡 𝑟𝐵𝑡 𝑑𝑡 [ ( )] √ = 𝑟Π𝑡 𝑑𝑡 + 𝜙𝑡 𝑆𝑡 (𝜇 − 𝑟)𝑑𝑡 + 𝜎𝑡 𝜌𝑑𝑊𝑡𝜎 + 1 − 𝜌2 𝑑𝑍𝑡 .

716

7.2.3 Stochastic Volatility

By substituting ̃ 𝜎 − 𝛾𝑡 𝑑𝑡 𝑑𝑊𝑡𝜎 = 𝑑 𝑊 𝑡 ̃𝑡 − 𝜆𝑡 𝑑𝑡 𝑑𝑍𝑡 = 𝑑 𝑍 into 𝑑Π𝑡 we have 𝑑Π𝑡 = 𝑟Π𝑡 𝑑𝑡

[ ( )] √ √ ̃ 𝜎 − 𝜌𝛾𝑡 𝑑𝑡 + 1 − 𝜌2 𝑑 𝑍 ̃𝑡 − 1 − 𝜌2 𝜆𝑡 𝑑𝑡 +𝜙𝑡 𝑆𝑡 (𝜇 − 𝑟)𝑑𝑡 + 𝜎𝑡 𝜌𝑑 𝑊 𝑡

= 𝑟Π𝑡 𝑑𝑡 +𝜙𝑡 𝑆𝑡

[(

( ) ) ( )] √ √ ̃ 𝜎 + 1 − 𝜌2 𝑑 𝑍 ̃𝑡 . 𝜇 − 𝑟 − 𝜌𝛾𝑡 + 1 − 𝜌2 𝜆𝑡 𝜎𝑡 𝑑𝑡 + 𝜎𝑡 𝜌𝑑 𝑊 𝑡

̃ 𝜎 and 𝑍 ̃𝑡 are ℚ-martingales, and in order for the discounted portfolio 𝑒−𝑟𝑡 Π𝑡 to Since 𝑊 𝑡 be a ℚ-martingale, 𝑑(𝑒−𝑟𝑡 Π𝑡 ) = −𝑟 𝑒−𝑟𝑡 Π𝑡 𝑑𝑡 + 𝑒−𝑟𝑡 𝑑Π𝑡 [( ( ) ) √ = 𝑒−𝑟𝑡 𝜙𝑡 𝑆𝑡 𝜇 − 𝑟 − 𝜌𝛾𝑡 + 1 − 𝜌2 𝜆𝑡 𝜎𝑡 𝑑𝑡 ( )] √ ̃ 𝜎 + 1 − 𝜌2 𝑑 𝑍 ̃𝑡 +𝜎𝑡 𝜌𝑑 𝑊 𝑡

we set 𝜌𝛾𝑡 +

√ 𝜇−𝑟 1 − 𝜌2 𝜆𝑡 = . 𝜎𝑡

Hence, by substituting ̃ 𝜎 − 𝛾𝑡 𝑑𝑡 𝑑𝑊𝑡𝜎 = 𝑑 𝑊 𝑡 ̃ 𝑑𝑍𝑡 = 𝑑 𝑍𝑡 − 𝜆𝑡 𝑑𝑡 √ 𝜇−𝑟 𝜌𝛾𝑡 + 1 − 𝜌2 𝜆𝑡 = 𝜎𝑡 ( ) √ into 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝜌𝑑𝑊𝑡𝜎 + 1 − 𝜌2 𝑑𝑍𝑡 , we have ( ) √ √ ̃ 𝜎 − 𝜌𝛾𝑡 𝑑𝑡 + 1 − 𝜌2 𝑑 𝑍 ̃𝑡 − 1 − 𝜌2 𝜆𝑡 𝑑𝑡 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝜌𝑑 𝑊 𝑡 ( ) ( ) √ √ ̃ 𝜎 + 1 − 𝜌2 𝑑 𝑍 ̃𝑡 − 𝜎𝑡 𝑆𝑡 𝜌𝛾𝑡 + 1 − 𝜌2 𝜆𝑡 𝑑𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝜌𝑑 𝑊 𝑡 ( ) ( ) √ 𝜇−𝑟 𝜎 2 ̃ ̃ = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝜌𝑑 𝑊𝑡 + 1 − 𝜌 𝑑 𝑍𝑡 − 𝜎𝑡 𝑆𝑡 𝑑𝑡 𝜎𝑡 ( ) √ ̃ 𝜎 + 1 − 𝜌2 𝑑 𝑍 ̃𝑡 = (𝑟 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝜌𝑑 𝑊 𝑡

=

̃𝑆 . (𝑟 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑 𝑊 𝑡

7.2.3 Stochastic Volatility

717

In contrast, by substituting ̃ 𝜎 − 𝛾𝑡 𝑑𝑡 𝑑𝑊𝑡𝜎 = 𝑑 𝑊 𝑡 into 𝑑𝜎𝑡 = 𝛼(𝜎𝑡 , 𝑡)𝑑𝑡 + 𝛽(𝜎𝑡 , 𝑡)𝑑𝑊𝑡𝜎 , the instantaneous volatility under the ℚ measure becomes ( ) ̃ 𝜎 − 𝛾𝑡 𝑑𝑡 𝑑𝜎𝑡 = 𝛼(𝜎𝑡 , 𝑡)𝑑𝑡 + 𝛽(𝜎𝑡 , 𝑡) 𝑑 𝑊 𝑡 ) ( ̃𝜎 = 𝛼(𝜎𝑡 , 𝑡) − 𝛾𝑡 𝛽(𝜎𝑡 , 𝑡) 𝑑𝑡 + 𝛽(𝜎𝑡 , 𝑡)𝑑 𝑊 𝑡 ̃𝜎 =𝛼 ̃(𝜎𝑡 , 𝑡)𝑑𝑡 + 𝛽(𝜎𝑡 , 𝑡)𝑑 𝑊 𝑡

where 𝛼 ̃(𝜎𝑡 , 𝑡) = 𝛼(𝜎𝑡 , 𝑡) − 𝛾𝑡 𝛽(𝜎𝑡 , 𝑡). To show that 𝑏

∫𝑎

𝑓 (𝑢) 𝑑𝑢 = 𝑓

(

) ( ) 𝑎+𝑏 (𝑏 − 𝑎) + 𝑂 (𝑏 − 𝑎)3 2

for a smooth function 𝑓 , see Problem 7.2.2.2 (see page 687). From ̃𝑆 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑 𝑊 𝑡 ̃𝜎 𝑑𝜎𝑡 = 𝛼 ̃(𝜎𝑡 , 𝑡)𝑑𝑡 + 𝛽(𝜎𝑡 , 𝑡)𝑑 𝑊 𝑡

̃𝑆 ⋅ 𝑑𝑊 ̃ 𝜎 = 𝜌𝑑𝑡 𝑑𝑊 𝑡 𝑡 and taking integrals 𝑡+Δ𝑡

∫𝑡 ∫𝑡

𝑡+Δ𝑡

𝑑𝑆𝑢 = 𝑑𝜎𝑢 =

𝑡+Δ𝑡

∫𝑡

𝑡+Δ𝑡

∫𝑡

(𝑟 − 𝐷)𝑆𝑢 𝑑𝑢 + 𝛼 ̃(𝜎𝑢 , 𝑢) 𝑑𝑢 +

𝑡+Δ𝑡

∫𝑡

𝑡+Δ𝑡

∫𝑡

̃𝑆 𝜎𝑢 𝑆𝑢 𝑑 𝑊 𝑢

̃𝜎 𝛽(𝜎𝑢 , 𝑢) 𝑑 𝑊 𝑢

we have 𝑆𝑡+Δ𝑡 − 𝑆𝑡 = 𝜎𝑡+Δ𝑡 − 𝜎𝑡 =

𝑡+Δ𝑡

∫𝑡

𝑡+Δ𝑡

∫𝑡

(𝑟 − 𝐷)𝑆𝑢 𝑑𝑢 + 𝛼 ̃(𝜎𝑢 , 𝑢) 𝑑𝑢 +

𝑡+Δ𝑡

∫𝑡

𝑡+Δ𝑡

∫𝑡

̃𝑆 𝜎𝑢 𝑆𝑢 𝑑 𝑊 𝑢

̃𝜎. 𝛽(𝜎𝑢 , 𝑢) 𝑑 𝑊 𝑢

Taking expectations, using the approximate integral formula and It¯o calculus, 𝔼

ℚ

[

𝑡+Δ𝑡

]

𝑆𝑡+Δ𝑡 − 𝑆𝑡 || ℱ𝑡 = (𝑟 − 𝐷)

∫𝑡

[ 𝑆𝑢 𝑑𝑢 + 𝔼

ℚ

∫𝑡

= (𝑟 − 𝐷)𝑆𝑡+ 1 Δ𝑡 Δ𝑡 + 𝑂((Δ𝑡) ) 3

2

𝑡+Δ𝑡

|

̃ 𝑆 || ℱ𝑡 𝜎𝑢 𝑆𝑢 𝑑 𝑊 𝑢 | |

]

718

7.2.3 Stochastic Volatility

𝔼

ℚ

[

]

𝑡+Δ𝑡

𝜎𝑡+Δ𝑡 − 𝜎𝑡 || ℱ𝑡 =

[ ℚ

𝑡+Δ𝑡

|

̃ 𝜎 || ℱ𝑡 𝛽(𝜎𝑢 , 𝑢)𝑑 𝑊 𝑢 |

𝛼 ̃(𝜎𝑢 , 𝑢) 𝑑𝑢 + 𝔼 ∫𝑡 ∫𝑡 ( ) 1 =𝛼 ̃ 𝜎𝑡+ 1 Δ𝑡 , 𝑡 + Δ𝑡 Δ𝑡 + 𝑂((Δ𝑡)3 ) 2 2

]

|

)2 ( 𝑡+Δ𝑡 [ ] | 𝑆𝑢 𝑑𝑢 𝔼ℚ (𝑆𝑡+Δ𝑡 − 𝑆𝑡 )2 | ℱ𝑡 = (𝑟 − 𝐷)2 | ∫𝑡 [ ] ( 𝑡+Δ𝑡 ) 𝑡+Δ𝑡 | ℚ 𝑆| ̃ +2(𝑟 − 𝐷) 𝑆𝑢 𝑑𝑢 𝔼 𝜎𝑢 𝑆𝑢 𝑑 𝑊𝑢 | ℱ𝑡 | ∫𝑡 ∫𝑡 | [( )2 | ] 𝑡+Δ𝑡 | ̃ 𝑆 | ℱ𝑡 +𝔼ℚ 𝜎𝑢 𝑆𝑢 𝑑 𝑊 | 𝑢 ∫𝑡 | | )2 ( 𝑡+Δ𝑡 𝑡+Δ𝑡 = (𝑟 − 𝐷)2 𝑆𝑢 𝑑𝑢 + 𝜎𝑢2 𝑆𝑢2 𝑑𝑢 ∫𝑡 ∫𝑡 = 𝜎2

𝑆2 Δ𝑡 + 𝑂((Δ𝑡)2 ) 𝑡+ 12 Δ𝑡 𝑡+ 21 Δ𝑡

)2 [ ] ( 𝑡+Δ𝑡 | 𝔼ℚ (𝜎𝑡+Δ𝑡 − 𝜎𝑡 )2 | ℱ𝑡 = 𝛼 ̃(𝜎𝑢 , 𝑢) 𝑑𝑢 | ∫𝑡 [ ] ( 𝑡+Δ𝑡 ) 𝑡+Δ𝑡 | | ℚ 𝜎 ̃ | ℱ𝑡 +2 𝛼 ̃(𝜎𝑢 , 𝑢) 𝑑𝑢 𝔼 𝛽(𝜎𝑢 , 𝑢) 𝑑 𝑊 𝑢 | ∫𝑡 ∫𝑡 | [( ] ) | 2 𝑡+Δ𝑡 | ̃ 𝜎 | ℱ𝑡 +𝔼ℚ 𝛽(𝜎𝑢 , 𝑢) 𝑑 𝑊 | 𝑢 ∫𝑡 | | ( 𝑡+Δ𝑡 )2 𝑡+Δ𝑡 = 𝛼 ̃(𝜎𝑢 , 𝑢) 𝑑𝑢 + 𝛽(𝜎𝑢 , 𝑢)2 𝑑𝑢 ∫𝑡 ∫𝑡 ( )2 1 = 𝛽 𝜎𝑡+ 1 Δ𝑡 , 𝑡 + Δ𝑡 Δ𝑡 + 𝑂((Δ𝑡)2 ) 2 2 [ ] 𝔼ℚ (𝑆𝑡+Δ𝑡 − 𝑆𝑡 )(𝜎𝑡+Δ𝑡 − 𝜎𝑡 )|| ℱ𝑡 ) ( 𝑡+Δ𝑡 ) ( 𝑡+Δ𝑡 𝑆𝑢 𝑑𝑢 𝛼 ̃(𝜎𝑢 , 𝑢) 𝑑𝑢 = (𝑟 − 𝐷) ∫𝑡 ∫𝑡 [ ] ( 𝑡+Δ𝑡 ) 𝑡+Δ𝑡 | | ℚ 𝜎 ̃ | ℱ𝑡 +(𝑟 − 𝐷) 𝑆𝑢 𝑑𝑢 𝔼 𝛽(𝜎𝑢 , 𝑢) 𝑑 𝑊 𝑢 | ∫𝑡 ∫𝑡 | [ ] ( 𝑡+Δ𝑡 ) 𝑡+Δ𝑡 | ̃ 𝑆 || ℱ𝑡 + 𝛼 ̃(𝜎𝑢 , 𝑢) 𝑑𝑢 𝔼ℚ 𝜎𝑢 𝑆𝑢 𝑑 𝑊 𝑢 | ∫𝑡 ∫𝑡 | [( ) ( 𝑡+Δ𝑡 )| ] 𝑡+Δ𝑡 ̃𝑆 ̃ 𝜎 || ℱ𝑡 +𝔼ℚ 𝜎𝑢 𝑆𝑢 𝑑 𝑊 𝛽(𝜎𝑢 , 𝑢) 𝑑 𝑊 𝑢 𝑢 | ∫𝑡 ∫𝑡 | ( 𝑡+Δ𝑡 ) ( 𝑡+Δ𝑡 ) = (𝑟 − 𝐷) 𝑆𝑢 𝑑𝑢 𝛼 ̃(𝜎𝑢 , 𝑢) 𝑑𝑢 ∫𝑡 ∫𝑡

7.2.3 Stochastic Volatility

[(

719

)| ] ̃𝑆 , 𝑊 ̃ 𝜎 ⟩𝑢 || ℱ𝑡 +𝔼ℚ 𝜎𝑢 𝛽(𝜎𝑢 , 𝑢)𝑆𝑢 𝑑⟨𝑊 | ∫𝑡 | ) ( 𝑡+Δ𝑡 ) ( 𝑡+Δ𝑡 𝑆𝑢 𝑑𝑢 𝛼 ̃(𝜎𝑢 , 𝑢) 𝑑𝑢 + 𝜌 = (𝑟 − 𝐷) ∫𝑡 ∫𝑡 ∫𝑡 ( ) 1 = 𝜌𝛽 𝜎𝑡+ 1 Δ𝑡 , 𝑡 + Δ𝑡 𝜎𝑡+ 1 Δ𝑡 𝑆𝑡+ 1 Δ𝑡 Δ𝑡 + 𝑂((Δ𝑡)2 ) 2 2 2 2 [( ] )3 | 𝔼ℚ 𝑆𝑡+Δ𝑡 − 𝑆𝑡 | ℱ𝑡 | )3 ( 𝑡+Δ𝑡 = (𝑟 − 𝐷)3 𝑆𝑢 𝑑𝑢 ∫𝑡 [ )2 ( 𝑡+Δ𝑡

𝑡+Δ𝑡

+3(𝑟 − 𝐷)

2

∫𝑡

( +3(𝑟 − 𝐷) ( +

𝑡+Δ𝑡

∫𝑡

𝑡+Δ𝑡

∫𝑡

𝑆𝑢 𝑑𝑢 )(

𝑆𝑢 𝑑𝑢 )

𝜎𝑢2 𝑆𝑢2 𝑑𝑢

𝔼

𝔼

ℚ

𝑡+Δ𝑡

ℚ

∫𝑡

𝑡+Δ𝑡

∫𝑡

[

𝑡+Δ𝑡

∫𝑡

𝑡+Δ𝑡

𝜎𝑢 𝛽(𝜎𝑢 , 𝑢)𝑆𝑢 𝑑𝑢

|

| 𝜎𝑢 𝑆𝑢 𝑑𝑊𝑢𝑆 | ℱ𝑡 | |

)

𝜎𝑢2 𝑆𝑢2 𝑑𝑢 |

| 𝜎𝑢 𝑆𝑢 𝑑𝑊𝑢𝑆 | ℱ𝑡 |

]

]

|

= 𝑂((Δ𝑡) ) 2

[( )2 ( )| ] 𝜎𝑡+Δ𝑡 − 𝜎𝑡 | ℱ𝑡 𝑆𝑡+Δ𝑡 − 𝑆𝑡 | )2 ( 𝑡+Δ𝑡 ) ( 𝑡+Δ𝑡 2 = (𝑟 − 𝐷) 𝑆𝑢 𝑑𝑢 𝛼 ̃(𝜎𝑢 , 𝑢) 𝑑𝑢 ∫𝑡 ∫𝑡 ( 𝑡+Δ𝑡 ) ( 𝑡+Δ𝑡 ) +2(𝑟 − 𝐷) 𝑆𝑢 𝑑𝑢 𝛼 ̃(𝜎𝑢 , 𝑢) 𝑑𝑢 ∫𝑡 ∫𝑡 [ ] 𝑡+Δ𝑡 | ℚ 𝑆| ×𝔼 𝜎𝑢 𝑆𝑢 𝑑𝑊𝑢 | ℱ𝑡 | ∫𝑡 | ( 𝑡+Δ𝑡 ) ( 𝑡+Δ𝑡 ) + 𝛼 ̃(𝜎𝑢 , 𝑢) 𝑑𝑢 𝜎𝑢2 𝑆𝑢2 𝑑𝑢 ∫𝑡 ∫𝑡 [( )2 (

𝔼ℚ

+(𝑟 − 𝐷)

2

𝑡+Δ𝑡

∫𝑡 (

+2(𝑟 − 𝐷) ∫𝑡 [(

𝑡+Δ𝑡

𝑆𝑢 𝑑𝑢

𝔼

ℚ

)

𝑡+Δ𝑡

∫𝑡

𝛽(𝜎𝑢 , 𝑢) 𝑑𝑊𝑢𝜎

𝑆𝑢 𝑑𝑢

)| ] | | ℱ𝑡 | |

)| ] | ×𝔼 | ℱ𝑡 | ∫𝑡 ∫𝑡 | [( ] ( 𝑡+Δ𝑡 ) ) 𝑡+Δ𝑡 | | + 𝜎𝑢2 𝑆𝑢2 𝑑𝑢 𝔼ℚ 𝛽(𝜎𝑢 , 𝑢) 𝑑𝑊𝑢𝜎 | ℱ𝑡 | ∫𝑡 ∫𝑡 | = 𝑂((Δ𝑡)2 ) ℚ

𝑡+Δ𝑡

𝜎𝑢 𝑆𝑢 𝑑𝑊𝑢𝑆

)(

𝑡+Δ𝑡

𝛽(𝜎𝑢 , 𝑢) 𝑑𝑊𝑢𝜎

720

7.2.3 Stochastic Volatility

[( )( )2 | ] 𝑆𝑡+Δ𝑡 − 𝑆𝑡 𝜎𝑡+Δ𝑡 − 𝜎𝑡 | ℱ𝑡 | ( 𝑡+Δ𝑡 ) ( 𝑡+Δ𝑡 )2 = (𝑟 − 𝐷) 𝑆𝑢 𝑑𝑢 𝛼 ̃(𝜎𝑢 , 𝑢) 𝑑𝑢 ∫𝑡 ∫𝑡 ( 𝑡+Δ𝑡 ) ( 𝑡+Δ𝑡 ) +2(𝑟 − 𝐷) 𝑆𝑢 𝑑𝑢 𝛼 ̃(𝜎𝑢 , 𝑢) 𝑑𝑢 ∫𝑡 ∫𝑡 [ ] 𝑡+Δ𝑡 | ℚ 𝜎| ×𝔼 𝛽(𝜎𝑢 , 𝑢) 𝑑𝑊𝑢 | ℱ𝑡 | ∫𝑡 | ( 𝑡+Δ𝑡 ) ( 𝑡+Δ𝑡 ) +(𝑟 − 𝐷) 𝑆𝑢 𝑑𝑢 𝛽(𝜎𝑢 , 𝑢)2 𝑑𝑢 ∫𝑡 ∫𝑡 [ ] ( 𝑡+Δ𝑡 )2 𝑡+Δ𝑡 | ℚ 𝑆| + 𝛼 ̃(𝜎𝑢 , 𝑢) 𝑑𝑢 𝔼 𝜎𝑢 𝑆𝑢 𝑑𝑊𝑢 | ℱ𝑡 | ∫𝑡 ∫𝑡 | ) ( 𝑡+Δ𝑡 𝛼 ̃(𝜎𝑢 , 𝑢) 𝑑𝑢 +2 ∫𝑡 [( ) ( 𝑡+Δ𝑡 )| ] 𝑡+Δ𝑡 ℚ 𝑆 𝜎 | ×𝔼 𝜎𝑢 𝑆𝑢 𝑑𝑊𝑢 𝛽(𝜎𝑢 , 𝑢) 𝑑𝑊𝑢 | ℱ𝑡 | ∫𝑡 ∫𝑡 | [ ] ( 𝑡+Δ𝑡 ) 𝑡+Δ𝑡 | | + 𝛽(𝜎𝑢 , 𝑢)2 𝑑𝑢 𝔼ℚ 𝜎𝑢 𝑆𝑢 𝑑𝑊𝑢𝑆 | ℱ𝑡 | ∫𝑡 ∫𝑡 | = 𝑂((Δ𝑡)2 ) 𝔼ℚ

and [ ] | 𝔼ℚ (𝜎𝑡+Δ𝑡 − 𝜎𝑡 )3 | ℱ𝑡 | ( 𝑡+Δ𝑡 )3 = 𝛼 ̃(𝜎𝑢 , 𝑢) 𝑑𝑢 ∫𝑡 [ ] ( 𝑡+Δ𝑡 )2 𝑡+Δ𝑡 | | +3 𝛼 ̃(𝜎𝑢 , 𝑢) 𝑑𝑢 𝔼ℚ 𝛽(𝜎𝑢 , 𝑢) 𝑑𝑊𝑢𝜎 | ℱ𝑡 | ∫𝑡 ∫𝑡 | ( 𝑡+Δ𝑡 ) ( 𝑡+Δ𝑡 ) +3 𝛼 ̃(𝜎𝑢 , 𝑢) 𝑑𝑢 𝛽(𝜎𝑢 , 𝑢)2 𝑑𝑢 ∫𝑡 ∫𝑡 [ ] ( 𝑡+Δ𝑡 ) 𝑡+Δ𝑡 | | + 𝛽(𝜎𝑢 , 𝑢)2 𝑑𝑢 𝔼ℚ 𝛽(𝜎𝑢 , 𝑢) 𝑑𝑊𝑢𝜎 | ℱ𝑡 | ∫𝑡 ∫𝑡 | = 𝑂((Δ𝑡)2 ). From the Chapman–Kolmogorov equation we have 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑆𝑇 , 𝑇 ) =

∞

∫0

∞

∫0

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡)𝑝(𝑥, 𝑦, 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥

7.2.3 Stochastic Volatility

721

and expanding 𝑝(𝑥, 𝑦, 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) using Taylor series centred on 𝑆𝑡 and 𝜎𝑡 , 𝑝(𝑥, 𝑦, 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) = 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕 +(𝑥 − 𝑆𝑡 ) 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝑆𝑡 𝜕 +(𝑦 − 𝜎𝑡 ) 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝜎𝑡 1 𝜕2 + (𝑥 − 𝑆𝑡 )2 2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 2 𝜕𝑆𝑡 𝜕2 1 + (𝑦 − 𝜎𝑡 )2 2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 2 𝜕𝜎𝑡 +(𝑥 − 𝑆𝑡 )(𝑦 − 𝜎𝑡 )

𝜕2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝜎𝑡 𝜕𝑆𝑡

+𝑂((𝑥 − 𝑆𝑡 )3 ) + 𝑂((𝑥 − 𝑆𝑡 )2 (𝑦 − 𝜎𝑡 )) +𝑂((𝑥 − 𝑆𝑡 )(𝑦 − 𝜎𝑡 )2 ) + 𝑂((𝑦 − 𝜎𝑡 )3 ). Substituting Taylor’s expansion into the Chapman–Kolmogorov equation yields 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑆𝑇 , 𝑇 ) = 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 )

∞

∫0

∞

∫0

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡) 𝑑𝑦𝑑𝑥

∞

+

𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) ∫0 𝜕𝑆𝑡

+

𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) ∫0 𝜕𝜎𝑡

+

1 𝜕2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) ∫0 2 𝜕𝑆𝑡2

+

1 𝜕2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) ∫0 2 𝜕𝜎𝑡2

+

𝜕2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝜎𝑡 𝜕𝑆𝑡 ∞

∞

∞

∫0 ∞

∞

∫0

(𝑥 − 𝑆𝑡 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡) 𝑑𝑦𝑑𝑥

(𝑦 − 𝜎𝑡 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡) 𝑑𝑦𝑑𝑥

∞

∞

∫0 ∞

∞

∫0

(𝑥 − 𝑆𝑡 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡) 𝑑𝑦𝑑𝑥

(𝑦 − 𝜎𝑡 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡) 𝑑𝑦𝑑𝑥

(𝑥 − 𝑆𝑡 )(𝑦 − 𝜎𝑡 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡) 𝑑𝑦𝑑𝑥 ∫0 ∫0 ( ∞ ∞ ) +𝑂 (𝑥 − 𝑆𝑡 )3 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡) 𝑑𝑦𝑑𝑥 ∫ 0 ∫0 ( ∞ ∞ ) +𝑂 (𝑥 − 𝑆𝑡 )2 (𝑦 − 𝜎𝑡 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡) 𝑑𝑦𝑑𝑥 ∫ 0 ∫0 ( ∞ ∞ ) 2 +𝑂 (𝑥 − 𝑆𝑡 )(𝑦 − 𝜎𝑡 ) 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡) 𝑑𝑦𝑑𝑥 ∫ 0 ∫0 ( ∞ ∞ ) 3 +𝑂 (𝑦 − 𝜎𝑡 ) 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡) 𝑑𝑦𝑑𝑥 . ∫ 0 ∫0 ×

722

7.2.3 Stochastic Volatility

Since ∞

∫0 ∞

∞

∫0

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡) 𝑑𝑦𝑑𝑥 = 1

∞

(𝑥 − 𝑆𝑡 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡) 𝑑𝑦𝑑𝑥 = (𝑟 − 𝐷)𝑆𝑡+ 1 Δ𝑡 Δ𝑡 2 ) ( +𝑂 (Δ𝑡)3 ( ) ∞ ∞ 1 (𝑦 − 𝜎𝑡 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡) 𝑑𝑦𝑑𝑥 = 𝛼 ̃ 𝜎𝑡+ 1 Δ𝑡 , 𝑡 + Δ𝑡 Δ𝑡 ∫0 ∫0 2 2 ) ( +𝑂 (Δ𝑡)2

∫0

∫0

∞

∞

∫0

∫0

(𝑥 − 𝑆𝑡 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡) 𝑑𝑦𝑑𝑥 = 𝜎 2

𝑆2 Δ𝑡 𝑡+ 12 Δ𝑡 𝑡+ 21 Δ𝑡 ) ( 2

+𝑂 (Δ𝑡) ( )2 1 (𝑦 − 𝜎𝑡 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡) 𝑑𝑦𝑑𝑥 = 𝛽 𝜎𝑡+ 1 Δ𝑡 , 𝑡 + Δ𝑡 Δ𝑡 ∫0 ∫0 2 2 ) ( 2 +𝑂 (Δ𝑡) ( ) ∞ ∞ 1 (𝑥 − 𝑆𝑡 )(𝑦 − 𝜎𝑡 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡) 𝑑𝑦𝑑𝑥 = 𝜌𝛽 𝜎𝑡+ 1 Δ𝑡 , 𝑡 + Δ𝑡 ∫0 ∫0 2 2 ×𝜎𝑡+ 1 Δ𝑡 𝑆𝑡+ 1 Δ𝑡 Δ𝑡 2 2 ) ( +𝑂 (Δ𝑡)2 ∞ ∞ ) ( (𝑥 − 𝑆𝑡 )3 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡) 𝑑𝑦𝑑𝑥 = 𝑂 (Δ𝑡)2 ∫0 ∫0 ∞ ∞ ) ( (𝑥 − 𝑆𝑡 )2 (𝑦 − 𝜎𝑡 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡) 𝑑𝑦𝑑𝑥 = 𝑂 (Δ𝑡)2 ∫0 ∫0 ∞ ∞ ) ( (𝑥 − 𝑆𝑡 )(𝑦 − 𝜎𝑡 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡) 𝑑𝑦𝑑𝑥 = 𝑂 (Δ𝑡)2 ∫0 ∫0 ∞ ∞ ) ( (𝑦 − 𝜎𝑡 )3 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑥, 𝑦, 𝑡) 𝑑𝑦𝑑𝑥 = 𝑂 (Δ𝑡)2 ∫0 ∫0 ∞

∞

then by taking limits Δ𝑡 → 0, 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡 − Δ𝑡; 𝑆𝑇 , 𝑇 ) − 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) Δ𝑡 𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) = (𝑟 − 𝐷) lim 𝑆𝑡+ 1 Δ𝑡 Δ𝑡→0 𝜕𝑆𝑡 2 ( ) 1 𝜕 ̃ 𝜎𝑡+ 1 Δ𝑡 , 𝑡 + Δ𝑡 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) + lim 𝛼 Δ𝑡→0 2 𝜕𝜎𝑡 2 lim

Δ𝑡→0

1 𝜕2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) lim 𝜎 2 1 𝑆 2 1 2 Δ𝑡→0 𝑡+ 2 Δ𝑡 𝑡+ 2 Δ𝑡 𝜕𝑆𝑡2 ( )2 2 1 1 𝜕 + lim 𝛽 𝜎𝑡+ 1 Δ𝑡 , 𝑡 + Δ𝑡 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 2 Δ𝑡→0 2 2 𝜕𝜎𝑡2 +

7.2.3 Stochastic Volatility

723

( ) 1 𝜕2 +𝜌 lim 𝛽 𝜎𝑡+ 1 Δ𝑡 , 𝑡 + Δ𝑡 𝜎𝑡+ 1 Δ𝑡 𝑆𝑡+ 1 Δ𝑡 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) Δ𝑡→0 2 𝜕𝑆𝑡 𝜕𝜎𝑡 2 2 2 + lim 𝑂(Δ𝑡) Δ𝑡→0

and hence −

𝜕 1 𝜕2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) = 𝜎𝑡2 𝑆𝑡2 2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝑡 2 𝜕𝑆𝑡

1 𝜕2 𝜕2 + 𝛽(𝜎𝑡 , 𝑡)2 2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) + 𝜌𝛽(𝜎𝑡 , 𝑡)𝜎𝑡 𝑆𝑡 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 2 𝜕𝜎𝑡 𝜕𝑆𝑡 𝜕𝜎𝑡 +(𝑟 − 𝐷)𝑆𝑡

𝜕 𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) + 𝛼 ̃(𝜎𝑡 , 𝑡) 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝑆𝑡 𝜕𝜎𝑡

or 1 𝜕2 𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) + 𝜎𝑡2 𝑆𝑡2 2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝑡 2 𝜕𝑆𝑡 1 𝜕2 𝜕2 + 𝛽(𝜎𝑡 , 𝑡)2 2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) + 𝜌𝛽(𝜎𝑡 , 𝑡)𝜎𝑡 𝑆𝑡 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 2 𝜕𝜎𝑡 𝜕𝑆𝑡 𝜕𝜎𝑡 +(𝑟 − 𝐷)𝑆𝑡

𝜕 𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) + 𝛼 ̃(𝜎𝑡 , 𝑡) 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) = 0 𝜕𝑆𝑡 𝜕𝜎𝑡

with boundary condition 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑡) = 𝛿(𝑆𝑡 − 𝑆𝑇 )𝛿(𝜎𝑡 − 𝜎𝑇 ), ∀𝑡. 3. Black–Scholes Equation – Stochastic Volatility Model. Under the risk-neutral measure ℚ, ̃ 𝜎 : 𝑡 ≥ 0} be two correlated ℚ-standard Wiener processes on the ̃ 𝑆 : 𝑡 ≥ 0} and {𝑊 let {𝑊 𝑡 𝑡 probability space (Ω, ℱ, ℚ) with correlation 𝜌 ∈ (−1, 1). Let the asset price 𝑆𝑡 under the ℚ measure follow a stochastic volatility model with the following dynamics ̃𝑆 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑 𝑊 𝑡 ̃𝜎 𝑑𝜎𝑡 = 𝛼 ̃(𝜎𝑡 , 𝑡)𝑑𝑡 + 𝛽(𝜎𝑡 , 𝑡)𝑑 𝑊 𝑡

̃𝑆 ⋅ 𝑑𝑊 ̃ 𝜎 = 𝜌𝑑𝑡 𝑑𝑊 𝑡 𝑡

where 𝑟 is the risk-free interest-rate parameter from the money-market account, 𝐷 is the continuous dividend yield, 𝜎𝑡 is the volatility process, 𝛼 ̃(𝜎𝑡 , 𝑡) and 𝛽(𝜎𝑡 , 𝑡) are continuous functions. Let 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) be the transition pdf of the asset price, where the asset price and volatility are 𝑆𝑡 and 𝜎𝑡 at time 𝑡, respectively, given that the asset price and volatility are 𝑆𝑇 and 𝜎𝑇 at time 𝑇 > 𝑡, respectively. We consider a European option written on 𝑆𝑡

724

7.2.3 Stochastic Volatility

with strike price 𝐾 > 0 expiring at time 𝑇 > 𝑡 with payoff Ψ(𝑆𝑇 ), where the option price at time 𝑡 under the risk-neutral measure ℚ is [ ] 𝑉 (𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ Ψ(𝑆𝑇 )|| ℱ𝑡 = 𝑒−𝑟(𝑇 −𝑡)

∞

∫0

∞

∫0

Ψ(𝑥)𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )𝑑𝑦𝑑𝑥.

By using the backward Kolmogorov equation, show that 𝑉 (𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) satisfies the two-dimensional Black–Scholes equation 1 𝜕2𝑉 𝜕2𝑉 𝜕2𝑉 1 𝜕𝑉 + 𝜎𝑡2 𝑆𝑡2 2 + 𝛽(𝜎𝑡 , 𝑡)2 2 + 𝜌𝛽(𝜎𝑡 , 𝑡)𝜎𝑡 𝑆𝑡 𝜕𝑡 2 2 𝜕𝜎𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝜎𝑡 +(𝑟 − 𝐷)𝑆𝑡

𝜕𝑉 𝜕𝑉 +𝛼 ̃(𝜎𝑡 , 𝑡) − 𝑟𝑉 (𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 𝜕𝑆𝑡 𝜕𝜎𝑡

with boundary condition 𝑉 (𝑆𝑇 , 𝜎𝑇 , 𝑇 ; 𝐾, 𝑇 ) = Ψ(𝑆𝑇 ). Solution: Given 𝑉 (𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡)

∞

∫0

∞

∫0

Ψ(𝑥)𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥

by taking first and second-order differentials, we have 𝜕𝑉 = 𝑟𝑒−𝑟(𝑇 −𝑡) ∫0 𝜕𝑡

∞

∞

∫0 ∞

+𝑒−𝑟(𝑇 −𝑡)

∫0

Ψ(𝑥)𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥

∞

Ψ(𝑥)

∫0

𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑡

= 𝑟𝑒−𝑟(𝑇 −𝑡) 𝑉 (𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) ∞

∞

𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑡 ∞ ∞ 𝜕 𝜕𝑉 = 𝑒−𝑟(𝑇 −𝑡) Ψ(𝑥) 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 ∫ ∫ 𝜕𝑆𝑡 𝜕𝑆𝑡 0 0 +𝑒−𝑟(𝑇 −𝑡)

∫0

∫0

∞

∞

𝜕𝑉 = 𝑒−𝑟(𝑇 −𝑡) ∫0 𝜕𝜎𝑡 𝜕2𝑉 = 𝑒−𝑟(𝑇 −𝑡) ∫0 𝜕𝑆𝑡2 𝜕2𝑉 = 𝑒−𝑟(𝑇 −𝑡) ∫0 𝜕𝜎𝑡2 𝜕2𝑉

𝜕𝜎𝑡 𝜕𝑆𝑡

= 𝑒−𝑟(𝑇 −𝑡)

∫0

∫0 ∞

Ψ(𝑥)

Ψ(𝑥)

𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝜎𝑡

Ψ(𝑥)

𝜕2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑆𝑡2

Ψ(𝑥)

𝜕2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝜎𝑡2

Ψ(𝑥)

𝜕2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥. 𝜕𝜎𝑡 𝜕𝑆𝑡

∞

∫0 ∞

∞

∫0 ∞

∞

∫0

7.2.3 Stochastic Volatility

725

From Problem 7.2.3.2 (page 713), the backward Kolmogorov equation satisfied by the transition probability 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) is 1 𝜕 𝜕2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) + 𝜎𝑡2 𝑆𝑡2 2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝑡 2 𝜕𝑆𝑡 1 𝜕2 𝜕2 + 𝛽(𝜎𝑡 , 𝑡)2 2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) + 𝜌𝛽(𝜎𝑡 , 𝑡)𝜎𝑡 𝑆𝑡 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 2 𝜕𝜎𝑡 𝜕𝑆𝑡 𝜕𝜎𝑡 +(𝑟 − 𝐷)𝑆𝑡

𝜕 𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) + 𝛼 ̃(𝜎𝑡 , 𝑡) 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) = 0 𝜕𝑆𝑡 𝜕𝜎𝑡

with boundary condition 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) = 𝛿(𝑆𝑡 − 𝑆𝑇 )𝛿(𝜎𝑡 − 𝜎𝑇 ) for all 𝑡. By multiplying the backward Kolmogorov equation with Ψ(𝑆𝑇 ) and taking double integrals we have ∞

∫0

∞

Ψ(𝑥)

∫0

1 + 𝜎𝑡2 𝑆𝑡2 ∫0 2

∞

∞

1 + 𝛽(𝜎𝑡 , 𝑡)2 ∫0 2 +𝜌𝛽(𝜎𝑡 , 𝑡)𝜎𝑡 𝑆𝑡 +(𝑟 − 𝐷)𝑆𝑡 +̃ 𝛼 (𝜎𝑡 , 𝑡)

∫0

𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )𝑑𝑦𝑑𝑥 𝜕𝑡

∫0

Ψ(𝑥)

∞

∞

∫0

Ψ(𝑥)

∞

∞

∫0

∫0

∞

∞

∫0

∫0

∞

∞

∫0

𝜕2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑆𝑡2

Ψ(𝑥)

Ψ(𝑥)

Ψ(𝑥)

𝜕2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝜎𝑡2

𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑆𝑡

𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 = 0. 𝜕𝜎𝑡

By writing the above equation in terms of [ 𝑒

−𝑟(𝑇 −𝑡)

𝜕2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝜎𝑡 𝜕𝑆𝑡

𝜕𝑉 𝜕𝑉 𝜕𝑉 𝜕 2 𝑉 𝜕 2 𝑉 𝜕2𝑉 , , , and we have , 2 2 𝜕𝑡 𝜕𝑆𝑡 𝜕𝜎𝑡 𝜕𝑆𝑡 𝜕𝜎𝑡 𝜕𝜎𝑡 𝜕𝑆𝑡

1 1 𝜕2𝑉 𝜕2𝑉 𝜕𝑉 − 𝑟𝑉 (𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝜎𝑡2 𝑆𝑡2 2 + 𝛽(𝜎𝑡 , 𝑡)2 2 𝜕𝑡 2 2 𝜕𝑆𝑡 𝜕𝜎𝑡

] 𝜕𝑉 𝜕2𝑉 𝜕𝑉 +𝜌𝛽(𝜎𝑡 , 𝑡)𝜎𝑡 𝑆𝑡 + (𝑟 − 𝐷)𝑆𝑡 +𝛼 ̃(𝜎𝑡 , 𝑡) =0 𝜕𝜎𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝜎𝑡

726

7.2.3 Stochastic Volatility

or 1 𝜕2𝑉 𝜕2𝑉 𝜕2𝑉 1 𝜕𝑉 + 𝜎𝑡2 𝑆𝑡2 2 + 𝛽(𝜎𝑡 , 𝑡)2 2 + 𝜌𝛽(𝜎𝑡 , 𝑡)𝜎𝑡 𝑆𝑡 𝜕𝑡 2 2 𝜕𝜎𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝜎𝑡 +(𝑟 − 𝐷)𝑆𝑡

𝜕𝑉 𝜕𝑉 +𝛼 ̃(𝜎𝑡 , 𝑡) − 𝑟𝑉 (𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) = 0. 𝜕𝑆𝑡 𝜕𝜎𝑡

Finally, for the boundary condition we note that 𝑉 (𝑆𝑇 , 𝜎𝑇 , 𝑇 ; 𝐾, 𝑇 ) =

∞

∫0

∞

∫0

Ψ(𝑥)𝑝(𝑆𝑇 , 𝜎𝑇 , 𝑇 ; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥

and from the definition of the Dirac delta function, 𝑉 (𝑆𝑇 , 𝜎𝑇 , 𝑇 ; 𝐾, 𝑇 ) =

∞

∫0

∞

∫0 ∞

∞

=

∫0 ∫0 = Ψ(𝑆𝑇 ).

Ψ(𝑥)𝑝(𝑆𝑇 , 𝜎𝑇 , 𝑇 ; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 Ψ(𝑥)𝛿(𝑆𝑇 − 𝑥)𝛿(𝜎𝑇 − 𝑦) 𝑑𝑦𝑑𝑥

4. Forward Kolmogorov Equation – Stochastic Volatility Model. Let {𝑊𝑡𝑆 : 𝑡 ≥ 0} and {𝑊𝑡𝜎 : 𝑡 ≥ 0} be two correlated ℙ-standard Wiener processes on the probability space (Ω, ℱ, ℙ) and let the asset price 𝑆𝑡 follow a stochastic volatility model with the following SDEs 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡𝑆 𝑆𝑡 𝑑𝜎𝑡 = 𝛼(𝜎𝑡 , 𝑡)𝑑𝑡 + 𝛽(𝜎𝑡 , 𝑡)𝑑𝑊𝑡𝜎

𝑑𝑊𝑡𝑆 ⋅ 𝑑𝑊𝑡𝜎 = 𝜌𝑑𝑡

where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎𝑡 is the volatility process, 𝛼(𝜎𝑡 , 𝑡) and 𝛽(𝜎𝑡 , 𝑡) are continuous functions, 𝜌 ∈ (−1, 1) is the correlation parameter, and let 𝑟 be the risk-free interest-rate parameter from the money-market account. By introducing {𝑍𝑡 : 𝑡 ≥ 0} as a standard Wiener process, independent of 𝑊𝑡𝜎 , show that we can write √ 𝑊𝑡𝑆 = 𝜌𝑊𝑡𝜎 + 1 − 𝜌2 𝑍𝑡 . Using the two-dimensional Girsanov’s theorem, show that under the risk-neutral measure ℚ, ̃𝑆 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑 𝑊 𝑡

̃𝜎 𝑑𝜎𝑡 = 𝛼 ̃(𝜎𝑡 , 𝑡)𝑑𝑡 + 𝛽(𝜎𝑡 , 𝑡)𝑑 𝑊 𝑡 √ ̃ 𝑆 = 𝜌𝑊 ̃ 𝜎 + 1 − 𝜌2 𝑍 ̃ 𝜎 and 𝑍 ̃𝑡 are ℚ-standard Wiener processes, 𝑊 ̃𝜎 ⟂ ̃𝑡 , 𝑊 where 𝑊 𝑡 𝑡 𝑡 𝑡 ⟂ ̃ ̃(𝜎𝑡 , 𝑡) = 𝛼(𝜎𝑡 , 𝑡) − 𝛾𝑡 𝛽(𝜎𝑡 , 𝑡) such that 𝛾𝑡 is the market price of volatility risk. 𝑍𝑡 and 𝛼

7.2.3 Stochastic Volatility

727

Let 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) be the transition pdf of the asset price, where the asset price and volatility are 𝑆𝑡 and 𝜎𝑡 at time 𝑡, respectively, given that the asset price and volatility are 𝑆𝑇 and 𝜎𝑇 at time 𝑇 > 𝑡, respectively. From the Chapman–Kolmogorov equation for Δ𝑇 > 0, 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 + Δ𝑇 ) ∞

=

∫0

∞

∫0

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 + Δ𝑇 ) 𝑑𝑦𝑑𝑥

show that in the limit Δ𝑇 → 0, 𝑇

𝜕 𝑝(𝑆𝑡 , 𝑡; 𝑆𝑇 , 𝑇 ) = − ∫0 ∫0 𝜕𝑇

∞

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

𝜕 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥. 𝜕𝑇

Finally, using the backward Kolmogorov equation, show that 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) satisfies the forward Kolmogorov equation ] 1 𝜕2 [ 2 2 𝜕 𝜎 𝑆 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) − 𝜕𝑇 2 𝜕𝑆 2 𝑇 𝑇 𝑇 −

] 1 𝜕2 [ 𝛽(𝜎𝑇 , 𝑇 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 2 2 𝜕𝜎 𝑇

−𝜌

[ ] 𝜕2 𝛽(𝜎𝑇 , 𝑇 )𝜎𝑇 𝑆𝑇 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝜎𝑇 𝜕𝑆𝑇

+(𝑟 − 𝐷)

] ] 𝜕 [ 𝜕 [ 𝑆𝑇 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) + 𝛼 ̃(𝜎𝑇 , 𝑇 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) = 0 𝜕𝑆𝑇 𝜕𝜎𝑇

with boundary condition 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) = 𝛿(𝑆𝑡 − 𝑆𝑇 )𝛿(𝜎𝑡 − 𝜎𝑇 ),

∀𝑡.

Solution: To show that under the risk-neutral measure ℚ, the asset price and the volatility follow the dynamics ̃𝑆 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑 𝑊 𝑡 ̃𝜎 𝑑𝜎𝑡 = 𝛼 ̃(𝜎𝑡 , 𝑡)𝑑𝑡 + 𝛽(𝜎𝑡 , 𝑡)𝑑 𝑊 𝑡

see Problem 7.2.3.2 (page 713). From the Chapman–Kolmogorov equation for Δ𝑇 > 0, 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 + Δ𝑇 ) ∞

=

∫0

∞

∫0

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 + Δ𝑇 ) 𝑑𝑦𝑑𝑥

728

7.2.3 Stochastic Volatility

we can write

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 + Δ𝑇 ) − 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) ∞

=

∞

∫0

∫0

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 + Δ𝑇 ) 𝑑𝑦𝑑𝑥

−𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) ∞

=

∞

∫0

∫0

∞

−

=

∞

∫0

∫0

∞

∞

∫0

∫0 ∞

−

∫0

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 + Δ𝑇 ) 𝑑𝑦𝑑𝑥

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 + Δ𝑇 ) 𝑑𝑦𝑑𝑥 ∞

∫0

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )𝛿(𝑥 − 𝑆𝑇 )𝛿(𝑦 − 𝜎𝑇 ) 𝑑𝑦𝑑𝑥.

By dividing the expression with Δ𝑇 and taking limits Δ𝑇 → 0,

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 + Δ𝑇 ) − 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) Δ𝑇 →0 Δ𝑇 lim

∞

= lim

Δ𝑡→0 ∫0

∞

∫0 ∞

− lim

Δ𝑇 →0 ∫0

∞

∫0 ∞{

∞

= lim

Δ𝑇 →0 ∫0

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 + Δ𝑇 ) 𝑑𝑦𝑑𝑥 Δ𝑇

∫0

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )𝛿(𝑥 − 𝑆𝑇 )𝛿(𝑦 − 𝜎𝑇 ) 𝑑𝑦𝑑𝑥 Δ𝑇 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

[

𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 + Δ𝑇 ) − 𝛿(𝑥 − 𝑆𝑇 )𝛿(𝑦 − 𝜎𝑇 ) × Δ𝑇 ∞{

∞

= lim

Δ𝑇 →0 ∫0

[

∫0

]} 𝑑𝑦𝑑𝑥

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 + Δ𝑇 ) − 𝑝(𝑥, 𝑦, 𝑇 + Δ𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 + Δ𝑇 ) × Δ𝑇

]} 𝑑𝑦𝑑𝑥.

7.2.3 Stochastic Volatility

729

Since lim

Δ𝑇 →0

=

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 + Δ𝑇 ) − 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) Δ𝑇

𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝑇

and 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 + Δ𝑇 ) − 𝑝(𝑥, 𝑦, 𝑇 + Δ𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 + Δ𝑇 ) Δ𝑇 𝜕 = − 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝑇 lim

Δ𝑇 →0

thus 𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝑇 ∞

=−

∫0

∞

∫0

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

𝜕 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥. 𝜕𝑇

From the backward Kolmogorov equation on 𝑝(𝑥, 𝑦, 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) for the stochastic volatility model 1 𝜕2 𝜕 𝑝(𝑥, 𝑦, 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) + 𝑦2 𝑥2 2 𝑝(𝑥, 𝑦, 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝑡 2 𝜕𝑥 1 𝜕2 𝜕2 + 𝛽(𝑦, 𝑡)2 2 𝑝(𝑥, 𝑦, 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) + 𝜌𝛽(𝑦, 𝑡)𝑦𝑥 𝑝(𝑥, 𝑦, 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 2 𝜕𝑦𝜕𝑥 𝜕𝑦 +(𝑟 − 𝐷)𝑥

𝜕 𝜕 ̃(𝑦, 𝑡) 𝑝(𝑥, 𝑦, 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) = 0 𝑝(𝑥, 𝑦, 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) + 𝛼 𝜕𝑥 𝜕𝑦

we have 𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝑇 ∞

=

∫0

∞

∫0

+̃ 𝛼 (𝑦, 𝑇 ) +

[ 𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) (𝑟 − 𝐷)𝑥 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝑥

𝜕 1 𝜕2 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) + 𝑦2 𝑥2 2 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝑦 2 𝜕𝑥

𝜕2 1 𝛽(𝑦, 𝑇 )2 2 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 2 𝜕𝑦

] 𝜕2 +𝜌𝛽(𝑦, 𝑇 )𝑦𝑥 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑦𝜕𝑥

730

7.2.3 Stochastic Volatility ∞

=

∫0

∞

∫0 ∞

+

∫0

∞

∫0 ∞

+

∫0 ∫0 ∫0

𝛼 ̃(𝑦, 𝑇 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

𝜕 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑦

𝜕2 1 2 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 2 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 2 𝜕𝑥

∞

𝜕2 1 𝛽(𝑦, 𝑇 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 2 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 2 𝜕𝑦

∫0 ∞

+

𝜕 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑥

∞

∫0 ∞

+

(𝑟 − 𝐷)𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

∞

∫0

𝜌𝛽(𝑦, 𝑇 )𝑦𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

𝜕2 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥. 𝜕𝑦𝜕𝑥

Integrating by parts for ∞

∞

∫0

∫0

(𝑟 − 𝐷)𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

𝜕 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑥

we let 𝑢 = (𝑟 − 𝐷)𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

and

𝑑𝑣 𝜕 = 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑥 𝜕𝑥

so that ] 𝑑𝑢 𝜕 [ (𝑟 − 𝐷)𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) = 𝑑𝑥 𝜕𝑥

and

𝑣 = 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ).

Thus, we have ∞

∞

∫0

∫0 ∞

=

∫0

(𝑟 − 𝐷)𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

|∞ (𝑟 − 𝐷)𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 )|| 𝑑𝑦 |0 ∞

−

∫0

∞

∫0 ∞

=−

∫0

𝜕 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑥

∞

∫0

𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 )

] 𝜕 [ (𝑟 − 𝐷)𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑥

𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 )

] 𝜕 [ (𝑟 − 𝐷)𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥. 𝜕𝑥

7.2.3 Stochastic Volatility

731

For the case of integration by parts for ∞

∫0

∞

∫0

𝛼 ̃(𝑦, 𝑇 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

𝜕 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑦

we let 𝑢=𝛼 ̃(𝑦, 𝑇 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

and

𝑑𝑣 𝜕 = 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦 𝜕𝑦

so that ] 𝜕 [ 𝑑𝑢 𝛼 ̃(𝑦, 𝑇 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) = 𝑑𝑦 𝜕𝑦

and

𝑣 = 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ).

Hence, ∞

∞

∫0 ∫0

∞

−

∫0

∞

] 𝜕 [ 𝛼 ̃(𝑦, 𝑇 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑦

∞

] 𝜕 [ 𝛼 ̃(𝑦, 𝑇 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥. 𝜕𝑦

∫0 ∞

=−

𝜕 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑦

|∞ 𝛼 ̃(𝑦, 𝑇 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 )|| 𝑑𝑥 |0

∞

=

𝛼 ̃(𝑦, 𝑇 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

∫0

∫0

∫0

For the case of ∞

∫0

∞

∫0

𝜕2 1 2 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 2 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 2 𝜕𝑥

and solving the integral by parts, we let 𝑢=

1 2 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 2

and

𝑑𝑣 𝜕2 = 2 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑥 𝜕𝑥

so that [ ] 𝑑𝑢 𝜕 1 2 2 = 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥 𝜕𝑥 2

and

𝑣=

𝜕 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ). 𝜕𝑥

732

7.2.3 Stochastic Volatility

Hence, ∞

∞

∫0

∫0

|∞ 𝜕 1 2 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 )|| 𝑑𝑥 2 𝜕𝑥 |0

∞

=

𝜕2 1 2 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 2 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 2 𝜕𝑥

∫0

∞

−

∞

∫0

∫0 ∞

=−

∫0

∞

∫0

[ ] 𝜕 1 2 2 𝜕 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑥 𝜕𝑥 2 [ ] 𝜕 1 2 2 𝜕 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥. 𝜕𝑥 𝜕𝑥 2

Integrating by parts again, we let 𝑢=

[ ] 𝜕 1 2 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝜕𝑥 2

and

𝑑𝑣 𝜕 = 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑥 𝜕𝑥

so that [ ] 𝑑𝑢 𝜕2 1 2 2 = 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥 𝜕𝑥 2

and

𝑣 = 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ).

Thus, we have ∞

∞

∫0

𝜕2 1 2 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 2 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 2 𝜕𝑥

∫0

[ ] |∞ 𝜕 1 2 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 )|| 𝑑𝑦 𝜕𝑥 2 |0

∞

=−

∫0 ∞

+

=

∞

∫0

∫0

∞

∞

∫0

∫0

[ ] 𝜕2 1 2 2 𝑥 𝑝(𝑆 , 𝜎 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 𝑦 𝑡 𝑡 𝜕𝑥2 2

[ ] 𝜕2 1 2 2 𝑥 𝑝(𝑆 , 𝜎 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥. 𝑦 𝑡 𝑡 𝜕𝑥2 2

Taking integration by parts of ∞

∫0

∞

∫0

𝜕2 1 𝛽(𝑦, 𝑇 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 2 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 )𝑑𝑦𝑑𝑥 2 𝜕𝑦

we let 𝑢=

1 𝛽(𝑦, 𝑇 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 2

and

𝑑𝑣 𝜕2 = 2 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦 𝜕𝑦

7.2.3 Stochastic Volatility

733

so that [ ] 𝑑𝑢 𝜕 1 = 𝛽(𝑦, 𝑇 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦 𝜕𝑦 2

𝑣=

and

𝜕 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ). 𝜕𝑦

Therefore, ∞

∞

∫0

∫0

|∞ 𝜕 1 𝛽(𝑦, 𝑇 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 )|| 𝑑𝑥 2 𝜕𝑦 |0

∞

=

𝜕2 1 𝛽(𝑦, 𝑇 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 2 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 2 𝜕𝑦

∫0

∞

−

∫0

∞

∫0 ∞

=−

∫0

∞

∫0

[ ] 𝜕 𝜕 1 𝛽(𝑦, 𝑇 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑦 2 𝜕𝑦 [ ] 𝜕 𝜕 1 𝛽(𝑦, 𝑇 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥. 𝜕𝑦 2 𝜕𝑦

Integrating by parts again, we let 𝑢=

[ ] 𝜕 1 𝛽(𝑦, 𝑇 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝜕𝑦 2

and

𝑑𝑣 𝜕 = 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦 𝜕𝑦

so that [ ] 𝜕2 1 𝑑𝑢 = 2 𝛽(𝑦, 𝑇 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦 𝜕𝑦 2

and

𝑣 = 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 )

and hence ∞

∞

𝜕2 1 𝛽(𝑦, 𝑇 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 2 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 2 𝜕𝑦

∫0

∫0

[ ] |∞ 𝜕 1 𝛽(𝑦, 𝑇 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 )|| 𝑑𝑥 𝜕𝑦 2 |0

∞

=−

∫0 ∞

+

=

∞

∫0

∫0

∞

∞

∫0

[ ] 𝜕2 1 2 𝑝(𝑆 , 𝜎 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 𝛽(𝑦, 𝑇 ) 𝑡 𝑡 𝜕𝑦2 2

∫0

[ ] 𝜕2 1 2 𝑝(𝑆 , 𝜎 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥. 𝛽(𝑦, 𝑇 ) 𝑡 𝑡 𝜕𝑦2 2

∞

∞

Finally, for

∫0

∫0

𝜌𝛽(𝑦, 𝑇 )𝑦𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

𝜕2 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑦𝜕𝑥

734

7.2.3 Stochastic Volatility

we let 𝑢 = 𝜌𝛽(𝑦, 𝑇 )𝑦𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

𝑑𝑣 𝜕2 = 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦 𝜕𝑦𝜕𝑥

and

so that ] 𝑑𝑢 𝜕 [ 𝜌𝛽(𝑦, 𝑇 )𝑦𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) = 𝑑𝑦 𝜕𝑦

and

𝑣=

𝜕 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ). 𝜕𝑥

Therefore, ∞

∞

∫0

∫0 ∞

=

𝜌𝛽(𝑦, 𝑇 )𝑦𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

𝜌𝛽(𝑦, 𝑇 )𝑦𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

∫0

∞

−

∫0 ∫0

|∞ 𝜕 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 )|| 𝑑𝑥 𝜕𝑥 |0

∞

] 𝜕 𝜕 [ 𝜌𝛽(𝑦, 𝑇 )𝑦𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑦 𝜕𝑥

∞

] 𝜕 𝜕 [ 𝜌𝛽(𝑦, 𝑇 )𝑦𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥. 𝜕𝑦 𝜕𝑥

∫0 ∞

=−

𝜕2 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑦𝜕𝑥

∫0

Using integration by parts again, we let 𝑢=

] 𝜕 [ 𝜌𝛽(𝑦, 𝑇 )𝑦𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝜕𝑦

and

𝑑𝑣 𝜕 = 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑥 𝜕𝑥

so that ] 𝑑𝑢 𝜕2 [ 𝜌𝛽(𝑦, 𝑇 )𝑦𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) = 𝑑𝑥 𝜕𝑦𝜕𝑥

and

𝑣 = 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 )

and therefore, ∞

∞

∫0

𝜌𝛽(𝑦, 𝑇 )𝑦𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

∫0

|∞ ] 𝜕 [ 𝜌𝛽(𝑦, 𝑇 )𝑦𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 )|| 𝑑𝑦 𝜕𝑦 |0

∞

=−

∫0 ∞

+

=

∞

∫0

∫0

∞

∞

∫0

𝜕2 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑦𝜕𝑥

∫0

] 𝜕2 [ 𝜌𝛽(𝑦, 𝑇 )𝑦𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 𝜕𝑦𝜕𝑥

] 𝜕2 [ 𝜌𝛽(𝑦, 𝑇 )𝑦𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥. 𝜕𝑦𝜕𝑥

7.2.3 Stochastic Volatility

735

Thus 𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝑇 ∞ ∞ ] 𝜕 [ (𝑟 − 𝐷)𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 =− ∫0 ∫0 𝜕𝑥 ∞ ∞ ] 𝜕 [ 𝛼 ̃(𝑦, 𝑇 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 − ∫0 ∫0 𝜕𝑦 ] ∞ ∞ 2 [ 1 2 2 𝜕 𝑥 𝑝(𝑆 , 𝜎 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 + 𝑦 𝑡 𝑡 ∫0 ∫0 𝜕𝑥2 2 ] ∞ ∞ 2 [ 1 𝜕 2 𝑝(𝑆 , 𝜎 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 𝛽(𝑦, 𝑇 ) + 𝑡 𝑡 ∫0 ∫0 𝜕𝑦2 2 ∞ ∞ ] 𝜕2 [ 𝜌𝛽(𝑦, 𝑇 )𝑦𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑝(𝑥, 𝑦, 𝑇 ; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑑𝑦𝑑𝑥 + ∫0 ∫0 𝜕𝑦𝜕𝑥 or 𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝑇 ∞ ∞ ] 𝜕 [ (𝑟 − 𝐷)𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝛿(𝑥 − 𝑆𝑇 )𝛿(𝑦 − 𝜎𝑇 ) 𝑑𝑦𝑑𝑥 =− ∫0 ∫0 𝜕𝑥 ∞ ∞ ] 𝜕 [ 𝛼 ̃(𝑦, 𝑇 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝛿(𝑥 − 𝑆𝑇 )𝛿(𝑦 − 𝜎𝑇 ) 𝑑𝑦𝑑𝑥 − ∫0 ∫0 𝜕𝑦 ] ∞ ∞ 2 [ 1 2 2 𝜕 𝑥 𝑝(𝑆 , 𝜎 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝛿(𝑥 − 𝑆𝑇 )𝛿(𝑦 − 𝜎𝑇 ) 𝑑𝑦𝑑𝑥 𝑦 + 𝑡 𝑡 ∫0 ∫0 𝜕𝑥2 2 ] ∞ ∞ 2 [ 1 𝜕 2 𝑝(𝑆 , 𝜎 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝛿(𝑥 − 𝑆𝑇 )𝛿(𝑦 − 𝜎𝑇 ) 𝑑𝑦𝑑𝑥 + 𝛽(𝑦, 𝑇 ) 𝑡 𝑡 ∫0 ∫0 𝜕𝑦2 2 ∞ ∞ ] 𝜕2 [ 𝜌𝛽(𝑦, 𝑇 )𝑦𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝛿(𝑥 − 𝑆𝑇 )𝛿(𝑦 − 𝜎𝑇 ) 𝑑𝑦𝑑𝑥. + ∫0 ∫0 𝜕𝑦𝜕𝑥 Hence, 𝜕 𝜕 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) = − 𝜕𝑇 𝜕𝑆𝑇 𝜕 − 𝜕𝜎𝑇 +

𝜕2 𝜕𝑆𝑇2

+

𝜕2 𝜕𝜎𝑇2

+

[

(𝑟 − 𝐷)𝑆𝑇 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 )

]

[ ] 𝛼 ̃(𝜎𝑇 , 𝑇 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) [ [

1 2 2 𝜎 𝑆 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 2 𝑇 𝑇

]

] 1 𝛽(𝜎𝑇 , 𝑇 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 2

[ ] 𝜕2 𝜌𝛽(𝜎𝑇 , 𝑇 )𝜎𝑇 𝑆𝑇 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝜎𝑇 𝜕𝑆𝑇

736

7.2.3 Stochastic Volatility

and rearranging terms we have the forward Kolmogorov equation for the stochastic volatility model ] 1 𝜕2 [ 2 2 𝜕 𝜎𝑇 𝑆𝑇 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) − 2 𝜕𝑇 2 𝜕𝑆 𝑇 −

] 1 𝜕2 [ 𝛽(𝜎𝑇 , 𝑇 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 2 2 𝜕𝜎 𝑇

−𝜌

[ ] 𝜕2 𝛽(𝜎𝑇 , 𝑇 )𝜎𝑇 𝑆𝑇 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝜎𝑇 𝜕𝑆𝑇

+(𝑟 − 𝐷)

] ] 𝜕 [ 𝜕 [ 𝑆𝑇 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) + 𝛼 ̃(𝜎𝑇 , 𝑇 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) = 0 𝜕𝑆𝑇 𝜕𝜎𝑇

with boundary condition 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑡) = 𝛿(𝑆𝑡 − 𝑆𝑇 )𝛿(𝜎𝑡 − 𝜎𝑇 ) for all 𝑡. ̃ 𝑆 : 𝑡 ≥ 0} and {𝑊 ̃ 𝜎 : 𝑡 ≥ 0} be two corre5. Under the risk-neutral measure ℚ, let {𝑊 𝑡 𝑡 lated ℚ-standard Wiener processes on the probability space (Ω, ℱ, ℚ) with correlation 𝜌 ∈ (−1, 1). Let the asset price 𝑆𝑡 under the ℚ measure follow a stochastic volatility model with the following dynamics ̃𝑆 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑 𝑊 𝑡 ̃𝜎 𝑑𝜎𝑡 = 𝛼 ̃(𝜎𝑡 , 𝑡)𝑑𝑡 + 𝛽(𝜎𝑡 , 𝑡)𝑑 𝑊 𝑡

̃𝑆 ⋅ 𝑑𝑊 ̃ 𝜎 = 𝜌𝑑𝑡 𝑑𝑊 𝑡 𝑡

where 𝑟 is the risk-free interest-rate parameter from the money-market account, 𝐷 is the ̃(𝜎𝑡 , 𝑡) and 𝛽(𝜎𝑡 , 𝑡) are continuous continuous dividend yield, 𝜎𝑡 is the volatility process, 𝛼 functions. Let 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) be the transition pdf of the asset price, where the asset price and volatility are 𝑆𝑡 and 𝜎𝑡 at time 𝑡, respectively, given that the asset price and volatility are 𝑆𝑇 and 𝜎𝑇 at time 𝑇 > 𝑡, respectively. We consider a European call option written on 𝑆𝑡 with strike price 𝐾 > 0 expiring at time 𝑇 > 𝑡 with payoff Ψ(𝑆𝑇 ) = max{𝑆𝑇 − 𝐾, 0} where, under the risk-neutral measure ℚ, the option price at time 𝑡 is [ ] 𝐶(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑆𝑇 − 𝐾, 0}|| ℱ𝑡 = 𝑒−𝑟(𝑇 −𝑡)

∞

∫0

∞

∫0

max{𝑥 − 𝐾, 0}𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥.

7.2.3 Stochastic Volatility

737

Show that the following identities are true 𝑥

𝜕 𝜕 (𝑥 − 𝐾) = 𝐾 (𝑥 − 𝐾) − (𝑥 − 𝐾) 𝜕𝐾 𝜕𝐾 𝜕 𝜕 (𝑥 − 𝐾) = − (𝑥 − 𝐾) 𝜕𝑥 𝜕𝐾 𝜕2 𝜕2 (𝑧 − 𝐾) = (𝑧 − 𝐾). 𝜕𝑧2 𝜕𝐾 2

By using the forward Kolmogorov equation and the above identities or otherwise, show that 𝐶(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) satisfies [ ] 𝜕𝐶 1 ℚ 2 | 𝜕𝐶 𝜕2𝐶 + (𝑟 − 𝐷)𝐾 − 𝔼 𝜎𝑇 | 𝑆𝑇 = 𝐾, 𝑆𝑡 , 𝜎𝑡 𝐾 2 + 𝐷𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 | 𝜕𝑇 2 𝜕𝐾 𝜕𝐾 2 with boundary condition 𝐶(𝑆𝑇 , 𝜎𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0}. Explain the significance of this equation. Deduce that for a European put option price 𝑃 (𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) written on 𝑆𝑡 at time 𝑡 with payoff Ψ(𝑆𝑇 ) = max{𝐾 − 𝑆𝑇 , 0} expiring at time 𝑇 > 𝑡, 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) satisfies [ ] 𝜕𝑃 𝜕𝑃 𝜕2𝑃 1 | + (𝑟 − 𝐷)𝐾 − 𝔼ℚ 𝜎𝑇2 | 𝑆𝑇 = 𝐾, 𝑆𝑡 , 𝜎𝑡 𝐾 2 + 𝐷𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 | 𝜕𝑇 2 𝜕𝐾 𝜕𝐾 2 with boundary condition 𝑃 (𝑆𝑇 , 𝜎𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝐾 − 𝑆𝑇 , 0}. Under what condition do both the stochastic volatility and local volatility models price European options equally? Solution: For the proof of the three identities, see Problem 7.2.2.5 (page 699). By definition, [ ] 𝐶(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ max{𝑆𝑇 − 𝐾, 0}|| ℱ𝑡 = 𝑒−𝑟(𝑇 −𝑡) = 𝑒−𝑟(𝑇 −𝑡)

∞

∫0

∞

∫0 ∞

∫𝐾 ∫0

∞

max{𝑥 − 𝐾, 0}𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 (𝑥 − 𝐾)𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥.

738

7.2.3 Stochastic Volatility

Differentiating 𝐶(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) with respect to 𝑇 and from the forward Kolmogorov equation for the stochastic volatility model ] 1 𝜕2 [ 2 2 𝜕 𝜎 𝑆 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) − 𝜕𝑇 2 𝜕𝑆 2 𝑇 𝑇 𝑇 −

] 1 𝜕2 [ 𝛽(𝜎𝑇 , 𝑇 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 2 2 𝜕𝜎 𝑇

−𝜌

[ ] 𝜕2 𝛽(𝜎𝑇 , 𝑇 )𝜎𝑇 𝑆𝑇 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) 𝜕𝜎𝑇 𝜕𝑆𝑇

+(𝑟 − 𝐷)

] ] 𝜕 [ 𝜕 [ 𝑆𝑇 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) + 𝛼 ̃(𝜎𝑇 , 𝑇 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 , 𝜎𝑇 , 𝑇 ) = 0 𝜕𝑆𝑇 𝜕𝜎𝑇

we have ∞

∞

𝜕𝐶 (𝑥 − 𝐾)𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 = −𝑟 𝑒−𝑟(𝑇 −𝑡) ∫𝐾 ∫0 𝜕𝑇 ∞ ∞ 𝜕 (𝑥 − 𝐾) 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 +𝑒−𝑟(𝑇 −𝑡) ∫𝐾 ∫0 𝜕𝑇 = −𝑟𝐶(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) ∞ ∞ ] 1 𝜕2 [ + 𝑒−𝑟(𝑇 −𝑡) (𝑥 − 𝐾) 2 𝑦2 𝑥2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 ∫𝐾 ∫0 2 𝜕𝑥 ∞ ∞ ] 𝜕2 [ 1 −𝑟(𝑇 −𝑡) (𝑥 − 𝐾) 2 𝛽(𝑦, 𝑇 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 + 𝑒 ∫𝐾 ∫0 2 𝜕𝑦 ∞ ∞ ] 𝜕2 [ 𝛽(𝑦, 𝑇 )𝑦𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 +𝜌𝑒−𝑟(𝑇 −𝑡) (𝑥 − 𝐾) ∫𝐾 ∫0 𝜕𝑦𝜕𝑥 ∞ ∞ ] 𝜕 [ 𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 (𝑥 − 𝐾) −(𝑟 − 𝐷)𝑒−𝑟(𝑇 −𝑡) ∫𝐾 ∫0 𝜕𝑥 ∞ ∞ [ ] 𝜕 𝛼 ̃(𝑦, 𝑇 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥. (𝑥 − 𝐾) −𝑒−𝑟(𝑇 −𝑡) ∫𝐾 ∫0 𝜕𝑦 Given that the payoff does not depend on 𝜎𝑇 , using integration by parts, all partial derivatives with respect to 𝜎𝑇 will vanish. Hence, ] 𝜕2 [ 𝛽(𝑦, 𝑇 )2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 = 0 2 ∫ 𝐾 ∫0 𝜕𝑦 ∞ ∞ 2 ] 𝜕 [ 𝛽(𝑦, 𝑇 )𝑦𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 = 0 (𝑥 − 𝐾) ∫𝐾 ∫0 𝜕𝑦𝜕𝑥 ∞ ∞ ] 𝜕 [ 𝛼 ̃(𝑦, 𝑇 )𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑦𝑑𝑥 = 0 (𝑥 − 𝐾) ∫ 𝐾 ∫0 𝜕𝑦 ∞

∞

(𝑥 − 𝐾)

7.2.3 Stochastic Volatility

739

and we can rewrite the equation as 𝜕𝐶 = −𝑟𝐶(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) 𝜕𝑇 1 + 𝑒−𝑟(𝑇 −𝑡) ∫0 2

∞

∞

∫𝐾

−(𝑟 − 𝐷)𝑒−𝑟(𝑇 −𝑡)

(𝑥 − 𝐾)

∞

∫0

∞

∫𝐾

] 𝜕2 [ 2 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥𝑑𝑦 2 𝜕𝑥

(𝑥 − 𝐾)

] 𝜕 [ 𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥𝑑𝑦. 𝜕𝑥

Integrating by parts for the inner integral ∞

∫𝐾

(𝑥 − 𝐾)

] 𝜕 [ 𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥 𝜕𝑥

we let 𝑢 = (𝑥 − 𝐾) and

] 𝜕 [ 𝑑𝑣 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) = 𝑑𝑥 𝜕𝑥

so that 𝜕 𝜕 𝑑𝑢 = (𝑥 − 𝐾) = − (𝑥 − 𝐾) and 𝑑𝑥 𝜕𝑥 𝜕𝐾

𝑣 = 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ).

Therefore, ∞

∫𝐾

(𝑥 − 𝐾)

] 𝜕 [ 𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥 𝜕𝑥

∞ = (𝑥 − 𝐾)𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )||𝐾 + ∞

∞

∫𝐾

𝑥𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

𝜕 (𝑥 − 𝐾) 𝑑𝑥 𝜕𝐾 ] ∞[ 𝜕 𝐾 (𝑥 − 𝐾) − (𝑥 − 𝐾) 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥 = ∫𝐾 𝜕𝐾

=

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )𝑧

∫𝐾

=𝐾

∞

∫𝐾 ∞

−

∫𝐾

] 𝜕 [ (𝑥 − 𝐾)𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥 𝜕𝐾

(𝑥 − 𝐾)𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥

𝜕 (𝑥 − 𝐾) 𝑑𝑥 𝜕𝐾

740

7.2.3 Stochastic Volatility

and hence, 𝑒−𝑟(𝑇 −𝑡)

∞

∞

∫0

∫𝐾

(𝑥 − 𝐾)

∞

= 𝐾 𝑒−𝑟(𝑇 −𝑡)

∫0

∞

∫𝐾

∞

−𝑒−𝑟(𝑇 −𝑡) =𝐾

∫0

∞

∫𝐾

] 𝜕 [ 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥𝑑𝑦 𝜕𝑥

] 𝜕 [ (𝑥 − 𝐾)𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥𝑑𝑦 𝜕𝐾

(𝑥 − 𝐾)𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥𝑑𝑦

𝜕𝐶 − 𝐶(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ). 𝜕𝐾

For the case of the inner integral ∞

∫𝐾

(𝑥 − 𝐾)

] 𝜕2 [ 2 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥 2 𝜕𝑧

we let 𝑢 = (𝑥 − 𝐾) and

] 𝜕2 [ 𝑑𝑣 = 2 𝑦2 𝑥2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥 𝜕𝑥

so that 𝜕 𝑑𝑢 = (𝑥 − 𝐾) and 𝑑𝑥 𝜕𝑥

𝑣=

] 𝜕 [ 2 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝜕𝑥

and therefore ] 𝜕2 [ 2 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥 2 ∫𝐾 𝜕𝑥 ]|∞ 𝜕 [ 2 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) || = (𝑥 − 𝐾) 𝜕𝑥 |𝐾 ∞ ] 𝜕 𝜕 [ 2 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥 − (𝑥 − 𝐾) ∫𝐾 𝜕𝑥 𝜕𝑥 ∞ ] 𝜕 𝜕 [ 2 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥. =− (𝑥 − 𝐾) ∫𝐾 𝜕𝑥 𝜕𝑥 ∞

(𝑥 − 𝐾)

To integrate ∞

−

∫𝐾

] 𝜕 𝜕 [ 2 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥 (𝑥 − 𝐾) 𝜕𝑥 𝜕𝑥

we set 𝑢=

𝜕 (𝑥 − 𝐾) and 𝜕𝑥

] 𝑑𝑣 𝜕 [ 2 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) = 𝑑𝑥 𝜕𝑥

7.2.3 Stochastic Volatility

741

so that 𝑑𝑢 𝜕2 = 2 (𝑥 − 𝐾) and 𝑑𝑥 𝜕𝑥

𝑣 = 𝑦2 𝑥2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

and hence ∞

−

∫𝐾

(𝑥 − 𝐾)

] 𝜕2 [ 2 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥 2 𝜕𝑥

= − 𝑦2 𝑥2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

∞ |∞ 𝜕 𝜕2 𝑦2 𝑥2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 2 (𝑥 − 𝐾) 𝑑𝑥 (𝑥 − 𝐾)|| + 𝜕𝑥 𝜕𝑥 |𝐾 ∫𝐾 ∞

= 𝑦2 𝐾 2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑦, 𝑇 ) +

∫𝐾

𝑦2 𝑥2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 )

𝜕2 (𝑥 − 𝐾) 𝑑𝑥 𝜕𝐾 2

= 𝑦2 𝐾 2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑦, 𝑇 ) since

𝜕 𝜕2 𝜕2 (𝑥 − 𝐾) = 0. (𝑥 − 𝐾) = 1 and 2 (𝑥 − 𝐾) = 𝜕𝑥 𝜕𝑥 𝜕𝐾 2

Therefore, 1 −𝑟(𝑇 −𝑡) 𝑒 ∫0 2

∞

∞

∫𝐾

(𝑥 − 𝐾)

] 𝜕2 [ 2 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥𝑑𝑦 𝜕𝑥2

∞

1 −𝑟(𝑇 −𝑡) 𝑦2 𝐾 2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑦, 𝑇 ) 𝑑𝑦 𝑒 ∫0 2 ] [ ∞ 1 𝑦2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑦, 𝑇 ) 𝑑𝑦 . = 𝐾 2 𝑒−𝑟(𝑇 −𝑡) ∫0 2

=

Given that ( ) 𝜕𝐶 𝜕2𝐶 𝜕 = 𝜕𝐾 𝜕𝐾 𝜕𝐾 2 ] [ ∞ ∞ ] 𝜕 𝜕 [ −𝑟(𝑇 −𝑡) (𝑥 − 𝐾)𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥𝑑𝑦 = 𝑒 ∫0 ∫𝐾 𝜕𝐾 𝜕𝐾 ∞[ 𝜕 −𝑝(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) (𝑥 − 𝐾) = 𝑒−𝑟(𝑇 −𝑡) ∫0 𝜕𝐾 ] ∞ 𝜕2 + (𝑥 − 𝐾)𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥 𝑑𝑦 ∫𝐾 𝜕𝐾 2 = 𝑒−𝑟(𝑇 −𝑡)

∞

∫0

𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑦, 𝑇 ) 𝑑𝑦

= 𝑒−𝑟(𝑇 −𝑡) 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 = 𝐾, 𝑇 )

742

7.2.3 Stochastic Volatility

and from the identity 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑦, 𝑇 ) = 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 = 𝐾, 𝑇 )𝑝(𝑆𝑇 = 𝐾, 𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑦) we have 1 −𝑟(𝑇 −𝑡) 𝑒 ∫0 2

∞

∞

∫𝐾

(𝑥 − 𝐾)

] 𝜕2 [ 2 2 𝑦 𝑥 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑥, 𝑦, 𝑇 ) 𝑑𝑥𝑑𝑦 2 𝜕𝑥

∞

1 −𝑟(𝑇 −𝑡) 𝑦2 𝐾 2 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑦, 𝑇 ) 𝑑𝑦 𝑒 ∫0 2 ] [ ∞ 1 2 −𝑟(𝑇 −𝑡) 2 𝑦 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑦, 𝑇 ) 𝑑𝑦 = 𝐾 𝑒 ∫0 2

=

] [ ∞ 1 2 −𝑟(𝑇 −𝑡) 2 = 𝐾 𝑒 𝑝(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑆𝑇 = 𝐾, 𝑇 ) 𝑦 𝑝(𝑆𝑇 = 𝐾, 𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑦) 𝑑𝑦 ∫0 2 ∞

=

1 2 𝜕2𝐶 𝐾 2 𝜕𝐾 2 ∫0

=

[ ] 1 ℚ 2| 𝜕2𝐶 . 𝔼 𝜎𝑇 | 𝑆𝑇 = 𝐾, 𝑆𝑡 , 𝜎𝑡 , 𝑡 𝐾 2 | 2 𝜕𝐾 2

𝑦2 𝑝(𝑆𝑇 = 𝐾, 𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝑦) 𝑑𝑦

Thus, [ ] 𝜕𝐶 1 𝜕2𝐶 | = −𝑟𝐶(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝔼ℚ 𝜎𝑇2 | 𝑆𝑇 = 𝐾, 𝑆𝑡 , 𝜎𝑡 , 𝑡 𝐾 2 | 𝜕𝑇 2 𝜕𝐾 2 [ ] 𝜕𝐶 −(𝑟 − 𝐷) 𝐾 − 𝐶(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) 𝜕𝐾 or [ ] 𝜕𝐶 𝜕2𝐶 𝜕𝐶 1 ℚ 2 | + (𝑟 − 𝐷)𝐾 − 𝔼 𝜎𝑇 | 𝑆𝑇 = 𝐾, 𝑆𝑡 , 𝜎𝑡 , 𝑡 𝐾 2 + 𝐷𝐶(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 | 𝜕𝑇 2 𝜕𝐾 𝜕𝐾 2 with boundary condition 𝐶(𝑆𝑇 , 𝜎𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0}. From the PDE expression, we can write [

]

| 𝔼ℚ 𝜎𝑇2 | 𝑆𝑇 = 𝐾, 𝑆𝑡 , 𝜎𝑡 , 𝑡 = |

( 2

𝜕𝐶 𝜕𝐶 + (𝑟 − 𝐷)𝐾 + 𝐷𝐶(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) 𝜕𝑇 𝜕𝐾 𝜕2 𝐶 𝐾2 𝜕𝐾 2

)

7.2.3 Stochastic Volatility

743

such that given the market prices of call options written on 𝑆𝑡 with arbitrary strikes 𝐾 and expiries 𝑇 , the expected volatility at time 𝑇 can be extracted given that the asset price is 𝑆𝑇 = 𝐾 at time 𝑇 . From the put–call parity 𝐶(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑃 (𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾 𝑒−𝑟(𝑇 −𝑡) we have 𝜕𝑃 𝜕𝐶 = − 𝐷𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) + 𝑟𝐾 𝑒−𝑟(𝑇 −𝑡) 𝜕𝑇 𝜕𝑇 𝜕𝐶 𝜕𝑃 = − 𝑒−𝑟(𝑇 −𝑡) 𝜕𝐾 𝜕𝐾 𝜕2𝑃 𝜕2𝐶 = . 𝜕𝐾 2 𝜕𝐾 2 Substituting the above information into the PDE for 𝐶(𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ), [ ] 2 𝜕𝑃 1 𝜕 𝑃 | − 𝐷𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) + 𝑟𝐾 𝑒−𝑟(𝑇 −𝑡) − 𝔼ℚ 𝜎𝑇2 | 𝑆𝑇 = 𝐾, 𝑆𝑡 , 𝜎𝑡 , 𝑡 | 𝜕𝑇 2 𝜕𝐾 2 ] [ ] [ 𝜕𝑃 +(𝑟 − 𝐷)𝐾 − 𝑒−𝑟(𝑇 −𝑡) + 𝐷 𝑃 (𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾 𝑒−𝑟(𝑇 −𝑡) = 0 𝜕𝐾 or [ ] 𝜕𝑃 𝜕𝑃 𝜕2𝑃 1 | + (𝑟 − 𝐷)𝐾 − 𝔼ℚ 𝜎𝑇2 | 𝑆𝑇 = 𝐾, 𝑆𝑡 , 𝜎𝑡 , 𝑡 𝐾 2 + 𝐷𝑃 (𝑆𝑡 , 𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 | 𝜕𝑇 2 𝜕𝐾 𝜕𝐾 2 with boundary condition 𝑃 (𝑆𝑇 , 𝜎𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝐾 − 𝑆𝑇 , 0}. Take note that if the volatility is a function of asset price and time, then [ [ ] ] | | 𝔼ℚ 𝜎𝑇2 | 𝑆𝑇 = 𝐾, 𝑆𝑡 , 𝜎𝑡 , 𝑡 = 𝔼ℚ 𝜎(𝑆𝑇 , 𝑇 )2 | 𝑆𝑇 = 𝐾, 𝑆𝑡 , 𝜎𝑡 , 𝑡 = 𝜎(𝐾, 𝑇 )2 | | and the PDE becomes the Dupire equation. Thus, a local volatility model and stochastic volatility model will price a European option equally provided the above relationship is satisfied. 6. Hull–White Model. Let {𝑊𝑡𝑆 : 𝑡 ≥ 0} and {𝑊𝑡𝜎 : 𝑡 ≥ 0} be two independent standard Wiener processes on the probability space (Ω, ℱ, ℙ). Suppose the asset price 𝑆𝑡 and its

744

7.2.3 Stochastic Volatility

instantaneous variance 𝜎𝑡2 have the following diffusion processes 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑𝑊𝑡𝑆

𝑑𝜎𝑡2 = 𝛼𝜎𝑡2 𝑑𝑡 + 𝜉𝜎𝑡2 𝑑𝑊𝑡𝜎

𝑑𝑊𝑡𝑆 ⋅ 𝑑𝑊𝑡𝜎 = 𝜌𝑑𝑡

where 𝜇, 𝐷, 𝛼 and 𝜉 are constants. In addition, let 𝐵𝑡 be the risk-free asset with the differential equation 𝑑𝐵𝑡 = 𝑟 𝐵𝑡 𝑑𝑡 where 𝑟 is the risk-free interest rate. By using the two-dimensional Girsanov’s theorem, show that under the risk-neutral measure ℚ, ̃𝑆 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑 𝑊 𝑡

̃𝜎 𝑑𝜎𝑡2 = (𝛼 − 𝜉𝛾)𝜎𝑡2 𝑑𝑡 + 𝜉𝜎𝑡2 𝑑 𝑊 𝑡 ̃ 𝑆 and 𝑊 ̃ 𝜎 are ℚ-standard Wiener processes, 𝑊 ̃𝑆 ⟂ ̃𝜎 where 𝑊 𝑡 𝑡 𝑡 ⟂ 𝑊𝑡 and 𝛾 is the market price of volatility risk. Is the market arbitrage free and complete under the ℚ measure? By letting 𝛼 ̃ = 𝛼 − 𝜉𝛾, and conditional on ℱ𝑡 and {𝜎𝑢 : 𝑡 ≤ 𝑢 ≤ 𝑇 }, show that under the risk-neutral measure ℚ )| { } | | ℱ𝑡 , 𝜎𝑢 : 𝑡 ≤ 𝑢 ≤ 𝑇 | | ) ] [( 1 2 2 (𝑡, 𝑇 ) (𝑇 − 𝑡), 𝜎RMS (𝑡, 𝑇 )(𝑇 − 𝑡) ∼  𝑟 − 𝐷 − 𝜎RMS 2 (

log

𝑆𝑇 𝑆𝑡

and conditional on ℱ𝑡 show that under the risk-neutral measure ℚ ( log

)| ) [( ] | 1 2 2 |ℱ ∼  𝛼 (𝑇 − 𝑡), 𝜉 ̃ − (𝑇 − 𝑡) 𝜉 𝑡 | 2 𝜎𝑡2 | |

𝜎𝑇2

where 2 (𝑡, 𝑇 ) = 𝜎RMS

1 𝑇 − 𝑡 ∫𝑡

𝑇

𝜎𝑢2 𝑑𝑢

is known as the mean variance over the time interval [𝑡, 𝑇 ]. Show that [ ] ( ) 𝜎𝑡2 | 2 𝑒𝛼̃(𝑇 −𝑡) − 1 (𝑡, 𝑇 )| ℱ𝑡 = 𝔼ℚ 𝜎RMS | 𝛼 ̃(𝑇 − 𝑡)

7.2.3 Stochastic Volatility

745

and [ ] 𝜎𝑡2 ⎡ 2𝑒(2̃𝛼+𝜉 2 )(𝑇 −𝑡) | 2 ⎢ (𝑡, 𝑇 )| ℱ𝑡 = Varℚ 𝜎RMS | (𝑇 − 𝑡)2 ⎢ (̃ 𝛼 + 𝜉 2 )(2̃ 𝛼 + 𝜉2) ⎣ )2 ( ) ( 𝛼̃(𝑇 −𝑡) 𝑒 −1 ⎤ 2 1 𝑒𝛼̃(𝑇 −𝑡) ⎥. + − − ⎥ 𝛼 ̃ 2̃ 𝛼 + 𝜉2 𝛼 ̃ + 𝜉2 𝛼 ̃2 ⎦ We consider a European option price 𝑉 (𝑆𝑡 , 𝜎𝑡2 , 𝑡; 𝐾, 𝑇 ) at time 𝑡 written on 𝑆𝑡 with expiry time 𝑇 > 𝑡, strike price 𝐾, continuous dividend yield 𝐷 and instantaneous variance 𝜎𝑡2 , [ ] 𝑉 (𝑆𝑡 , 𝜎𝑡2 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ Ψ(𝑆𝑇 )|| ℱ𝑡 where the payoff ⎧ max {𝑆 − 𝐾, 0} for call option 𝑇 ⎪ Ψ(𝑆𝑇 ) = ⎨ } { ⎪ max 𝐾 − 𝑆𝑇 , 0 for put option. ⎩ Using the tower property, show that the price of a European option at time 𝑡 is [ ( )| ] 2 𝑉 (𝑆𝑡 , 𝜎𝑡2 , 𝑡; 𝐾, 𝑇 ) = 𝔼ℚ 𝑉𝑏𝑠 𝑆𝑡 , 𝜎RMS (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 | ℱ𝑡 | 2 where 𝑉𝑏𝑠 (𝑆𝑡 , 𝜎RMS (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) denotes the Black–Scholes formula for a European option at time 𝑡 with spot price 𝑆𝑡 , strike price 𝐾, time-dependent (or term-structure) vari2 (𝑡, 𝑇 ) and option expiry time 𝑇 > 𝑡. ance 𝜎RMS 2 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) about its expected value By expanding 𝑉𝑏𝑠 (𝑆𝑡 , 𝜎RMS

[ ] | 2 𝜎 2RMS (𝑡, 𝑇 ) = 𝔼ℚ 𝜎RMS (𝑡, 𝑇 )| ℱ𝑡 | up to second order, show that the option price can be approximated by 𝑉 (𝑆𝑡 , 𝜎𝑡2 , 𝑡; 𝐾, 𝑇 ) ≈ 𝑉𝑏𝑠 (𝑆𝑡 , 𝜎 2RMS (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) √ 1 RMS 2 1 𝑇 −𝑡 + 2 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑒− 2 (𝑑+ ) 2𝜋 4𝜎 RMS (𝑡, 𝑇 ) [ ] ( RMS RMS ) | 2 × 𝑑+ 𝑑− − 1 Varℚ 𝜎RMS (𝑡, 𝑇 )| ℱ𝑡 | where 𝑑±RMS

log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎 2RMS (𝑡, 𝑇 ))(𝑇 − 𝑡) . = √ 𝜎 RMS (𝑡, 𝑇 ) 𝑇 − 𝑡

746

7.2.3 Stochastic Volatility

Solution: We first define ̃𝑆 = 𝑊 𝑆 + 𝑊 𝑡 𝑡 ̃𝜎 = 𝑊 𝜎 + 𝑊 𝑡 𝑡

𝑡

∫0

𝑡

∫0

𝜆𝑢 𝑑𝑢 𝛾𝑢 𝑑𝑢

where 𝜆𝑡 is the market price of asset risk and 𝛾𝑡 is the market price of volatility risk. Since ̃𝑆 ⟂ ̃𝜎 𝑊𝑡𝑆 ⟂ ⟂ 𝑊𝑡𝜎 , we can deduce 𝑊 𝑡 ⟂ 𝑊𝑡 . Let the portfolio Π𝑡 be defined as Π𝑡 = 𝜙𝑡 𝑆𝑡 + 𝜓𝑡 𝐵𝑡 where 𝜙𝑡 units are invested in risky asset 𝑆𝑡 and 𝜓𝑡 units are invested in risk-free asset 𝐵𝑡 . Given that the holder of the portfolio will receive 𝐷𝑆𝑡 𝑑𝑡 for every risky asset held, ( ) 𝑑Π𝑡 = 𝜙𝑡 𝑑𝑆𝑡 + 𝐷𝑆𝑡 𝑑𝑡 + 𝜓𝑡 𝑑𝐵𝑡 ( ) ( ) = 𝜙𝑡 𝜇𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑𝑊𝑡𝑆 + 𝜓𝑡 𝑟 𝐵𝑡 𝑑𝑡 [ ] = 𝑟Π𝑡 𝑑𝑡 + 𝜓𝑡 𝑆𝑡 (𝜇 − 𝑟)𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡𝑆 . Substituting ̃ 𝑆 − 𝜆𝑡 𝑑𝑡 𝑑𝑊𝑡𝑆 = 𝑑 𝑊 𝑡 ̃ 𝜎 − 𝛾𝑡 𝑑𝑡 𝑑𝑊𝑡𝜎 = 𝑑 𝑊 𝑡

into 𝑑Π𝑡 we have [ ( )] ̃ 𝑆 − 𝜆𝑡 𝑑𝑡 𝑑Π𝑡 = 𝑟Π𝑡 𝑑𝑡 + 𝜓𝑡 𝑆𝑡 (𝜇 − 𝑟)𝑑𝑡 + 𝜎𝑡 𝑑 𝑊 𝑡 [( ] ) ̃𝑆 . = 𝑟Π𝑡 𝑑𝑡 + 𝜓𝑡 𝑆𝑡 𝜇 − 𝑟 − 𝜆𝑡 𝜎𝑡 𝑑𝑡 + 𝜎𝑡 𝑑 𝑊 𝑡 ̃ 𝑆 is a ℚ-martingale, and in order for the discounted portfolio 𝑒−𝑟𝑡 Π𝑡 to be a ℚSince 𝑊 𝑡 martingale, 𝑑(𝑒−𝑟𝑡 Π𝑡 ) = −𝑟 𝑒−𝑟𝑡 Π𝑡 𝑑𝑡 + 𝑒−𝑟𝑡 𝑑Π𝑡 [( ] ) ̃𝑆 = 𝑒−𝑟𝑡 𝜓𝑡 𝑆𝑡 𝜇 − 𝑟 − 𝜆𝑡 𝜎𝑡 𝑑𝑡 + 𝜎𝑡 𝑑 𝑊 𝑡 we set 𝜆𝑡 =

𝜇−𝑟 . 𝜎𝑡

7.2.3 Stochastic Volatility

747

Hence, by substituting ̃ 𝑆 − 𝜆𝑡 𝑑𝑡 𝑑𝑊𝑡𝑆 = 𝑑 𝑊 𝑡 into 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑𝑊𝑡𝑆 , the asset price dynamics under the ℚ measure becomes ( ) ̃ 𝑆 − 𝜆𝑡 𝑑𝑡 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑 𝑊 𝑡 ( ( ) ) 𝜇−𝑟 𝑆 ̃ 𝑑𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑 𝑊𝑡 − 𝜎𝑡 or ̃𝑆 . 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑 𝑊 𝑡 As for the case of the instantaneous variance, given that 𝜎𝑡2 is not a traded security, we can set 𝛾𝑡 = 𝛾 where 𝛾 is a constant. By substituting ̃ 𝜎 − 𝛾𝑑𝑡 𝑑𝑊𝑡𝜎 = 𝑑 𝑊 𝑡 into 𝑑𝜎𝑡2 = 𝛼𝜎𝑡2 𝑑𝑡 + 𝜉𝜎𝑡2 𝑑𝑊𝑡𝜎 , the instantaneous variance under the ℚ measure is ( ) ̃ 𝜎 − 𝛾𝑑𝑡 𝑑𝜎𝑡2 = 𝛼𝜎𝑡2 𝑑𝑡 + 𝜉𝜎𝑡2 𝑑 𝑊 𝑡 or ̃𝜎. 𝑑𝜎𝑡2 = (𝛼 − 𝜉𝛾) 𝜎𝑡2 𝑑𝑡 + 𝜉𝜎𝑡2 𝑑 𝑊 𝑡 The market is arbitrage free since we can construct a risk-neutral measure ℚ on the filtration ℱ𝑠 , 0 ≤ 𝑠 ≤ 𝑡. However, the market is not complete as ℚ is not unique, since 𝜎𝑡2 is not a traded security. Expanding 𝑑 log 𝑆𝑡 and applying It¯o’s lemma, 𝑑 log 𝑆𝑡 =

𝑑𝑆𝑡 1 − 𝑆𝑡 2

(

𝑑𝑆𝑡 𝑆𝑡

)2 +…

̃ 𝑆 − 1 𝜎 2 𝑑𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑡 𝑑 𝑊 𝑡 2 𝑡 ) ( 1 ̃𝑆 . = 𝑟 − 𝐷 − 𝜎𝑡2 𝑑𝑡 + 𝜎𝑡 𝑑 𝑊 𝑡 2

748

7.2.3 Stochastic Volatility

Taking integrals, ) 𝑇 1 ̃𝑆 𝑟 − 𝐷 − 𝜎𝑢2 𝑑𝑢 + 𝜎𝑢 𝑑 𝑊 𝑢 ∫𝑡 ∫𝑡 ∫𝑡 2 ( ) ( ) 𝑇 𝑆𝑇 1 2 ̃𝑆 log (𝑡, 𝑇 ) (𝑇 − 𝑡) + 𝜎𝑢 𝑑 𝑊 = 𝑟 − 𝐷 − 𝜎RMS 𝑢 ∫𝑡 𝑆𝑡 2 𝑇

𝑇

𝑑 log 𝑆𝑢 =

(

where 2 𝜎RMS (𝑡, 𝑇 ) =

1 𝑇 − 𝑡 ∫𝑡

𝑇

𝜎𝑢2 𝑑𝑢.

From the properties of the It¯o integral, [ 𝔼

ℚ

( log

𝑆𝑇 𝑆𝑡

] )| ) ( { } 1 2 | (𝑡, 𝑇 ) (𝑇 − 𝑡) = 𝑟 − 𝐷 − 𝜎RMS | ℱ𝑡 , 𝜎𝑢 : 𝑡 ≤ 𝑢 ≤ 𝑇 | 2 |

and [ Var

ℚ

( log [

= Varℚ

𝑆𝑇 𝑆𝑡

𝑇

∫𝑡

[(

] )| { } | | ℱ𝑡 , 𝜎𝑢 : 𝑡 ≤ 𝑢 ≤ 𝑇 | |

| { } ̃ 𝑆 || ℱ𝑡 , 𝜎𝑢 : 𝑡 ≤ 𝑢 ≤ 𝑇 𝜎𝑢 𝑑 𝑊 𝑢 | |

]

] )2 | | { } ̃ 𝑆 | ℱ𝑡 , 𝜎𝑢 : 𝑡 ≤ 𝑢 ≤ 𝑇 𝜎𝑢 𝑑 𝑊 = 𝔼ℚ | 𝑢 ∫𝑡 | | { [ ]}2 𝑇 | { } ℚ 𝑆| ̃ 𝜎𝑢 𝑑 𝑊𝑢 | ℱ𝑡 , 𝜎𝑢 : 𝑡 ≤ 𝑢 ≤ 𝑇 − 𝔼 | ∫𝑡 | [ ] 𝑇 | { } | = 𝔼ℚ 𝜎𝑢2 𝑑𝑢| ℱ𝑡 , 𝜎𝑢 : 𝑡 ≤ 𝑢 ≤ 𝑇 | ∫𝑡 | 𝑇

2 (𝑡, 𝑇 )(𝑇 − 𝑡). = 𝜎RMS

Thus, conditional on ℱ𝑡 and {𝜎𝑢 : 𝑡 ≤ 𝑢 ≤ 𝑇 }, we can easily deduce that (

) 𝑆𝑇 || log | ℱ , {𝜎 : 𝑡 ≤ 𝑢 ≤ 𝑇 } 𝑆𝑡 || 𝑡 𝑢 ) ] [( 1 2 2 (𝑇 − 𝑡), 𝜎𝑅𝑀𝑆 (𝑇 − 𝑡) . ∼  𝑟 − 𝐷 − 𝜎𝑅𝑀𝑆 2

7.2.3 Stochastic Volatility

749

In contrast, by expanding 𝑑 log 𝜎𝑡2 and applying It¯o’s lemma, 𝑑

log 𝜎𝑡2

=

𝑑𝜎𝑡2 𝜎𝑡2

( 1 − 2

𝑑𝜎𝑡2

)2 +…

𝜎𝑡2

̃ 𝜎 − 1 𝜉 2 𝑑𝑡 =𝛼 ̃𝑑𝑡 + 𝜉𝑑 𝑊 𝑡 2 ) ( 1 2 ̃𝜎 = 𝛼 ̃ − 𝜉 𝑑𝑡 + 𝜉𝑑 𝑊 𝑡 2 and taking integrals, ) 𝑇 ( 𝑇 1 ̃𝜎 𝛼 ̃ − 𝜉 2 𝑑𝑢 + 𝑑 log 𝜎𝑢2 = 𝜉𝑑 𝑊 𝑢 ∫𝑡 ∫𝑡 ∫𝑡 2 ( 2) ) ( 𝜎𝑇 1 2 ̃𝜎 . (𝑇 − 𝑡) + 𝜉 𝑊 log 𝜉 = 𝛼 ̃ − 𝑇 −𝑡 2 𝜎𝑡2 𝑇

Thus, using It¯o integral properties, ( log

)| ) [( ] | 1 2 2 |ℱ ∼  𝛼 (𝑇 − 𝑡), 𝜉 ̃ − (𝑇 − 𝑡) . 𝜉 𝑡 | 2 𝜎𝑡2 | |

𝜎𝑇2

[ ] | 2 To find 𝔼ℚ 𝜎RMS (𝑡, 𝑇 )| ℱ𝑡 we note that | 𝔼

ℚ

[

| 2 𝜎RMS (𝑡, 𝑇 )| ℱ𝑡 |

]

[ 1 𝑇 − 𝑡 ∫𝑡

ℚ

=𝔼 =

𝑇

1 𝑇 − 𝑡 ∫𝑡

𝔼ℚ

𝑇

[

|

| 𝜎𝑢2 𝑑𝑢| ℱ𝑡 |

]

| ] | 𝜎𝑢2 | ℱ𝑡 𝑑𝑢 |

𝑇

1 𝜎𝑡2 𝑒𝛼̃(𝑢−𝑡) 𝑑𝑢 𝑇 − 𝑡 ∫𝑡 ( ) 𝜎𝑡2 = 𝑒𝛼̃(𝑇 −𝑡) − 1 . 𝛼 ̃(𝑇 − 𝑡)

=

[ ] | 2 (𝑡, 𝑇 )| ℱ𝑡 , we first note For the case of Varℚ 𝜎RMS | 𝔼ℚ

[

] | 4 𝜎RMS (𝑡, 𝑇 )| ℱ𝑡 = 𝔼ℚ |

[ 1 (𝑇 − 𝑡)2

𝑢=𝑇

=

1 (𝑇 − 𝑡)2 ∫𝑢=𝑡

)| ] | 𝜎𝑢2 𝑑𝑢 𝜎𝑣2 𝑑𝑣 | ℱ𝑡 | ∫𝑡 ∫𝑡 | [ ] 𝑣=𝑇 | 𝔼ℚ 𝜎𝑢2 𝜎𝑣2 | ℱ𝑡 𝑑𝑣𝑑𝑢. | ∫𝑣=𝑡

(

𝑇

)(

𝑇

750

7.2.3 Stochastic Volatility

Since 1 2 ̃𝜎 )(𝑢−𝑡)+𝜉 𝑊 𝑢−𝑡

𝜎𝑢2 = 𝜎𝑡2 𝑒(̃𝛼− 2 𝜉

1 2 ̃𝜎 )(𝑣−𝑡)+𝜉 𝑊 𝑣−𝑡

𝜎𝑣2 = 𝜎𝑡2 𝑒(̃𝛼− 2 𝜉

and

we have 1 2 ̃ 𝜎 +𝑊 ̃𝜎 ) )(𝑢+𝑣−2𝑡)+𝜉(𝑊 𝑢−𝑡 𝑣−𝑡

𝜎𝑢2 𝜎𝑣2 = 𝜎𝑡4 𝑒(̃𝛼− 2 𝜉 where

̃ 𝜎 ∼  (0, 𝑢 + 𝑣 − 2𝑡 + 2 min{𝑢 − 𝑡, 𝑣 − 𝑡}) ̃𝜎 + 𝑊 𝑊 𝑢−𝑡 𝑣−𝑡 since ) ( ̃ 𝜎 = min{𝑢 − 𝑡, 𝑣 − 𝑡}. ̃𝜎 , 𝑊 Cov 𝑊 𝑢−𝑡 𝑣−𝑡 Thus, [ ] 𝜎 2 𝑒−2̃𝛼𝑡 𝑢=𝑇 𝑣=𝑇 2 | 4 (𝑡, 𝑇 )| ℱ𝑡 = 𝑡 𝑒𝛼̃(𝑢+𝑣)+𝜉 min{𝑢−𝑡,𝑣−𝑡} 𝑑𝑣𝑑𝑢 𝔼ℚ 𝜎RMS 2 | ∫ ∫ (𝑇 − 𝑡) 𝑢=𝑡 𝑣=𝑡 =

𝜎𝑡2 𝑒−2̃𝛼𝑡

(𝑇 − 𝑡)2 ∫𝑢=𝑡 +

=

𝑢=𝑇

𝜎𝑡2 𝑒−2̃𝛼𝑡

∫𝑣=𝑡 𝑢=𝑇

(𝑇

2 )𝑡

∫𝑣=𝑢 𝑢=𝑇

∫𝑢=𝑡

− 𝑡)2

𝜎𝑡2 𝑒−(2̃𝛼+𝜉

2 )𝑡

(𝑇 − 𝑡)2

𝑒𝛼̃(𝑢+𝑣)+𝜉

𝑣=𝑇

(𝑇 − 𝑡)2 ∫𝑢=𝑡

𝜎𝑡2 𝑒−(2̃𝛼+𝜉

+

𝑣=𝑢

𝑒𝛼̃(𝑢+𝑣)+𝜉

𝑣=𝑢

∫𝑣=𝑡 𝑢=𝑇

∫𝑢=𝑡

2 (𝑣−𝑡)

𝑒(̃𝛼+𝜉

𝑣=𝑇

∫𝑣=𝑢

𝑑𝑣𝑑𝑢

2 (𝑢−𝑡)

2 )𝑣+̃ 𝛼𝑢

𝑒(̃𝛼+𝜉

𝑑𝑣𝑑𝑢

𝑑𝑣𝑑𝑢

2 )𝑢+̃ 𝛼𝑣

𝑑𝑣𝑑𝑢.

Since 𝑢=𝑇

∫𝑢=𝑡

𝑣=𝑢

∫𝑣=𝑡

𝑒(̃𝛼+𝜉

2 )𝑣+̃ 𝛼𝑢

𝑒(2̃𝛼+𝜉 )𝑇 𝑒𝛼̃𝑇 𝑒(̃𝛼+𝜉 )𝑡 − (̃ 𝛼 + 𝜉 2 )(2̃ 𝛼 + 𝜉2) 𝛼 ̃(̃ 𝛼 + 𝜉2) 2

𝑑𝑣𝑑𝑢 =

2

𝑒(2̃𝛼+𝜉 )𝑡 𝑒(2̃𝛼+𝜉 )𝑡 + 2 2 (̃ 𝛼 + 𝜉 )(2̃ 𝛼+𝜉 ) 𝛼 ̃(̃ 𝛼 + 𝜉2) 2

− 𝑢=𝑇

∫𝑢=𝑡

𝑣=𝑇

∫𝑣=𝑢

𝑒(̃𝛼+𝜉

2 )𝑢+̃ 𝛼𝑣

𝑒(2̃𝛼+𝜉 )𝑇 𝑒(2̃𝛼+𝜉 )𝑇 𝑒𝛼̃𝑇 𝑒(̃𝛼+𝜉 )𝑡 − − 𝛼 ̃(̃ 𝛼 + 𝜉2) 𝛼 ̃(2̃ 𝛼 + 𝜉2) 𝛼 ̃(̃ 𝛼 + 𝜉2) 2

𝑑𝑣𝑑𝑢 =

2

2

𝑒(2̃𝛼+𝜉 )𝑡 𝛼 ̃(2̃ 𝛼 + 𝜉2) 2

+

2

7.2.3 Stochastic Volatility

751

and after some algebraic manipulations, we eventually arrive at [ 2 [ ] 𝜎𝑡2 2𝑒(2̃𝛼+𝜉 )(𝑇 −𝑡) | 4 (𝑡, 𝑇 )| ℱ𝑡 = 𝔼ℚ 𝜎RMS | (𝑇 − 𝑡)2 (̃ 𝛼 + 𝜉 2 )(2̃ 𝛼 + 𝜉2) ( )] 2 1 𝑒𝛼̃(𝑇 −𝑡) + − . 𝛼 ̃ 2̃ 𝛼 + 𝜉2 𝛼 ̃ + 𝜉2 Thus,

[ ] [ ] { [ ]}2 | | | 2 4 2 (𝑡, 𝑇 )| ℱ𝑡 = 𝔼ℚ 𝜎RMS (𝑡, 𝑇 )| ℱ𝑡 − 𝔼ℚ 𝜎RMS (𝑡, 𝑇 )| ℱ𝑡 Varℚ 𝜎RMS | | | ⎡ 2 2 (2̃ 𝛼 +𝜉 )(𝑇 −𝑡) 𝜎𝑡 ⎢ 2𝑒 = (𝑇 − 𝑡)2 ⎢ (̃ 𝛼 + 𝜉 2 )(2̃ 𝛼 + 𝜉2) ⎣ )2 ( ) ( 𝛼̃(𝑇 −𝑡) 𝑒 −1 ⎤ 2 1 𝑒𝛼̃(𝑇 −𝑡) ⎥. + − − ⎥ 𝛼 ̃ 2̃ 𝛼 + 𝜉2 𝛼 ̃ + 𝜉2 𝛼 ̃2 ⎦

Conditional on ℱ𝑡 and {𝜎𝑢 : 𝑡 ≤ 𝑢 ≤ 𝑇 }, 𝑆𝑇 is log normal with initial value 𝑆𝑡 and from the tower property, [ ] 𝑉 (𝑆𝑡 , 𝜎𝑡2 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ Ψ(𝑆𝑇 )|| ℱ𝑡 [ [ ]| ] = 𝔼ℚ 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ Ψ(𝑆𝑇 )|| ℱ𝑡 , {𝜎𝑢 : 𝑡 ≤ 𝑢 ≤ 𝑇 } | ℱ𝑡 | [ ] | ℚ 2 = 𝔼 𝑉𝑏𝑠 (𝑆𝑡 , 𝜎RMS (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 )| ℱ𝑡 | where the inner expectation is the Black–Scholes formula with initial value 𝑆𝑡 and time𝑇 1 2 dependent (or term-structure) variance 𝜎RMS (𝑡, 𝑇 ) = 𝜎𝑢2 𝑑𝑢. Thus, the option 𝑇 − 𝑡 ∫𝑡 price under stochastic volatility is the average value over all possible volatility paths. 2 By expanding 𝑉𝑏𝑠 (𝑆𝑡 , 𝜎RMS (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) about its expected value [ ] | 2 𝜎 2RMS (𝑡, 𝑇 ) = 𝔼ℚ 𝜎RMS (𝑡, 𝑇 )| ℱ𝑡 | up to second order, 2 𝑉𝑏𝑠 (𝑆𝑡 , 𝜎RMS (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 )

≈ 𝑉𝑏𝑠 (𝑆𝑡 , 𝜎 2RMS (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) +

(

| | | 2 𝜕(𝜎RMS (𝑡, 𝑇 )) ||𝜎 2 𝜕𝑉𝑏𝑠

RMS

+

| | | 2 𝜕(𝜎RMS (𝑡, 𝑇 ))2 ||𝜎 2

(𝑡,𝑇 )=𝜎 2RMS (𝑡,𝑇 )

(

𝜕 2 𝑉𝑏𝑠

RMS

2 𝜎RMS (𝑡, 𝑇 ) − 𝜎 2RMS (𝑡, 𝑇 )

(𝑡,𝑇 )=𝜎 2RMS (𝑡,𝑇 )

)

)2 2 𝜎RMS (𝑡, 𝑇 ) − 𝜎 2RMS (𝑡, 𝑇 ) .

752

7.2.3 Stochastic Volatility

From Problems 2.2.2.6 (page 105) and 2.2.4.7 (page 226), we have 𝜕𝑉 = 𝜕𝜎 𝜕2𝑉 = 𝜕𝜎 2

√ √

1 2 𝑇 −𝑡 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑒− 2 𝑑+ 2𝜋 1 2 𝑑 𝑑 𝑇 −𝑡 + − 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑒− 2 𝑑+ 2𝜋 𝜎

where log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝑡) . 𝑑± = √ 𝜎 𝑇 −𝑡 By setting 𝑣 = 𝜎 2 , 𝜕𝑉 𝜕𝜎 𝜕𝑉 = 𝜕𝜎 𝜕𝑣 𝜕𝜎 2 √ 1 2 1 𝑇 −𝑡 = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑒− 2 𝑑+ 2𝜎 2𝜋 ( ) 2 𝜕 𝜕𝑉 𝜕𝜎 𝜕 𝑉 = 𝜕𝜎 𝜕𝜎 𝜕𝑣 𝜕(𝜎 2 )2 𝜕 2 𝑉 𝜕𝜎 𝜕𝑉 𝜕 2 𝜎 + 𝜕𝜎 𝜕𝜎𝜕𝑣 𝜕𝜎 2 𝜕𝑣 √ 1 1 𝑇 −𝑡 −𝐷(𝑇 −𝑡) − 2 𝑑+2 = 𝑒 𝑒 𝑑+ 𝑑− 𝑆 𝑡 2𝜋 2𝜎 2 √ 1 2 1 𝑇 −𝑡 − 2 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑒− 2 𝑑+ 2𝜋 2𝜎 √ 1 2 ( ) 1 𝑇 −𝑡 = 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑒− 2 𝑑+ 𝑑+ 𝑑− − 1 . 2 2𝜋 2𝜎

=

Thus, | | | 2 𝜕(𝜎RMS (𝑡, 𝑇 )) ||𝜎 2 𝜕𝑉𝑏𝑠

RMS

| | | 2 𝜕(𝜎RMS (𝑡, 𝑇 ))2 ||𝜎 2

= (𝑡,𝑇 )=𝜎 2RMS (𝑡,𝑇 )

𝜕 2 𝑉𝑏𝑠

RMS

= (𝑡,𝑇 )=𝜎 2RMS (𝑡,𝑇 )

1 2𝜎 2RMS (𝑡, 𝑇 ) 1 2𝜎 2RMS (𝑡, 𝑇 )

√

√

1 RMS 2 𝑇 −𝑡 𝑆 𝑒−𝐷(𝑇 −𝑡) 𝑒− 2 (𝑑+ ) 2𝜋 𝑡 1 RMS 2 𝑇 −𝑡 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑒− 2 (𝑑+ ) 2𝜋

( ) × 𝑑+RMS 𝑑−RMS − 1 .

7.2.3 Stochastic Volatility

753

Therefore, 𝑉 (𝑆𝑡 , 𝜎𝑡2 , 𝑡; 𝐾, 𝑇 ) ≈ 𝑉𝑏𝑠 (𝑆𝑡 , 𝜎 2RMS (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) √ [ ] 1 RMS 2 1 𝑇 −𝑡 | 2 + 2 (𝑡, 𝑇 ) − 𝜎 2RMS (𝑡, 𝑇 )| ℱ𝑡 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑒− 2 (𝑑+ ) 𝔼ℚ 𝜎RMS | 2𝜋 2𝜎 RMS (𝑡, 𝑇 ) √ 1 RMS 2 ( ) 1 𝑇 −𝑡 + 2 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑒− 2 (𝑑+ ) 𝑑+RMS 𝑑−RMS − 1 2𝜋 4𝜎 RMS (𝑡, 𝑇 ) [( )2 | ] 2 ℚ 2 𝜎RMS (𝑡, 𝑇 ) − 𝜎 RMS (𝑡, 𝑇 ) || ℱ𝑡 ×𝔼 | = 𝑉𝑏𝑠 (𝑆𝑡 , 𝜎 2RMS (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) √ 1 RMS 2 ( ) 1 𝑇 −𝑡 + 2 𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑒− 2 (𝑑+ ) 𝑑+RMS 𝑑−RMS − 1 2𝜋 4𝜎 RMS (𝑡, 𝑇 ) [ ] | 2 (𝑡, 𝑇 )| ℱ𝑡 . ×Varℚ 𝜎RMS | 7. Heston Model. Let {𝑊𝑡𝑆 : 𝑡 ≥ 0} and {𝑊𝑡𝜎 : 𝑡 ≥ 0} be two standard Wiener processes on the probability space (Ω, ℱ, ℙ) with correlation 𝜌 ∈ (−1, 1). Suppose that the asset price 𝑆𝑡 and its instantaneous variance 𝜎𝑡2 have the following dynamics 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑𝑊𝑡𝑆

𝑑𝜎𝑡2 = 𝜅(𝜃 − 𝜎𝑡2 )𝑑𝑡 + 𝛼𝜎𝑡 𝑑𝑊𝑡𝜎

𝑑𝑊𝑡𝑆 ⋅ 𝑑𝑊𝑡𝜎 = 𝜌𝑑𝑡

where 𝜇, 𝐷, 𝜅, 𝜃 and 𝛼 are constants. In addition, let 𝐵𝑡 be the risk-free asset with differential equation 𝑑𝐵𝑡 = 𝑟 𝐵𝑡 𝑑𝑡 where 𝑟 is the risk-free interest rate. By introducing {𝑍𝑡 : 𝑡 ≥ 0} as a standard Wiener process, independent of 𝑊𝑡𝜎 , show that we can write √ 𝑊𝑡𝑆 = 𝜌𝑊𝑡𝜎 + 1 − 𝜌2 𝑍𝑡 . Using the two-dimensional Girsanov’s theorem, show that under the risk-neutral measure ℚ, ( ) √ ̃ 𝜎 + 1 − 𝜌2 𝑑 𝑍 ̃𝑡 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝜌𝑑 𝑊 𝑡 ) ( 2 2 ̃𝜎 𝑑𝜎𝑡 = 𝜅(𝜃 − 𝜎𝑡 ) − 𝛼𝛾𝑡 𝜎𝑡 𝑑𝑡 + 𝛼𝜎𝑡 𝑑 𝑊 𝑡

754

7.2.3 Stochastic Volatility

̃ 𝜎 and 𝑍 ̃𝑡 are ℚ-standard Wiener processes, 𝑊 ̃𝜎 ⟂ ̃ where 𝑊 𝑡 𝑡 ⟂ 𝑍𝑡 and 𝛾𝑡 is the market price of volatility risk. Is the market arbitrage free and complete under the ℚ measure? By assuming the market price of volatility risk is proportional to the instantaneous volatility 𝜎𝑡 , 𝛾𝑡 = 𝑐𝜎𝑡 where 𝑐 is a positive constant, show that the dynamics of the model can be described by ( ) √ ̃ 𝜎 + 1 − 𝜌2 𝑑 𝑍 ̃𝑡 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝜌𝑑 𝑊 𝑡 ̃𝜎 ̃ 𝜃̃ − 𝜎𝑡2 )𝑑𝑡 + 𝛼𝜎𝑡 𝑑 𝑊 𝑑𝜎𝑡2 = 𝜅( 𝑡 such that 𝜅̃ = 𝜅 + 𝛼𝑐 and 𝜃̃ = 𝜅𝜃(𝜅 + 𝛼𝑐)−1 . Hence, conditional on ℱ𝑡 and {𝜎𝑢 : 𝑡 ≤ 𝑢 ≤ 𝑇 } show that (

)| { } | log | ℱ𝑡 , 𝜎𝑢 : 𝑡 ≤ 𝑢 ≤ 𝑇 | | ) ] [( 1 2 2 (𝑡, 𝑇 ) (𝑇 − 𝑡), 𝜎RMS (𝑡, 𝑇 )(𝑇 − 𝑡) ∼  𝑟 − 𝐷 − 𝜎RMS 2 𝑆𝑇 ∕𝜉𝑇 𝑆𝑡 ∕𝜉𝑡

where 𝑠

𝜉𝑠 = 𝑒𝜌 ∫0

𝑠

̃ 𝜎 − 1 𝜌2 ∫ 𝜎 2 𝑑𝑢 𝜎𝑢 𝑑 𝑊 𝑢 0 𝑢 2

and

2 𝜎RMS (𝑡, 𝑇 ) =

1 − 𝜌2 𝑇 − 𝑡 ∫𝑡

𝑇

𝜎𝑢2 𝑑𝑢.

We consider a European option price 𝑉 (𝑆𝑡 , 𝜎𝑡2 , 𝑡; 𝐾, 𝑇 ) at time 𝑡 written on 𝑆𝑡 with expiry time 𝑇 > 𝑡, strike price 𝐾, continuous dividend yield 𝐷 and instantaneous variance 𝜎𝑡 , [ ] 𝑉 (𝑆𝑡 , 𝜎𝑡2 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ Ψ(𝑆𝑇 )|| ℱ𝑡 where the payoff is ⎧ max {𝑆 − 𝐾, 0} for call option 𝑇 ⎪ Ψ(𝑆𝑇 ) = ⎨ } { ⎪ max 𝐾 − 𝑆𝑇 , 0 for put option. ⎩ Using the tower property, show that the price of a European option at time 𝑡 can be expressed by 𝑉 (𝑆𝑡 , 𝜎𝑡2 , 𝑡; 𝐾, 𝑇 )

=𝔼

ℚ

[

( )| 𝜉𝑇 2 𝑉𝑏𝑠 𝑆𝑡 , 𝜎RMS (𝑡, 𝑇 ), 𝑡; 𝐾∕(𝜉𝑇 ∕𝜉𝑡 ), 𝑇 || ℱ𝑡 𝜉𝑡 |

]

7.2.3 Stochastic Volatility

755

2 where 𝑉𝑏𝑠 (𝑆𝑡 , 𝜎RMS (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) denotes the Black–Scholes formula for a European 2 (𝑡, 𝑇 ) and option expiry option at time 𝑡 with spot price 𝑆𝑡 , strike price 𝐾, variance 𝜎RMS time 𝑇 > 𝑡. Finally, by denoting ℚ𝜉 as a new measure where 𝜉𝑡 is used as num´eraire, show that 𝜉

𝑉 (𝑆𝑡 , 𝜎𝑡2 , 𝑡; 𝐾, 𝑇 ) = 𝔼ℚ

[

( )| ] 2 𝑉𝑏𝑠 𝑆𝑡 , 𝜎RMS (𝑡, 𝑇 ), 𝑡; 𝐾∕(𝜉𝑇 ∕𝜉𝑡 ), 𝑇 | ℱ𝑡 . |

Solution: For the first two results, see Problem 7.2.2.2 (page 687). The market is arbitrage free since we can construct a risk-neutral measure ℚ on the filtration ℱ𝑠 , 0 ≤ 𝑠 ≤ 𝑡. However, the market is not complete since 𝜎𝑡2 is not a traded asset and therefore ℚ is not unique. Thus, the risk-neutral measure ℚ is an equivalent market measure. By setting 𝛾 = 𝑐𝜎𝑡 , ) ( ̃𝜎 𝑑𝜎𝑡2 = 𝜅(𝜃 − 𝜎𝑡2 ) − 𝛼𝑐𝜎𝑡2 𝑑𝑡 + 𝛼𝜎𝑡 𝑑 𝑊 𝑡 ) ( ̃𝜎 = 𝜅𝜃 − (𝜅 + 𝛼𝑐)𝜎𝑡2 𝑑𝑡 + 𝛼𝜎𝑡 𝑑 𝑊 𝑡 ̃𝜎 = 𝜅( ̃ 𝜃̃ − 𝜎𝑡2 )𝑑𝑡 + 𝛼𝜎𝑡 𝑑 𝑊 𝑡

such that 𝜅̃ = 𝜅 + 𝛼𝑐 and 𝜃̃ = 𝜅𝜃(𝜅 + 𝛼𝑐)−1 . To show that (

)| { } | log | ℱ𝑡 , 𝜎𝑢 : 𝑡 ≤ 𝑢 ≤ 𝑇 | | ) ] [( 1 2 2 (𝑡, 𝑇 ) (𝑇 − 𝑡), 𝜎RMS (𝑡, 𝑇 )(𝑇 − 𝑡) ∼  𝑟 − 𝐷 − 𝜎RMS 2 𝑆𝑇 ∕𝜉𝑇 𝑆𝑡 ∕𝜉𝑡

where 𝑠

𝜉𝑠 = 𝑒𝜌 ∫0

̃ 𝜎 − 1 𝜌2 ∫ 𝑠 𝜎 2 𝑑𝑢 𝜎𝑢 𝑑 𝑊 𝑢 0 𝑢 2

and

2 𝜎RMS (𝑡, 𝑇 ) =

1 − 𝜌2 𝑇 − 𝑡 ∫𝑡

𝑇

𝜎𝑢2 𝑑𝑢

see Problem 3.2.3.10 of Problems and Solutions in Mathematical Finance, Volume 1: Stochastic Calculus. Conditional on ℱ𝑡 and {𝜎𝑢 : 𝑡 ≤ 𝑢 ≤ 𝑇 }, 𝑆𝑇 is log normal with initial value 𝑆𝑡 𝜉𝑇 ∕𝜉𝑡 and using the tower property, [ ] 𝑉 (𝑆𝑡 , 𝜎𝑡2 , 𝑡; 𝐾, 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ Ψ(𝑆𝑇 )|| ℱ𝑡 [ [ ]| ] = 𝔼ℚ 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ Ψ(𝑆𝑇 )|| ℱ𝑡 , {𝜎𝑢 : 𝑡 ≤ 𝑢 ≤ 𝑇 } | ℱ𝑡 | such that the inner expectation is the Black–Scholes formula with initial value 𝑆𝑡 𝜉𝑇 ∕𝜉𝑡 and 1 − 𝜌2 𝑇 2 2 (𝑡, 𝑇 ) = 𝜎𝑢 𝑑𝑢. term-structure variance 𝜎RMS 𝑇 − 𝑡 ∫𝑡

756

7.2.3 Stochastic Volatility

Thus, [ 𝑉 (𝑆𝑡 , 𝜎𝑡2 , 𝑡; 𝐾, 𝑇 )

=𝔼

ℚ

( 𝑉𝑏𝑠

𝑆𝑡 𝜉𝑇 2 , 𝜎RMS (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 𝜉𝑡

)| ] | | ℱ𝑡 . | |

For a constant value 𝜋 > 0, the Black–Scholes formula with a constant or time-dependent volatility 𝜎 has the following identity (see Problem 2.2.2.8, page 109) ( ) 𝑉𝑏𝑠 𝜋𝑆𝑡 , 𝜎 2 , 𝑡; 𝐾, 𝑇 = 𝜋𝑉𝑏𝑠 (𝑆𝑡 , 𝜎 2 , 𝑡; 𝐾∕𝜋, 𝑇 ) and therefore 𝑉 (𝑆𝑡 , 𝜎𝑡2 , 𝑡; 𝐾, 𝑇 )

=𝔼

ℚ

[

] ( )| 𝜉𝑇 2 | 𝑉 𝑆 ,𝜎 (𝑡, 𝑇 ), 𝑡; 𝐾∕(𝜉𝑇 ∕𝜉𝑡 ), 𝑇 | ℱ𝑡 . 𝜉𝑡 𝑏𝑠 𝑡 RMS |

Under the change of num´eraire for a payoff 𝑋𝑇 [ (1) 𝑁𝑡(1) 𝔼ℚ

[ | ] | ] 𝑋𝑇 || 𝑋𝑇 || (2) ℚ(2) | ℱ𝑡 = 𝑁𝑡 𝔼 | ℱ𝑡 𝑁𝑇(1) || 𝑁𝑇(2) ||

where for 𝑖 = 1, 2, 𝑁 (𝑖) is a num´eraire and ℚ(𝑖) is the measure under which the asset prices discounted by 𝑁 (𝑖) are ℚ(𝑖) -martingales. Under the risk-neutral measure ℚ we have 𝑁𝑡(1) = 1 and

𝑁𝑇(1) = 1

and under the measure ℚ𝜉 𝑁𝑡(2) = 𝜉𝑡 By setting 𝑋𝑇 = is

and

𝑁𝑇(2) = 𝜉𝑇 .

( ) 𝜉𝑇 2 𝑉𝑏𝑠 𝑆𝑡 , 𝜎RMS (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 , the option price under the measure ℚ𝜉 𝜉𝑡 [

𝑉 (𝑆𝑡 , 𝜎𝑡2 , 𝑡; 𝐾, 𝑇 )

] ( ) 2 𝜉𝑇 𝑉𝑏𝑠 𝑆𝑡 , 𝜎RMS (𝑡, 𝑇 ), 𝑡; 𝐾∕(𝜉𝑇 ∕𝜉𝑡 ), 𝑇 || = 𝜉𝑡 𝔼 | ℱ𝑡 | 𝜉𝑡 𝜉𝑇 | [ ] ( )| ℚ𝜉 2 𝑉𝑏𝑠 𝑆𝑡 , 𝜎RMS (𝑡, 𝑇 ), 𝑡; 𝐾∕(𝜉𝑇 ∕𝜉𝑡 ), 𝑇 | ℱ𝑡 . =𝔼 | ℚ𝜉

8. Heston Model – Black–Scholes Equation. Let (Ω, ℱ, ℙ) be a probability space and let {𝑊𝑡𝑆 : 𝑡 ≥ 0} and {𝑊𝑡𝑣 : 𝑡 ≥ 0} be two correlated Wiener processes. Suppose that the

7.2.3 Stochastic Volatility

757

asset price 𝑆𝑡 and the instantaneous variance 𝑣𝑡 follow the diffusion processes √

𝑣𝑡 𝑆𝑡 𝑑𝑊𝑡𝑆 √ 𝑑𝑣𝑡 = 𝜅(𝜃 − 𝑣𝑡 )𝑑𝑡 + 𝜉 𝑣𝑡 𝑑𝑊𝑡𝑣

𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝑑𝑊𝑡𝑆 ⋅ 𝑑𝑊𝑡𝑣 = 𝜌𝑑𝑡

where 𝜇 is the drift, 𝐷 is the continuous dividend yield, 𝜅, 𝜃, 𝜉 > 0 are constant parameters, 𝜌 is the correlation coefficient such that 𝜌 ∈ (−1, 1) and let 𝑟 be the risk-free interest rate from a money-market account. By considering a hedging portfolio consisting of two European-style options 𝑉 (𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝑇1 ) and 𝑉 (𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝑇2 ) of different expiry dates 𝑇1 and 𝑇2 and the underlying asset 𝑆𝑡 , show that 1 𝜕2𝑉 𝜕2𝑉 𝜕2𝑉 1 𝜕𝑉 + 𝑣𝑡 𝑆𝑡2 2 + 𝜉 2 𝑣𝑡 2 + 𝜌𝜉𝑣𝑡 𝑆𝑡 𝜕𝑡 2 2 𝜕𝑆𝑡 𝜕𝑣𝑡 𝜕𝑆𝑡 𝜕𝑣𝑡 +(𝑟 − 𝐷)𝑆𝑡

( ) 𝜕𝑉 𝜕𝑉 + 𝜅(𝜃 − 𝑣𝑡 ) − 𝜆(𝑆𝑡 , 𝑣𝑡 ) − 𝑟𝑉 (𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝑇 ) = 0 𝜕𝑆𝑡 𝜕𝑣𝑡

where 𝜆(𝑆𝑡 , 𝑣𝑡 ) is the market price of volatility risk independent of the expiry time T. Solution: Let 𝑉 (𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝑇 ) denote the price of an option at time 𝑡 < 𝑇 with expiry time 𝑇 . By applying It¯o’s lemma, 𝑑𝑉 is given by 𝑑𝑉 =

𝜕𝑉 1 𝜕2𝑉 1 𝜕2𝑉 𝜕𝑉 𝜕𝑉 2 𝑑𝑆𝑡 + 𝑑𝑣𝑡 + (𝑑𝑆 ) + (𝑑𝑣𝑡 )2 𝑑𝑡 + 𝑡 𝜕𝑡 𝜕𝑆𝑡 𝜕𝑣𝑡 2 𝜕𝑆𝑡2 2 𝜕𝑣2𝑡

𝜕2𝑉 (𝑑𝑆𝑡 )(𝑑𝑣𝑡 ) + … 𝜕𝑆𝑡 𝜕𝑣𝑡 ) √ 𝜕𝑉 𝜕𝑉 ( (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝑣𝑡 𝑆𝑡 𝑑𝑊𝑡𝑆 = 𝑑𝑡 + 𝜕𝑡 𝜕𝑆𝑡 ) √ 𝜕𝑉 ( 𝜅(𝜃 − 𝑣𝑡 )𝑑𝑡 + 𝜉 𝑣𝑡 𝑑𝑊𝑡𝑣 + 𝜕𝑣𝑡 ) 1 𝜕2𝑉 2 𝜕2𝑉 1 𝜕2𝑉 ( 𝑣𝑡 𝑆𝑡2 𝑑𝑡 + (𝜉 𝑣𝑡 𝑑𝑡) + (𝜌𝜉𝑣𝑡 𝑆𝑡 𝑑𝑡) + 2 2 2 𝜕𝑆𝑡 2 𝜕𝑣𝑡 𝜕𝑆𝑡 𝜕𝑣𝑡 ( 𝜕𝑉 1 𝜕2 𝑉 𝜕2𝑉 𝜕2 𝑉 𝜕𝑉 1 = + (𝜇 − 𝐷)𝑆𝑡 + 𝑣𝑡 𝑆𝑡2 2 + 𝜉 2 𝑣𝑡 2 + 𝜌𝜉𝑣𝑡 𝑆𝑡 𝜕𝑡 2 2 𝜕𝑆 𝜕𝑣 𝜕𝑆 𝜕𝑆𝑡 𝜕𝑣𝑡 𝑡 𝑡 𝑡 ) √ 𝜕𝑉 √ 𝜕𝑉 𝜕𝑉 + 𝜅(𝜃 − 𝑣𝑡 ) 𝑑𝑊𝑡𝑆 + 𝜉 𝑣𝑡 𝑑𝑊𝑡𝑣 . 𝑑𝑡 + 𝑣𝑡 𝑆𝑡 𝜕𝑣𝑡 𝜕𝑆𝑡 𝜕𝑣𝑡 +

Let the portfolio Π𝑡 consist of buying one option with expiry date 𝑇1 , 𝑉 (𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝑇1 ), selling Δ1 units of asset 𝑆𝑡 and selling Δ2 units of the option with expiry date 𝑇2 , 𝑉 (𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝑇2 ),

758

7.2.3 Stochastic Volatility

where 𝑇1 ≠ 𝑇2 . By setting 𝑉1 (𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝑇1 ) = 𝑉 (𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝑇1 ) 𝑉2 (𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝑇2 ) = 𝑉 (𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝑇2 ) at time 𝑡, the value of the portfolio is Π𝑡 = 𝑉1 (𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝑇1 ) − Δ1 𝑆𝑡 − Δ2 𝑉2 (𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝑇2 ). Since we receive 𝐷𝑆𝑡 𝑑𝑡 for every asset held, the change in portfolio Π𝑡 is 𝑑Π𝑡 ( ) = 𝑑𝑉1 − Δ1 𝑑𝑆𝑡 + 𝐷𝑆𝑡 𝑑𝑡 − Δ2 𝑑𝑉2 ( =

𝜕𝑉1 1 𝜕 2 𝑉1 1 2 𝜕 2 𝑉1 𝜕 2 𝑉1 𝜕𝑉 + 𝑣 + 𝜌𝜉𝑣 𝑆 + (𝜇 − 𝐷)𝑆𝑡 1 + 𝑣𝑡 𝑆𝑡2 𝜉 𝑡 𝑡 𝑡 2 2 𝜕𝑡 2 2 𝜕𝑆𝑡 𝜕𝑣𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑣𝑡

) ( 𝜕𝑉1 𝜕𝑉2 1 𝜕 2 𝑉2 1 2 𝜕 2 𝑉2 𝜕 2 𝑉2 +𝜅(𝜃 − 𝑣𝑡 ) + 𝑣 + 𝜌𝜉𝑣 𝑆 𝑑𝑡 − Δ2 + 𝑣𝑡 𝑆𝑡2 𝜉 𝑡 𝑡 𝑡 𝜕𝑣𝑡 𝜕𝑡 2 2 𝜕𝑆𝑡 𝜕𝑣𝑡 𝜕𝑆𝑡2 𝜕𝑣2𝑡 ) 𝜕𝑉2 𝜕𝑉2 +(𝜇 − 𝐷)𝑆𝑡 + 𝜅(𝜃 − 𝑣𝑡 ) 𝑑𝑡 − Δ1 𝜇𝑆𝑡 𝑑𝑡 𝜕𝑆𝑡 𝜕𝑣𝑡 ( +

𝜕𝑉 𝜕𝑉1 − Δ1 − Δ2 2 𝜕𝑆𝑡 𝜕𝑆𝑡

)

√ 𝑣𝑡 𝑆𝑡 𝑑𝑊𝑡𝑆 +

(

𝜕𝑉1 𝜕𝑉 − Δ2 2 𝜕𝑆𝑡 𝜕𝑣𝑡

)

√ 𝜉 𝑣𝑡 𝑑𝑊𝑡𝑣 .

To eliminate both the asset and volatility risk, we have 𝜕𝑉 𝜕𝑉1 = Δ1 + Δ2 2 𝜕𝑆𝑡 𝜕𝑆𝑡

𝜕𝑉1 𝜕𝑉 = Δ2 2 𝜕𝑆𝑡 𝜕𝑣𝑡

and

or Δ1 =

𝜕𝑉1 − 𝜕𝑆𝑡

(

𝜕𝑉1 𝜕𝑉2 𝜕𝑆𝑡 𝜕𝑆𝑡

)/

𝜕𝑉2 𝜕𝑣𝑡

and

Δ2 =

𝜕𝑉1 𝜕𝑆𝑡

/

𝜕𝑉2 . 𝜕𝑣𝑡

Under the no-arbitrage condition, the return on the portfolio Π𝑡 invested in a risk-free interest rate would see a growth of 𝑑Π𝑡 = 𝑟Π𝑡 𝑑𝑡

7.2.3 Stochastic Volatility

759

and therefore we have 𝜕𝑉 𝜕 2 𝑉1 1 2 𝜕 2 𝑉1 𝜕 2 𝑉1 𝜕𝑉 𝜕𝑉1 1 + 𝑣 + 𝜌𝜉𝑣 𝑆 + (𝜇 − 𝐷)𝑆𝑡 1 + 𝜅(𝜃 − 𝑣𝑡 ) 1 + 𝑣𝑡 𝑆𝑡2 𝜉 𝑡 𝑡 𝑡 2 2 𝜕𝑡 2 2 𝜕𝑆𝑡 𝜕𝑣𝑡 𝜕𝑆𝑡 𝜕𝑣𝑡 𝜕𝑆𝑡 𝜕𝑣𝑡 ( ( / ) 𝜕𝑉1 𝜕 2 𝑉2 1 2 𝜕 2 𝑉2 𝜕 2 𝑉2 𝜕𝑉2 𝜕𝑉2 1 − + 𝑣 + 𝜌𝜉𝑣 𝑆 + 𝑣𝑡 𝑆𝑡2 𝜉 𝑡 𝑡 𝑡 𝜕𝑆𝑡 𝜕𝑣𝑡 𝜕𝑡 2 2 𝜕𝑆𝑡 𝜕𝑣𝑡 𝜕𝑆𝑡2 𝜕𝑣2𝑡 ( ) ( )/ ) 𝜕𝑉2 𝜕𝑉1 𝜕𝑉2 𝜕𝑉2 𝜕𝑉1 𝜕𝑉2 +(𝜇 − 𝐷)𝑆𝑡 + 𝜅(𝜃 − 𝑣𝑡 ) − − 𝜇𝑆𝑡 𝜕𝑆𝑡 𝜕𝑣𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑣𝑡 [ ( ( )/ ) 𝜕𝑉1 𝜕𝑉1 𝜕𝑉2 𝜕𝑉2 = 𝑟 𝑉1 (𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝑇1 ) − − 𝑆𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑆𝑡 𝜕𝑣𝑡 ( ] / ) 𝜕𝑉1 𝜕𝑉2 − 𝑉2 (𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝑇2 ) . 𝜕𝑆𝑡 𝜕𝑣𝑡

By gathering together all 𝑉1 terms on the left-hand side and all 𝑉2 terms on the right-hand side ⎡ 𝜕𝑉1 𝜕 2 𝑉1 1 2 𝜕 2 𝑉1 𝜕 2 𝑉1 ⎤ 1 ⎢ ⎥ + 𝑣 + 𝜌𝜉𝑣 𝑆 + 𝑣𝑡 𝑆𝑡2 𝜉 𝑡 𝑡 𝑡 ⎢ 𝜕𝑡 2 2 𝜕𝑆𝑡 𝜕𝑣𝑡 ⎥ 𝜕𝑆𝑡2 𝜕𝑣2𝑡 ⎢ ⎢ ⎢ 𝜕𝑉1 ⎢ ⎢ +(𝑟 − 𝐷)𝑆𝑡 ⎣ 𝜕𝑆𝑡

+ 𝜅(𝜃

⎥ ⎥ ⎥ 𝜕𝑉1 ⎥ − 𝑣𝑡 ) − 𝑟𝑉1 (𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝑇1 ) ⎥ ⎦ 𝜕𝑣𝑡

𝜕𝑉1 𝜕𝑣𝑡

=

2 2 ⎡ 𝜕𝑉 𝜕 2 𝑉2 ⎤ 1 2 𝜕 𝑉2 1 2 2 𝜕 𝑉2 ⎢ ⎥ 𝑆 + 𝑣 + 𝜌𝜉𝑣 𝑆 + 𝑣 𝜉 𝑡 𝑡 𝑡 𝑡 𝑡 ⎢ 𝜕𝑡 2 2 𝜕𝑆𝑡 𝜕𝑣𝑡 ⎥⎥ 𝜕𝑆𝑡2 𝜕𝑣2𝑡 ⎢ ⎢ ⎥ ⎢ ⎥ 𝜕𝑉2 𝜕𝑉2 ⎢ + 𝜅(𝜃 − 𝑣𝑡 ) − 𝑟𝑉2 (𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝑇2 ) ⎥⎥ ⎢ +(𝑟 − 𝐷)𝑆𝑡 𝜕𝑆𝑡 𝜕𝑣𝑡 ⎣ ⎦

𝜕𝑉2 𝜕𝑣𝑡

.

Since the left-hand side is a function of 𝑇1 and the right-hand side is a function of 𝑇2 , the only way for the equality to hold is for both sides to be equal to a common function 𝜆(𝑆𝑡 , 𝑣𝑡 ) independent of the expiry time 𝑇 . By dropping the subscripts we have ⎡ 𝜕𝑉 1 𝜕2 𝑉 𝜕2𝑉 𝜕 2 𝑉 ⎤⎥ 1 ⎢ + 𝑣𝑡 𝑆𝑡2 2 + 𝜉 2 𝑣𝑡 2 + 𝜌𝜉𝑣𝑡 𝑆𝑡 ⎢ 𝜕𝑡 2 2 𝜕𝑆𝑡 𝜕𝑣𝑡 ⎥ 𝜕𝑆𝑡 𝜕𝑣𝑡 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 𝜕𝑉 𝜕𝑉 ⎢ +(𝑟 − 𝐷)𝑆 + 𝜅(𝜃 − 𝑣𝑡 ) − 𝑟𝑉 (𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝑇1 ) ⎥⎥ 𝑡 ⎢ 𝜕𝑆𝑡 𝜕𝑣𝑡 ⎣ ⎦

𝜕𝑉 𝜕𝑣𝑡

=

𝜆(𝑆𝑡 , 𝑣𝑡 ).

760

7.2.3 Stochastic Volatility

Thus, 1 𝜕2𝑉 𝜕2 𝑉 𝜕2𝑉 1 𝜕𝑉 + 𝑣𝑡 𝑆𝑡2 2 + 𝜉 2 𝑣𝑡 2 + 𝜌𝜉𝑣𝑡 𝑆𝑡 𝜕𝑡 2 2 𝜕𝑆𝑡 𝜕𝑣𝑡 𝜕𝑆𝑡 𝜕𝑣𝑡 +(𝑟 − 𝐷)𝑆𝑡

( ) 𝜕𝑉 𝜕𝑉 + 𝜅(𝜃 − 𝑣𝑡 ) − 𝜆(𝑆𝑡 , 𝑣𝑡 ) − 𝑟𝑉 (𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝑇 ) = 0. 𝜕𝑆𝑡 𝜕𝑣𝑡

9. Heston Model – European Option Price. Let {𝑊𝑡𝑆 : 𝑡 ≥ 0} and {𝑊𝑡𝑣 : 𝑡 ≥ 0} be two standard Wiener processes on the probability space (Ω, ℱ, ℙ) with correlation 𝜌 ∈ (−1, 1). Suppose that the asset price 𝑆𝑡 and its instantaneous variance 𝑣𝑡 follow √

𝑣𝑡 𝑆𝑡 𝑑𝑊𝑡𝑆 √ 𝑑𝑣𝑡 = 𝜅(𝜃 − 𝑣𝑡 )𝑑𝑡 + 𝜉 𝑣𝑡 𝑑𝑊𝑡𝑣

𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝑑𝑊𝑡𝑆 ⋅ 𝑑𝑊𝑡𝑣 = 𝜌𝑑𝑡

where 𝜇 is the asset drift, 𝐷 is the continuous dividend yield, 𝜅 is the variance meanreversion rate, 𝜃 is the variance long-term mean, 𝜉 is the vol-of-vol and 𝜌 ∈ (−1, 1) is the correlation coefficient. In addition, we let 𝑟 be the risk-free interest rate from a moneymarket account. Let 𝐶(𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) denote the price of a European call option at time 𝑡 with strike price 𝐾 and expiry time 𝑇 > 𝑡, satisfying the Black–Scholes equation 𝜕𝐶 1 𝜕2𝐶 1 𝜕2𝐶 𝜕2𝐶 + 𝑣𝑡 𝑆𝑡2 2 + 𝜉 2 𝑣𝑡 2 + 𝜌𝜉𝑣𝑡 𝑆𝑡 𝜕𝑡 2 2 𝜕𝑆𝑡 𝜕𝑣𝑡 𝜕𝑆𝑡 𝜕𝑣𝑡 +(𝑟 − 𝐷)𝑆𝑡

) 𝜕𝐶 𝜕𝐶 ( + 𝜅(𝜃 − 𝑣𝑡 ) − 𝜆(𝑆𝑡 , 𝑣𝑡 ) − 𝑟𝐶(𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 𝜕𝑆𝑡 𝜕𝑣𝑡

such that 𝜆(𝑆𝑡 , 𝑣𝑡 ) is the market price of volatility risk, with boundary condition 𝐶(𝑆𝑇 , 𝑣𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0}. By introducing 𝑥𝑡 as the logarithm of the spot price 𝑥𝑡 = log 𝑆𝑡 and assuming 𝜆(𝑆𝑡 , 𝑣𝑡 ) = 𝜆𝑣𝑡 where 𝜆 is a constant value, show that 𝐶(𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) satisfies the following PDE 𝜕𝐶 1 𝜕 2 𝐶 1 2 𝜕 2 𝐶 𝜕2𝐶 + 𝑣𝑡 2 + 𝜉 𝑣𝑡 2 + 𝜌𝜉𝑣𝑡 𝜕𝑡 2 𝜕𝑥𝑡 2 𝜕𝑥𝑡 𝜕𝑣𝑡 𝜕𝑣𝑡 ) ( ) 𝜕𝐶 𝜕𝐶 ( 1 + 𝑟 − 𝐷 − 𝑣𝑡 + 𝜅(𝜃 − 𝑣𝑡 ) − 𝜆𝑣𝑡 − 𝑟𝐶(𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 2 𝜕𝑥𝑡 𝜕𝑣𝑡

7.2.3 Stochastic Volatility

761

with boundary condition { } 𝐶(𝑥𝑇 , 𝑣𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max 𝑒𝑥𝑇 − 𝐾, 0 . By analogy with the Black–Scholes formula, let the solution of the call price take the form 𝐶(𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒𝑥𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑃1 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝐾𝑒−𝑟(𝑇 −𝑡) 𝑃2 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) where 𝑃1 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑃2 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) are the risk-neutral probabilities that the call option is ITM at expiry time 𝑇 . By substituting the proposed solution into the PDE, show that 𝑃1 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) and 𝑃2 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) satisfy the PDEs 2 𝜕 2 𝑃𝑗 𝜕 2 𝑃𝑗 ) 𝜕𝑃𝑗 ( 1 𝜕 𝑃𝑗 1 + 𝑟 − 𝐷 + 𝑢𝑗 𝑣𝑡 + 𝑣𝑡 2 + 𝜉 2 𝑣𝑡 2 + 𝜌𝜉𝑣𝑡 𝜕𝑡 2 𝜕𝑥𝑡 2 𝜕𝑥𝑡 𝜕𝑣𝑡 𝜕𝑥𝑡 𝜕𝑣𝑡

𝜕𝑃𝑗

) 𝜕𝑃𝑗 ( + 𝑎 − 𝑏𝑗 𝑣𝑡 =0 𝜕𝑣𝑡 for 𝑗 = 1, 2 where 𝑢1 =

1 , 2

1 𝑢2 = − , 2

𝑎 = 𝜅𝜃,

𝑏1 = 𝜅 + 𝜆 − 𝜌𝜉,

𝑏2 = 𝜅 + 𝜆

subject to the boundary condition 𝑃𝑗 (𝑥𝑇 , 𝑣𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 1I𝑥𝑇 ≥log 𝐾 . Let 𝑃̂𝑗 (𝑚, 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) be the Fourier transform of 𝑃𝑗 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ), where 𝑃̂𝑗 (𝑚, 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) =

∞

∫−∞

𝑒−𝑖𝑚𝑥𝑡 𝑃𝑗 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) 𝑑𝑥𝑡 ,

𝑗 = 1, 2

and show that 𝑃̂𝑗 (𝑚, 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) satisfies 𝜕 2 𝑃̂𝑗 ( ) 𝜕 𝑃̂𝑗 1 + 𝜉 2 𝑣𝑡 2 + 𝑎 − 𝑏𝑗 𝑣𝑡 + 𝑖𝑚𝜌𝜉𝑣𝑡 𝜕𝑡 2 𝜕𝑣𝑡 𝜕𝑣𝑡

𝜕 𝑃̂𝑗

) ( 1 + (𝑟 − 𝐷 + 𝑢𝑗 𝑣𝑡 )𝑖𝑚 − 𝑚2 𝑣𝑡 𝑃̂𝑗 (𝑚, 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) = 2 with boundary condition 𝑒−𝑖𝑚 log 𝐾 𝑃̂𝑗 (𝑚, 𝑣𝑇 , 𝑇 ; 𝐾, 𝑇 ) = . 𝑖𝑚

0

762

7.2.3 Stochastic Volatility

By seeking an exponential affine solution of the form 𝑃̂𝑗 (𝑚, 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒𝐴(𝑡,𝑇 )+𝐵(𝑡,𝑇 )𝑣𝑡 𝑃̂𝑗 (𝑚, 𝑣𝑇 , 𝑇 ; 𝐾, 𝑇 ) with 𝐴(𝑇 , 𝑇 ) = 0 and 𝐵(𝑇 , 𝑇 ) = 0, show that )( ) ( 𝜕𝐵 = 𝛾 𝐵(𝑡, 𝑇 ) − 𝜋+ 𝐵(𝑡, 𝑇 ) − 𝜋− 𝜕𝑡 𝜕𝐴 = 𝛿𝐵(𝑡, 𝑇 ) + 𝜀 𝜕𝑡 where 𝛼=

𝛿 = −1,

1 2 𝑚 − 𝑖𝑚𝑢𝑗 , 2

𝛽 = 𝑏𝑗 − 𝑖𝑚𝜌𝜉,

𝜀 = −(𝑟 − 𝐷)𝑖𝑚,

𝜋± =

1 𝛾 = − 𝜉2 2

−𝛽 ±

√

𝛽 2 − 4𝛼𝛾 . 2𝛾

Solving the two differential equations, show that 𝐴(𝑡, 𝑇 ) = 𝐵(𝑡, 𝑇 ) =

) ( ) ] [( 𝛿𝜋− 1 − 𝜋̃𝑒𝜂(𝑇 −𝑡) 𝜋̃ − 1 log − 𝜂(𝑇 − 𝑡) − 𝜀(𝑇 − 𝑡) 𝜂 𝜋̃ 1 − 𝜋̃ ( ) 𝜂(𝑇 −𝑡) 𝜋− 1 − 𝑒 1 − 𝜋̃𝑒𝜂(𝑇 −𝑡)

where 𝜋̃ = 𝜋− ∕𝜋+

and

𝜂=

√ 𝛽 2 − 4𝛼𝛾.

Finally, by taking the Fourier inversion of 𝑃̂𝑗 (𝑚, 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ), 𝑃𝑗 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) =

∞

1 𝑒𝑖𝑚𝑥𝑡 𝑃̂𝑗 (𝑚, 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) 𝑑𝑚 2𝜋 ∫−∞

show that the European call option at time 𝑡 is 𝐶(𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒𝑥𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑃1 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝐾𝑒−𝑟(𝑇 −𝑡) 𝑃2 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) where 𝑃𝑗 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) =

1 𝑒𝑖𝑚(𝑥𝑡 −log 𝐾)+𝐴(𝑡,𝑇 )+𝐵(𝑡,𝑇 )𝑣𝑡 𝑑𝑚, ∫ 2𝜋 −∞ 𝑖𝑚 ∞

𝑗 = 1, 2.

7.2.3 Stochastic Volatility

763

Solution: From the PDE satisfied by 𝐶(𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ), 𝜕2 𝐶 1 𝜕2𝐶 𝜕2𝐶 𝜕𝐶 1 + 𝑣𝑡 𝑆𝑡2 2 + 𝜉 2 𝑣𝑡 2 + 𝜌𝜉𝑣𝑡 𝑆𝑡 𝜕𝑡 2 2 𝜕𝑆𝑡 𝜕𝑣𝑡 𝜕𝑆𝑡 𝜕𝑣𝑡 +(𝑟 − 𝐷)𝑆𝑡

) 𝜕𝐶 𝜕𝐶 ( + 𝜅(𝜃 − 𝑣𝑡 ) − 𝜆(𝑆𝑡 , 𝑣𝑡 ) − 𝑟𝐶(𝑆𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 𝜕𝑆𝑡 𝜕𝑣𝑡

such that 𝜆(𝑆𝑡 , 𝑣𝑡 ) is the market price of volatility risk, with boundary condition 𝐶(𝑆𝑇 , 𝑣𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0} and by substituting 𝜆(𝑆𝑡 , 𝑣𝑡 ) = 𝜆𝑣𝑡 and 𝑥𝑡 = log 𝑆𝑡 , we have 𝜕𝐶 𝜕𝐶 𝜕𝑥𝑡 1 𝜕𝐶 = = 𝜕𝑆𝑡 𝜕𝑥𝑡 𝜕𝑆𝑡 𝑆𝑡 𝜕𝑥𝑡

𝜕 𝜕2𝐶 = 2 𝜕𝑆𝑡 𝜕𝑆𝑡

(

1 𝜕𝐶 𝑆𝑡 𝜕𝑥𝑡

)

1 𝜕 1 𝜕𝐶 + =− 2 𝑆𝑡 𝜕𝑥𝑡 𝑆𝑡 𝜕𝑆𝑡

(

𝜕𝐶 𝜕𝑥𝑡

)

( 1 = 2 𝑆𝑡

𝜕 2 𝐶 𝜕𝐶 − 𝜕𝑥𝑡 𝜕𝑥2𝑡

)

and 1 𝜕 𝜕2 𝐶 = 𝜕𝑆𝑡 𝜕𝑣𝑡 𝑆𝑡 𝜕𝑣𝑡

(

𝜕𝐶 𝜕𝑥𝑡

) =

1 𝜕2𝐶 . 𝑆𝑡 𝜕𝑥𝑡 𝜕𝑣𝑡

Hence, by substituting the above equations into the Black–Scholes PDE we have 𝜕𝐶 1 𝜕 2 𝐶 1 2 𝜕 2 𝐶 𝜕2 𝐶 + 𝑣𝑡 2 + 𝜉 𝑣𝑡 2 + 𝜌𝜉𝑣𝑡 𝜕𝑡 2 𝜕𝑥𝑡 2 𝜕𝑥𝑡 𝜕𝑣𝑡 𝜕𝑣𝑡 ) ( ) 𝜕𝐶 𝜕𝐶 ( 1 + 𝑟 − 𝐷 − 𝑣𝑡 + 𝜅(𝜃 − 𝑣𝑡 ) − 𝜆𝑣𝑡 − 𝑟𝐶(𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 2 𝜕𝑥𝑡 𝜕𝑣𝑡 with boundary condition { } ( ) 𝐶(𝑥𝑇 , 𝑣𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max 𝑒𝑥𝑇 − 𝐾, 0 = 𝑒𝑥𝑇 − 𝐾 1I𝑥𝑇 ≥log 𝐾 . Setting the solution of the call price in the form 𝐶(𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒𝑥𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑃1 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝐾𝑒−𝑟(𝑇 −𝑡) 𝑃2 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 )

764

7.2.3 Stochastic Volatility

we have 𝜕𝑃 𝜕𝐶 = 𝐷𝑒𝑥𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑃1 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑒𝑥𝑡 𝑒−𝐷(𝑇 −𝑡) 1 𝜕𝑡 𝜕𝑡 −𝑟(𝑇 −𝑡) −𝑟(𝑇 −𝑡) 𝜕𝑃2 𝑃2 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝐾𝑒 −𝑟𝐾𝑒 𝜕𝑡 𝜕𝑃 𝜕𝐶 𝑥𝑡 −𝐷(𝑇 −𝑡) 𝑥𝑡 −𝐷(𝑇 −𝑡) 𝜕𝑃1 =𝑒 𝑒 𝑃1 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) + 𝑒 𝑒 − 𝐾𝑒−𝑟(𝑇 −𝑡) 2 𝜕𝑥𝑡 𝜕𝑥𝑡 𝜕𝑥𝑡 2 𝜕2𝐶 𝑥𝑡 −𝐷(𝑇 −𝑡) 𝑥𝑡 −𝐷(𝑇 −𝑡) 𝜕𝑃1 𝑥𝑡 −𝐷(𝑇 −𝑡) 𝜕 𝑃1 = 𝑒 𝑒 𝑃 (𝑥 , 𝑣 , 𝑡; 𝐾, 𝑇 ) + 2𝑒 𝑒 + 𝑒 𝑒 1 𝑡 𝑡 𝜕𝑥𝑡 𝜕𝑥2𝑡 𝜕𝑥2𝑡

−𝐾𝑒−𝑟(𝑇 −𝑡)

𝜕 2 𝑃2 𝜕𝑥2𝑡

𝜕𝑃 𝜕 2 𝑃1 𝜕 2 𝑃2 𝜕2𝐶 = 𝑒𝑥𝑡 𝑒−𝐷(𝑇 −𝑡) 1 + 𝑒𝑥𝑡 𝑒−𝐷(𝑇 −𝑡) − 𝐾𝑒−𝑟(𝑇 −𝑡) 𝜕𝑣𝑡 𝜕𝑥𝑡 𝜕𝑣𝑡 𝜕𝑣𝑡 𝜕𝑥𝑡 𝜕𝑣𝑡 𝜕𝑥𝑡 𝜕𝑃 𝜕𝑃 𝜕𝐶 = 𝑒𝑥𝑡 𝑒−𝐷(𝑇 −𝑡) 1 − 𝐾𝑒−𝑟(𝑇 −𝑡) 2 𝜕𝑣𝑡 𝜕𝑣𝑡 𝜕𝑣𝑡 2 2 𝜕2𝐶 𝑥𝑡 −𝐷(𝑇 −𝑡) 𝜕 𝑃1 −𝑟(𝑇 −𝑡) 𝜕 𝑃2 = 𝑒 𝑒 − 𝐾𝑒 . 𝜕𝑣2𝑡 𝜕𝑣2𝑡 𝜕𝑣2𝑡

At expiry time 𝑇 , ( ) 𝐶(𝑆𝑇 , 𝑣𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝑒𝑥𝑇 − 𝐾, 0} = 𝑒𝑥𝑇 − 𝐾 1I𝑥𝑇 ≥log 𝐾 which is equivalent to 𝑒𝑥𝑇 𝑃1 (𝑥𝑇 , 𝑣𝑇 , 𝑇 ; 𝐾, 𝑇 ) − 𝐾𝑃2 (𝑥𝑇 , 𝑣𝑇 , 𝑇 ; 𝑘, 𝑇 ) = 𝑒𝑥𝑇 1I𝑥𝑇 ≥log 𝐾 − 𝐾1I𝑥𝑇 ≥log 𝐾 . Hence, the boundary condition becomes 𝑃1 (𝑥𝑇 , 𝑣𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 1I𝑥𝑇 ≥log 𝐾

and

𝑃2 (𝑥𝑇 , 𝑣𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 1I𝑥𝑇 ≥log 𝐾 .

Substituting the above equations into the PDE satisfied by 𝐶(𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ), 𝑃1 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) will satisfy ) 𝜕𝑃 ( 𝜕𝑃1 1 𝜕 2 𝑃1 1 2 𝜕 2 𝑃1 𝜕 2 𝑃1 1 1 + 𝑟 − 𝐷 + 𝑣𝑡 + 𝑣𝑡 2 + 𝜉 𝑣𝑡 2 + 𝜌𝜉𝑣𝑡 𝜕𝑡 2 𝜕𝑥𝑡 2 𝜕𝑥𝑡 𝜕𝑣𝑡 2 𝜕𝑥𝑡 𝜕𝑣𝑡 ) 𝜕𝑃1 ( + 𝜅𝜃 − (𝜅 + 𝜆 − 𝜌𝜉)𝑣𝑡 =0 𝜕𝑣𝑡 with boundary condition 𝑃1 (𝑥𝑇 , 𝑣𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 1I𝑥𝑇 ≥log 𝐾 .

7.2.3 Stochastic Volatility

765

In contrast, 𝑃2 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) will satisfy ) 𝜕𝑃 ( 𝜕 2 𝑃2 𝜕𝑃2 1 𝜕 2 𝑃2 1 2 𝜕 2 𝑃2 1 2 + 𝑟 − 𝐷 − 𝑣𝑡 + 𝑣𝑡 2 + 𝜉 𝑣𝑡 2 + 𝜌𝜉𝑣𝑡 𝜕𝑡 2 𝜕𝑥𝑡 2 𝜕𝑥𝑡 𝜕𝑣𝑡 2 𝜕𝑥𝑡 𝜕𝑣𝑡 ) 𝜕𝑃2 ( + 𝜅𝜃 − (𝜅 + 𝜆)𝑣𝑡 =0 𝜕𝑣𝑡 with boundary condition 𝑃2 (𝑥𝑇 , 𝑣𝑇 , 𝑇 ; 𝐾, 𝑇 ) = 1I𝑥𝑇 ≥log 𝐾 . From the definition of the Fourier transform, 𝑃̂𝑗 (𝑚, 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) =

∞

∫−∞

𝑒−𝑖𝑚𝑥𝑡 𝑃𝑗 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) 𝑑𝑥𝑡 , 𝑗 = 1, 2

and taking Fourier transforms of the PDE satisfied by 𝑃𝑗 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ), 𝑗 = 1, 2 we have ∞ ∞ 𝜕 2 𝑃𝑗 𝜕 2 𝑃𝑗 1 1 𝑒−𝑖𝑚𝑥𝑡 2 𝑑𝑥𝑡 + 𝜉 2 𝑣𝑡 𝑒−𝑖𝑚𝑥𝑡 2 𝑑𝑥𝑡 𝑑𝑥𝑡 + 𝑣𝑡 ∫−∞ ∫−∞ 𝜕𝑡 2 ∫−∞ 2 𝜕𝑥𝑡 𝜕𝑣𝑡 2𝑃 ∞ ∞ 𝜕 𝜕𝑃 ) ( 𝑗 𝑗 +𝜌𝜉𝑣𝑡 𝑒−𝑖𝑚𝑥𝑡 𝑑𝑥 + 𝑟 − 𝐷 + 𝑢𝑗 𝑣𝑡 𝑒−𝑖𝑚𝑥𝑡 𝑑𝑥𝑡 ∫−∞ ∫−∞ 𝜕𝑥𝑡 𝜕𝑣𝑡 𝑡 𝜕𝑥𝑡 ) ∞ −𝑖𝑚𝑥 𝜕𝑃𝑗 ( 𝑡 + 𝑎 − 𝑏𝑗 𝑣𝑡 𝑒 𝑑𝑥𝑡 = 0. ∫−∞ 𝜕𝑣𝑡 ∞

𝑒−𝑖𝑚𝑥𝑡

𝜕𝑃𝑗

Using integration by parts for partial derivatives with respect to 𝑥𝑡 , ∞

∫−∞

𝑒−𝑖𝑚𝑥𝑡

∞

∫−∞

𝜕𝑃𝑗 𝜕𝑥𝑡

𝑑𝑥𝑡 = 𝑖𝑚

𝑒−𝑖𝑚𝑥𝑡

∞

∫−∞

𝜕 2 𝑃𝑗 𝜕𝑥2𝑡

∞

∫−∞

𝑒−𝑖𝑚𝑥𝑡 𝑃𝑗 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) 𝑑𝑥𝑡 = 𝑖𝑚𝑃̂𝑗 (𝑚, 𝑣𝑡 , 𝑡; 𝐾, 𝑇 )

𝑑𝑥𝑡 = 𝑖𝑚

𝑒−𝑖𝑚𝑥𝑡

𝜕 2 𝑃𝑗 𝜕𝑥𝑡 𝜕𝑣𝑡

∞

∫−∞

𝑒−𝑖𝑚𝑥𝑡

𝑑𝑥𝑡 = 𝑖𝑚

𝜕𝑃𝑗 𝜕𝑥𝑡

∞

∫−∞

𝑑𝑥𝑡 = −𝑚2 𝑃̂𝑗 (𝑚, 𝑣𝑡 , 𝑡; 𝐾, 𝑇 )

𝑒−𝑖𝑚𝑥𝑡

𝜕𝑃𝑗 𝜕𝑣𝑡

𝑑𝑥𝑡 = 𝑖𝑚

𝜕 𝑃̂𝑗 𝜕𝑣𝑡

and since ∞

∫−∞

𝑒−𝑖𝑚𝑥𝑡

𝜕𝑃𝑗 𝜕𝑡

𝑑𝑥𝑡 =

𝜕 𝑃̂𝑗 𝜕𝑡

∞

,

∫−∞

𝑒−𝑖𝑚𝑥𝑡

𝜕𝑃𝑗 𝜕𝑣𝑡

𝑑𝑥𝑡 =

𝜕 𝑃̂𝑗 𝜕𝑣𝑡

766

7.2.3 Stochastic Volatility

and ∞

∫−∞

𝑒−𝑖𝑚𝑥𝑡

𝜕 2 𝑃𝑗 𝜕𝑣2𝑡

𝑑𝑥𝑡 =

𝜕 2 𝑃̂𝑗 𝜕𝑣2𝑡

by substituting them into the Fourier-transformed PDE, we eventually have 2̂ ) 𝜕 𝑃̂𝑗 1 2 𝜕 𝑃𝑗 ( + 𝜉 𝑣𝑡 2 + 𝑎 − 𝑏𝑗 𝑣𝑡 + 𝑖𝑚𝜌𝜉𝑣𝑡 𝜕𝑡 2 𝜕𝑣𝑡 𝜕𝑣𝑡 ) ( 1 + (𝑟 − 𝐷 + 𝑢𝑗 𝑣𝑡 )𝑖𝑚 − 𝑚2 𝑣𝑡 𝑃̂𝑗 (𝑚, 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 2

𝜕 𝑃̂𝑗

with boundary condition 𝑃̂𝑗 (𝑚, 𝑣𝑇 , 𝑇 ; 𝐾, 𝑇 ) =

∞

∫−∞

𝑒−𝑖𝑚𝑥𝑇 1I{𝑥𝑇 ≥log 𝐾} 𝑑𝑥𝑇

∞

=

= − =

𝑒−𝑖𝑚𝑥𝑇 𝑑𝑥𝑇

∫log 𝐾

𝑒−𝑖𝑚𝑥𝑇 𝑖𝑚

|∞ | | |log 𝐾

𝑒−𝑖𝑚 log 𝐾 𝑖𝑚

for 𝑗 = 1, 2. Setting the solution of the Fourier-transformed PDE as 𝑃̂𝑗 (𝑚, 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒𝐴(𝑡,𝑇 )+𝐵(𝑡,𝑇 )𝑣𝑡 𝑃̂𝑗 (𝑚, 𝑣𝑇 , 𝑇 ; 𝐾, 𝑇 ) and in order to satisfy the boundary condition, we therefore have 𝐴(𝑇 , 𝑇 ) = 0 and

𝐵(𝑇 , 𝑇 ) = 0.

Substituting 𝜕 𝑃̂𝑗 𝜕𝑡

( =

) 𝜕𝐴 𝜕𝐵 + 𝑣𝑡 𝑃̂𝑗 (𝑚, 𝑣𝑇 , 𝑇 ; 𝐾, 𝑇 ), 𝜕𝑡 𝜕𝑡

𝜕 𝑃̂𝑗 𝜕𝑣𝑡

= 𝐵(𝑡, 𝑇 )𝑃̂𝑗 (𝑚, 𝑣𝑡 , 𝑡; 𝐾, 𝑇 )

and 𝜕 2 𝑃̂𝑗 𝜕𝑣2𝑡

= 𝐵(𝑡, 𝑇 )2 𝑃̂𝑗 (𝑚, 𝑣𝑡 , 𝑡; 𝐾, 𝑇 )

7.2.3 Stochastic Volatility

767

into the Fourier-transformed PDE and removing the common factor 𝑃̂𝑗 (𝑚, 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ), ) ( 1 𝜕𝐴 𝜕𝐵 + 𝑣𝑡 + 𝜉 2 𝑣𝑡 𝐵(𝑡, 𝑇 )2 + 𝑎 − 𝑏𝑗 𝑣𝑡 + 𝑖𝑚𝜌𝜉𝑣𝑡 𝐵(𝑡, 𝑇 ) 𝜕𝑡 𝜕𝑡 2 1 +(𝑟 − 𝐷 + 𝑢𝑗 𝑣𝑡 )𝑖𝑚 − 𝑚2 𝑣𝑡 = 0. 2 Thus, by equating coefficients we have 𝜕𝐵 1 1 + 𝑖 𝑚𝑢𝑗 − 𝑚2 + (𝑖 𝑚𝜌𝜉 − 𝑏𝑗 )𝐵(𝑡, 𝑇 ) + 𝜉 2 𝐵(𝑡, 𝑇 )2 = 0 𝜕𝑡 2 2 𝜕𝐴 + 𝑎𝐵(𝑡, 𝑇 ) + (𝑟 − 𝐷)𝑖 𝑚 = 0 𝜕𝑡 with boundary conditions 𝐴(𝑇 , 𝑇 ) = 0

and

𝐵(𝑇 , 𝑇 ) = 0.

By setting 𝛼= 𝛿 = −1,

1 2 𝑚 − 𝑖 𝑚𝑢𝑗 , 2

1 𝛾 = − 𝜉2 2 √ −𝛽 ± 𝛽 2 − 4𝛼𝛾 𝜋± = 2𝛾

𝛽 = 𝑏𝑗 − 𝑖 𝑚𝜌𝜉,

𝜀 = −(𝑟 − 𝐷)𝑖 𝑚,

the above differential equations can be written as 𝜕𝐵 = 𝛼 + 𝛽𝐵(𝑡, 𝑇 ) + 𝛾𝐵(𝑡, 𝑇 )2 = 𝛾(𝐵(𝑡, 𝑇 ) − 𝜋+ )(𝐵(𝑡, 𝑇 ) − 𝜋− ) 𝜕𝑡 𝜕𝐴 = 𝛿𝐵(𝑡, 𝑇 ) + 𝜀 𝜕𝑡 with boundary conditions 𝐴(𝑇 , 𝑇 ) = 0

and

𝐵(𝑇 , 𝑇 ) = 0.

Solving the differential equation for 𝐵(𝑡, 𝑇 ), 𝑑𝐵 = 𝛾 𝑑𝑡 ∫ (𝐵(𝑡, 𝑇 ) − 𝜋+ )(𝐵(𝑡, 𝑇 ) − 𝜋− ) ∫ [ ] 𝑑𝐵 𝑑𝐵 1 − = 𝛾 𝑑𝑡 ∫ 𝜋− − 𝜋+ ∫ 𝐵 − 𝜋− ∫ 𝐵 − 𝜋+ ( ) 𝐵 − 𝜋− 1 log = 𝛾𝑡 + 𝐶 𝜋− − 𝜋+ 𝐵 − 𝜋+ where 𝐶 is a constant.

768

7.2.3 Stochastic Volatility

At 𝑡 = 𝑇 , 𝐵(𝑇 , 𝑇 ) = 0 and so 1 log 𝐶= 𝜋− − 𝜋+

(

𝜋− 𝜋+

) − 𝛾𝑇 .

Thus, 𝐵(𝑡, 𝑇 ) =

( ) 𝜋− 1 − 𝑒(𝜋+ −𝜋− )𝛾(𝑇 −𝑡) 1 − 𝜋̃𝑒(𝜋+ −𝜋− )𝛾(𝑇 −𝑡)

=

( ) 𝜋− 1 − 𝑒𝜂(𝑇 −𝑡) 1 − 𝜋̃𝑒𝜂(𝑇 −𝑡)

where 𝜋̃ = 𝜋− ∕𝜋+

𝜂 = (𝜋+ − 𝜋− )𝛾 =

and

√

𝛽 2 − 4𝛼𝛾.

Finally, for the case of 𝐴(𝑡, 𝑇 ) we can rewrite 𝐵(𝑡, 𝑇 ) as ( ) 𝜋− 1 − 𝑒𝜂(𝑇 −𝑡)

𝐵(𝑡, 𝑇 ) =

1 − 𝜋̃𝑒𝜂(𝑇 −𝑡)

[ = 𝜋−

𝑒−𝜂(𝑇 −𝑡) 𝑒𝜂(𝑇 −𝑡) − −𝜂(𝑇 −𝑡) 𝑒 − 𝜋̃ 1 − 𝜋̃𝑒𝜂(𝑇 −𝑡)

]

and solving for 𝐴(𝑡, 𝑇 ), 𝐵(𝑡, 𝑇 ) 𝑑𝑡 + 𝜀 𝑑𝑡 ∫ ] [ 𝑒−𝜂(𝑇 −𝑡) 𝑒𝜂(𝑇 −𝑡) 𝑑𝑡 + 𝜀 𝑑𝑡 𝑑𝑡 − = 𝛿𝜋− ∫ ∫ 𝑒−𝜂(𝑇 −𝑡) − 𝜋̃ ∫ 1 − 𝜋̃𝑒𝜂(𝑇 −𝑡) )] ( ) 1 𝛿𝜋 [ ( 𝐴(𝑡, 𝑇 ) = − log 𝑒−𝜂(𝑇 −𝑡) − 𝜋̃ − log 1 − 𝜋̃𝑒𝜂(𝑇 −𝑡) + 𝜀𝑡 + 𝐶 𝜂 𝜋̃ ∫

𝑑𝐴 =

∫

where 𝐶 is a constant. From the boundary condition 𝐴(𝑇 , 𝑇 ) = 0, we have { 𝐶=−

𝛿𝜋− 𝜂

[(

𝜋̃ − 1 𝜋̃

)

] } log(1 − 𝜋̃) + 𝜀𝑇

and therefore 𝐴(𝑡, 𝑇 ) =

𝛿𝜋− 𝜂

[(

𝜋̃ − 1 𝜋̃

)

( log

1 − 𝜋̃𝑒𝜂(𝑇 −𝑡) 1 − 𝜋̃

)

] − 𝜂(𝑇 − 𝑡) − 𝜀(𝑇 − 𝑡).

Hence, for 𝑗 = 1, 2, 𝑒𝐴(𝑡,𝑇 )+𝐵(𝑡,𝑇 )𝑣𝑡 −𝑖 𝑚 log 𝐾 𝑃̃𝑗 (𝑚, 𝑣𝑡 .𝑡; 𝐾, 𝑇 ) = 𝑖𝑚

7.2.4 Volatility Derivatives

769

and from the Fourier inversion theorem ∞

1 𝑒𝑖𝑚𝑥𝑡 𝑃̃𝑗 (𝑚, 𝑣𝑡 .𝑡; 𝐾, 𝑇 ) 𝑑𝑚 2𝜋 ∫−∞ ∞ 𝑖𝑚(𝑥𝑡 −log 𝐾)+𝐴(𝑡,𝑇 )+𝐵(𝑡,𝑇 )𝑣𝑡 𝑒 1 𝑑𝑚, 𝑗 = 1, 2 = 2𝜋 ∫−∞ 𝑖𝑚

𝑃𝑗 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) =

such that the European call option price at time 𝑡 is 𝐶(𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒𝑥𝑡 𝑒−𝐷(𝑇 −𝑡) 𝑃1 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ) − 𝐾𝑒−𝑟(𝑇 −𝑡) 𝑃2 (𝑥𝑡 , 𝑣𝑡 , 𝑡; 𝐾, 𝑇 ).

7.2.4

Volatility Derivatives

1. Variance Swap. Let {𝑊𝑡 : 𝑡 ≥ 0} be a standard Wiener process on the probability space (Ω, ℱ, ℙ). Suppose that the asset price 𝑆𝑡 has the following dynamics 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡 𝑆𝑡 where 𝜇, 𝐷 are constants and the volatility 𝜎𝑡 is a continuous (possibly stochastic) process. In addition, let 𝑟 be the risk-free interest rate from a money-market account. By considering a European call option 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) written at time 𝑡 on 𝑆𝑡 with strike price 𝐾 and expiry time 𝑇 (𝑇 > 𝑡), state the Dupire equation and payoff satisfied by 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) as a function of strike 𝐾 and time 𝑡. Sketch the call option payoff as a function of strike 𝐾. Assume that for European call options, all strikes are available and we wish to replicate a payoff Ψ(𝑆𝑇 ) by synthesising from the following ∞

Ψ(𝑆𝑇 ) =

∫0

𝜙(𝐾) max{𝑆𝑇 − 𝐾, 0} 𝑑𝐾

where 𝜙(𝐾) is a density function of European call option payoffs with strike 𝐾. Show that ′′

𝜙(𝐾) = Ψ (𝐾). Under what conditions can the payoff Ψ(𝑆𝑇 ) be synthesised using the above expression? Hence, deduce that the payoff Ψ(𝑆𝑇 ) can be constructed using a combination of European put, call, asset and cash of the form 𝐾0

Ψ(𝑆𝑇 ) =

∫0

𝜙𝑃 (𝐾) max{𝐾 − 𝑆𝑇 , 0}𝑑𝐾 +

+𝛼𝑆𝑇 + 𝛽

∞

∫𝐾0

𝜙𝐶 (𝐾) max{𝑆𝑇 − 𝐾, 0} 𝑑𝐾

770

7.2.4 Volatility Derivatives

where 𝛼, 𝛽 and 𝐾0 are constants, 𝜙𝑃 (𝐾) and 𝜙𝐶 (𝐾) are density functions of European put and call option payoffs with strike 𝐾, respectively. Find the replicating portfolio for the payoff Ψ(𝑆𝑇 ) = log 𝑆𝑇 in terms of 𝐾0 . A variance swap is a contract with payoff 𝑇

Ψ(𝜎𝑇 ) =

∫𝑡

𝜎𝑢2 𝑑𝑢 − 𝐾𝑣𝑎𝑟

in which the constant 𝐾𝑣𝑎𝑟 is chosen so that there is no upfront fee to be paid when the contract is initiated at time 𝑡. Show that 𝑇

∫𝑡

[ 𝜎𝑢2 𝑑𝑢 = 2

𝑇

∫𝑡

𝑑𝑆𝑢 𝑆 − log( 𝑇 ) 𝑆𝑢 𝑆𝑡

]

and using the replicating portfolio for log 𝑆𝑇 or otherwise, deduce that this contract can be replicated with a combination of European call, put, cash and asset. Solution: The Dupire equation is 𝜕𝐶 𝜕𝐶 1 2 2 𝜕 2 𝐶 + (𝐷 − 𝑟)𝐾 + 𝜎𝑡 𝐾 − 𝐷𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 0 𝜕𝑇 2 𝜕𝐾 𝜕𝐾 2 with payoff 𝐶(𝑆𝑇 , 𝑇 ; 𝐾, 𝑇 ) = max{𝑆𝑇 − 𝐾, 0}. The call option payoff as a function of strike 𝐾 is illustrated in Figure 7.1. Given that ∞

Ψ(𝑆𝑇 ) =

Figure 7.1

∫0

𝜙(𝐾) max{𝑆𝑇 − 𝐾, 0} 𝑑𝐾

Call option payoff as a function of strike 𝐾.

7.2.4 Volatility Derivatives

771

by differentiating the expression with respect to 𝑆𝑇 , ∞

𝜕 𝜙(𝐾) max{𝑆𝑇 − 𝐾, 0} 𝑑𝐾 𝜕𝑆𝑇 ∫0 [ 𝑆𝑇 𝜕 = 𝜙(𝐾) max{𝑆𝑇 − 𝐾, 0} 𝑑𝐾 𝜕𝑆𝑇 ∫0 ] ∞ + 𝜙(𝐾) max{𝑆𝑇 − 𝐾, 0} 𝑑𝐾 ∫ 𝑆𝑇

Ψ′ (𝑆𝑇 ) =

=

𝜕 𝜕𝑆𝑇 ∫0 𝑆𝑇

=

∫0

𝑆𝑇

𝜙(𝐾)(𝑆𝑇 − 𝐾) 𝑑𝐾

𝜙(𝐾) 𝑑𝐾.

Differentiating it again Ψ′′ (𝑆𝑇 ) =

𝜕 𝜕𝑆𝑇 ∫0

𝑆𝑇

𝜙(𝐾) 𝑑𝐾

= 𝜙(𝑆𝑇 ). Therefore, Ψ′′ (𝐾) = 𝜙(𝐾). From the above information, the conditions under which the payoff Ψ(𝑆𝑇 ) can be synthesised are s Ψ(𝑆 ) has gradient continuity 𝑇 s Ψ′′ (𝑆 ) is bounded for small or large 𝑆 values. 𝑇 𝑇 If Ψ(𝑆𝑇 ) has a gradient discontinuity say at 𝑆𝑇 = 𝐾0 , then lim Ψ′ (𝑆𝑇 ) ≠

𝑆𝑇 →𝐾0−

lim Ψ′ (𝑆𝑇 )

𝑆𝑇 →𝐾0+

and hence Ψ′ (𝑆𝑇 ) and subsequently Ψ′′ (𝑆𝑇 ) is not continuous at 𝑆𝑇 = 𝐾0 . Thus, 𝜙(𝑆𝑇 ) is also not defined at 𝑆𝑇 = 𝐾0 . In contrast, if Ψ′′ (𝑆𝑇 ) is not bounded for small or large 𝑆𝑇 , the integral will not converge (i.e., trading options with unbounded strikes). Since 𝜙(𝑆𝑇 ) = Ψ′′ (𝑆𝑇 ) there is a unique 𝜙(𝑆𝑇 ) corresponding to each Ψ(𝑆𝑇 ). Therefore, depending on the position of the arbitrary constant 𝐾0 with respect to 𝑆𝑇 , the synthesising portfolio can also accommodate a put option and also a linear function of 𝑆𝑇 as

772

7.2.4 Volatility Derivatives

well. To show how this portfolio is synthesised, for a constant 𝐾0 we can set Ψ′′ (𝑆𝑇 ) = 𝜙(𝑆𝑇 ) ⎧ 𝜙 (𝑆 ) ⎪ 𝐶 𝑇 =⎨ ⎪ 𝜙𝑃 (𝑆𝑇 ) ⎩

for 0 < 𝐾0 < 𝑆𝑇 < ∞ for 0 < 𝑆𝑇 < 𝐾0 < ∞

where 𝜙𝐶 (𝑆𝑇 ) and 𝜙𝑃 (𝑆𝑇 ) are density functions of European call and put option payoffs with strike 𝐾 = 𝑆𝑇 , respectively. Taking integrals, we can write it as ⎧ 𝑆𝑇 𝜙𝐶 (𝐾) 𝑑𝐾 + 𝛼 ⎪∫ 𝐾0 ⎪ Ψ′ (𝑆𝑇 ) = ⎨ 𝐾0 ⎪ 𝜙𝑃 (𝐾) 𝑑𝐾 + 𝛼 ⎪− ⎩ ∫𝑆𝑇

for 0 < 𝐾0 < 𝑆𝑇 < ∞ for 0 < 𝑆𝑇 < 𝐾0 < ∞

where 𝛼 is a constant. Integrating again, we can write it as ⎧ 𝑆𝑇 𝜙𝐶 (𝐾)(𝑆𝑇 − 𝐾) 𝑑𝐾 + 𝛼𝑆𝑇 + 𝛽 ⎪∫ ⎪ 𝐾0 Ψ(𝑆𝑇 ) = ⎨ ⎪ 𝐾0 𝜙𝑃 (𝐾)(𝐾 − 𝑆𝑇 ) 𝑑𝐾 + 𝛼𝑆𝑇 + 𝛽 ⎪ ⎩ ∫𝑆 𝑇

for 0 < 𝐾0 < 𝑆𝑇 < ∞ for 0 < 𝑆𝑇 < 𝐾0 < ∞

where 𝛽 is a constant. Hence, ⎧ ∞ ⎪ ∫ 𝜙𝐶 (𝐾) max{𝑆𝑇 − 𝐾, 0} 𝑑𝐾 + 𝛼𝑆𝑇 + 𝛽 ⎪ 𝐾0 Ψ(𝑆𝑇 ) = ⎨ ⎪ 𝐾0 𝜙𝑃 (𝐾) max{𝐾 − 𝑆𝑇 , 0} 𝑑𝐾 + 𝛼𝑆𝑇 + 𝛽 ⎪ ⎩ ∫0

for 0 < 𝐾0 < 𝑆𝑇 < ∞ for 0 < 𝑆𝑇 < 𝐾0 < ∞

or in general, 𝐾0

Ψ(𝑆𝑇 ) =

∫0

𝜙𝑃 (𝐾) max{𝐾 − 𝑆𝑇 , 0} 𝑑𝐾 +

+𝛼𝑆𝑇 + 𝛽.

∞

∫𝐾0

𝜙𝐶 (𝐾) max{𝑆𝑇 − 𝐾, 0} 𝑑𝐾

To find the replicating strategy for Ψ(𝑆𝑇 ) = log 𝑆𝑇 , we consider two cases.

7.2.4 Volatility Derivatives

773

Case 1: 0 < 𝐾0 < 𝑆𝑇 < ∞. 𝐾0

Ψ(𝑆𝑇 ) =

∫0

𝜙𝑃 (𝐾) max{𝐾 − 𝑆𝑇 , 0} 𝑑𝐾 +

+𝛼𝑆𝑇 + 𝛽 ∞

=

∫ 𝐾0 𝑆𝑇

=

∫𝐾0

𝑆𝑇

∫ 𝐾0

∫𝐾0

𝜙𝐶 (𝐾) max{𝑆𝑇 − 𝐾, 0} 𝑑𝐾

𝜙𝐶 (𝐾) max{𝑆𝑇 − 𝐾, 0} 𝑑𝐾 + 𝛼𝑆𝑇 + 𝛽 𝜙𝐶 (𝐾) max{𝑆𝑇 − 𝐾, 0} 𝑑𝐾 +

+𝛼𝑆𝑇 + 𝛽 =

∞

∞

∫𝑆𝑇

𝜙𝐶 (𝐾) max{𝑆𝑇 − 𝐾, 0} 𝑑𝐾

𝜙𝐶 (𝐾)(𝑆𝑇 − 𝐾) 𝑑𝐾 + 𝛼𝑆𝑇 + 𝛽.

By setting Ψ(𝑆𝑇 ) = log 𝑆𝑇 and differentiating with respect to 𝑆𝑇 up to second order, Ψ(𝑆𝑇 ) = log 𝑆𝑇 = Ψ′ (𝑆𝑇 ) =

𝑆𝑇

∫ 𝐾0

𝜙𝐶 (𝐾)(𝑆𝑇 − 𝐾) 𝑑𝐾 + 𝛼𝑆𝑇 + 𝛽

𝑆

𝑇 1 = 𝜙𝐶 (𝐾) 𝑑𝐾 + 𝛼 ∫𝐾0 𝑆𝑇

Ψ′′ (𝑆𝑇 ) = −

1 = 𝜙𝐶 (𝑆𝑇 ). 𝑆𝑇2

At 𝑆𝑇 = 𝐾0 log 𝐾0 = 𝛼𝐾0 + 𝛽 1 = 𝛼. 𝐾0 Thus, 𝛼=

1 , 𝐾0

𝛽 = log 𝐾0 − 1

and

𝜙𝐶 (𝐾) = −

1 . 𝐾2

Case 2: 0 < 𝑆𝑇 < 𝐾0 < ∞. 𝐾0

Ψ(𝑆𝑇 ) =

∫0

𝜙𝑃 (𝐾) max{𝐾 − 𝑆𝑇 , 0} 𝑑𝐾 +

+𝛼𝑆𝑇 + 𝛽 𝐾0

=

∫0

∞

∫𝐾0

𝜙𝐶 (𝐾) max{𝑆𝑇 − 𝐾, 0} 𝑑𝐾

𝜙𝑃 (𝐾) max{𝐾 − 𝑆𝑇 , 0} 𝑑𝐾 + 𝛼𝑆𝑇 + 𝛽

774

7.2.4 Volatility Derivatives 𝑆𝑇

=

∫0

𝜙𝑃 (𝐾) max{𝐾 − 𝑆𝑇 , 0} 𝑑𝐾 +

+𝛼𝑆𝑇 + 𝛽 𝐾0

=

∫ 𝑆𝑇

𝐾0

∫𝑆 𝑇

𝜙𝑃 (𝐾) max{𝐾 − 𝑆𝑇 , 0} 𝑑𝐾

𝜙𝑃 (𝐾)(𝐾 − 𝑆𝑇 ) 𝑑𝐾 + 𝛼𝑆𝑇 + 𝛽.

By setting Ψ(𝑆𝑇 ) = log 𝑆𝑇 and differentiating with respect to 𝑆𝑇 up to second order, Ψ(𝑆𝑇 ) = log 𝑆𝑇 =

𝐾0

∫𝑆𝑇

𝜙𝑃 (𝐾)(𝐾 − 𝑆𝑇 ) 𝑑𝐾 + 𝛼𝑆𝑇 + 𝛽

𝐾

0 1 =− 𝜙𝑃 (𝐾) 𝑑𝐾 + 𝛼 ∫𝑆𝑇 𝑆𝑇 1 Ψ′′ (𝑆𝑇 ) = − 2 = 𝜙𝑃 (𝑆𝑇 ). 𝑆𝑇

Ψ′ (𝑆𝑇 ) =

At 𝑆𝑇 = 𝐾0 log 𝐾0 = 𝛼𝐾0 + 𝛽 1 = 𝛼. 𝐾0 Thus, 1 , 𝐾0

𝛼=

𝛽 = log 𝐾0 − 1 and

𝜙𝑃 (𝐾) = −

1 . 𝐾2

Combining case 1 and case 2 results we finally have the replicating formula for log 𝑆𝑇 in terms of 𝐾0 log 𝑆𝑇 = − +

𝐾0

∫0

∞

1 1 max{𝐾 − 𝑆𝑇 , 0} 𝑑𝐾 − max{𝑆𝑇 − 𝐾, 0} 𝑑𝐾 2 2 ∫ 𝐾 𝐾0 𝐾

𝑆𝑇 + log 𝐾0 − 1. 𝐾0

Under the risk-neutral measure ℚ, 𝑑𝑆𝑡 ̃𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑡 𝑑 𝑊 𝑆𝑡 ̃𝑡 = 𝑊𝑡 + where 𝑊

𝑡

∫0

𝜇−𝑟 𝑑𝑢 is a ℚ-standard Wiener process. 𝜎𝑢

7.2.4 Volatility Derivatives

775

From Taylor’s expansion and subsequently applying It¯o’s lemma, 𝑑𝑆𝑡 1 − 𝑑 log 𝑆𝑡 = 𝑆𝑡 2

(

𝑑𝑆𝑡 𝑆𝑡

)2 +…

̃𝑡 − 1 𝜎 2 𝑑𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑡 𝑑 𝑊 2 𝑡 𝑑𝑆𝑡 1 2 − 𝜎𝑡 𝑑𝑡. = 𝑆𝑡 2 Taking integrals, 𝑇

∫𝑡

𝑑 log 𝑆𝑢 = (

log

𝑆𝑇 𝑆𝑡

) =

𝑇

∫𝑡

𝑇

∫𝑡

𝑑𝑆𝑢 1 − 𝑆𝑢 2 ∫𝑡 𝑑𝑆𝑢 1 − 𝑆𝑢 2 ∫𝑡

𝑇

𝑇

𝜎𝑢2 𝑑𝑢 𝜎𝑢2 𝑑𝑢

or [

𝑇

𝜎𝑢2 𝑑𝑢

∫𝑡

=2

𝑇

∫𝑡

𝑑𝑆𝑢 − log 𝑆𝑢

(

𝑆𝑇 𝑆𝑡

)] .

For the case of a variance swap payoff 𝑇

Ψ(𝜎𝑇 ) =

∫𝑡

𝜎𝑢2 𝑑𝑢 − 𝐾𝑣𝑎𝑟

let the value of the swap contract at time 𝑡 be [ 𝑉 (𝜎𝑡 , 𝑡; 𝐾𝑣𝑎𝑟 , 𝑇 ) = 𝑒

−𝑟(𝑇 −𝑡) ℚ

𝔼

𝑇

∫𝑡

|

| 𝜎𝑢2 𝑑𝑢 − 𝐾𝑣𝑎𝑟 | ℱ𝑡 | |

] .

Since there is no cash flow at initiation of the contract at time 𝑡, 𝑉 (𝜎𝑡 , 𝑡; 𝐾𝑣𝑎𝑟 , 𝑇 ) = 0 we can then express [ 𝐾𝑣𝑎𝑟 = 𝔼ℚ {

𝑇

∫𝑡 𝔼

=2 { =2

[

ℚ

[ 𝔼ℚ

| | 𝜎𝑢2 𝑑𝑢| ℱ𝑡 | | 𝑇

∫𝑡 𝑇

∫𝑡

]

( )| ]} 𝑑𝑆𝑢 𝑆𝑇 | − log |ℱ 𝑆𝑢 𝑆𝑡 || 𝑡 ] [ ( )| ]} 𝑑𝑆𝑢 || 𝑆𝑇 | ℚ log . | ℱ𝑡 − 𝔼 |ℱ | 𝑆𝑢 | 𝑆𝑡 || 𝑡

776

7.2.4 Volatility Derivatives

From the dynamics 𝑑𝑆𝑡 ̃𝑡 = (𝑟 − 𝐷)𝑑𝑡 + 𝜎𝑡 𝑑 𝑊 𝑆𝑡 then

[ 𝔼

ℚ

𝑇

∫𝑡

𝑑𝑆𝑢 || |ℱ 𝑆𝑢 || 𝑡

]

[ =𝔼

ℚ

∫𝑡

(𝑟 − 𝐷) 𝑑𝑢 +

𝑇

∫𝑡

= (𝑟 − 𝐷)(𝑇 − 𝑡)

since

[ 𝔼

𝑇

ℚ

𝑇

∫𝑡

| ̃𝑢 || ℱ𝑡 𝜎𝑢 𝑑 𝑊 | |

| ̃𝑢 || ℱ𝑡 𝜎𝑢 𝑑 𝑊 | |

]

] = 0.

As for log(𝑆𝑇 ∕𝑆𝑡 ), for an arbitrary constant 𝐾0 we can write ( ) ( ) ( ) 𝐾0 𝑆𝑇 𝑆𝑇 log = log + log 𝑆𝑡 𝐾0 𝑆𝑡 and from the results of log 𝑆𝑇 , ( log

𝑆𝑇 𝑆𝑡

)

𝐾0

∞

1 1 max{𝐾 − 𝑆𝑇 , 0} 𝑑𝐾 − max{𝑆𝑇 − 𝐾, 0} 𝑑𝐾 ∫𝐾0 𝐾 2 ∫0 𝐾 2 ( ) 𝐾0 𝑆𝑇 + − 1 + log . 𝐾0 𝑆𝑡

=−

Since [ ] 𝔼ℚ max{𝑆𝑇 − 𝐾, 0}|| ℱ𝑡 = 𝑒𝑟(𝑇 −𝑡) 𝐶(𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) [ ] 𝔼ℚ max{𝐾 − 𝑆𝑇 , 0}|| ℱ𝑡 = 𝑒𝑟(𝑇 −𝑡) 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) [ ] 𝔼ℚ 𝑆𝑇 || ℱ𝑡 = 𝑆𝑡 𝑒𝑟(𝑇 −𝑡) the parameter 𝐾𝑣𝑎𝑟 becomes {

𝐾𝑣𝑎𝑟

[

] [ ( )| ]} | 𝑑𝑆 𝑆𝑇 | | 𝑢 = 2 𝔼ℚ | ℱ − 𝔼ℚ log |ℱ ∫ 𝑡 𝑆𝑢 | 𝑡 𝑆𝑡 || 𝑡 | { [ ] 𝐾0 | 1 | ℚ = 2 (𝑟 − 𝐷)(𝑇 − 𝑡) + 𝔼 max{𝐾 − 𝑆𝑇 , 0} 𝑑𝐾 | ℱ𝑡 | ∫0 𝐾 2 | [ ] 𝐾0 | 1 1 ℚ[ | ] | +𝔼ℚ max{𝑆𝑇 − 𝐾, 0} 𝑑𝐾 | ℱ𝑡 − 𝔼 𝑆𝑇 | ℱ𝑡 | ∫0 𝐾 2 𝐾0 | )]} [ ( 𝐾0 −1 − log 𝑆𝑡 𝑇

7.2.4 Volatility Derivatives

777

{

𝐾0

1 𝑃 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) 𝑑𝐾 𝐾2 ∞ 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) 1 𝐶(𝑆 , 𝑡; 𝐾, 𝑇 ) 𝑑𝐾 − +𝑒(𝑟−𝐷)(𝑇 −𝑡) 𝑡 ∫𝐾0 𝐾 2 𝐾0 [ ( )]} 𝐾0 − log −1 . 𝑆𝑡

=2

(𝑟 − 𝐷)(𝑇 − 𝑡) + 𝑒(𝑟−𝐷)(𝑇 −𝑡)

∫0

2. Let the asset price 𝑆𝑡 and its instantaneous variance 𝜎𝑡2 have the following dynamics 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑𝑊𝑡𝑆

𝑑𝜎𝑡2 = 𝜅(𝜃 − 𝜎𝑡2 )𝑑𝑡 + 𝛼𝜎𝑡 𝑑𝑊𝑡𝜎

𝑑𝑊𝑡𝑆 ⋅ 𝑑𝑊𝑡𝜎 = 𝜌𝑑𝑡

where 𝜇, 𝐷, 𝜅, 𝜃 and 𝛼 are constants, {𝑊𝑡𝑆 : 𝑡 ≥ 0} and {𝑊𝑡𝜎 : 𝑡 ≥ 0} are two standard Wiener processes on the probability space (Ω, ℱ, ℙ) with correlation 𝜌 ∈ (−1, 1). In addition, we let 𝐵𝑡 be a risk-free asset with the differential equation 𝑑𝐵𝑡 = 𝑟 𝐵𝑡 𝑑𝑡 where 𝑟 is the risk-free interest rate. We consider a variance swap with payoff 𝑇

Ψ(𝜎𝑇 ) =

∫𝑡

𝜎𝑢2 𝑑𝑢 − 𝐾𝑣𝑎𝑟

where 𝐾𝑣𝑎𝑟 is chosen so that no cash flows are exchanged when the contract is initiated at time 𝑡. Assuming that the market price of volatility risk is zero, calculate the fair value of 𝐾𝑣𝑎𝑟 . Solution: Following Problem 7.2.3.7 (page 753), it can be shown that under the riskneutral measure ℚ, ̃𝑆 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑 𝑊 𝑡 ) ( 2 2 ̃𝜎 𝑑𝜎𝑡 = 𝜅(𝜃 − 𝜎𝑡 ) − 𝛼𝛾𝑡 𝜎𝑡 𝑑𝑡 + 𝛼𝜎𝑡 𝑑 𝑊 𝑡 √ ̃ 𝑆 = 𝜌𝑊 ̃ 𝜎 + 1 − 𝜌2 𝑍 ̃ 𝜎 and 𝑍 ̃𝑡 are ℚ-standard Wiener processes, 𝑊 ̃𝜎 ⟂ ̃𝑡 , 𝑊 where 𝑊 𝑡 𝑡 𝑡 𝑡 ⟂ ̃𝑡 and 𝛾𝑡 is the market price of volatility risk. 𝑍 By assuming that the market price of volatility risk is zero, ̃𝑆 𝑑𝑆𝑡 = (𝑟 − 𝐷)𝑆𝑡 𝑑𝑡 + 𝜎𝑡 𝑆𝑡 𝑑 𝑊 𝑡

̃𝜎. 𝑑𝜎𝑡2 = 𝜅(𝜃 − 𝜎𝑡2 )𝑑𝑡 + 𝛼𝜎𝑡 𝑑 𝑊 𝑡

778

7.2.4 Volatility Derivatives

Let the variance swap payoff at expiry time 𝑇 be 𝑇

Ψ(𝜎𝑇 ) =

∫𝑡

𝜎𝑢2 𝑑𝑢 − 𝐾𝑣𝑎𝑟

then under the risk-neutral measure ℚ, the value of the contract at time 𝑡 is [ ] 𝑉 (𝜎𝑡 , 𝑡; 𝐾𝑣𝑎𝑟 , 𝑇 ) = 𝑒−𝑟(𝑇 −𝑡) 𝔼ℚ Ψ(𝜎𝑇 )|| ℱ𝑡 [ =𝑒

−𝑟(𝑇 −𝑡) ℚ

𝔼

𝑇

∫𝑡

|

| 𝜎𝑢2 𝑑𝑢 − 𝐾𝑣𝑎𝑟 | ℱ𝑡 |

]

|

.

Since there is no cash flow when the contract is initiated, we can set 𝑉 (𝜎𝑡 , 𝑡; 𝐾𝑣𝑎𝑟 , 𝑇 ) = 0 and the fair value of 𝐾𝑣𝑎𝑟 is defined as [ 𝐾𝑣𝑎𝑟 = 𝔼

ℚ

𝑇

∫𝑡

|

| 𝜎𝑢2 𝑑𝑢| ℱ𝑡 |

]

|

.

From the stochastic volatility model under the risk-neutral measure ℚ, ̃𝜎 𝑑𝜎𝑡2 = 𝜅(𝜃 − 𝜎𝑡2 )𝑑𝑡 + 𝛼𝜎𝑡 𝑑 𝑊 𝑡 and taking integrals on both sides we have 𝑇

∫𝑡

𝑑𝜎𝑢2 =

𝑇

∫𝑡

𝜅(𝜃 − 𝜎𝑢2 ) 𝑑𝑢 +

𝑇

∫𝑡

̃𝜎 𝛼𝜎𝑢 𝑑 𝑊 𝑢

Taking expectations under the filtration ℱ𝑡 , [ 𝔼

ℚ

∫𝑡 𝑇

∫𝑡

𝑇

|

| 𝑑𝜎𝑢2 | ℱ𝑡 |

[

|

]

[ =𝔼

]

ℚ

| 𝔼ℚ 𝑑𝜎𝑢2 | ℱ𝑡 = | ∫𝑡 [

𝑇

𝑇

∫𝑡

𝜅(𝜃

|

| − 𝜎𝑢2 ) 𝑑𝑢| ℱ𝑡 | |

[

| 𝔼ℚ 𝜅(𝜃 − 𝜎𝑢2 ) 𝑑𝑢| ℱ𝑡 |

]

]

[ +𝔼

ℚ

𝑇

∫𝑡

| ̃ 𝜎 || ℱ𝑡 𝛼𝜎𝑢 𝑑 𝑊 𝑢 | |

] | | ̃ 𝜎 | ℱ𝑡 = 0. since 𝔼ℚ 𝛼𝜎𝑢 𝑑 𝑊 𝑢 | ∫𝑡 | Simplifying further, 𝑇

𝑇

∫𝑡

[ ] | 𝑑𝔼ℚ 𝜎𝑢2 | ℱ𝑡 = | ∫𝑡

𝑇

[ [ ]] | 𝜅𝜃 − 𝜅𝔼ℚ 𝜎𝑢2 | ℱ𝑡 𝑑𝑢. |

]

7.2.4 Volatility Derivatives

779

By differentiating with respect to 𝑇 , [ ] [ ] 𝑑 ℚ 2| | 𝔼 𝜎𝑇 | ℱ𝑡 = 𝜅𝜃 − 𝜅𝔼ℚ 𝜎𝑇2 | ℱ𝑡 | | 𝑑𝑇 or [ ] [ ] 𝑑 ℚ 2| | 𝔼 𝜎𝑇 | ℱ𝑡 + 𝜅𝔼ℚ 𝜎𝑇2 | ℱ𝑡 = 𝜅𝜃. | | 𝑑𝑇 To solve the first-order differential equation, we set the particular integral 𝐼 = 𝑒∫

𝜅𝑑𝑇

= 𝑒𝜅𝑇

and multiplying 𝐼 with the differential equation [ [ ]] 𝑑 | 𝑒𝜅𝑇 𝔼ℚ 𝜎𝑇2 | ℱ𝑡 = 𝜅𝜃𝑒𝜅𝑇 | 𝑑𝑇 and then taking integrals, [ ] | 𝑒𝜅𝑇 𝔼ℚ 𝜎𝑇2 | ℱ𝑡 = 𝜃𝑒𝜅𝑇 + 𝐶 | where 𝐶 is a constant. When the contract is initialised at time 𝑡, [ ] | 𝔼ℚ 𝜎𝑡2 | ℱ𝑡 = 𝜎𝑡2 | so that 𝐶 = 𝑒𝜅𝑡 (𝜎𝑡2 − 𝜃). Therefore, [ ] | 𝔼ℚ 𝜎𝑇2 | ℱ𝑡 = 𝜃 + 𝑒−𝜅(𝑇 −𝑡) (𝜎𝑡2 − 𝜃) | and finally the fair value 𝐾𝑣𝑎𝑟 takes the form 𝐾𝑣𝑎𝑟 =

∫𝑡

𝑇

[ ] | 𝔼ℚ 𝜎𝑢2 | ℱ𝑡 𝑑𝑢 |

𝑇

[

] 𝜃 + 𝑒−𝜅(𝑢−𝑡) (𝜎𝑡2 − 𝜃) 𝑑𝑢 ∫𝑡 ( ) 𝜎𝑡2 − 𝜃 ( ) 1 − 𝑒−𝜅(𝑇 −𝑡) . = 𝜃(𝑇 − 𝑡) + 𝜅

=

780

7.2.4 Volatility Derivatives

3. Let {𝑊𝑡 : 𝑡 ≥ 0} be a standard Wiener process on the probability space (Ω, ℱ, ℙ). Suppose that the asset price 𝑆𝑡 has the following dynamics 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡 𝑆𝑡 where 𝜇, 𝐷 are constants and the volatility 𝜎𝑡 is a continuous (possibly stochastic) process. In addition, let 𝑟 be the risk-free interest rate from a money-market account. By dividing the time interval [𝑡, 𝑇 ] into 𝑛 equal intervals with corresponding asset price values 𝑆𝑡0 , 𝑆𝑡1 , …, 𝑆𝑡𝑛 , 𝑡𝑖 = 𝑡 + 𝑖Δ𝑡, Δ𝑡 = (𝑇 − 𝑡)∕𝑛, 𝑡 = 𝑡0 < 𝑡1 < ⋯ < 𝑡𝑛−1 < 𝑡𝑛 = 𝑇 , 𝑇

show that the realised integrated variance

∫𝑡

approximated by 𝑇

∫𝑡

𝜎𝑢2 𝑑𝑢

= lim

𝑛→∞

𝑛−1 ∑

𝜎𝑢2 𝑑𝑢 and volatility

(

𝑆𝑡𝑖+1 − 𝑆𝑡𝑖

𝑇

∫𝑡

𝜎𝑢 𝑑𝑢 can be

)2

𝑆 𝑡𝑖

𝑖=0

and 𝑇

∫𝑡

√ 𝜎𝑢 𝑑𝑢 = lim

𝑛→∞

𝜋 2

(

) 𝑛−1 𝑇 − 𝑡 ∑ || 𝑆𝑡𝑖+1 − 𝑆𝑡𝑖 || | | | | 𝑛 𝑆 𝑡𝑖 | 𝑖=0 |

respectively. We consider a call option on the realised integrated variance having payoff { Ψ(𝜎𝑇 ) = max

∫𝑡

𝑇

} 𝜎𝑢2 𝑑𝑢 − 𝐾, 0

.

Explain how to value the call option at time 𝑡 as a discretely sampled Asian option. Solution: From the diffusion process 𝑑𝑆𝑡 = (𝜇 − 𝐷) 𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡 𝑆𝑡 and using It¯o’s lemma (

𝑑𝑆𝑡 𝑆𝑡

)2

]2 [ = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑡 𝑑𝑊𝑡 = (𝜇 − 𝐷)2 (𝑑𝑡)2 + 2(𝜇 − 𝐷)𝜎𝑡 𝑑𝑊𝑡 𝑡 + 𝜎𝑡2 (𝑑𝑊𝑡 )2 = 𝜎𝑡2 𝑑𝑡.

7.2.4 Volatility Derivatives

781

Thus, for a small time interval Δ𝑡, ( 𝜎𝑡2 Δ𝑡

=

𝑆𝑡𝑖+1 − 𝑆𝑡𝑖

)2

𝑆 𝑡𝑖

and the realised integrated variance is 𝑇

∫𝑡

𝜎𝑢2 𝑑𝑢 ≈

𝑛−1 ∑

(

𝑆𝑡𝑖+1 − 𝑆𝑡𝑖

)2

𝑆 𝑡𝑖

𝑖=0

.

Therefore, as 𝑛 → ∞, 𝑇

∫𝑡

𝜎𝑢2 𝑑𝑢

= lim

𝑛→∞

𝑛−1 ∑

(

𝑆𝑡𝑖+1 − 𝑆𝑡𝑖 𝑆 𝑡𝑖

𝑖=0

)2 .

From It¯o’s lemma ( 𝜎𝑡2 (𝑑𝑊𝑡 )2

=

𝑑𝑆𝑡 𝑆𝑡

)2

we can set | 𝑑𝑆 | 𝜎𝑡 |𝑑𝑊𝑡 | = || 𝑡 || . | 𝑆𝑡 | Since |𝑑𝑊𝑡 | follows a folded normal distribution |𝑑𝑊𝑡 | ∼ 𝑓 (0, 𝑑𝑡) with mean √ 𝔼(|𝑑𝑊𝑡 |) =

2𝑑𝑡 𝜋

informally we can write √ |𝑑𝑊𝑡 | =

2𝑑𝑡 . 𝜋

For small Δ𝑡, √ 𝜎𝑡

2Δ𝑡 || 𝑆𝑡𝑖+1 − 𝑆𝑡𝑖 || =| | | | 𝜋 𝑆 𝑡𝑖 | |

or √ 𝜎𝑡 Δ𝑡 =

𝜋Δ𝑡 || 𝑆𝑡𝑖+1 − 𝑆𝑡𝑖 || |. | | 2 || 𝑆 𝑡𝑖 |

782

7.2.4 Volatility Derivatives

Hence, the realised integrated volatility can be approximated by 𝑇

∫𝑡

√ 𝜎𝑢 𝑑𝑢 ≈

(

𝜋 2

) 𝑛−1 𝑇 − 𝑡 ∑ || 𝑆𝑡𝑖+1 − 𝑆𝑡𝑖 || | | | | 𝑛 𝑆 𝑡𝑖 | 𝑖=0 |

and in the limit 𝑛 → ∞, 𝑇

∫𝑡

√ 𝜎𝑢 𝑑𝑢 = lim

𝑛→∞

𝜋 2

(

) 𝑛−1 𝑇 − 𝑡 ∑ || 𝑆𝑡𝑖+1 − 𝑆𝑡𝑖 || | |. | | 𝑛 𝑆 𝑡𝑖 | 𝑖=0 |

By definition, the call option price at time 𝑡 under the risk-neutral measure ℚ is [ 𝐶𝑣𝑎𝑟 (𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝑒

−𝑟(𝑇 −𝑡) ℚ

𝔼

{ max

𝑇

∫𝑡

}| ] | | ℱ𝑡 | |

𝜎𝑢2 𝑑𝑢 − 𝐾, 0

and since we can approximate 𝑇

∫𝑡

𝜎𝑢2 𝑑𝑢

≈

𝑛−1 ∑

(

𝑆𝑡𝑖+1 − 𝑆𝑡𝑖

𝑖=0

)2

𝑆 𝑡𝑖

the payoff can be written as )2 ⎧𝑛−1 ( ⎫ ⎪∑ 𝑆𝑡𝑖+1 − 𝑆𝑡𝑖 ⎪ Ψ(𝜎𝑇 ) ≈ max ⎨ − 𝐾, 0⎬ . 𝑆 𝑡𝑖 ⎪ 𝑖=0 ⎪ ⎩ ⎭ To value the option as a discretely sampled Asian option, we let 2 ⎧𝑛−1 ⎛ (𝑗) ⎫ 𝑆𝑡 − 𝑆𝑡(𝑗) ⎞ ∑ ⎪ ⎪ 𝑖+1 𝑖 ⎟ (𝑗) ⎜ Ψ(𝜎𝑇 ) ≈ max ⎨ − 𝐾, 0⎬ , (𝑗) ⎜ ⎟ 𝑆𝑡 ⎪ 𝑖=0 ⎝ ⎪ ⎠ 𝑖 ⎩ ⎭

where Ψ(𝜎𝑇(𝑗) ) is the 𝑗th realisation of the payoff.

𝑗 = 1, 2, … , 𝑚

7.2.4 Volatility Derivatives

783

Thus, the call option price at time 𝑡, which is the discounted expected payoff, is approximated by ⎡ 𝑚 ⎧𝑛−1 ⎛ (𝑗) ⎫⎤ (𝑗) 2 ∑ ∑ 𝑆𝑡𝑖+1 − 𝑆𝑡𝑖 ⎞ ⎢ ⎪ ⎪⎥ 1 −𝑟(𝑇 −𝑡) ⎜ ⎟ max ⎨ − 𝐾, 0⎬⎥ 𝐶𝑣𝑎𝑟 (𝜎𝑡 , 𝑡; 𝐾, 𝑇 ) ≈ 𝑒 ⎢𝑚 (𝑗) ⎜ ⎟ 𝑆𝑡 ⎢ 𝑗=1 ⎪ 𝑖=0 ⎝ ⎪⎥ ⎠ 𝑖 ⎣ ⎩ ⎭⎦ =𝑒

−𝑟(𝑇 −𝑡)

⎧ 𝑚 𝑛−1 ⎛ (𝑗) ⎫ (𝑗) ⎞2 ⎪ 1 ∑ ∑ ⎜ 𝑆𝑡𝑖+1 − 𝑆𝑡𝑖 ⎟ ⎪ max ⎨ − 𝐾, 0⎬ . (𝑗) ⎜ ⎟ 𝑚 𝑆𝑡 ⎪ 𝑗=1 𝑖=0 ⎝ ⎪ ⎠ 𝑖 ⎩ ⎭

̃ 𝑆 : 𝑡 ≥ 0} and {𝑊 ̃ 𝑉 : 𝑡 ≥ 0} be 4. Timer Option. Under the risk-neutral measure ℚ, let {𝑊 𝑡 𝑡 two correlated ℚ-standard Wiener processes on the probability space (Ω, ℱ, ℚ) with correlation 𝜌 ∈ (−1, 1). Let the asset price 𝑆𝑡 under the ℚ measure follow a stochastic volatility model with the following dynamics √ 𝑑𝑆𝑡 ̃𝑆 = (𝑟 − 𝐷)𝑑𝑡 + 𝑉𝑡 𝑑 𝑊 𝑡 𝑆𝑡 ̃𝑉 𝑑𝑉𝑡 = 𝛼(𝑉𝑡 , 𝑡)𝑑𝑡 + 𝛽(𝑉𝑡 , 𝑡)𝑑 𝑊 𝑡

̃𝑆 ⋅ 𝑑𝑊 ̃ 𝑉 = 𝜌𝑑𝑡 𝑑𝑊 𝑡 𝑡

where 𝑟 is the risk-free interest-rate parameter from the money-market account, 𝐷 is the √ continuous dividend yield, 𝑉𝑡 is the volatility process, 𝛼(𝑉𝑡 , 𝑡) and 𝛽(𝑉𝑡 , 𝑡) are continuous functions. We denote 𝜏 as the random expiry time of a call option written on 𝑆𝑡 at time 𝑡 with strike price 𝐾 > 0, 𝜏 = inf

{ 𝑢>𝑡:

𝑢

∫𝑡

𝑉𝑠 𝑑𝑠 = 𝑉̃

}

where it is the first hitting time of the realised integrated variance to the variance budget 𝑉̃ . Show that the timer call option price at time 𝑡 is equal to [ }| ] { 1̃ 𝐶𝑡𝑖𝑚𝑒𝑟 (𝑆𝑡 , 𝑉𝑡 , 𝑡; 𝐾, 𝜏) = 𝔼ℚ max 𝑆𝑡 𝑒−𝐷(𝜏−𝑡)− 2 𝑉 +𝐵𝑉̃ − 𝐾𝑒−𝑟(𝜏−𝑡) , 0 || ℱ𝑡 | where 𝐵𝑉 =

𝜏

∫𝑡

) √ √ ( ̃ 𝑉 + 1 − 𝜌2 𝑑 𝑍 ̃𝑢 𝑉𝑢 𝜌𝑑 𝑊 𝑢

̃𝑡 ⟂ ̃𝑉 . ̃𝑡 is a ℚ-standard Wiener process and 𝑍 ⟂𝑊 such that 𝑍 𝑡 Is there an explicit solution for the above timer call option?

784

7.2.4 Volatility Derivatives

Discuss under what conditions we can expect the timer call option to have closed-form solutions. Finally, show that the put–call parity of timer options is 𝐶𝑡𝑖𝑚𝑒𝑟 (𝑆𝑡 , 𝑉𝑡 , 𝑡; 𝐾, 𝜏) − 𝑃𝑡𝑖𝑚𝑒𝑟 (𝑆𝑡 , 𝑉𝑡 , 𝑡; 𝐾, 𝜏) [ ] [ ] | | = 𝔼ℚ 𝑆𝜏 𝑒−𝑟(𝜏−𝑡) | ℱ𝑡 − 𝐾𝔼ℚ 𝑒−𝑟(𝜏−𝑡) | ℱ𝑡 | | where 𝑃𝑡𝑖𝑚𝑒𝑟 (𝑆𝑡 , 𝑉𝑡 , 𝑡; 𝐾, 𝜏) is the timer put option written on 𝑆𝑡 at time 𝑡 with strike price 𝐾 and random expiry time 𝜏. Solution: By expanding 𝑑 log 𝑆𝑡 using Taylor’s series and subsequently applying It¯o’s lemma, (

) 𝑑𝑆𝑡 2 +… 𝑆𝑡 √ 1 = (𝑟 − 𝐷)𝑑𝑡 + 𝑉𝑡 𝑑𝑊𝑡𝑆 − 𝑉𝑡 𝑑𝑡 2 ) ( √ 1 = 𝑟 − 𝐷 − 𝑉𝑡 𝑑𝑡 + 𝑉𝑡 𝑑𝑊𝑡𝑆 . 2

𝑑 log 𝑆𝑡 =

𝑑𝑆𝑡 1 − 𝑆𝑡 2

Since we can set ̃𝑉 + 𝑑𝑊𝑡𝑆 = 𝜌𝑑 𝑊 𝑡

√ ̃𝑡 1 − 𝜌2 𝑑 𝑍

̃𝑉 ⟂ ̃ where 𝑊 𝑡 ⟂ 𝑍𝑡 , we can set ) ) ( √ √ ( 1 ̃ 𝑉 + 1 − 𝜌2 𝑑 𝑍 ̃𝑡 . 𝑑 log 𝑆𝑡 = 𝑟 − 𝐷 − 𝑉𝑡 𝑑𝑡 + 𝑉𝑡 𝜌𝑑 𝑊 𝑡 2 Taking integrals, 𝜏

∫𝑡

𝑑 log 𝑆𝑢 =

𝜏

∫𝑡 +

( log

𝑆𝜏 𝑆𝑡

)

𝜏

1 𝑉 𝑑𝑢 2 ∫𝑡 𝑢 ( ) 𝜏√ √ ̃ 𝑉 + 1 − 𝜌2 𝑑 𝑍 ̃𝑢 𝑉𝑢 𝜌𝑑 𝑊 𝑢

(𝑟 − 𝐷) 𝑑𝑢 −

∫𝑡

1 = (𝑟 − 𝐷)(𝜏 − 𝑡) − 𝑉̃ + 𝐵𝑉̃ 2

or 1

̃

𝑆𝜏 = 𝑆𝑡 𝑒(𝑟−𝐷)(𝜏−𝑡)− 2 𝑉 +𝐵𝑉̃ .

7.2.4 Volatility Derivatives

785

Given that the payoff of a timer call option is paid at random time 𝜏, the price of a timer call option at time 𝑡 is [ { }| ] 𝐶𝑡𝑖𝑚𝑒𝑟 (𝑆𝑡 , 𝑉𝑡 , 𝑡; 𝐾, 𝜏) = 𝔼ℚ 𝑒−𝑟(𝜏−𝑡) max 𝑆𝜏 − 𝐾, 0 | ℱ𝑡 | [ { }| ] 1̃ = 𝔼ℚ 𝑒−𝑟(𝜏−𝑡) max 𝑆𝑡 𝑒(𝑟−𝐷)(𝜏−𝑡)+𝐵𝑉̃ − 2 𝑉 − 𝐾, 0 || ℱ𝑡 | [ }| ] { 1̃ = 𝔼ℚ max 𝑆𝑡 𝑒−𝐷(𝜏−𝑡)− 2 𝑉 +𝐵𝑉̃ − 𝐾𝑒−𝑟(𝜏−𝑡) , 0 || ℱ𝑡 . | From the timer call option price expression, there is no explicit solution since 𝐵𝑉̃ and the | random expiry time 𝜏 can be correlated and the conditional distribution of 𝐵𝑉̃ | 𝜏 is not a | normal distribution. Conditions where we expect the timer call option to have closed-form solutions are as follows. (a) 𝐾 = 0 and 𝐷 = 0 By setting 𝐾 = 0 and 𝐷 = 0, the timer call option is reduced to a timer share contract with random expiry time 𝜏. Hence, we can set [ ] | 𝐶𝑡𝑖𝑚𝑒𝑟 (𝑆𝑡 , 𝑉𝑡 , 𝑡; 𝐾, 𝜏) = 𝔼ℚ 𝑒−𝑟(𝜏−𝑡) 𝑆𝜏 | ℱ𝑡 | = 𝑆𝑡 by assuming, under the general stochastic volatility model, that 𝑒𝑟𝑡 𝑆𝑡 is a martingale (i.e., putting restrictions on 𝛼(𝑉𝑡 , 𝑡) and 𝛽(𝑉𝑡 , 𝑡)). (b) 𝑟 = 0 and 𝐷 = 0 By setting 𝑟 = 0 and 𝐷 = 0, we have 𝐶𝑡𝑖𝑚𝑒𝑟 (𝑆𝑡 , 𝑉𝑡 , 𝑡; 𝐾, 𝜏) = 𝔼

ℚ

[

{

max 𝑆𝑡 𝑒

− 21 𝑉̃ +𝐵𝑉̃

}| ] − 𝐾, 0 || ℱ𝑡 . |

Note that the difference between the timer option and a European option is only the expiry time 𝜏, which is random when 𝑟 = 𝐷 = 0, then the exact date when the cash flow occurs is no longer applicable. Thus, the timer option is not dependent on 𝜏 and following the Black–Scholes formula we have 𝐶𝑡𝑖𝑚𝑒𝑟 (𝑆𝑡 , 𝑉𝑡 , 𝑡; 𝐾) = 𝑆𝑡 Φ(𝑑+ ) − 𝐾Φ(𝑑− ) where log(𝑆𝑡 ∕𝐾) ± 12 𝑉̃ 𝑑± = √ 𝑉̃ and Φ(⋅) is the cdf of a standard normal.

786

7.2.4 Volatility Derivatives

(c) 𝛽(𝑉𝑡 , 𝑡) = 0 When 𝛽(𝑉𝑡 , 𝑡) = 0, the instantaneous variance process is deterministic and hence we know exactly when to exercise the timer call option. By solving the first-order PDE 𝑑𝑉𝑡 = 𝛼(𝑉𝑡 , 𝑡) 𝑑𝑡 to find the solution 𝑉𝑡 , let 𝑇𝑉̃ be the time for the realised integrated variance to reach the variance budget 𝑉̃ , i.e. 𝑇𝑉̃

∫𝑡

𝑉𝑢 𝑑𝑢 = 𝑉̃ .

Thus, the solution of the timer call option reduces to the Black–Scholes formula 𝐶𝑡𝑖𝑚𝑒𝑟 (𝑆𝑡 , 𝑉𝑡 , 𝑡; 𝐾) = 𝑆𝑡 𝑒−𝐷(𝑇𝑉̃ −𝑡) Φ(𝑑+ ) − 𝐾𝑒−𝑟(𝑇𝑉̃ −𝑡) Φ(𝑑− ) where 𝑑± =

log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷)(𝑇𝑉̃ − 𝑡) ± 12 𝑉̃ √ 𝑉̃

and Φ(⋅) is the cdf of a standard normal. For the put–call parity relationship, from the identity max{𝑆𝜏 − 𝐾, 0} − max{𝐾 − 𝑆𝜏 , 0} = 𝑆𝜏 − 𝐾. Taking expectations under the risk-neutral measure ℚ, [ ] [ ] | | 𝔼ℚ 𝑒−𝑟(𝜏−𝑡) max{𝑆𝜏 − 𝐾, 0}| ℱ𝑡 − 𝔼ℚ 𝑒−𝑟(𝜏−𝑡) max{𝑆𝜏 − 𝐾, 0}| ℱ𝑡 | | [ ( )| ] ℚ −𝑟(𝜏−𝑡) 𝑆𝜏 − 𝐾 | ℱ𝑡 =𝔼 𝑒 | or 𝐶𝑡𝑖𝑚𝑒𝑟 (𝑆𝑡 , 𝑉𝑡 , 𝑡; 𝐾, 𝜏) − 𝑃𝑡𝑖𝑚𝑒𝑟 (𝑆𝑡 , 𝑉𝑡 , 𝑡; 𝐾, 𝜏) [ ] [ ] | | = 𝔼ℚ 𝑒−𝑟(𝜏−𝑡) 𝑆𝜏 | ℱ𝑡 − 𝐾𝔼ℚ 𝑒−𝑟(𝜏−𝑡) | ℱ𝑡 . | | From the put–call parity, we can see that the first term on the right-hand side is the price of a timer share contract at time 𝑡, whilst the second term is the price of a timer cash contract at time 𝑡.

Appendix A Mathematics Formulae Indices 𝑥𝑎 = 𝑥𝑎−𝑏 , 𝑥𝑏

𝑥𝑎 𝑥𝑏 = 𝑥𝑎+𝑏 ,

𝑥

Surds

−𝑎

(𝑥𝑎 )𝑏 = (𝑥𝑏 )𝑎 = 𝑥𝑎𝑏

( )𝑎 𝑥 𝑥𝑎 = 𝑎, 𝑦 𝑦

1 = 𝑎, 𝑥

𝑥0 = 1.

√ 𝑎 √ √ √ 𝑥 √ 𝑎 𝑎 𝑥𝑦 = 𝑎 𝑥 𝑎 𝑦, 𝑥∕𝑦 = √ 𝑎 𝑦 √√ ( √ )𝑎 ( √ )𝑏 √ 𝑏 √ 𝑎 𝑎 𝑏 𝑎 𝑎 𝑥 = 𝑥, 𝑥 = 𝑎𝑏 𝑥, 𝑥 = 𝑥𝑏 = 𝑥 𝑎 .

√ 𝑥 = 𝑎 𝑥, 1 𝑎

Exponential and Natural Logarithm 𝑒𝑥 𝑒𝑦 = 𝑒𝑥+𝑦 , log(𝑥𝑦) = log 𝑥 + log 𝑦,

(𝑒𝑥 )𝑦 = (𝑒𝑦 )𝑥 = 𝑒𝑥𝑦 ,

𝑒0 = 1

( ) 𝑥 log = log 𝑥 − log 𝑦, 𝑦

log 𝑒𝑥 = 𝑥,

𝑒log 𝑥 = 𝑥,

log 𝑥𝑦 = 𝑦 log 𝑥

𝑒𝑎 log 𝑥 = 𝑥𝑎 .

Quadratic Equation For constants 𝑎, 𝑏 and 𝑐, the roots of a quadratic equation 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 are 𝑥=

Binomial Formula

−𝑏 ±

( ) 𝑛 𝑛! = , 𝑘 𝑘!(𝑛 − 𝑘)! (𝑥 + 𝑦)𝑛 =

√

𝑏2 − 4𝑎𝑐 . 2𝑎

( ) ( ) ( ) 𝑛 𝑛 𝑛+1 + = 𝑘 𝑘+1 𝑘+1

𝑛 ( ) 𝑛 ∑ 𝑛! 𝑛 𝑛−𝑘 𝑘 ∑ 𝑥𝑛−𝑘 𝑦𝑘 . 𝑥 𝑦 = 𝑘!(𝑛 − 𝑘)! 𝑘 𝑘=0 𝑘=0

788

Appendix A

Series Arithmetic: For initial term 𝑎 and common difference 𝑑, the 𝑛th term is 𝑇𝑛 = 𝑎 + (𝑛 − 1)𝑑 and the sum of 𝑛 terms is 𝑆𝑛 =

1 𝑛 [2𝑎 + (𝑛 − 1)𝑑] . 2

Geometric: For initial term 𝑎 and common ratio 𝑟, the 𝑛th term is 𝑇𝑛 = 𝑎𝑟𝑛−1 the sum of 𝑛 terms is 𝑆𝑛 =

𝑎(1 − 𝑟𝑛 ) 1−𝑟

and the sum of infinite terms is lim 𝑆𝑛 =

𝑛→∞

𝑎 , 1−𝑟

|𝑟| < 1.

Summation For 𝑛 ∈ ℤ+ , 𝑛 ∑

𝑛 ∑

1 𝑘 = 𝑛(𝑛 + 1), 2 𝑘=1

1 𝑘 = 𝑛(𝑛 + 1)(2𝑛 + 1), 6 𝑘=1 2

𝑛 ∑

𝑘3 =

𝑘=1

[

1 𝑛(𝑛 + 1) 2

Let 𝑎1 , 𝑎2 , … be a sequence of numbers. ∑ s If 𝑎𝑛 < ∞ ⟹ lim 𝑎𝑛 = 0. 𝑛→∞ ∑ s If lim 𝑎 ≠ 0 ⟹ 𝑎 = ∞. 𝑛 𝑛 𝑛→∞

Trigonometric Functions sin(−𝑥) = − sin 𝑥, csc 𝑥 =

1 , sin 𝑥

cos2 𝑥 + sin2 𝑥 = 1,

cos(−𝑥) = cos 𝑥, sec 𝑥 =

1 , cos 𝑥

tan2 𝑥 + 1 = sec2 𝑥,

tan 𝑥 =

cot 𝑥 =

sin 𝑥 cos 𝑥

1 tan 𝑥

cot 2 𝑥 + 1 = csc2 𝑥

sin(𝑥 ± 𝑦) = sin 𝑥 cos 𝑦 ± cos 𝑥 sin 𝑦

]2

.

Appendix A

789

cos(𝑥 ± 𝑦) = cos 𝑥 cos 𝑦 ∓ sin 𝑥 sin 𝑦 tan 𝑥 ± tan 𝑦 1 ∓ tan 𝑥 tan 𝑦

tan(𝑥 ± 𝑦) = ( ) 1 𝜋 = , sin 6 2

sin(0) = 0,

√ ( ) 3 𝜋 cos = , 6 2

cos(0) = 1,

tan(0) = 0,

tan

√ ( ) 3 𝜋 = sin , 3 2

( ) 1 𝜋 =√ , sin 4 2

cos

( ) 1 𝜋 =√ , 4 2

cos

tan

( ) 𝜋 = 1, 4

tan

( ) √ 𝜋 = 3, 3

cosh 𝑥 =

𝑒𝑥 + 𝑒−𝑥 , 2

tanh 𝑥 =

( ) 𝜋 1 =√ , 6 3

sin

( ) 1 𝜋 = , 3 2

( ) 𝜋 =1 2

cos

tan

( ) 𝜋 =0 2

( ) 𝜋 = ∞. 2

Hyperbolic Functions sinh 𝑥 =

𝑒𝑥 − 𝑒−𝑥 , 2 csch𝑥 =

1 , sinh 𝑥

sinh(−𝑥) = − sinh 𝑥, cosh2 𝑥 − sinh2 𝑥 = 1,

1 , cosh 𝑥

sech𝑥 =

sinh 𝑥 𝑒𝑥 − 𝑒−𝑥 = 𝑥 cosh 𝑥 𝑒 + 𝑒−𝑥

coth 𝑥 =

cosh(−𝑥) = cosh 𝑥,

1 tanh 𝑥

tanh(−𝑥) = − tanh 𝑥

coth2 𝑥 − 1 = csch2 𝑥,

1 − tanh2 𝑥 = sech2 𝑥

sinh(𝑥 ± 𝑦) = sinh 𝑥 cosh 𝑦 ± cosh 𝑥 sinh 𝑦 cosh(𝑥 ± 𝑦) = cosh 𝑥 cosh 𝑦 ± sinh 𝑥 sinh 𝑦 tanh(𝑥 ± 𝑦) =

tanh 𝑥 ± tanh 𝑦 . 1 ± tanh 𝑥 tanh 𝑦

Complex Numbers Let 𝑤 = 𝑢 + 𝑖𝑣 and 𝑧 = 𝑥 + 𝑖𝑦 where 𝑢, 𝑣, 𝑥, 𝑦 ∈ ℝ, 𝑖 = 𝑤 ± 𝑧 = (𝑢 ± 𝑥) + (𝑣 ± 𝑦)𝑖, 𝑤 = 𝑧 𝑧̄ = 𝑥 − 𝑖𝑦,

𝑧̄ = 𝑧,

(

𝑢𝑥 + 𝑣𝑦 𝑥2 + 𝑦2

√

−1 and 𝑖2 = −1, then

𝑤𝑧 = (𝑢𝑥 − 𝑣𝑦) + (𝑣𝑥 + 𝑢𝑦)𝑖 )

( +

𝑤 + 𝑧 = 𝑤̄ + 𝑧, ̄

𝑣𝑥 − 𝑢𝑦 𝑥2 + 𝑦2

) 𝑖

𝑤𝑧 = 𝑤̄ 𝑧, ̄

(

𝑤 𝑧

) =

𝑤̄ . 𝑧̄

790

Appendix A

De Moivre’s Formula: Let 𝑧 = 𝑥 + 𝑖𝑦 where 𝑥, 𝑦 ∈ ℝ and we can write 𝑧 = 𝑟(cos 𝜃 + 𝑖 sin 𝜃),

𝑟=

√ 𝑥2 + 𝑦2 ,

𝜃 = tan−1

(𝑦) 𝑥

.

For 𝑛 ∈ ℤ [𝑟(cos 𝜃 + 𝑖 sin 𝜃)]𝑛 = 𝑟𝑛 [cos(𝑛𝜃) + 𝑖 sin(𝑛𝜃)] . Euler’s Formula: For 𝜃 ∈ ℝ 𝑒𝑖𝜃 = cos 𝜃 + 𝑖 sin 𝜃. Derivatives If 𝑓 (𝑥) and 𝑔(𝑥) are differentiable functions of 𝑥 and 𝑎 and 𝑏 are constants Sum Rule: 𝑑 (𝑎𝑓 (𝑥) + 𝑏𝑔(𝑥)) = 𝑎𝑓 ′ (𝑥) + 𝑏𝑔 ′ (𝑥) 𝑑𝑥 Product/Chain Rule: 𝑑 (𝑓 (𝑥)𝑔(𝑥)) = 𝑓 (𝑥)𝑔 ′ (𝑥) + 𝑓 ′ (𝑥)𝑔(𝑥) 𝑑𝑥 Quotient Rule: 𝑑 𝑑𝑥 where

(

𝑓 (𝑥) 𝑔(𝑥)

) =

𝑓 ′ (𝑥)𝑔(𝑥) − 𝑓 (𝑥)𝑔 ′ (𝑥) , 𝑔(𝑥)2

𝑔(𝑥) ≠ 0

𝑑 𝑑 𝑓 (𝑥) = 𝑓 ′ (𝑥) and 𝑔(𝑥) = 𝑔 ′ (𝑥). 𝑑𝑥 𝑑𝑥

If 𝑓 (𝑧) is a differentiable function of 𝑧 and 𝑧 = 𝑧(𝑥) is a differentiable function of 𝑥, then 𝑑 𝑓 (𝑧(𝑥)) = 𝑓 ′ (𝑧(𝑥))𝑧′ (𝑥). 𝑑𝑥 If 𝑥 = 𝑥(𝑠), 𝑦 = 𝑦(𝑠) and 𝐹 (𝑠) = 𝑓 (𝑥(𝑠), 𝑦(𝑠)), then 𝜕𝑓 𝜕𝑥 𝜕𝑓 𝜕𝑦 𝑑 𝐹 (𝑠) = ⋅ + ⋅ . 𝑑𝑠 𝜕𝑥 𝜕𝑠 𝜕𝑦 𝜕𝑠 If 𝑥 = 𝑥(𝑢, 𝑣), 𝑦 = 𝑦(𝑢, 𝑣) and 𝐹 (𝑢, 𝑣) = 𝑓 (𝑥(𝑢, 𝑣), 𝑦(𝑢, 𝑣)), then 𝜕𝑓 𝜕𝑥 𝜕𝑓 𝜕𝑦 𝜕𝐹 𝜕𝑓 𝜕𝑥 𝜕𝑓 𝜕𝑦 𝜕𝐹 = ⋅ + ⋅ , = ⋅ + ⋅ . 𝜕𝑢 𝜕𝑥 𝜕𝑢 𝜕𝑦 𝜕𝑢 𝜕𝑣 𝜕𝑥 𝜕𝑣 𝜕𝑦 𝜕𝑣

Appendix A

791

Standard Differentiations If 𝑓 (𝑥) and 𝑔(𝑥) are differentiable functions of 𝑥 and 𝑎 and 𝑏 are constants 𝑑 𝑎 = 0, 𝑑𝑥 𝑑 𝑓 (𝑥) = 𝑓 ′ (𝑥)𝑒𝑓 (𝑥) , 𝑒 𝑑𝑥

𝑑 [𝑓 (𝑥)]𝑛 = 𝑛[𝑓 (𝑥)]𝑛−1 𝑓 ′ (𝑥) 𝑑𝑥

𝑓 ′ (𝑥) 𝑑 log 𝑓 (𝑥) = , 𝑑𝑥 𝑓 (𝑥)

𝑑 𝑓 (𝑥) = 𝑓 ′ (𝑥)𝑎𝑓 (𝑥) log 𝑎 𝑎 𝑑𝑥

𝑑 sin(𝑎𝑥) = 𝑎 cos 𝑥, 𝑑𝑥

𝑑 cos(𝑎𝑥) = −𝑎 sin(𝑎𝑥), 𝑑𝑥

𝑑 tan(𝑎𝑥) = 𝑎 sec2 𝑥 𝑑𝑥

𝑑 sinh(𝑎𝑥) = 𝑎 cosh(𝑎𝑥), 𝑑𝑥

𝑑 cosh(𝑎𝑥) = 𝑎 sinh(𝑎𝑥), 𝑑𝑥

𝑑 tanh(𝑎𝑥) = 𝑎 sech2 (𝑎𝑥) 𝑑𝑥

where 𝑓 ′ (𝑥) =

𝑑 𝑓 (𝑥). 𝑑𝑥

Taylor Series If 𝑓 (𝑥) is an analytic function of 𝑥, then for small ℎ 𝑓 (𝑥0 + ℎ) = 𝑓 (𝑥0 ) + 𝑓 ′ (𝑥0 )ℎ +

1 ′′ 1 𝑓 (𝑥0 )ℎ2 + 𝑓 ′′′ (𝑥0 )ℎ3 + … 2! 3!

If 𝑓 (𝑥, 𝑦) is an analytic function of 𝑥 and 𝑦, then for small Δ𝑥, Δ𝑦 𝜕𝑓 (𝑥0 , 𝑦0 ) 𝜕𝑓 (𝑥0 , 𝑦0 ) 𝑓 (𝑥0 + Δ𝑥, 𝑦0 + Δ𝑦) = 𝑓 (𝑥0 , 𝑦0 ) + Δ𝑥 + Δ𝑦 𝜕𝑥 𝜕𝑦 [ 2 ] 𝜕 2 𝑓 (𝑥0 , 𝑦0 ) 𝜕 2 𝑓 (𝑥0 , 𝑦0 ) 1 𝜕 𝑓 (𝑥0 , 𝑦0 ) 2 2 (Δ𝑥) + 2 (Δ𝑦) Δ𝑥Δ𝑦 + + 2! 𝜕𝑥𝜕𝑦 𝜕𝑥2 𝜕𝑦2 [ 3 𝜕 3 𝑓 (𝑥0 , 𝑦0 ) 1 𝜕 𝑓 (𝑥0 , 𝑦0 ) 3 + (Δ𝑥) + 3 (Δ𝑥)2 Δ𝑦 3! 𝜕𝑥3 𝜕𝑥2 𝜕𝑦 ] 𝜕 3 𝑓 (𝑥0 , 𝑦0 ) 𝜕 3 𝑓 (𝑥0 , 𝑦0 ) 2 3 Δ𝑥(Δ𝑦) + (Δ𝑦) + … +3 𝜕𝑥𝜕𝑦2 𝜕𝑦3 Maclaurin Series Taylor series expansion of a function about 𝑥0 = 0: 1 = 1 − 𝑥 + 𝑥2 − 𝑥3 + … , 1+𝑥

|𝑥| < 1

1 = 1 + 𝑥 + 𝑥2 + 𝑥3 + … , 1−𝑥

|𝑥| < 1

𝑒𝑥 = 1 + 𝑥 +

1 2 1 3 𝑥 + 𝑥 + …, 2! 3!

for all 𝑥

792

Appendix A

𝑒−𝑥 = 1 − 𝑥 +

1 2 1 3 𝑥 − 𝑥 + …, 2! 3!

for all 𝑥

1 1 1 log(1 + 𝑥) = 𝑥 − 𝑥2 + 𝑥3 − 𝑥4 + … , 2 3 4 1 1 1 log(1 − 𝑥) = −𝑥 − 𝑥2 − 𝑥3 − 𝑥4 + … , 2 3 4

𝑥 ∈ (−1, 1] |𝑥| < 1

sin 𝑥 = 𝑥 −

1 3 1 5 1 7 𝑥 + 𝑥 − 𝑥 + …, 3! 5! 7!

for all 𝑥

cos 𝑥 = 1 −

1 2 1 4 1 6 𝑥 + 𝑥 − 𝑥 + …, 2! 4! 6!

for all 𝑥

2𝑥5 17 7 1 + 𝑥 + …, tan 𝑥 = 𝑥 + 𝑥3 + 3 15 315

|𝑥|
0 such that |𝑓 (𝑥)| ≤ 𝐾|𝑔(𝑥)| for |𝑥 − 𝑥0 | < 𝛿 s 𝑓 (𝑥) = 𝑜 (𝑔(𝑥)) if lim 𝑓 (𝑥) = 0 𝑥→𝑥0 𝑔(𝑥) s 𝑓 (𝑥) ∼ 𝑔(𝑥) if lim 𝑓 (𝑥) = 1. 𝑥→𝑥0 𝑔(𝑥) L’Ĥopital’s Rule Let 𝑓 and 𝑔 be differentiable on 𝑎 ∈ ℝ such that 𝑔 ′ (𝑥) ≠ 0 in an interval around 𝑎, except possibly at 𝑎 itself. Suppose that lim 𝑓 (𝑥) = lim 𝑔(𝑥) = 0

𝑥→𝑎

𝑥→𝑎

or lim 𝑓 (𝑥) = lim 𝑔(𝑥) = ±∞

𝑥→𝑎

𝑥→𝑎

Appendix A

793

then 𝑓 (𝑥) 𝑓 ′ (𝑥) = lim ′ . 𝑥→𝑎 𝑔(𝑥) 𝑥→𝑎 𝑔 (𝑥) lim

Indefinite Integrals If 𝐹 (𝑥) is a differentiable function and 𝑓 (𝑥) is its derivative, then ∫

𝑓 (𝑥) 𝑑𝑥 = 𝐹 (𝑥) + 𝑐

𝑑 𝐹 (𝑥) = 𝑓 (𝑥) and 𝑐 is an arbitrary constant. where 𝐹 ′ (𝑥) = 𝑑𝑥 If 𝑓 (𝑥) is a continuous function then 𝑑 𝑓 (𝑥) 𝑑𝑥 = 𝑓 (𝑥). 𝑑𝑥 ∫

Standard Indefinite Integrals If 𝑓 (𝑥) is a differentiable function of 𝑥 and 𝑎 and 𝑏 are constants 𝑎 𝑑𝑥 = 𝑎𝑥 + 𝑐,

∫

∫

𝑓 ′ (𝑥) 𝑑𝑥 = log |𝑓 (𝑥)| + 𝑐, ∫ 𝑓 (𝑥) ∫

∫

∫

𝑒𝑓 (𝑥) 𝑑𝑥 =

log(𝑎𝑥) 𝑑𝑥 = 𝑥 log(𝑎𝑥) − 𝑎𝑥 + 𝑐,

1 sin(𝑎𝑥) 𝑑𝑥 = − cos(𝑎𝑥) + 𝑐, 𝑎

∫

(𝑎𝑥 + 𝑏)𝑛+1 + 𝑐, 𝑎(𝑛 + 1)

(𝑎𝑥 + 𝑏)𝑛 𝑑𝑥 =

sinh(𝑎𝑥) 𝑑𝑥 =

1 cosh(𝑎𝑥) + 𝑐, 𝑎

∫ ∫

∫

𝑛 ≠ −1

1 𝑓 (𝑥) +𝑐 𝑒 𝑓 ′ (𝑥)

𝑎𝑥 𝑑𝑥 =

cos(𝑎𝑥) 𝑑𝑥 = cosh(𝑎𝑥) 𝑑𝑥 =

𝑎𝑥 +𝑐 log 𝑎

1 sin(𝑎𝑥) + 𝑐 𝑎 1 sinh(𝑎𝑥) + 𝑐 𝑎

where 𝑐 is an arbitrary constant. Definite Integrals If 𝐹 (𝑥) is a differentiable function and 𝑓 (𝑥) is its derivative and is continuous on a closed interval [𝑎, 𝑏], then 𝑏

∫𝑎 where 𝐹 ′ (𝑥) =

𝑑 𝐹 (𝑥) = 𝑓 (𝑥). 𝑑𝑥

𝑓 (𝑥) 𝑑𝑥 = 𝐹 (𝑏) − 𝐹 (𝑎)

794

Appendix A

If 𝑓 (𝑥) and 𝑔(𝑥) are integrable functions then 𝑎

∫𝑎 𝑏

∫𝑎

[𝛼 𝑓 (𝑥) + 𝛽𝑔(𝑥)] 𝑑𝑥 = 𝛼 𝑏

∫𝑎

𝑏

𝑓 (𝑥) 𝑑𝑥 = 0,

𝑓 (𝑥) 𝑑𝑥 =

𝑏

∫𝑎 𝑐

∫𝑎

∫𝑎

𝑓 (𝑥) 𝑑𝑥 = −

𝑓 (𝑥) 𝑑𝑥 + 𝛽

𝑓 (𝑥) 𝑑𝑥 +

𝑏

∫𝑐

𝑏

∫𝑎

𝑎

∫𝑏

𝑓 (𝑥) 𝑑𝑥

𝑔(𝑥) 𝑑𝑥,

𝑓 (𝑥) 𝑑𝑥,

𝛼, 𝛽 are constants

𝑐 ∈ [𝑎, 𝑏].

Derivatives of Definite Integrals If 𝑓 (𝑡) is a continuous function of 𝑡 and 𝑎(𝑥) and 𝑏(𝑥) are continuous functions of 𝑥 𝑏(𝑥)

𝑑 𝑑 𝑑 𝑓 (𝑡) 𝑑𝑡 = 𝑓 (𝑏(𝑥)) 𝑏(𝑥) − 𝑓 (𝑎(𝑥)) 𝑎(𝑥) 𝑑𝑥 ∫𝑎(𝑥) 𝑑𝑥 𝑑𝑥 𝑏(𝑥)

𝑑 𝑑 𝑑 𝑑𝑡 = 𝑏(𝑥) − 𝑎(𝑥). 𝑑𝑥 ∫𝑎(𝑥) 𝑑𝑥 𝑑𝑥 If 𝑔(𝑥, 𝑡) is a differentiable function of two variables then 𝑏(𝑥) 𝑏(𝑥) 𝜕𝑔(𝑥, 𝑡) 𝑑 𝑑 𝑑 𝑔(𝑥, 𝑡) 𝑑𝑡 = 𝑔 (𝑥, 𝑏(𝑥)) 𝑏(𝑥) − 𝑔 (𝑥, 𝑎(𝑥)) 𝑎(𝑥) + 𝑑𝑡. ∫𝑎(𝑥) 𝑑𝑥 ∫𝑎(𝑥) 𝑑𝑥 𝑑𝑥 𝜕𝑥

Integration by Parts For definite integrals 𝑏

∫𝑎 where 𝑢′ (𝑥) =

𝑏 |𝑏 | 𝑢(𝑥)𝑣 (𝑥) 𝑑𝑥 = 𝑢(𝑥)𝑣(𝑥)| − 𝑣(𝑥)𝑢′ (𝑥) 𝑑𝑥 | ∫𝑎 |𝑎 ′

𝑑 𝑑 𝑢(𝑥) and 𝑣′ (𝑥) = 𝑣(𝑥). 𝑑𝑥 𝑑𝑥

Integration by Substitution If 𝑓 (𝑥) is a continuous function of 𝑥 and 𝑔 ′ is continuous on the closed interval [𝑎, 𝑏], then 𝑔(𝑎)

∫𝑔(𝑏)

𝑓 (𝑥) 𝑑𝑥 =

𝑏

∫𝑎

𝑓 (𝑔(𝑢))𝑔 ′ (𝑢) 𝑑𝑢.

Appendix A

795

Gamma Function The gamma function is defined as ∞

Γ(𝑧) =

∫0

𝑡𝑧−1 𝑒−𝑡 𝑑𝑡

such that Γ(𝑧 + 1) = 𝑧Γ(𝑧),

Γ

( ) √ 1 = 𝜋, 2

Γ(𝑛) = (𝑛 − 1)! for 𝑛 ∈ ℕ.

Beta Function The beta function is defined as 𝐵(𝑥, 𝑦) =

1

∫0

𝑡𝑥−1 (1 − 𝑡)𝑦−1 𝑑𝑡 for 𝑥 > 0, 𝑦 > 0

such that 𝐵(𝑥, 𝑦) = 𝐵(𝑦, 𝑥),

𝐵(𝑥, 𝑦) =

Γ(𝑥)Γ(𝑦) . Γ(𝑥 + 𝑦)

In addition 𝑢

∫0

𝑡𝑥−1 (𝑢 − 𝑡)𝑦−1 𝑑𝑡 = 𝑢𝑥+𝑦−1 𝐵(𝑥, 𝑦).

Convex Function A set Ω in a vector space over ℝ is called a convex set if for 𝑥, 𝑦 ∈ Ω, 𝑥 ≠ 𝑦 and for any 𝜆 ∈ (0, 1), 𝜆𝑥 + (1 − 𝜆)𝑦 ∈ Ω. Let Ω be a convex set in a vector space over ℝ. A function 𝑓 : Ω ⟼ ℝ is called a convex function if for 𝑥, 𝑦 ∈ Ω, 𝑥 ≠ 𝑦 and for any 𝜆 ∈ (0, 1), 𝑓 (𝜆𝑥 + (1 − 𝜆)𝑦) ≤ 𝜆𝑓 (𝑥) + (1 − 𝜆) 𝑓 (𝑦). s If the inequality is strict then 𝑓 is strictly convex. s If 𝑓 is convex and differentiable on ℝ then 𝑓 (𝑥) ≥ 𝑓 (𝑦) + 𝑓 ′ (𝑦)(𝑥 − 𝑦). s If 𝑓 is a twice continuously differentiable function on ℝ then 𝑓 is convex if and only if 𝑓 ′′ ≥ 0. If 𝑓 ′′ > 0 then 𝑓 is strictly convex. s 𝑓 is a (strictly) concave function if −𝑓 is a (strictly) convex function.

796

Appendix A

Dirac Delta Function The Dirac delta function is defined as { 𝛿(𝑥) =

0 ∞

𝑥≠0 𝑥=0

and for a continuous function 𝑓 (𝑥) and a constant 𝑎 we have ∞

∫−∞

𝛿(𝑥) 𝑑𝑥 = 1,

∞

∫−∞

𝑓 (𝑥)𝛿(𝑥) 𝑑𝑥 = 𝑓 (0),

∞

∫−∞

𝑓 (𝑥)𝛿(𝑥 − 𝑎) 𝑑𝑥 = 𝑓 (𝑎).

Heaviside Step Function The Heaviside step function 𝐻(𝑥) is defined as the integral of the Dirac delta function given as { 𝑥 0 𝑥 0. ∫−∞ Fubini’s Theorem Suppose that 𝑓 (𝑥, 𝑦) is 𝐴 × 𝐵 measurable and if ( ∫𝐴×𝐵

𝑓 (𝑥, 𝑦) 𝑑(𝑥, 𝑦) =

∫𝐴 ∫𝐵

∫𝐴×𝐵

|𝑓 (𝑥, 𝑦)| 𝑑(𝑥, 𝑦) < ∞, then

) 𝑓 (𝑥, 𝑦) 𝑑𝑦 𝑑𝑥 =

( ∫𝐵 ∫𝐴

) 𝑓 (𝑥, 𝑦) 𝑑𝑥 𝑑𝑦.

Appendix B Probability Theory Formulae Probability Concepts Let 𝐴 and 𝐵 be events of the sample space Ω with probabilities ℙ(𝐴) ∈ [0, 1] and ℙ(𝐵) ∈ [0, 1], then Complement: ℙ(𝐴𝑐 ) = 1 − ℙ(𝐴). Conditional: ℙ(𝐴|𝐵) =

ℙ(𝐴 ∩ 𝐵) . ℙ(𝐵)

Independence: The events 𝐴 and 𝐵 are independent if and only if ℙ(𝐴 ∩ 𝐵) = ℙ(𝐴) ⋅ ℙ(𝐵). Mutually Exclusive: The events 𝐴 and 𝐵 are mutually exclusive if and only if ℙ(𝐴 ∩ 𝐵) = 0. Addition: ℙ(𝐴 ∪ 𝐵) = ℙ(𝐴) + ℙ(𝐵) − ℙ(𝐴 ∩ 𝐵). Multiplication: ℙ(𝐴 ∩ 𝐵) = ℙ(𝐴|𝐵)ℙ(𝐵) = ℙ(𝐵|𝐴)ℙ(𝐴). Partition: ℙ(𝐴) = ℙ(𝐴 ∩ 𝐵) + ℙ(𝐴 ∩ 𝐵 𝑐 ) = ℙ(𝐴|𝐵)ℙ(𝐵) + ℙ(𝐴|𝐵 𝑐 )ℙ(𝐵 𝑐 ). Bayes’ Rule Let 𝐴 and 𝐵 be events of the sample space Ω with probabilities ℙ(𝐴) ∈ [0, 1] and ℙ(𝐵) ∈ [0, 1], then ℙ(𝐴|𝐵) =

ℙ(𝐵|𝐴)ℙ(𝐴) . ℙ(𝐵)

798

Appendix B

Indicator Function The indicator function 1I𝐴 of an event 𝐴 of a sample space Ω is a function 1I𝐴 : Ω ⟼ ℝ defined as { 1 if 𝜔 ∈ 𝐴 1I𝐴 (𝜔) = 0 if 𝜔 ∈ 𝐴𝑐 . Properties: For events 𝐴 and 𝐵 of the sample space Ω 1I𝐴𝑐 = 1 − 1I𝐴 , 𝔼(1I𝐴 ) = ℙ(𝐴),

1I𝐴∩𝐵 = 1I𝐴 1I𝐵 ,

Var(1I𝐴 ) = ℙ(𝐴)ℙ(𝐴𝑐 ),

1I𝐴∪𝐵 = 1I𝐴 + 1I𝐵 − 1I𝐴 1I𝐵 Cov(1I𝐴 , 1I𝐵 ) = ℙ(𝐴 ∩ 𝐵) − ℙ(𝐴)ℙ(𝐵).

Discrete Random Variables Univariate Case Let 𝑋 be a discrete random variable whose possible values are 𝑥 = 𝑥1 , 𝑥2 , … and let 𝑃 (𝑋 = 𝑥) be the probability mass function. Total Probability of All Possible Values: ∞ ∑ 𝑘=1

ℙ(𝑋 = 𝑥𝑘 ) = 1.

Cumulative Distribution Function: ℙ(𝑋 ≤ 𝑥𝑛 ) =

𝑛 ∑ 𝑘=1

ℙ(𝑋 = 𝑥𝑘 ).

Expectation: 𝔼(𝑋) = 𝜇 =

∞ ∑ 𝑘=1

𝑥𝑘 ℙ(𝑋 = 𝑥𝑘 ).

Variance: Var(𝑋) = 𝜎 2 ] [ = 𝔼 (𝑋 − 𝜇)2 = =

∞ ∑ 𝑘=1 ∞ ∑ 𝑘=1

(𝑥𝑘 − 𝜇)2 ℙ(𝑋 = 𝑥𝑘 ) 𝑥2𝑘 ℙ(𝑋 = 𝑥𝑘 ) − 𝜇2

= 𝔼(𝑋 2 ) − [𝔼(𝑋)]2 .

Appendix B

799

Moment-Generating Function: ∞ ( ) ∑ 𝑀𝑋 (𝑡) = 𝔼 𝑒𝑡𝑋 = 𝑒𝑡𝑥𝑘 ℙ(𝑋 = 𝑥𝑘 ),

𝑡 ∈ ℝ.

𝑘=1

Characteristic Function: ∞ ) ∑ ( 𝑒𝑖𝑡𝑥𝑘 ℙ(𝑋 = 𝑥𝑘 ), 𝜑𝑋 (𝑡) = 𝔼 𝑒𝑖𝑡𝑋 =

𝑖=

√

−1 and 𝑡 ∈ ℝ.

𝑘=1

Bivariate Case Let 𝑋 and 𝑌 be discrete random variables whose possible values are 𝑥 = 𝑥1 , 𝑥2 , … and 𝑦 = 𝑦1 , 𝑦2 , …, respectively, and let 𝑃 (𝑋 = 𝑥, 𝑌 = 𝑦) be the joint probability mass function. Total Probability of All Possible Values: ∞ ∞ ∑ ∑ 𝑗=1 𝑘=1

ℙ(𝑋 = 𝑥𝑗 , 𝑌 = 𝑦𝑘 ) = 1.

Joint Cumulative Distribution Function: ℙ(𝑋 ≤ 𝑥𝑛 , 𝑌 ≤ 𝑦𝑚 ) =

𝑛 ∑ 𝑚 ∑ 𝑗=1 𝑘=1

ℙ(𝑋 = 𝑥𝑗 , 𝑌 = 𝑦𝑘 ).

Marginal Probability Mass Function:

ℙ(𝑋 = 𝑥) =

∞ ∑ 𝑘=1

ℙ(𝑋 = 𝑥, 𝑌 = 𝑦𝑘 ),

ℙ(𝑌 = 𝑦) =

∞ ∑ 𝑗=1

ℙ(𝑋 = 𝑥𝑗 , 𝑌 = 𝑦).

Conditional Probability Mass Function: ℙ(𝑋 = 𝑥|𝑌 = 𝑦) =

ℙ(𝑋 = 𝑥, 𝑌 = 𝑦) , ℙ(𝑌 = 𝑦)

ℙ(𝑌 = 𝑦|𝑋 = 𝑥) =

ℙ(𝑋 = 𝑥, 𝑌 = 𝑦) . ℙ(𝑋 = 𝑥)

Conditional Expectation:

𝔼(𝑋|𝑌 ) = 𝜇𝑥|𝑦 =

𝑛 ∑ 𝑗=1

𝑥𝑗 ℙ(𝑋 = 𝑥𝑗 |𝑌 = 𝑦),

𝔼(𝑌 |𝑋) = 𝜇𝑦|𝑥 =

𝑛 ∑ 𝑘=1

𝑦𝑘 ℙ(𝑌 = 𝑦𝑘 |𝑋 = 𝑥).

800

Appendix B

Conditional Variance: 2 Var(𝑋|𝑌 ) = 𝜎𝑥|𝑦 [ ] | = 𝔼 (𝑋 − 𝜇𝑥|𝑦 )2 | 𝑌 | ∞ ∑ (𝑥𝑗 − 𝜇𝑥|𝑦 )2 ℙ(𝑋 = 𝑥𝑗 |𝑌 = 𝑦) = 𝑗=1

∞ ∑

=

𝑗=1

2 𝑥2𝑗 ℙ(𝑋 = 𝑥𝑗 |𝑌 = 𝑦) − 𝜇𝑥|𝑦

2 Var(𝑌 |𝑋) = 𝜎𝑦|𝑥 [ ] | = 𝔼 (𝑌 − 𝜇𝑦|𝑥 )2 | 𝑋 | ∞ ∑ (𝑦𝑘 − 𝜇𝑦|𝑥 )2 ℙ(𝑌 = 𝑦𝑘 |𝑋 = 𝑥) =

=

𝑘=1 ∞ ∑ 𝑘=1

2 𝑦2𝑘 ℙ(𝑌 = 𝑦𝑘 |𝑋 = 𝑥) − 𝜇𝑦|𝑥 .

Covariance: For 𝔼(𝑋) = 𝜇𝑥 and 𝔼(𝑌 ) = 𝜇𝑦 [ ] Cov(𝑋, 𝑌 ) = 𝔼 (𝑋 − 𝜇𝑥 )(𝑌 − 𝜇𝑦 ) =

=

∞ ∑ ∞ ∑ 𝑗=1 𝑘=1 ∞ ∑ ∞ ∑ 𝑗=1 𝑘=1

(𝑥𝑗 − 𝜇𝑥 )(𝑦𝑘 − 𝜇𝑦 )ℙ(𝑋 = 𝑥𝑗 , 𝑌 = 𝑦𝑘 ) 𝑥𝑗 𝑦𝑘 ℙ(𝑋 = 𝑥𝑗 , 𝑌 = 𝑦𝑘 ) − 𝜇𝑥 𝜇𝑦

= 𝔼(𝑋𝑌 ) − 𝔼(𝑋)𝔼(𝑌 ). Joint Moment-Generating Function: For 𝑠, 𝑡 ∈ ℝ ∞ ∑ ∞ ) ∑ ( 𝑒𝑠𝑥𝑗 +𝑡𝑦𝑘 ℙ(𝑋 = 𝑥𝑗 , 𝑌 = 𝑦𝑘 ). 𝑀𝑋𝑌 (𝑠, 𝑡) = 𝔼 𝑒𝑠𝑋+𝑡𝑌 = 𝑗=1 𝑘=1

Joint Characteristic Function: √ For 𝑖 = −1 and 𝑠, 𝑡 ∈ ℝ ∞ ∑ ∞ ) ∑ ( 𝜑𝑋𝑌 (𝑠, 𝑡) = 𝔼 𝑒𝑖𝑠𝑋+𝑖𝑡𝑌 = 𝑒𝑖𝑠𝑥𝑗 +𝑖𝑡𝑦𝑘 ℙ(𝑋 = 𝑥𝑗 , 𝑌 = 𝑦𝑘 ). 𝑗=1 𝑘=1

Appendix B

801

Independence: 𝑋 and 𝑌 are independent if and only if s ℙ(𝑋 = 𝑥, 𝑌 = 𝑦) = ℙ(𝑋 = 𝑥)ℙ(𝑌 = 𝑦). s 𝑀 (𝑠, 𝑡) = 𝔼 (𝑒𝑠𝑋+𝑡𝑌 ) = 𝔼 (𝑒𝑠𝑋 ) 𝔼 (𝑒𝑡𝑌 ) = 𝑀 (𝑠)𝑀 (𝑡). 𝑋𝑌 𝑋 𝑌 s 𝜑 (𝑠, 𝑡) = 𝔼 (𝑒𝑖𝑠𝑋+𝑖𝑡𝑌 ) = 𝔼 (𝑒𝑖𝑠𝑋 ) 𝔼 (𝑒𝑖𝑡𝑌 ) = 𝜑 (𝑠)𝜑 (𝑡). 𝑋𝑌 𝑋 𝑌 Continuous Random Variables Univariate Case Let 𝑋 be a continuous random variable whose values 𝑥 ∈ ℝ and let 𝑓𝑋 (𝑥) be the probability density function. Total Probability of All Possible Values: ∞

∫−∞

𝑓𝑋 (𝑥) 𝑑𝑥 = 1.

Evaluating Probability: ℙ(𝑎 ≤ 𝑋 ≤ 𝑏) =

𝑏

∫𝑎

𝑓𝑋 (𝑥) 𝑑𝑥.

Cumulative Distribution Function: 𝑥

𝐹𝑋 (𝑥) = ℙ(𝑋 ≤ 𝑥) =

∫−∞

𝑓𝑋 (𝑥) 𝑑𝑥.

Probability Density Function: 𝑓𝑋 (𝑥) =

𝑑 𝐹 (𝑥). 𝑑𝑥 𝑋

Expectation: ∞

𝔼(𝑋) = 𝜇 =

∫−∞

𝑥𝑓𝑋 (𝑥) 𝑑𝑥.

Variance: Var(𝑋) = 𝜎 2 =

∞

∫−∞

(𝑥 − 𝜇)2 𝑓𝑋 (𝑥) 𝑑𝑥 =

∞

∫−∞

𝑥2 𝑓𝑋 (𝑥) 𝑑𝑥 − 𝜇 2 = 𝔼(𝑋 2 ) − [𝔼(𝑋)]2 .

Moment-Generating Function: ( ) 𝑀𝑋 (𝑡) = 𝔼 𝑒𝑡𝑋 =

∞

∫−∞

𝑒𝑡𝑥 𝑓𝑋 (𝑥) 𝑑𝑥, 𝑡 ∈ ℝ.

802

Appendix B

Characteristic Function: ) ( 𝜑𝑋 (𝑡) = 𝔼 𝑒𝑖𝑡𝑋 =

∞

∫−∞

𝑒𝑖𝑡𝑥 𝑓𝑋 (𝑥) 𝑑𝑥,

𝑖=

√

−1 and 𝑡 ∈ ℝ.

Probability Density Function of a Dependent Variable: Let the random variable 𝑌 = 𝑔(𝑋). If 𝑔 is monotonic then the probability density function of 𝑌 is |−1 ( )| 𝑑 𝑓𝑌 (𝑦) = 𝑓𝑋 𝑔 −1 (𝑦) || 𝑔 −1 (𝑦)|| | 𝑑𝑦 | where 𝑔 −1 denotes the inverse function. Bivariate Case Let 𝑋 and 𝑌 be two continuous random variables whose values 𝑥 ∈ ℝ and 𝑦 ∈ ℝ, and let 𝑓𝑋𝑌 (𝑥, 𝑦) be the joint probability density function. Total Probability of All Possible Values: ∞

∞

∫−∞ ∫−∞

∞

𝑓𝑋𝑌 (𝑥, 𝑦) 𝑑𝑥 𝑑𝑦 =

∞

∫−∞ ∫−∞

𝑓𝑋𝑌 (𝑥, 𝑦) 𝑑𝑦 𝑑𝑥 = 1.

Joint Cumulative Distribution Function: 𝐹𝑋𝑌 (𝑥, 𝑦) = ℙ(𝑋 ≤ 𝑥, 𝑌 ≤ 𝑦) =

𝑥

𝑦

𝑓𝑋𝑌 (𝑥, 𝑦) 𝑑𝑦 𝑑𝑥 =

∫−∞ ∫−∞

𝑦

𝑥

∫−∞ ∫−∞

𝑓𝑋𝑌 (𝑥, 𝑦) 𝑑𝑥𝑑𝑦.

Evaluating Joint Probability: ℙ(𝑥𝑎 ≤ 𝑋 ≤ 𝑥𝑏 , 𝑦𝑎 ≤ 𝑌 ≤ 𝑦𝑏 ) =

𝑥𝑏

∫ 𝑥 𝑎 ∫ 𝑦𝑎 𝑦𝑏

=

𝑦𝑏 𝑥𝑏

∫𝑦𝑎 ∫𝑥𝑎

𝑓𝑋𝑌 (𝑥, 𝑦) 𝑑𝑦 𝑑𝑥 𝑓𝑋𝑌 (𝑥, 𝑦) 𝑑𝑥 𝑑𝑦

= 𝐹𝑋𝑌 (𝑥𝑏 , 𝑦𝑏 ) − 𝐹𝑋𝑌 (𝑥𝑏 , 𝑦𝑎 ) − 𝐹𝑋𝑌 (𝑥𝑎 , 𝑦𝑏 ) + 𝐹𝑋𝑌 (𝑥𝑎 , 𝑦𝑎 ). Joint Probability Density Function: 𝑓𝑋𝑌 (𝑥, 𝑦) =

𝜕2 𝜕2 𝐹𝑋𝑌 (𝑥, 𝑦) = 𝐹 (𝑥, 𝑦). 𝜕𝑥𝜕𝑦 𝜕𝑦𝜕𝑥 𝑋𝑌

Marginal Probability Density Function: 𝑓𝑋 (𝑥) =

∞

∫−∞

𝑓𝑋𝑌 (𝑥, 𝑦) 𝑑𝑦,

𝑓𝑌 (𝑦) =

∞

∫−∞

𝑓𝑋𝑌 (𝑥, 𝑦) 𝑑𝑥.

Appendix B

803

Conditional Probability Density Function: 𝑓𝑋|𝑌 (𝑥|𝑦) =

𝑓𝑋𝑌 (𝑥, 𝑦) , 𝑓𝑌 (𝑦)

𝑓𝑌 |𝑋 (𝑦|𝑥) =

𝑓𝑋𝑌 (𝑥, 𝑦) . 𝑓𝑋 (𝑥)

Conditional Expectation: 𝔼(𝑋|𝑌 ) = 𝜇𝑥|𝑦 =

∞

∫−∞

𝑥𝑓𝑋|𝑌 (𝑥|𝑦) 𝑑𝑥,

𝔼(𝑌 |𝑋) = 𝜇𝑦|𝑥 =

∞

∫−∞

𝑦𝑓𝑌 |𝑋 (𝑦|𝑥) 𝑑𝑦.

Conditional Variance: 2 Var(𝑋|𝑌 ) = 𝜎𝑥|𝑦 [ ] | = 𝔼 (𝑋 − 𝜇𝑥|𝑦 )2 | 𝑌 | ∞

=

∫−∞

(𝑥 − 𝜇𝑥|𝑦 )2 𝑓𝑋|𝑌 (𝑥|𝑦) 𝑑𝑥

∞

=

∫−∞

2 𝑥2 𝑓𝑋|𝑌 (𝑥|𝑦) 𝑑𝑥 − 𝜇𝑥|𝑦

2 Var(𝑌 |𝑋) = 𝜎𝑦|𝑥 ] [ | = 𝔼 (𝑌 − 𝜇𝑦|𝑥 )2 | 𝑋 | ∞

=

∫−∞ ∞

=

∫−∞

(𝑦 − 𝜇𝑦|𝑥 )2 𝑓𝑌 |𝑋 (𝑦|𝑥) 𝑑𝑦 2 𝑦2 𝑓𝑌 |𝑋 (𝑦|𝑥) 𝑑𝑦 − 𝜇𝑦|𝑥 .

Covariance: For 𝔼(𝑋) = 𝜇𝑥 and 𝔼(𝑌 ) = 𝜇𝑦 [ ] Cov(𝑋, 𝑌 ) = 𝔼 (𝑋 − 𝜇𝑥 )(𝑌 − 𝜇𝑦 ) ∞

=

∞

∫−∞ ∫−∞ ∞

(𝑥 − 𝜇𝑥 )(𝑦 − 𝜇𝑦 )𝑓𝑋𝑌 (𝑥, 𝑦) 𝑑𝑦 𝑑𝑥

∞

𝑥𝑦𝑓𝑋𝑌 (𝑥, 𝑦) 𝑑𝑦 𝑑𝑥 − 𝜇𝑥 𝜇𝑦 ∫−∞ ∫−∞ = 𝔼(𝑋𝑌 ) − 𝔼(𝑋)𝔼(𝑌 ). =

Joint Moment-Generating Function: For 𝑡, 𝑠 ∈ ℝ ) ( 𝑀𝑋𝑌 (𝑠, 𝑡) = 𝔼 𝑒𝑠𝑋+𝑡𝑌 =

∞

∞

∫−∞ ∫−∞

𝑒𝑠𝑥+𝑡𝑦 𝑓𝑋𝑌 (𝑥, 𝑦) 𝑑𝑦 𝑑𝑥.

804

Appendix B

Joint Characteristic Function: √ For 𝑖 = −1 and 𝑡, 𝑠 ∈ ℝ ) ( 𝜑𝑋𝑌 (𝑠, 𝑡) = 𝔼 𝑒𝑖𝑠𝑋+𝑖𝑡𝑌 =

∞

∞

∫−∞ ∫−∞

𝑒𝑖𝑠𝑥+𝑖𝑡𝑦 𝑓𝑋𝑌 (𝑥, 𝑦) 𝑑𝑦𝑑𝑥.

Independence: 𝑋 and 𝑌 are independent if and only if s 𝑓 (𝑥, 𝑦) = 𝑓 (𝑥)𝑓 (𝑦). 𝑋𝑌 𝑋 𝑌 s 𝑀 (𝑠, 𝑡) = 𝔼 (𝑒𝑠𝑋+𝑡𝑌 ) = 𝔼 (𝑒𝑠𝑋 ) 𝔼 (𝑒𝑡𝑌 ) = 𝑀 (𝑠)𝑀 (𝑡). 𝑋𝑌 𝑋 𝑌 ( s 𝜑 (𝑠, 𝑡) = 𝔼 𝑒𝑖𝑠𝑋+𝑖𝑡𝑌 ) = 𝔼 (𝑒𝑖𝑠𝑋 ) 𝔼 (𝑒𝑖𝑡𝑌 ) = 𝜑 (𝑠)𝜑 (𝑡). 𝑋𝑌 𝑋 𝑌 Joint Probability Density Function of Dependent Variables: Let the random variables 𝑈 = 𝑔(𝑋, 𝑌 ), 𝑉 = ℎ(𝑋, 𝑌 ). If 𝑢 = 𝑔(𝑥, 𝑦) and 𝑣 = ℎ(𝑥, 𝑦) can be uniquely solved for 𝑥 and 𝑦 in terms of 𝑢 and 𝑣 with solutions given by, say, 𝑥 = 𝑝(𝑢, 𝑣) and 𝑦 = 𝑞(𝑢, 𝑣) and the functions 𝑔 and ℎ have continuous partial derivatives at all points (𝑥, 𝑦) such that the determinant | 𝜕𝑔 | | 𝜕𝑥 | 𝐽 (𝑥, 𝑦) = || | 𝜕ℎ | | 𝜕𝑥 |

𝜕𝑔 || 𝜕𝑦 || 𝜕𝑔 𝜕ℎ 𝜕𝑔 𝜕ℎ |= − ≠0 | 𝜕ℎ || 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑥 𝜕𝑦 ||

then the joint probability density function of 𝑈 and 𝑉 is 𝑓𝑈 𝑉 (𝑢, 𝑣) = 𝑓𝑋𝑌 (𝑥, 𝑦) |𝐽 (𝑥, 𝑦)|−1 where 𝑥 = 𝑝(𝑢, 𝑣) and 𝑦 = 𝑞(𝑢, 𝑣). Properties of Expectation and Variance Let 𝑋 and 𝑌 be two random variables and for constants 𝑎 and 𝑏 𝔼(𝑎𝑋 + 𝑏) = 𝑎𝔼(𝑋) + 𝑏, Var(𝑎𝑋 + 𝑏) = 𝑎2 Var(𝑋) 𝔼(𝑎𝑋 + 𝑏𝑌 ) = 𝑎𝔼(𝑋) + 𝑏𝔼(𝑌 ),

Var(𝑎𝑋 + 𝑏𝑌 ) = 𝑎2 Var(𝑋) + 𝑏2 Var(𝑌 ) + 2𝑎𝑏Cov(𝑋, 𝑌 ).

Properties of Moment-Generating and Characteristic Functions If a random variable 𝑋 has moments up to 𝑘th order where 𝑘 is a non-negative integer, then 𝔼(𝑋 𝑘 ) = where 𝑖 =

√ −1.

| | 𝑑𝑘 𝑑𝑘 𝑀𝑋 (𝑡)|| = 𝑖−𝑘 𝑘 𝜑𝑋 (𝑡)|| 𝑘 𝑑𝑡 𝑑𝑡 |𝑡=0 |𝑡=0

Appendix B

805

If the bivariate random variables 𝑋 and 𝑌 have moments up to 𝑚 + 𝑛 = 𝑘 where 𝑚, 𝑛 and 𝑘 are non-negative integers, then | | 𝑑𝑘 𝑑𝑘 −𝑘 | | 𝑀 (𝑠, 𝑡) = 𝑖 𝜑 (𝑠, 𝑡) 𝑋𝑌 𝑋𝑌 | | 𝑑𝑠𝑚 𝑑𝑡𝑛 𝑑𝑠𝑚 𝑑𝑡𝑛 |𝑠=0,𝑡=0 |𝑠=0,𝑡=0

𝔼(𝑋 𝑚 𝑌 𝑛 ) = where 𝑖 =

√ −1.

Correlation Coefficient Let 𝑋 and 𝑌 be two random variables with means 𝜇𝑥 and 𝜇𝑦 and variances 𝜎𝑥2 and 𝜎𝑦2 . The correlation coefficient 𝜌𝑥𝑦 between 𝑋 and 𝑌 is defined as Cov(𝑋, 𝑌 )

= 𝜌𝑥𝑦 = √ Var(𝑋)Var(𝑌 )

[ ] 𝔼 (𝑋 − 𝜇𝑥 )(𝑌 − 𝜇𝑦 ) 𝜎𝑥 𝜎𝑦

.

Important information: s s s s s

𝜌𝑥𝑦 measures only the linear dependency between 𝑋 and 𝑌 . −1 ≤ 𝜌𝑥𝑦 ≤ 1. If 𝜌𝑥𝑦 = 0 then 𝑋 and 𝑌 are uncorrelated. If 𝑋 and 𝑌 are independent then 𝜌𝑥𝑦 = 0. However the converse is not true. If 𝑋 and 𝑌 are jointly normally distributed then 𝑋 and 𝑌 are independent if and only if 𝜌𝑋𝑌 = 0.

Convolution If 𝑋 and 𝑌 are independent discrete random variables with probability mass functions ℙ(𝑋 = 𝑥) and ℙ(𝑌 = 𝑦), respectively, then the probability mass function for 𝑍 = 𝑋 + 𝑌 is ℙ(𝑍 = 𝑧) =

∑

ℙ(𝑋 = 𝑥)ℙ(𝑌 = 𝑧 − 𝑥) =

𝑥

∑

ℙ(𝑋 = 𝑧 − 𝑦)ℙ(𝑌 = 𝑦).

𝑦

If 𝑋 and 𝑌 are independent continuous random variables with probability density functions 𝑓𝑋 (𝑥) and 𝑓𝑌 (𝑦), respectively, then the probability density function for 𝑍 = 𝑋 + 𝑌 is 𝑓𝑍 (𝑧) =

∞

∫−∞

𝑓𝑋 (𝑥)𝑓𝑌 (𝑧 − 𝑥) 𝑑𝑥 =

∞

∫−∞

𝑓𝑋 (𝑧 − 𝑦)𝑓𝑌 (𝑦) 𝑑𝑦.

Discrete Distributions Bernoulli: A random variable 𝑋 is said to follow a Bernoulli distribution, 𝑋 ∼ Bernoulli(𝑝) where 𝑝 ∈ [0, 1] is the probability of success and the probability mass function is given as 𝑃 (𝑋 = 𝑥) = 𝑝𝑥 (1 − 𝑝)1−𝑥 ,

𝑥 = 0, 1

806

Appendix B

where 𝔼(𝑋) = 𝑝 and Var(𝑋) = 𝑝 − 𝑝2 . The moment-generating function is 𝑀𝑋 (𝑡) = 1 − 𝑝 + 𝑝𝑒𝑡 ,

𝑡∈ℝ

and the corresponding characteristic function is 𝜑𝑋 (𝑡) = 1 − 𝑝 + 𝑝𝑒𝑖𝑡 ,

𝑖=

√

−1 and 𝑡 ∈ ℝ.

Geometric: A random variable 𝑋 is said to follow a geometric distribution, 𝑋 ∼ Geometric(𝑝) where 𝑝 ∈ [0, 1] is the probability of success and the probability mass function is given as ℙ(𝑋 = 𝑥) = 𝑝(1 − 𝑝)𝑥−1 , where 𝔼(𝑋) =

𝑥 = 1, 2, …

1−𝑝 1 and Var(𝑋) = 2 . The moment-generating function is 𝑝 𝑝 𝑀𝑋 (𝑡) =

𝑝 , 1 − (1 − 𝑝)𝑒𝑡

𝑡∈ℝ

and the corresponding characteristic function is 𝜑𝑋 (𝑡) =

𝑝 , 1 − (1 − 𝑝)𝑒𝑖𝑡

𝑖=

√

−1 and 𝑡 ∈ ℝ.

Binomial: A random variable 𝑋 is said to follow a binomial distribution, 𝑋 ∼ Binomial(𝑛, 𝑝), 𝑝 ∈ [0, 1] where 𝑝 ∈ [0, 1] is the probability of success and 𝑛 ∈ ℕ0 is the number of trials and the probability mass function is given as ( ) 𝑛 𝑥 ℙ(𝑋 = 𝑥) = 𝑝 (1 − 𝑝)𝑛−𝑥 , 𝑥

𝑥 = 0, 1, 2, … , 𝑛

where 𝔼(𝑋) = 𝑛𝑝 and Var(𝑋) = 𝑛𝑝(1 − 𝑝). The moment-generating function is 𝑀𝑋 (𝑡) = (1 − 𝑝 + 𝑝𝑒𝑡 )𝑛 ,

𝑡∈ℝ

and the corresponding characteristic function is 𝜑𝑋 (𝑡) = (1 − 𝑝 + 𝑝𝑒𝑖𝑡 )𝑛 ,

𝑖=

√

−1 and 𝑡 ∈ ℝ.

Negative Binomial: A random variable 𝑋 is said to follow a negative binomial distribution, 𝑋 ∼ NB(𝑟, 𝑝) where 𝑝 ∈ [0, 1] is the probability of success and 𝑟 is the number of successes accumulated and the probability mass function is given as ( ℙ(𝑋 = 𝑥) =

) 𝑥−1 𝑟 𝑝 (1 − 𝑝)𝑛−𝑟 , 𝑟−1

𝑥 = 𝑟, 𝑟 + 1, 𝑟 + 2, …

Appendix B

where 𝔼(𝑋) =

807

𝑟(1 − 𝑝) 𝑟 . The moment-generating function is and Var(𝑋) = 𝑝 𝑝2 ( 𝑀𝑋 (𝑡) =

1−𝑝 1 − 𝑝𝑒𝑡

)𝑟

𝑡 < − log 𝑝

,

and the corresponding characteristic function is ( 𝑀𝑋 (𝑡) =

)𝑟

1−𝑝 1 − 𝑝𝑒𝑖𝑡

𝑖=

,

√

−1 and 𝑡 ∈ ℝ.

Poisson: A random variable 𝑋 is said to follow a Poisson distribution, 𝑋 ∼ Poisson(𝜆), 𝜆 > 0 with probability mass function given as ℙ(𝑋 = 𝑥) =

𝑒−𝜆 𝜆𝑥 , 𝑥!

𝑥 = 0, 1, 2, …

where 𝔼(𝑋) = 𝜆 and Var(𝑋) = 𝜆. The moment-generating function is 𝑡

𝑀𝑋 (𝑡) = 𝑒𝜆(𝑒 −1) ,

𝑡∈ℝ

and the corresponding characteristic function is 𝑖𝑡 −1)

𝜑𝑋 (𝑡) = 𝑒𝜆(𝑒

,

𝑖=

√

−1 and 𝑡 ∈ ℝ.

Continuous Distributions Uniform: A random variable 𝑋 is said to follow a uniform distribution, 𝑋 ∼  (𝑎, 𝑏), 𝑎 < 𝑏 with probability density function given as 𝑓𝑋 (𝑥) = where 𝔼(𝑋) =

1 , 𝑏−𝑎

𝑎 0 with probability density function given as 1 √

𝑓𝑋 (𝑥) =

𝑥𝜎 2𝜋

1 2

𝑒

− 12

(

) log(𝑥)−𝜇 2 𝜎

𝑥>0

,

where 𝔼(𝑋) = 𝑒𝜇+ 2 𝜎 and Var(𝑋) = (𝑒𝜎 − 1)𝑒2𝜇+𝜎 . The moment-generating function is 2

𝑀𝑋 (𝑡) =

2

∞ 𝑛 ∑ 𝑡 𝑛𝜇+ 21 𝑛2 𝜎 2 , 𝑒 𝑛! 𝑛=0

𝑡≤0

and the corresponding characteristic function is 𝜑𝑋 (𝑡) =

∞ ∑ (𝑖𝑡)𝑛 𝑛=0

𝑛!

1 2 2 𝜎

𝑒𝑛𝜇+ 2 𝑛

𝑖=

,

√

−1 and 𝑡 ∈ ℝ.

Exponential: A random variable 𝑋 is said to follow an exponential distribution, 𝑋 ∼ Exp(𝜆), 𝜆 > 0 with probability density function 𝑓𝑋 (𝑥) = 𝜆𝑒−𝜆𝑥 , where 𝔼(𝑋) =

𝑥≥0

1 1 and Var(𝑋) = 2 . The moment-generating function is 𝜆 𝜆 𝑀𝑋 (𝑡) =

𝜆 , 𝜆−𝑡

𝑡 0 with probability density function given as 𝑓𝑋 (𝑥) =

𝜆𝑒−𝜆𝑥 (𝜆𝑥)𝛼−1 , Γ(𝛼)

𝑥≥0

such that ∞

Γ(𝛼) = where 𝔼(𝑋) =

∫0

𝑒−𝑥 𝑥𝛼−1 𝑑𝑥

𝛼 𝛼 and Var(𝑋) = 2 . The moment-generating function is 𝜆 𝜆 𝑀𝑋 (𝑡) =

(

𝜆 𝜆−𝑡

)𝛼

𝑡 𝑠 ≥ 1 then { } { } 𝑟 𝑠 𝑋𝑛 ⟶ 𝑋 ⟹ 𝑋𝑛 ⟶ 𝑋 . Dominated Convergence Theorem 𝑎.𝑠

If 𝑋𝑛 ⟶ 𝑋 and for any 𝑛 ∈ ℕ we have |𝑋𝑛 | < 𝑌 for some 𝑌 such that 𝔼(|𝑌 |) < ∞, then 𝔼(|𝑋𝑛 |) < ∞ and lim 𝔼(𝑋𝑛 ) = 𝔼(𝑋).

𝑛→∞

Monotone Convergence Theorem 𝑎.𝑠

If 0 ≤ 𝑋𝑛 ≤ 𝑋𝑛+1 and 𝑋𝑛 ⟶ 𝑋 for any 𝑛 ∈ ℕ then lim 𝔼(𝑋𝑛 ) = 𝔼(𝑋).

𝑛→∞

812

Appendix B

The Weak Law of Large Numbers Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 be a sequence of independent and identically distributed random variables with common mean 𝜇 ∈ ℝ. Then for any 𝜀 > 0, (

)

| 𝑋 + 𝑋2 + … + 𝑋𝑛 | ℙ lim || 1 − 𝜇 || ≥ 𝜀 𝑛→∞ | 𝑛 |

= 0.

The Strong Law of Large Numbers Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 be a sequence of independent and identically distributed random variables with common mean 𝜇 ∈ ℝ. Then (

𝑋1 + 𝑋2 + … + 𝑋𝑛 =𝜇 𝑛→∞ 𝑛

ℙ

lim

) = 1.

The Central Limit Theorem Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 be a sequence of independent and identically distributed random variables with common mean 𝜇 ∈ ℝ and variance 𝜎 2 > 0 and we denote the sample mean as 𝑋= where 𝔼(𝑋) = 𝜇 and Var(𝑋) =

𝑋1 + 𝑋2 + … + 𝑋𝑛 𝑛

𝜎2 . By defining 𝑛 𝑋 − 𝔼(𝑋) 𝑋 − 𝜇 𝑍𝑛 = √ = √ 𝜎∕ 𝑛 Var(𝑋)

then for 𝑛 → ∞, 𝑋−𝜇 𝐷 √ ⟶  (0, 1) 𝑛→∞ 𝜎∕ 𝑛

lim 𝑍 𝑛 = lim

𝑛→∞

that is 𝑍𝑛 follows a standard normal distribution asymptotically.

Appendix C Differential Equations Formulae Separable Equations The form 𝑑𝑦 = 𝑓 (𝑥)𝑔(𝑦) 𝑑𝑥 has a solution 1 𝑑𝑦 = 𝑓 (𝑥) 𝑑𝑥. ∫ ∫ 𝑔(𝑦) If 𝑔(𝑦) is a linear equation and if 𝑦1 and 𝑦2 are two solutions, then 𝑦3 = 𝑎𝑦1 + 𝑏𝑦2 is also a solution for constant 𝑎 and 𝑏. First-Order Ordinary Differential Equations General Linear Equation: The general form of a first-order ordinary differential equation 𝑑𝑦 + 𝑓 (𝑥)𝑦 = 𝑔(𝑥) 𝑑𝑥 has a solution 𝑦 = 𝐼(𝑥)−1 where 𝐼(𝑥) = 𝑒∫

𝑓 (𝑥)𝑑𝑥

∫

𝐼(𝑢)𝑔(𝑢) 𝑑𝑢 + 𝐶

is the integrating factor and 𝐶 is a constant.

Bernoulli Differential Equation: For 𝑛 ≠ 1, the Bernoulli differential equation has the form 𝑑𝑦 + 𝑃 (𝑥)𝑦 = 𝑄(𝑥)𝑦𝑛 𝑑𝑥 which, by setting 𝑤 = tion of the form

1 𝑦𝑛−1

, can be transformed to a general linear ordinary differential equa-

𝑑𝑤 + (1 − 𝑛)𝑃 (𝑥)𝑤 = (1 − 𝑛)𝑄(𝑥) 𝑑𝑥

814

Appendix C

with a particular solution 𝑤 = (1 − 𝑛)𝐼(𝑥)−1 where 𝐼(𝑥) = 𝑒(1−𝑛) ∫ equation becomes

𝑃 (𝑥)𝑑𝑥

∫

𝐼(𝑢)𝑄(𝑢) 𝑑𝑢

is the integrating factor. The solution to the Bernoulli differential }−

{ 𝑦=

(1 − 𝑛)𝐼(𝑥)

−1

∫

𝐼(𝑢)𝑄(𝑢) 𝑑𝑢

1 𝑛−1

+𝐶

where 𝐶 is a constant value. Second-Order Ordinary Differential Equations General Linear Equation: For a homogeneous equation 𝑎

𝑑𝑦 𝑑2𝑦 +𝑏 + 𝑐𝑦 = 0 2 𝑑𝑥 𝑑𝑥

by setting 𝑦 = 𝑒𝑢𝑥 the differential equation has a general solution based on the characteristic equation 𝑎𝑢2 + 𝑏𝑢 + 𝑐 = 0 such that 𝑚1 and 𝑚2 are the roots of the quadratic equation, and if s 𝑚 , 𝑚 ∈ ℝ, 𝑚 ≠ 𝑚 then 𝑦 = 𝐴𝑒𝑚1 𝑥 + 𝐵𝑒𝑚2 𝑥 1 2 1 2 s 𝑚 , 𝑚 ∈ ℝ, 𝑚 = 𝑚 = 𝑚 then 𝑦 = 𝑒𝑚𝑥 (𝐴 + 𝐵𝑥) 1 2 1 2 s 𝑚 , 𝑚 ∈ ℂ, 𝑚 = 𝛼 + 𝑖𝛽, 𝑚 = 𝛼 − 𝑖𝛽 then 𝑦 = 𝑒𝛼𝑥 [𝐴 cos(𝛽𝑥) + 𝐵 sin(𝛽𝑥)] 1 2 1 2 where 𝐴, 𝐵 are constants. Cauchy–Euler Equation: For a homogeneous equation 𝑎𝑥2

𝑑𝑦 𝑑2𝑦 + 𝑏𝑥 + 𝑐𝑦 = 0 2 𝑑𝑥 𝑑𝑥

by setting 𝑦 = 𝑥𝑢 the Cauchy–Euler equation has a general solution based on the characteristic equation 𝑎𝑢2 + (𝑏 − 𝑎)𝑢 + 𝑐 = 0 such that 𝑚1 and 𝑚2 are the roots of the quadratic equation, and if s 𝑚 , 𝑚 ∈ ℝ, 𝑚 ≠ 𝑚 then 𝑦 = 𝐴𝑥𝑚1 + 𝐵𝑥𝑚2 1 2 1 2 s 𝑚 , 𝑚 ∈ ℝ, 𝑚 = 𝑚 = 𝑚 then 𝑦 = 𝑥𝑚 (𝐴 + 𝐵 log 𝑥) 1 2 1 2 s 𝑚 , 𝑚 ∈ ℂ, 𝑚 = 𝛼 + 𝑖𝛽, 𝑚 = 𝛼 − 𝑖𝛽 then 𝑦 = 𝑥𝛼 [𝐴 cos(𝛽 log 𝑥) + 𝐵 sin(𝛽 log 𝑥)] 1 2 1 2 where 𝐴, 𝐵 are constants.

Appendix C

815

Variation of Parameters: For a general non-homogeneous second-order differential equation 𝑎(𝑥)

𝑑𝑦 𝑑2𝑦 + 𝑏(𝑥) + 𝑐(𝑥) = 𝑓 (𝑥) 2 𝑑𝑥 𝑑𝑥

has the solution 𝑦 = 𝑦𝑐 + 𝑦𝑝 where 𝑦𝑐 , the complementary function, satisfies the homogeneous equation 𝑎(𝑥)

𝑑 2 𝑦𝑐 𝑑𝑥2

+ 𝑏(𝑥)

𝑑𝑦𝑐 + 𝑐(𝑥) = 0 𝑑𝑥

and 𝑦𝑝 , the particular integral, satisfies 𝑎(𝑥)

𝑑 2 𝑦𝑝 𝑑𝑥2

+ 𝑏(𝑥)

𝑑𝑦𝑝 𝑑𝑥

+ 𝑐(𝑥) = 𝑓 (𝑥).

(2) Let 𝑦𝑐 = 𝐶1 𝑦(1) 𝑐 (𝑥) + 𝐶2 𝑦𝑐 (𝑥) where 𝐶1 and 𝐶2 are constants, then the particular solution to the non-homogeneous second-order differential equation is

𝑦𝑝 = −𝑦(1) 𝑐 (𝑥)

𝑦(2) 𝑐 (𝑥)𝑓 (𝑥)

∫ 𝑎(𝑥)𝑊 (𝑦(1) (𝑥), 𝑦(2) (𝑥)) 𝑐 𝑐

𝑑𝑥 + 𝑦(2) 𝑐 (𝑥)

𝑦(1) 𝑐 (𝑥) 𝑓 (𝑥)

∫ 𝑎(𝑥)𝑊 (𝑦(1) (𝑥), 𝑦(2) (𝑥)) 𝑐 𝑐

𝑑𝑥

(2) where 𝑊 (𝑦(1) 𝑐 (𝑥), 𝑦𝑐 (𝑥)) is the Wronskian defined as

| 𝑦(1) (𝑥) | 𝑐 | | (2) | 𝑑 𝑊 (𝑦(1) (𝑥), 𝑦 (𝑥)) = | 𝑐 𝑐 𝑦(1) (𝑥) | | 𝑑𝑥 𝑐 | |

| 𝑦(2) 𝑐 (𝑥) | | | 𝑑 (2) 𝑑 (1) (1) (2) 𝑑 (2) || = 𝑦𝑐 (𝑥) 𝑦𝑐 (𝑥) − 𝑦𝑐 (𝑥) 𝑦𝑐 (𝑥) ≠ 0. 𝑦𝑐 (𝑥) | 𝑑𝑥 𝑑𝑥 | 𝑑𝑥 | |

Homogeneous Heat Equations Initial Value Problem on an Infinite Interval: The diffusion equation of the form 𝜕𝑢 𝜕2𝑢 = 𝛼 2, 𝜕𝑡 𝜕𝑥

𝛼 > 0,

−∞ < 𝑥 < ∞,

𝑡>0

with initial condition 𝑢(𝑥, 0) = 𝑓 (𝑥) has a solution ∞

(𝑥−𝑧)2 1 𝑓 (𝑧)𝑒− 4𝛼𝑡 𝑑𝑧. 𝑢(𝑥, 𝑡) = √ 2 𝜋𝛼𝑡 ∫−∞

816

Appendix C

Initial Value Problem on a Semi-Infinite Interval: The diffusion equation of the form 𝜕𝑢 𝜕2𝑢 = 𝛼 2, 𝜕𝑡 𝜕𝑥

𝛼 > 0,

0 ≤ 𝑥 < ∞,

𝑡>0

with s initial condition 𝑢(𝑥, 0) = 𝑓 (𝑥) and boundary condition 𝑢(0, 𝑡) = 0 has a solution 1 𝑢(𝑥, 𝑡) = √ 2 𝜋𝛼𝑡 ∫0

∞

[ ] (𝑥−𝑧)2 (𝑥+𝑧)2 𝑓 (𝑧) 𝑒− 4𝛼𝑡 − 𝑒− 4𝛼𝑡 𝑑𝑧

s initial condition 𝑢(𝑥, 0) = 𝑓 (𝑥) and boundary condition 𝑢 (0, 𝑡) = 0 has a solution 𝑥 1 𝑢(𝑥, 𝑡) = √ 2 𝜋𝛼𝑡 ∫0

∞

[ ] (𝑥−𝑧)2 (𝑥+𝑧)2 𝑓 (𝑧) 𝑒− 4𝛼𝑡 + 𝑒− 4𝛼𝑡 𝑑𝑧

s initial condition 𝑢(𝑥, 0) = 0 and boundary condition 𝑢(0, 𝑡) = 𝑔(𝑡) has a solution 𝑡

2

𝑥 1 − 𝑥 𝑢(𝑥, 𝑡) = √ 𝑔(𝑤)𝑒 4𝛼(𝑡−𝑤) 𝑑𝑤. √ ∫ 2 𝜋𝛼 0 𝑡−𝑤 Stochastic Differential Equations Suppose that 𝑋𝑡 , 𝑌𝑡 and 𝑍𝑡 are It¯o processes satisfying the following stochastic differential equations: 𝑑𝑋𝑡 = 𝜇(𝑋𝑡 , 𝑡)𝑑𝑡 + 𝜎(𝑋𝑡 , 𝑡)𝑑𝑊𝑡𝑥 𝑑𝑌𝑡 = 𝜇(𝑌𝑡 , 𝑡)𝑑𝑡 + 𝜎(𝑌𝑡 , 𝑡))𝑑𝑊𝑡𝑦 𝑑𝑍𝑡 = 𝜇(𝑍𝑡 , 𝑡)𝑑𝑡 + 𝜎(𝑍𝑡 , 𝑡)𝑑𝑊𝑡𝑧

where 𝑊𝑡𝑥 , 𝑊𝑡𝑦 and 𝑊𝑡𝑧 are standard Wiener processes. Reciprocal: (

) 1 ( ) 𝑋𝑡 𝑑𝑋𝑡 𝑑𝑋𝑡 2 + . ( ) =− 𝑋𝑡 𝑋𝑡 1 𝑋𝑡

𝑑

Product: 𝑑(𝑋𝑡 𝑌𝑡 ) 𝑑𝑋𝑡 𝑑𝑌𝑡 𝑑𝑋𝑡 𝑑𝑌𝑡 = + + . 𝑋𝑡 𝑌𝑡 𝑋𝑡 𝑌𝑡 𝑋𝑡 𝑌𝑡

Appendix C

817

Quotient: (

) 𝑋𝑡 𝑑 ( ) 𝑌𝑡 𝑑𝑋𝑡 𝑑𝑌𝑡 𝑑𝑋𝑡 𝑑𝑌𝑡 𝑑𝑌𝑡 2 − − + . ( ) = 𝑋𝑡 𝑌𝑡 𝑋𝑡 𝑌𝑡 𝑌𝑡 𝑋𝑡 𝑌𝑡 Product and Quotient I: ( ) 𝑋𝑡 𝑌𝑡 𝑑 ( ) 𝑍𝑡 𝑑𝑋𝑡 𝑑𝑌𝑡 𝑑𝑍𝑡 𝑑𝑋𝑡 𝑑𝑌𝑡 𝑑𝑋𝑡 𝑑𝑍𝑡 𝑑𝑌𝑡 𝑑𝑍𝑡 𝑑𝑍𝑡 2 + − + − − + . ( ) = 𝑋𝑡 𝑌𝑡 𝑍𝑡 𝑋𝑡 𝑌𝑡 𝑋𝑡 𝑍𝑡 𝑌𝑡 𝑍𝑡 𝑍𝑡 𝑋𝑡 𝑌𝑡 𝑍𝑡 Product and Quotient II: ( ) 𝑋𝑡 𝑑 𝑌𝑡 𝑍𝑡 𝑑𝑋𝑡 𝑑𝑌𝑡 𝑑𝑍𝑡 𝑑𝑋𝑡 𝑑𝑌𝑡 𝑑𝑋𝑡 𝑑𝑍𝑡 𝑑𝑌𝑡 𝑑𝑍𝑡 − − − − + . ( ) = 𝑋𝑡 𝑌𝑡 𝑍𝑡 𝑋𝑡 𝑌𝑡 𝑋𝑡 𝑍𝑡 𝑌𝑡 𝑍𝑡 𝑋𝑡 𝑌𝑡 𝑍𝑡 Black–Scholes Model Black–Scholes Equation (Continuous Dividend Yield): At time 𝑡, let the asset price 𝑆𝑡 follow a geometric Brownian motion 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter and 𝑊𝑡 is a standard Wiener process. For a European-style derivative 𝑉 (𝑆𝑡 , 𝑡) written on the asset 𝑆𝑡 , it satisfies the Black–Scholes equation with continuous dividend yield 𝜕𝑉 𝜕2𝑉 𝜕𝑉 1 − 𝑟𝑉 (𝑆𝑡 , 𝑡) = 0 + 𝜎 2 𝑆𝑡2 2 + (𝑟 − 𝐷)𝑆𝑡 𝜕𝑡 2 𝜕𝑆 𝜕𝑆𝑡 𝑡 where 𝑟 is the risk-free interest rate. The parameters 𝜇, 𝑟, 𝐷 and 𝜎 can be either constants, deterministic functions or stochastic processes. European Options: For a European option having the payoff Ψ(𝑆𝑇 ) = max{𝛿(𝑆𝑇 − 𝐾), 0} where 𝛿 ∈ {−1, 1}, 𝐾 is the strike price and 𝑇 is the option expiry time, and if 𝑟, 𝐷 and 𝜎 are constants the European option price at time 𝑡 < 𝑇 is 𝑉 (𝑆𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝛿𝑆𝑡 𝑒−𝐷(𝑇 −𝑡) Φ(𝛿𝑑+ ) − 𝛿𝐾 𝑒−𝑟(𝑇 −𝑡) Φ(𝛿𝑑− )

818

Appendix C

log(𝑆𝑡 ∕𝐾) + (𝑟 − 𝐷 ± 12 𝜎 2 )(𝑇 − 𝑡) where 𝑑± = and Φ(⋅) is the cumulative distribution func√ 𝜎 𝑇 −𝑡 tion of a standard normal. Reflection Principle: If 𝑉 (𝑆𝑡 , 𝑡) is a solution of the Black–Scholes equation then for a constant 𝐵 > 0, the function ( 𝑈 (𝑆𝑡 , 𝑡) =

𝑆𝑡 𝐵

(

)2𝛼 𝑉

) 𝐵2 ,𝑡 , 𝑆𝑡

( 1 𝛼= 2

1−

𝑟−𝐷

)

1 2 𝜎 2

also satisfies the Black–Scholes equation. Black Model Black Equation: At time 𝑡, let the asset price 𝑆𝑡 follow a geometric Brownian motion 𝑑𝑆𝑡 = (𝜇 − 𝐷)𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑆𝑡 where 𝜇 is the drift parameter, 𝐷 is the continuous dividend yield, 𝜎 is the volatility parameter and 𝑊𝑡 is a standard Wiener process. Consider the price of a futures contract maturing at time 𝑇 > 𝑡 on the asset 𝑆𝑡 as 𝐹 (𝑡, 𝑇 ) = 𝑆𝑡 𝑒(𝑟−𝐷)(𝑇 −𝑡) where 𝑟 is the risk-free interest rate. For a European option on futures 𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡) written on a futures contract 𝐹 (𝑡, 𝑇 ), it satisfies the Black equation 𝜕2𝑉 1 𝜕𝑉 + 𝜎 2 𝐹 (𝑡, 𝑇 )2 2 − 𝑟𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡) = 0. 𝜕𝑡 2 𝜕𝐹 The parameters 𝜇, 𝑟, 𝐷 and 𝜎 can be either constants, deterministic functions or stochastic processes. European Options on Futures: For a European option on futures having the payoff Ψ(𝐹 (𝑇 , 𝑇 )) = max{𝛿(𝐹 (𝑇 , 𝑇 ) − 𝐾), 0} where 𝛿 ∈ {−1, 1}, 𝐾 is the strike price and 𝑇 is the option expiry time, and if 𝑟 and 𝜎 are constants then the price of a European option on futures at time 𝑡 < 𝑇 is [ ] 𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡; 𝐾, 𝑇 ) = 𝛿𝑒−𝑟(𝑇 −𝑡) 𝐹 (𝑡, 𝑇 )Φ(𝛿𝑑+ ) − 𝐾Φ(𝛿𝑑− ) log (𝐹 (𝑡, 𝑇 )∕𝐾) ± 12 𝜎 2 (𝑇 − 𝑡) and Φ(⋅) is the cumulative distribution function of where 𝑑± = √ 𝜎 𝑇 −𝑡 a standard normal.

Appendix C

819

Reflection Principle: If 𝑉 (𝐹 (𝑡, 𝑇 ), 𝑡) is a solution of the Black equation then for a constant 𝐵 > 0 the function ( ) 𝐹 (𝑡, 𝑇 ) 𝐵2 𝑉 ,𝑡 𝑈 (𝐹 (𝑡, 𝑇 ), 𝑡) = 𝐵 𝐹 (𝑡, 𝑇 ) also satisfies the Black equation. Garman–Kohlhagen Model Garman–Kohlhagen Equation: At time 𝑡, let the foreign-to-domestic exchange rate 𝑋𝑡 follow a geometric Brownian motion 𝑑𝑋𝑡 = 𝜇𝑑𝑡 + 𝜎𝑑𝑊𝑡 𝑋𝑡 where 𝜇 is the drift parameter, 𝜎 is the volatility parameter and 𝑊𝑡 is a standard Wiener process. For a European-style derivative 𝑉 (𝑋𝑡 , 𝑡) which depends on 𝑋𝑡 , it satisfies the Garman– Kohlhagen equation 𝜕2𝑉 𝜕𝑉 1 𝜕𝑉 + (𝑟𝑑 − 𝑟𝑓 )𝑋𝑡 − 𝑟𝑑 𝑉 (𝑋𝑡 , 𝑡) = 0 + 𝜎 2 𝑋𝑡2 2 𝜕𝑡 2 𝜕𝑋 𝜕𝑋𝑡 𝑡 where 𝑟𝑑 and 𝑟𝑓 are the domestic and foreign currencies’ risk-free interest rates. The parameters 𝜇, 𝑟𝑑 , 𝑟𝑓 and 𝜎 can be either constants, deterministic functions or stochastic processes. European Options: For a European option having the payoff Ψ(𝑋𝑇 ) = max{𝛿(𝑋𝑇 − 𝐾), 0} where 𝛿 ∈ {−1, 1}, 𝐾 is the strike price and 𝑇 is the option expiry time, and if 𝑟𝑑 , 𝑟𝑓 and 𝜎 are constants then the European option price (domestic currency in one unit of foreign currency) at time 𝑡 < 𝑇 is 𝑉 (𝑋𝑡 , 𝑡; 𝐾, 𝑇 ) = 𝛿𝑋𝑡 𝑒−𝑟𝑓 (𝑇 −𝑡) Φ(𝛿𝑑+ ) − 𝛿𝐾 𝑒−𝑟𝑑 (𝑇 −𝑡) Φ(𝛿𝑑− ) log(𝑋𝑡 ∕𝐾) + (𝑟𝑑 − 𝑟𝑓 ± 12 𝜎 2 )(𝑇 − 𝑡) and Φ(⋅) is the cumulative distribution where 𝑑± = √ 𝜎 𝑇 −𝑡 function of a standard normal. Reflection Principle: If 𝑉 (𝑋𝑡 , 𝑡) is a solution of the Garman–Kohlhagen equation then for a constant 𝐵 > 0, the function ) ( ( )2𝛼 ( 2 ) 𝑟𝑑 − 𝑟𝑓 𝑋𝑡 𝐵 1 𝑉 ,𝑡 , 𝛼 = 1− 1 𝑈 (𝑋𝑡 , 𝑡) = 𝐵 𝑋𝑡 2 𝜎2 2

also satisfies the Garman–Kohlhagen equation.

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Notation SET NOTATION ∈ ∉ Ω ℰ ∅ 𝐴 𝐴𝑐 |𝐴| ℕ ℕ0 ℤ ℤ+ ℝ ℝ+ ℂ 𝐴×𝐵 𝑎∼𝑏 ⊆ ⊂ ∩ ∪ ⧵ △ sup inf [𝑎, 𝑏] [𝑎, 𝑏) (𝑎, 𝑏] (𝑎, 𝑏) ℱ , 𝒢, ℋ

is an element of is not an element of sample space universal set empty set subset of Ω complement of set 𝐴 cardinality of 𝐴 set of natural numbers, {1, 2, 3, …} set of natural numbers including zero, {0, 1, 2, …} set of integers, {0, ±1, ±2, ±3, …} set of positive integers, {1, 2, 3, …} set of real numbers set of positive real numbers, {𝑥 ∈ ℝ : 𝑥 > 0} set of complex numbers cartesian product of sets 𝐴 and 𝐵, 𝐴 × 𝐵 = {(𝑎, 𝑏) : 𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵} 𝑎 is equivalent to 𝑏 subset proper subset intersection union difference symmetric difference supremum or least upper bound infimum or greatest lower bound the closed interval {𝑥 ∈ ℝ : 𝑎 ≤ 𝑥 ≤ 𝑏} the interval {𝑥 ∈ ℝ : 𝑎 ≤ 𝑥 < 𝑏} the interval {𝑥 ∈ ℝ : 𝑎 < 𝑥 ≤ 𝑏} the open interval {𝑥 ∈ ℝ : 𝑎 < 𝑥 < 𝑏} 𝜎-algebra (or 𝜎-fields)

826

Notation

MATHEMATICAL NOTATION 𝑥+ 𝑥− ⌊𝑥⌋ ⌈𝑥⌉ 𝑥∨𝑦 𝑥∧𝑦 𝑖 ∞ ∃ ∃! ∀ ≈ 𝑝⟹𝑞 𝑝⟸𝑞 𝑝⟺𝑞 𝑓 :𝑋↦𝑌 𝑓 (𝑥) lim 𝑓 (𝑥) 𝑥→𝑎 𝛿𝑥, Δ𝑥 𝑓 −1 (𝑥) 𝑓 ′ (𝑥), 𝑓 ′′ (𝑥) 𝑑𝑦 𝑑 2 𝑦 , 𝑑𝑥 𝑑𝑥2 𝑦 𝑑𝑥,

∫ ∫𝑎 𝜕𝑓 𝜕 2 𝑓 , 𝜕𝑥𝑖 𝜕𝑥2 𝑖 𝜕2𝑓 𝜕𝑥𝑖 𝜕𝑥𝑗 log𝑎 𝑥 log 𝑥 𝑛 ∑ 𝑎𝑖 𝑖=1 𝑛 ∏ 𝑖=1

𝑎𝑖

|𝑎| ( √ )𝑚 𝑛 𝑎 𝑛! ( ) 𝑛 𝑘

𝑏

max{𝑥, 0} min{𝑥, 0} largest integer not greater than or equal to 𝑥, max{𝑚 ∈ ℤ|𝑚 ≤ 𝑥} smallest integer greater than or equal to 𝑥, min{𝑛 ∈ ℤ|𝑛 ≥ 𝑥} max{𝑥, 𝑦} min{𝑥, 𝑦} √ −1 infinity there exists there exists a unique for all approximately equal to 𝑝 implies 𝑞 𝑝 is implied by 𝑞 𝑝 implies and is implied by 𝑞 𝑓 is a function where every element of 𝑋 has an image in 𝑌 the value of the function 𝑓 at 𝑥 limit of 𝑓 (𝑥) as 𝑥 tends to 𝑎 increment of 𝑥 the inverse function of the function 𝑓 (𝑥) first and second-order derivative of the function 𝑓 (𝑥) first and second-order derivative of 𝑦 with respect to 𝑥 𝑦 𝑑𝑥

the indefinite and definite integral of 𝑦 with respect to 𝑥 first and second-order partial derivative of 𝑓 with respect to 𝑥𝑖 where 𝑓 is a function on (𝑥1 , 𝑥2 , … , 𝑥𝑛 ) second-order partial derivative of 𝑓 with respect to 𝑥𝑖 and 𝑥𝑗 where 𝑓 is a function on (𝑥1 , 𝑥2 , … , 𝑥𝑛 ) logarithm of 𝑥 to the base of 𝑎 natural logarithm of 𝑥 𝑎1 + 𝑎2 + … + 𝑎𝑛 𝑎 1 × 𝑎2 × … × 𝑎 𝑛 modulus of 𝑎 𝑚

𝑎𝑛 𝑛 factorial 𝑛! for 𝑛, 𝑘 ∈ ℤ+ 𝑘!(𝑛 − 𝑘)!

Notation

𝛿(𝑥) 𝐻(𝑥) Γ(𝑡) 𝐵(𝑥, 𝑦) a |a| a⋅b a×b M M𝑇 M−1 |M|

827

Dirac delta function Heaviside step function gamma function beta function a vector a magnitude of a vector a scalar or dot product of vectors a and b vector or cross product of vectors a and b a matrix M transpose of a matrix M inverse of a square matrix M determinant of a square matrix M

PROBABILITY NOTATION 𝐴, 𝐵, 𝐶 1I𝐴 ℙ, ℚ ℙ(𝐴) ℙ(𝐴|𝐵) 𝑋, 𝑌 , 𝑍 X, Y, Z 𝑃 (𝑋 = 𝑥) 𝑓𝑋 (𝑥)

∼ ≁ ∻

events indicator of the event 𝐴 probability measures probability of event 𝐴 probability of event A conditional on event 𝐵 random variables random vectors probability mass function of a discrete random variable 𝑋 probability density function of a continuous random variable 𝑋 cumulative distribution function of random variable 𝑋 moment-generating function of a random variable 𝑋 characteristic function of a random variable 𝑋 joint probability mass function of discrete variables 𝑋 and 𝑌 joint probability density function of continuous random variables 𝑋 and 𝑌 joint cumulative distribution function of random variables 𝑋 and 𝑌 joint moment generating function of random variables 𝑋 and 𝑌 joint characteristic function of random variables 𝑋 and 𝑌 transition probability density of 𝑦 at time 𝑇 starting at time 𝑡 at point 𝑥 is distributed as is not distributed as is approximately distributed as

⟶

converges almost surely

⟶

converges in the 𝑟-th mean

𝐹𝑋 (𝑥), ℙ(𝑋 ≤ 𝑥) 𝑀𝑋 (𝑡) 𝜑𝑋 (𝑡) 𝑃 (𝑋 = 𝑥, 𝑌 = 𝑦) 𝑓𝑋𝑌 (𝑥, 𝑦) 𝐹𝑋𝑌 (𝑥, 𝑦), ℙ(𝑋 ≤ 𝑥, 𝑌 ≤ 𝑦) 𝑀𝑋𝑌 (𝑠, 𝑡) 𝜑𝑋𝑌 (𝑠, 𝑡) 𝑝(𝑥, 𝑡; 𝑦, 𝑇 )

𝑎.𝑠 𝑟

828 𝑃

Notation

⟶

converges in probability

⟶

converges in distribution

𝑋=𝑌 𝑋⟂ ⟂𝑌 ∕ 𝑌 ⟂ 𝑋⟂ 𝔼(𝑋) 𝔼ℚ (𝑋)

𝑋 and 𝑌 are identically distributed random variables 𝑋 and 𝑌 are independent random variables 𝑋 and 𝑌 are not independent random variables expectation of random variable 𝑋 expectation of random variable 𝑋 under the probability measure ℚ expectation of 𝑔(𝑋) conditional expectation of 𝑋 variance of random variable 𝑋 conditional variance of 𝑋 covariance of random variables 𝑋 and 𝑌 correlation between random variables 𝑋 and 𝑌 Bernoulli distribution with mean 𝑝 and variance 𝑝(1 − 𝑝) geometric distribution with mean 𝑝−1 and variance (1 − 𝑝)𝑝−2 binomial distribution with mean 𝑛𝑝 and variance 𝑛𝑝(1 − 𝑝) negative binomial distribution with mean 𝑟𝑝−1 and variance 𝑟(1 − 𝑝)𝑝−2 Poisson distribution with mean 𝜆 and variance 𝜆 exponential distribution with mean 𝜆−1 and variance 𝜆−2 gamma distribution with mean 𝛼𝜆−1 and variance 𝛼𝜆−2 1 uniform distribution with mean 12 (𝑎 + 𝑏) and variance 12 (𝑏 − 𝑎)2 normal distribution with mean 𝜇 and variance 𝜎 2

𝐷

𝑑

𝔼[𝑔(𝑋)] 𝔼(𝑋| ℱ) Var(𝑋) Var(𝑋| ℱ) Cov(𝑋, 𝑌 ) 𝜌𝑥𝑦 Bernoulli(𝑝) Geometric(𝑝) Binomial(𝑛, 𝑝) BN(𝑛, 𝑟) Poisson(𝜆) Exp(𝜆) Gamma(𝛼, 𝜆)  (𝑎, 𝑏)  (𝜇, 𝜎 2 ) log- (𝜇, 𝜎 2 ) 𝜒 2 (𝑘) 𝑛 (𝜇, Σ) Φ(⋅), Φ(𝑥) 𝚽(𝑥, 𝑦, 𝜌𝑥𝑦 ) 𝑊𝑡 𝑁𝑡

1 2

lognormal distribution with mean 𝑒𝜇+ 2 𝜎 and variance (𝑒𝜎 − 2 1)𝑒2𝜇+𝜎 chi-square distribution with mean 𝑘 and variance 2𝑘 multivariate normal distribution with 𝑛-dimensional mean vector 𝜇 and 𝑛 × 𝑛 covariance matrix Σ cumulative distribution function of a standard normal cumulative distribution function of a standard bivariate normal with correlation coefficient 𝜌𝑥𝑦 standard Wiener process, 𝑊𝑡 ∼  (0, 𝑡) Poisson process, 𝑁𝑡 ∼ Poisson(𝜆𝑡) 2

Index 3-period binomial tree models barrier-style options 432–5 time-dependent options 310–14, 328–31 tree-based methods 211–17 ABM see arithmetic Brownian motion accumulators 428–9 American options 267–349 asymptotic optimal exercise boundary 316–21 Barone-Adesi and Whaley formula 324–7 basic properties 267, 271–92, 274–6, 276–9, 279–88, 289–91, 290–2 binomial option pricing 270–1 Black approximation 322–4 definition 2 digital options 331–49 early exercise 290–1, 321 free boundary formulation 268–9 linear complementarity 269–70 lookback options 613–17, 623–7 optimal stopping time 268 perpetual options 294–305 pricing formulations 268–70 put-call parity 267, 282–4 smooth pasting condition 314–16 time-dependent options 305–49 time-independent options 292–305 upper bound of price 321–2 approximation integration formula 689, 717 arbitrage 306–7 arbitrage opportunities American options 272–3, 275–7, 280, 290, 292 discrete dividends 116, 122 European options 65, 74–6, 80, 85, 116, 122 self-financing trading strategy 65 tree-based methods 192–3 arithmetic average options average rate options 440, 442, 445–7, 457–79, 487–91, 497, 525–9

average strike options 441, 442, 448–9, 491–6, 497, 525–9 continuous sampling 440–1, 487–500, 525–9 discrete sampling 440–1, 457–79 probabilistic approach 496–500 put-call parity 496–500 symmetry of arithmetic average rate/ strike 525–9 arithmetic average rate options 440, 457–79 arithmetic-geometric average rate identity 445–7 PDE approach 487–91 put-call relation 442, 497 symmetry of arithmetic average rate/strike 525–9 arithmetic average strike options 441 arithmetic-geometric average strike identity 448–9 PDE approach 491–6 put-call relation 442, 497 symmetry of arithmetic average rate/strike 525–9 arithmetic averaging 439–41, 443 arithmetic Brownian motion (ABM) 151–5, 554 arithmetic-geometric average rate identity 445–7 arithmetic-geometric average strike identity 448–9 Asian options 439–529 arithmetic averaging 439–41, 443 average rate option payoff 440–1 average strike option payoff 441 continuous sampling 440–1, 480–529 discrete sampling 440–1, 443–79 geometric averaging 439–41, 443–4 put-call parity 442–3, 496–500 volatility derivatives 780, 782 asset risk 573 asset-or-nothing options 139–44 Black–Scholes model 139–44 Greeks 258, 260 PDE approach 139–41 probabilistic approach 141–4 asset-paying dividends 72 assets, non-dividend paying 163–7

830 asymptotic optimal exercise boundaries 316–21 asymptotic property of binomial distribution 207 at-the-money (ATM) Brenner–Subrahmanyam approximation 662 Chambers–Nawalkha approximation 683 equity derivatives 3, 24 Li ATM volatility approximation 663–8 Li non-ATM volatility approximation 668–73 average options see Asian options average rate option payoff 440–1 see also arithmetic average rate options; geometric average rate options average strike option payoff 441 see also arithmetic average strike options; geometric average strike options backward Kolmogorov equation Black–Scholes equation 724, 725 forward Kolmogorov equation 727, 729 local volatility 687–92, 692–3, 695, 697 stochastic volatility 713–25, 727, 729 Barone-Adesi and Whaley formula 324–7 barrier options 351–437 barrier-style options 408–37 in-out parity 352 lookback options 609 pricing 353–407 probabilistic approach 353–4, 357–86 reflection principle 386–407 bear spreads 8, 37–9 Bermudan options 2, 270–1, 327–31 binomial distribution asymptotic property 207 binomial models continuous limit of 308–9 continuous-time limit 69–71, 202–4, 308–9 discrete 194 tree-based methods 194, 202–4 see also binomial tree models binomial option pricing 270–1 binomial tree models 204–17 3-period 211–17, 310–14, 328–31, 432–5 barrier-style options 432–5 continuous limit of 308–9 continuous-time limit of 202–4, 308–9 delta hedging 242–4 time-dependent options 308–14, 327–31 tree-based methods 204–17 bivariate normal distribution Curran approximation 469–70, 474–5 geometric average strike options 515, 518, 520 rainbow options 559, 565, 566, 568–70 Black approximation 322–4 Black equation 422–4

Index Black, Fischer 63 see also Black…; Black–Scholes… Black formula 422–3 Black model 144–6 Black–Scholes differential operator 247, 249, 256 Black–Scholes equation asset-or-nothing options 140 asymptotic optimal exercise boundaries 317, 320 Barone-Adesi and Whaley formula 324–6 barrier options pricing 354–6 continuous sampling 480–1 continuous-time limit of binomial model 70, 202 cross-currency options 572–9 delta hedging strategy 64 digital options 134–5 discrete dividends 116, 118, 122 down-and-out/in options 394–6, 413, 418 European option pricing 184, 189 European option valuation 95–7 exchange options 542 futures contracts 146–9 geometric average options 502, 506 Greeks 245, 247–9, 252, 254, 256, 260 Heston model 756–60, 763 higher derivatives property 110–12 immediate-touch options 339–42 in-out parity 389, 408 invariance properties 108–9 linear complementarity 307 local volatility 685, 687, 692–4 lookback options 604–9, 618–22 market price of risk 124–5 one-touch options 331–3, 334 reflection principle 386–8, 421–2 self-financing trading strategy 66 smooth pasting condition 314–15 solutions of 111–15 stochastic volatility 723–6, 756–60, 763 stock paying continuous dividend yield 89–92 stock paying discrete dividends 93–5 transaction costs 128 two-dimensional 537–9, 540, 724 up-and-out/in options 390–2, 410, 415 Black–Scholes formula 63 Brenner–Subrahmanyam approximation 661 Chambers–Nawalkha approximation 683–4 down-and-out/in options 415 European options 85, 109–10, 648–9, 676–7 generalised formula 161–3 Greeks 71, 220–4, 226, 228, 230–1, 237 Heston model 755, 756, 761 Hull–White model 745, 751 implied volatility 648–9, 706

Index invariance properties 109–10 Li non-ATM volatility approximation 668 local volatility 685, 706 Manaster–Koehler method 677 reflection principle 393, 397 theoretical 647, 649 timer options 785, 786 up-and-out/in options 412 Black–Scholes framework 430, 442, 647 Black–Scholes inequality 268 Black–Scholes methodology 353 Black–Scholes model 89–190 asset-or-nothing options 139–44 criticisms 73 digital options 134–9 European options 72–3, 89–190 extension of 72–3 local volatility 649–50 stochastic volatility 651 with transaction costs 73, 125–8 Black–Scholes operator 269 Black–Scholes options see European options Black–Scholes price 668, 673 Black–Scholes theoretical price 663–4, 666–7, 669 Black–Scholes world delta hedging 237 down-and-out/in options 412, 418 in-out parity with rebate at expiry 408 knock-out equity accumulator 428 knock-out/knock-in options 424 reflection principle 386, 389, 390, 421–2 time-dependent options 308 tree-based methods 202–11 up-and-out/in options 410, 415 bonds see zero-coupon bonds Bos–Vandermark model 120–1, 233–7 box spreads 39–40 Boyle method 197–9 Boyle–Emmanuel method 260–2 Brenner–Subrahmanyam approximation 661–3, 673 bull spreads 7, 33–6, 39 butterfly spreads 8, 54–8 call-on-a-put options 596, 599 call/put-on-a-call options 591–2, 595 capped options 532–3 Cauchy–Euler equation 616 Chambers–Nawalkha approximation 683–4 Chapman–Kolmogorov equation backward Kolmogorov equation 714, 720, 721 forward Kolmogorov equation 727 local volatility 688, 691, 694, 696

831 chooser options 531, 600–4 complex 601–4 simple 600–1 cliquet options 588–91 closed-form solutions Asian options 439 barrier options pricing 354 one-touch options 337 rainbow options 560 timer options 784, 785 collars 7, 41–4 company defaults 158–61, 429–31 complex chooser options 601–4 compo options 575, 580 compound options 591–9 conditional Jensen’s inequality 475 condor spreads 58–61 contingent claims 2 continuous arithmetic average 440 continuous dividend yield 75–8, 305–6 American options 271, 274, 277, 281, 292, 298, 301, 316, 319 arithmetic average rate options 446, 487, 525 arithmetic average strike options 448, 492, 525 backward Kolmogorov equation 687, 713 Barone-Adesi and Whaley formula 324 binomial tree models 308, 310, 327, 432 Black–Scholes equation 480, 537, 723 Black–Scholes formula 676 Black–Scholes model 72, 89–95 Boyle method 197 Boyle–Emmanuel method 260 Brenner–Subrahmanyam approximation 661 capped options 532 Chambers–Nawalkha approximation 683 chooser options 600, 601 cliquet options 588 compound options 591, 596 Corrado–Miller–Hallerbach approximation 673 corridor options 533 Cox–Ross–Rubinstein method 193 cross-currency options 572, 575–80 Curran approximation 468 delta hedging 237, 245, 248, 250, 253, 255, 258 down-and-out/in options 364, 380, 394, 400, 405, 412 Dupire equation 699 exchange options 540, 543 forward Kolmogorov equation 694, 726 forward start options 586 futures 13–14 geometric average options 446, 448, 450, 500, 504, 508, 514

832 continuous dividend yield (Continued) Greeks 218, 220–32, 237, 245, 248, 250, 253, 255, 258, 260 Heston model 754, 757, 760 Hull–White model 745 immediate-touch options 339, 345 in-out parity 389, 408 Jarrow–Rudd method 196 Kamrad–Ritchken method 199 Levy approximation 457 local volatility 650–1 lookback options 604, 613, 617–18, 623, 627, 634, 640, 643 one-touch options 331, 336 PDE approach 354 power options 534 put-call parity 442, 496, 522 rachet options 588 rainbow options 559, 569 reflection principle 386 risk-neutral approach 190 self-financing trading strategy 191, 192 similarity reduction 482, 483, 485 spread options 547 stop-loss options 609 time-dependent 72 time-independent options 292, 298, 301 timer options 783 tree-based methods 190–3, 196–7, 199 two-dimensional Black–Scholes equation 537 up-and-out/in options 357, 372, 390, 397, 403, 410 continuous geometric average 440 continuous limit of binomial model 308–9 continuous risk-neutral random walk 194 continuous sampling 440–1, 480–529 arithmetic average options 440–1, 487–96, 525–9 Black–Scholes equation 480–1 geometric average options 441, 500–25 PDE approach 487–96, 500–8 probabilistic approach 496–500, 508–22 put-call parity 496–500, 522–5 similarity reduction 482–7 continuous-time limit of binomial model 69–71, 202–4, 308–9 contradiction 314, 315, 318, 320 Corrado–Miller–Hallerbach approximation 673–6 corridor options 533–4 covered calls 7, 27, 32–3 covered puts 7, 28–30 Cox–Ross–Rubinstein model barrier-style options 432

Index time-dependent options 327–31 tree-based methods 193–6 cross-currency options 572–86 Black–Scholes equation 572–9 PDE approach 575–9 probabilistic approach 579–86 cubic equations, depressed 663–7 cumulative standard normal distribution function 237 Curran approximation 467–79 currency see cross-currency options De Moivre’s formula 665 defaulting companies 158–61, 429–31 delivery date, definition 2 delta 71, 220–3, 260, 264–5 see also delta hedging; Dirac delta function delta and gamma-neutral portfolios 264–5 delta hedging Black and Scholes assumptions 63 generalised perpetual American options 292 Greeks 237–60, 262–4 lookback options 607, 610, 620–1 strategy 63–4 time-dependent options 306 depressed cubic equations 663–7 derivatives, volatility 769–86 differential equations see ordinary differential equations; partial differential equations; stochastic differential equations diffusion processes cross-currency options 580, 581, 583 European option prices 167, 180 exchange options 540, 543–4 generalised Black–Scholes formula 161 generalised perpetual American options 292 generalised stochastic volatility model 710 Greeks 255 Heston model 757 Hull–White model 744 Merton model 132 non-dividend-paying asset price as num´eraire 163–4, 166 rainbow options 559, 569 spread options 547–50 time-dependent options 305 two-dimensional Black–Scholes equation 537 volatility derivatives 780 digital options 134–9 American options 331–49 Black–Scholes model 134–9 corridor options 533 down-and-out/in options 365, 381, 413–14

Index European options 134–9, 331, 337, 410–14 Greeks 253, 258, 259–60 knock-out/knock-in options 427 PDE approach 134–7 probabilistic approach 137–9 up-and-out/in options 357–8, 373, 410–11 Dirac delta function 235, 691, 726 discontinuous jumps 73 discounting, definition 2 discrete arithmetic average rate options 440, 457–79 Curran approximation 467–79 Levy approximation 457–67 discrete arithmetic average strike options 441 discrete arithmetic averaging 439–41, 443 discrete binomial model 194 discrete dividends American options 273, 274, 278, 281 Black–Scholes model 115–24 Bos–Vandermark model 120–1 escrowed model 115–18 European options 77–8 forward model 118–19 futures 12 yields 121–4 discrete geometric average rate options 440, 450–7 discrete geometric average strike options 441 discrete geometric averaging 439–41, 443–4 discrete sampling 440–1, 443–79 arithmetic-geometric average rate identity 445–7 arithmetic-geometric average strike identity 448–9 Asian options 780, 782 Curran approximation 467–79 discrete arithmetic average rate options 440, 457–79 discrete geometric average rate options 440–1, 450–7 Levy approximation 457–67 limit of discrete arithmetic average 443 limit of discrete geometric average 443–4 dividend yield discrete 121–4 down-and-out/in options 418 knock-out equity accumulator 428 knock-out/knock-in options 424 reflection principle 421, 422 up-and-out/in options 415 see also continuous dividend yield dividends forward model 118–19 futures 11–12, 14–15 see also discrete dividends domestic risk-neutral measure 580, 581, 583, 584

833 down-and-out/in barrier options 352 with immediate rebates 418–20 PDE pricing approach 356 probabilistic approach 364–72, 380–6 with rebates at expiry 412–15 reflection principle 394–7, 400–2, 405–7, 413–14 dual delta 222–3 dual gamma 223–4 dual problem 322 Dupire equation 699–705, 706 local volatility 685, 699–705, 706 stochastic volatility 743 variance swaps 769, 770 Dupire formula 650–1 Emmanuel see Boyle–Emmanuel method equity accumulator, knock-out 428–9 equity derivative theory 1–61 definitions 1–2 forwards 1, 4–5, 8–15, 17–18, 20–1 futures 1, 2, 5–6, 8–15, 10–11 hedging strategies 6–8, 27–61 options 1–4, 15–27 equity options, foreign 575–86 equivalent martingale measure 67–8, 132 escrowed dividend model 115–18 estimation GBM parameters 652–4 geometric mean-reversion process 658–60 maximum-likelihood 647–8, 652–61 moment-matching 653–4 Ornstein–Uhlenbeck process 654–8 Euler see Cauchy–Euler equation European options 63–265 American options 271–6, 277–8, 281–2, 284, 291 Barone-Adesi and Whaley formula 324 basic properties 74–89 binomial tree models 432 Black approximation 323 Black–Scholes equation 95–7, 184, 189, 692, 694, 723–4 Black–Scholes formula 648–9, 676, 677 Black–Scholes model 72–3, 89–190 Brenner–Subrahmanyam approximation 661–3 capped options 532 Chambers–Nawalkha approximation 683–4 chooser options 600, 602 cliquet options 589–91 compound options 591, 596 continuous-time limit of binomial model 69–71 Corrado–Miller–Hallerbach approximation 673 corridor options 533

834 European options (Continued) cross-currency options 576, 578, 579 definition 2 delta hedging strategy 63–4 digital options 331, 337 down-and-out/in options 364–5, 380–1, 394–5, 397, 400–1, 413–15, 418, 420 Dupire equation 699 exchange options 542 forward start options 587 Greeks 71–2, 218–65 Greek values 245 hedging strategies 40–1 Heston model 754–5, 760–9 Hull–White model 745 implied volatility 648–9 in-out parity 352, 389, 408–9 knock-out/knock-in options 425 Li ATM volatility approximation 664, 666–7 Li non-ATM volatility approximation 668–9 local volatility 651, 685 lookback options 627–45 Manaster–Koehler method 677 martingale pricing theory 67–9 Merton model 128–34, 158–61, 430 options theory 21–7 price formula 501, 505, 535–6, 547, 554 prices under stochastic interest rate 167–90 probabilistic approach 101–5, 137–9, 141–4, 353 put-call parity 267, 277–8, 280–1, 283, 284 rachet options 589–91 reflection principle 386, 421, 422 self-financing trading strategy 64–7 stochastic volatility 723–4, 736–7, 743, 745, 754–5, 760–9 time-dependent options 307, 323, 324, 331, 337 timer options 785 tree-based methods 190–217 up-and-in call options 435–6 up-and-out/in options 357–8, 372–3, 390–3, 403, 410–12, 415, 417 valuation 95–105, 131–4 variance swaps 769, 770, 772 European-style options Asian options 442 cross-currency options 575 Heston model 757 local volatility 650 lookback options 604, 618 exchange options 531, 540–7 formula 522 payoffs 586

Index PDE approach 540–2 probabilistic approach 543–7 exchange rate risk 573 exercise date, definition 2 exercise price, definition 2 exotic options 531–645 definition 351 path-dependent options 531, 586–645 path-independent options 531–86 see also Asian options; barrier options expiry date, definition 2 Feynman–Kac formula 67, 101, 169, 174 fixed strike lookback options 613–17, 640–5 see also average rate… floating strike lookback options 623–7, 640–5 see also average strike… folded normal distribution 781 foreign equity options 575–86 foreign exchange (FX) options 576, 580 foreign-to-domestic exchange rate 572, 575, 580 forward dividend model 118–19 forward Kolmogorov equation 694–9, 700–1, 726–38 forward start options 531, 586–8, 588 forwards 4–5 definition 1 options theory 17–18, 20–1 synthetic 7 Fourier inversion 762, 769 Fourier transforms 761, 765–7 free boundary formulation 268–9 futures 5–6 American options 267 Black model 144–6 Black–Scholes model 144–51 definition 1 initial margin 6 knock out/in options 424–8 knock-out equity accumulator 428 settlement price 6 stock index 2, 10–11 FX see foreign exchange options gamma 71, 221–4, 244, 248, 252, 264–5 GBM see geometric Brownian motion generalised Black–Scholes formula 161–3 generalised historical volatility 660–1 generalised perpetual American options 292–4

Index generalised stochastic volatility model 710–13 geometric average options average rate options 440–3, 450–7, 500–4, 508–14, 522–5 average strike options 441, 443, 504–8, 514–22 continuous sampling 441, 500–25 discrete sampling 440–1, 450–7 probabilistic approach 508–25 put-call parity 522–5 geometric average rate options 440–1, 450–7 PDE approach 500–4 probabilistic approach 508–14, 522–5 put-call relation 442–3 geometric average strike options 441 PDE approach 504–8 probabilistic approach 514–25 put-call relation 443 geometric averaging 439–41, 443–4 geometric Brownian motion (GBM) 155 arithmetic-geometric average rate identity 445 arithmetic-geometric average strike identity 448 asset-or-nothing options 139, 141 Black model 144 Black–Scholes equation 89, 91, 93, 480 Black–Scholes model 125 Boyle method 197 capped options 532 chooser options 600, 601 continuous-time limit of binomial model 69 corridor options 533 Cox–Ross–Rubinstein method 193 Curran approximation 467 delta hedging strategy 64 digital options 134, 137 down-and-out/in options 364, 380, 400, 405 European option price under stochastic interest rate 179 European option valuation 95, 101 forward start options 586 Greeks 233, 248, 253, 258, 260 immediate-touch options 339, 345 Jarrow–Rudd method 196 Kamrad–Ritchken method 199 Levy approximation 457 lookback options 604, 613, 618, 623, 627, 634, 640, 643 martingale pricing theory 67 Merton model 133, 160 one-touch options 331, 336 parameter estimation 652–4 perpetual American options 301 power options 534 risk-neutral approach 190

835 stop-loss options 609 tree-based methods 190, 193, 196–7, 199 up-and-out/in options 357, 372, 397, 402 volatility models 647 geometric mean-reversion process 658–60 Girsanov’s theorem 67–8, 156 arithmetic Brownian motion 152 asset-or-nothing options 142 backward Kolmogorov equation 689 compound options 596 cross-currency options 582, 585 digital options 138 discrete geometric average rate Asian option 452 European option valuation 102 European options under stochastic interest rate 168, 171, 177 exchange options 544 forward Kolmogorov equation 694, 695 forward start options 587 geometric average options 452, 518, 523 knock-out/knock-in options 425 lookback options 635 non-dividend-paying asset price as num´eraire 164 perpetual American options 302 power options 535 put-call parity 497, 523 spread options 549 symmetry of arithmetic average rate/strike 526 time-dependent options 337, 346–7 see also two-dimensional Girsanov’s theorem Greeks 218–65 Bos–Vandermark model 233–7 Boyle–Emmanuel method 260–2 delta 220–3, 260, 264–5 delta hedging 237–60, 262–4 dual delta 222–3 dual gamma 223–4 European options 71–2, 218–65 gamma 221–4, 244, 248, 252, 264–5 psi 231–3 rho 230–1 theta 228–30 vega 224–6, 244, 252 vomma 226–8 Hallerbach see Corrado–Miller–Hallerbach approximation heat equations 140, 332, 340 hedge, definition 2 hedging Boyle–Emmanuel method 260–2 strategies 6–8, 27–61, 310–14 see also delta hedging

836 hedging portfolios cross-currency options 573 Heston model 757–8 lookback options 605, 607, 610, 618, 620–1 two-dimensional Black–Scholes equation 537 Heston model 753–69 Black–Scholes equation 756–60, 763 European options 754–5, 760–9 higher derivatives property 110–12 historical volatility 647–9, 652–84 Brenner–Subrahmanyam approximation 661–3, 673 Chambers–Nawalkha approximation 683–4 Corrado–Miller–Hallerbach approximation 673–6 GBM parameter estimation 652–4 generalised 660–1 geometric mean-reversion process 658–60 Li ATM volatility approximation 663–8 Li non-ATM volatility approximation 668–73 Manaster–Koehler method 677–83 Ornstein–Uhlenbeck process 654–8 hitting-time distribution 303–4 Hull–White model 743–53 immediate rebates 415–20 immediate-touch options 339–49 American options 268 down-and-out/in options 418 PDE approach 339–45 probabilistic approach 345–9 time-dependent options 339–49 up-and-out/in options 416 implied volatility 648–9, 652–84 Brenner–Subrahmanyam approximation 661–3, 673 Chambers–Nawalkha approximation 683–4 Corrado–Miller–Hallerbach approximation 673–6 GBM parameter estimation 652–4 geometric mean-reversion process 658–60 historical volatility 648–9 Li ATM volatility approximation 663–8 Li non-ATM volatility approximation 668–73 and local volatility 706–10 Manaster–Koehler method 677–83 Ornstein–Uhlenbeck process 654–8 surfaces 649 in-out parity 352 PDE pricing approach 356 with rebate at expiry 408–9, 412 reflection principle 389–90 in-the-money (ITM) 3, 24, 27

Index independent increment property 453, 517 induction, mathematical 205–6, 682 infimum 322 instantaneous variance 745, 747 integration by parts Dupire equation 702 forward Kolmogorov equation 697, 698, 730–4 geometric average rate options 509, 510 Heston model 765 lookback options 631, 632, 637, 638 stochastic volatility 730–4, 738–9, 765 interest rates, stochastic 167–90 intermediate value theorem 626 intrinsic value 1, 74–5 invariance properties 108–10 iterative methods/analysis 117, 119 ITM see in-the-money It¯o calculus 689, 717 It¯o integrals 162, 715, 748, 749 It¯o’s formula European options 132, 168, 170, 173, 176, 179, 181, 184 GBM parameter estimation 652 generalised stochastic volatility model 711, 712 geometric average strike options 515, 518 geometric mean-reversion process 659 Merton model 132 Ornstein–Uhlenbeck process 654 rainbow options 566, 570 It¯o’s lemma 155–6 arithmetic Brownian motion 151–2 asset-or-nothing options 142 Black model 145 Black–Scholes equation 90, 480, 538 chooser options 603 compound options 596 cross-currency options 573, 581 digital options 138 discrete geometric average rate Asian option 452 down-and-out/in options 366, 382 European option price under stochastic interest rate 183 European option valuation 102 exchange options 545, 546 forward start options 587 generalised Black–Scholes formula 162 generalised perpetual American options 292 geometric average rate options 511 Heston model 757 Hull–White model 747, 749 knock out/knock in options 425 lookback options 607, 621, 629, 636 martingale property 106

Index Merton model 129 non-dividend-paying asset price as num´eraire 166 power options 535, 536 put-call parity 498 spread options 550, 552–4 symmetry of arithmetic average rate/strike 527 time-dependent options 306, 338, 347 timer options 784 two-dimensional Black–Scholes equation 538 up-and-out/in options 358, 374 variance swaps 775 volatility derivatives 780, 781, 784–5 Jarrow–Rudd method 196–7 Jensen’s inequality, conditional 475 jump diffusion process 132 jump-diffusion models 651 jumps, discontinuous 73 Kac see Feynman–Kac formula Kamrad–Ritchken method 199–202 Kirk’s approximation 549, 554 knock out/knock in time 415–16, 418 knock-in and knock-out parity relationship 398, 400, 404, 406 knock-out equity accumulator 428–9 knock-out/in barrier options 351–2 futures 424–8 PDE approach 355 probabilistic approach 353 Koehler see Manaster–Koehler method Kolmogorov see backward Kolmogorov equation; Chapman–Kolmogorov equation; forward Kolmogorov equation Laplace transforms 301–2, 304 lattice approach see binomial… Levy approximation 457–67 L’Hˆopital’s rule arithmetic average options 491, 496 binomial tree models 211 Greeks 259 Levy approximation 464, 466 Li ATM volatility approximation 663–8 Li non-ATM volatility approximation 668–73 linear complementarity 269–70, 307–8 local volatility 649–51, 685–710 backward Kolmogorov equation 687–92 Black–Scholes equation 685, 687, 692–4 Dupire equation 685, 699–705, 706 forward Kolmogorov equation 694–9 and implied volatility 706–10

837 stochastic volatility model 743 time-dependent volatility 705–6 long position 2, 4–5 long side 4 long straddles 44–5 long strangles 47–8 long straps 52–3 long strips 49–51 lookback options 531, 604–45 Black–Scholes equation 604–9, 618–22 European fixed strike 627–40 European floating strike 640–5 perpetual American options 613–17, 623–7 stop-loss options 609–13 MacLaurin series 445, 674 maintenance margin 6 Manaster–Koehler method 677–83 Margrabe’s formula 544 market price of risk 124–5, 185 marking-to-market 6 Markov process 648 Marsaglia’s formula 234–5 martingale pricing 63, 67–9 see also probabilistic approach martingale property 105–8 martingales arbitrary 322 backward Kolmogorov equation 716 European option price under stochastic interest rate 172, 176, 178 exchange options 543, 545 forward Kolmogorov equation 694 generalised stochastic volatility model 712 Heston model 756 Hull–White model 746 non-dividend-paying asset price as num´eraire 166–7 Ornstein–Uhlenbeck process 655 perpetual American options 301 simple chooser options 600, 601 spread options 548, 551, 552 symmetry of arithmetic average rate/strike 526–7 timer options 785 mathematical induction 205–6, 682 maximum-likelihood estimation (mle) GBM parameter estimation 652–3 generalised historical volatility 660–1 geometric mean-reversion process 659–60 historical volatility 647–8, 660–1 Ornstein–Uhlenbeck process 654, 655–7 mean reversion process, geometric 658–60

838

Index

Merton model Black–Scholes model 128–34, 158–61 company default 158–61, 429–31 European option valuation 131–4 Miller see Corrado–Miller–Hallerbach approximation mle see maximum-likelihood estimation moment-generating function 475, 518, 565 moment-matching 458, 653–4 money-market accounts 547, 548, 552, 554 monotonicity 627, 648, 676–7, 680

see also American…; Asian options; barrier options; European…; exotic options ordinary differential equations (ODEs) arithmetic average rate options 488 Black–Scholes equation 113–15 immediate-touch options 340–1 lookback options 611–12, 625 perpetual American options 292–5, 299 ordinary least squares method (OLS) 657–8, 660 Ornstein–Uhlenbeck process 167, 180, 654–8 out-of-the-money (OTM) 3, 24–5, 27

Nawalkha see Chambers–Nawalkha approximation Newton–Raphson method 678, 680, 682 Nikod´ym see Radon–Nikod´ym derivative/process no-arbitrage condition American options 290 Black–Scholes equation 90, 92, 94, 481, 529 cross-currency options 574 discrete dividends 115, 118, 120, 121 European option price under stochastic interest rate 184 futures contracts 149 Greeks 250, 252, 261 Heston model 758 lookback options 608, 622 two-dimensional Black–Scholes equation 539 non-dividend-paying assets 79–80, 163–7 normal distribution 558, 781 see also bivariate normal distribution num´eraires European options 163–7, 169, 178, 179 exchange options 543, 545 Heston model 755, 756 non-dividend-paying asset price 163–7 spread options 548, 551 symmetry of arithmetic average rate/strike 526–7

partial differential equations (PDEs) arithmetic average options 487–96 asset-or-nothing options 139–41 barrier options pricing 354–6 Black–Scholes equation 692, 694 cross-currency options 573, 575–9 digital options 134–7 European option price under stochastic interest rate 181, 184, 185 European option valuation 95–101 exchange options 540–2 geometric average options 500–8 Greeks 255 Heston model 760–1, 763–7 immediate-touch options 339–45 knock out/in options 355 local volatility 650 lookback options 605, 607, 610, 611, 620, 625 one-touch options 331–6 similarity reduction 482–3, 484, 485, 487 stochastic volatility 742–3, 760–1, 763–7 timer options 786 two-dimensional 573, 575–9 path-dependent options 586–645 Asian options 480, 482, 484, 485 chooser options 600–4 cliquet options 588–91 compound options 591–9 exotic options 531 forward start options 586–8 lookback options 604–45 rachet options 588–91 stop-loss options 609–13 path-independent options 531, 532–86 Black–Scholes equation 537–9, 540, 572–9 capped options 532–3 corridor options 533–4 cross-currency options 572–86 exchange options 531, 540–7 PDE approach 540–2 power options 534–7

ODEs see ordinary differential equations OLS see ordinary least squares method one-for-two stock splits 110 one-touch options 331–9 PDE approach 331–6 probabilistic approach 336–9 time-dependent options 331–9 optimal stopping theorem 301 optimal stopping time formulation 268 options definition 1 stock index 1–2 theory 1–4, 15–27 trading 2–4

Index probabilistic approach 543–7, 579–86 rainbow options 531, 558–75 spread options 531, 547–58 two-dimensional Black–Scholes equation 537–9, 540 payoffs/payoff diagrams average rate option payoff 440–1 average strike option payoff 441 capped options 532–3 corridor options 533–4 cross-currency options 575–86 definitions of payoff 1–4 down-and-out/in options 400, 401, 405, 407 exotic options 531, 575–86 knock-out equity accumulator 429 Merton model 430–1 path-dependent options 531 path-independent options 531 terminal payoffs 409, 533, 584, 586 up-and-out/in options 398, 399, 403, 404, 434–7 PDEs see partial differential equations perpetual American options 292–305 Barone-Adesi and Whaley formula 326 call options 294–8, 301–4 generalised 292–4 lookback options 613–17, 623–7 put options 298–305 perpetual barrier lookback options 609 perpetual call problem 339–40 Poisson processes 128, 131 polynomial approximation 238, 240, 241 portfolios delta and gamma-neutral 264–5 see also hedging portfolios power options 534–7 price/pricing American options 268–71, 321–2 barrier options 353–407 binomial option 270–1 Black Scholes price 663–4, 666–9, 673 European options price formula 501, 505, 535–6, 547, 554 European options under stochastic interest rate 167–90 European “partial barrier” option price 435–7 market price of risk 124–5, 185 non-dividend-paying asset price as num´eraire 163–7 settlement price 6 strike/exercise price 2 see also martingale pricing; spot price volatility primal problems 321

839 probabilistic approach arithmetic average options 496–500 asset-or-nothing options 141–4 barrier options 353–4, 357–86 cross-currency options 579–86 digital options 137–9 down-and-out/in options 364–72, 380–6 European option valuation 101–5 exchange options 543–7 geometric average options 508–25 immediate-touch options 345–9 one-touch options 336–9 up-and-out/in options 357–64, 372–80 profit 237, 262–4 protective calls 6–7, 30–1 protective puts 6–7, 32, 33 psi 231–3 purchased collars 41–2 put-call parity American options 267, 277–8, 280–4 arithmetic average options 496–500 arithmetic Brownian motion 152, 155 Asian options 442–3, 496–500 asset-or-nothing options 142, 144 digital options 137 down-and-out/in options 413–14 Dupire equation 704–5 equity derivatives theory 21–4, 25–7, 39 European options 76–8, 84, 101, 105, 150–2, 155, 161, 267, 277–8, 280–4 futures contracts 150 geometric average options 522–5 Greeks 230, 231, 233, 246 Merton model 161 simple chooser options 600, 601 stochastic volatility 743 timer options 784, 786 up-and-out/in options 410 put-on-a-call options 592, 595 put/call-on-a-put options 596, 597, 599 quanto options 575, 580, 583 rachet options 588–91 Radon–Nikod´ym derivative/process cross-currency options 582, 585 European option price under stochastic interest rate 171, 177 forward Kolmogorov equation 694 generalised stochastic volatility model 712 rainbow options 531, 558–75 random walks 194 Raphson see Newton–Raphson method

840 realised integrated variance 780–3, 786 rebates at expiry 408–15 immediate 415–20 reflection principle 386–407 barrier options pricing 353, 354 Black equation 422–4 Black–Scholes equation 386–8, 421–2 down-and-out/in options 394–7, 400–2, 405–7, 413–14 Merton model 430 reflected standard Wiener process 528 up-and-out/in options 390–3, 397–400, 402–5, 410–11 Wiener process 304 rho 72, 230–1 risk 124–5, 185, 237, 573 risk-neutral approach 190–1, 192–3 risk-neutral expectation strategy 353 risk-neutral measure 155 American options 268, 302 arithmetic average options 497, 499 backward Kolmogorov equation 688, 689, 713, 714 Black–Scholes equation 692, 723, 724 Boyle method 197 capped options 532 chooser options 600–2 cliquet options 589 compound options 592, 595, 596, 599 Cox–Ross–Rubinstein method 193 cross-currency options 580, 581, 583, 584 Curran approximation 468, 471 digital options 138 discrete geometric average rate Asian option 450–2 domestic measure 580, 581, 583, 584 down-and-out/in options 365, 372, 381–2, 386, 401–2, 406, 420 Dupire equation 699 European option valuation 102 European options under stochastic interest rate 168, 172, 174, 176, 178–9 exchange options 543–5 forward Kolmogorov equation 694, 695, 726, 727 forward start options 587 generalised Black–Scholes formula 161–2 generalised stochastic volatility model 711–13 geometric average options 509, 511, 513, 515–16, 518, 522–5 Greeks 259 Heston model 753, 755, 756 Hull–White model 744, 747

Index in-out parity 390 Jarrow–Rudd method 196 Kamrad–Ritchken method 199–200 knock out/knock in options 425 Levy approximation 458 local volatility 650 lookback options 627, 628, 635, 640–4 martingale pricing theory 69 martingale property 105 Merton model 430–1 non-dividend-paying asset price as num´eraire 163–5 power options 535 put-call parity 442, 497, 499, 523–5 rachet options 589 rainbow options 559, 565, 570 spread options 548–9, 551, 554–5 stochastic volatility 711–14, 723–4, 726–7, 736, 744, 747, 753, 755–6 symmetry of arithmetic average rate/strike 525, 526 time-dependent options 321, 336–7, 345, 346–7 timer options 783, 786 up-and-out/in options 357–8, 364, 373–4, 380, 398, 402, 404, 417 variance swaps 774, 777, 778 volatility derivatives 774, 777–8, 782–3, 786 risk-neutral pricing 163 risk-neutral probability measure 190 risk-neutral valuation arithmetic average options 497 arithmetic Brownian motion 152 Black–Scholes model 101, 141 compound options 596 put-call parity 497 spread options 555 see also valuation Ritchken see Kamrad–Ritchken method Ross see Cox–Ross–Rubinstein method Rubinstein see Cox–Ross–Rubinstein method Rudd see Jarrow–Rudd method Russian options 617 Scholes, Myron 63 see also Black–Scholes… SDEs see stochastic differential equations self-financing trading strategy 64–9, 191–3 short position 2, 4–5 short side 4 short straddles 44–6 short strangles 48–9 short straps 53–4 short strips 51–2

Index similarity reduction 482–7 simple chooser options 600–1 smooth functions 688, 713, 717 smooth pasting condition American options 269, 295, 297–8, 301, 314–16, 325 Barone-Adesi and Whaley formula 325 time-dependent options 314–16 spot price volatility delta 220 dual delta 223 dual gamma 224 gamma 221 psi 232 rho 230 theta 228 vega 224 vomma 226 spreads 531, 547–58 bear 8, 37–9 box 39–40 bull 7, 33–6, 39 butterfly 8, 54–8 condor 58–61 standard Wiener processes 155–6 arithmetic average options 446, 448, 487, 491, 496, 497 arithmetic Brownian motion 151 arithmetic-geometric average rate identity 446 arithmetic-geometric average strike identity 448 asset-or-nothing options 139, 141, 142 backward Kolmogorov equation 687, 689, 713, 715 Black equation 422 Black model 144 Black–Scholes equation 89, 91, 93, 386, 421, 480, 692, 723 Black–Scholes model with transaction costs 125 capped options 532 chooser options 60, 601, 603 cliquet options 588, 589 compound options 591, 592, 596 corridor options 533 cross-currency options 572, 575, 579, 581–5 Curran approximation 468, 471 digital options 134, 137, 138 discrete geometric average rate Asian option 450, 452–3 down-and-out/in options 364, 366, 380, 382, 394, 400, 405, 412, 418 Dupire equation 699 European option price under stochastic interest rate 167, 169, 171, 177–80

841 European option valuation 95, 101–2 exchange options 540, 543–5 forward Kolmogorov equation 694, 695, 726 forward start options 586, 587 GBM parameter estimation 652 generalised Black–Scholes formula 161–2 generalised stochastic volatility model 710–12 geometric average options 446, 448, 450, 452–3, 500, 504, 508, 510–11, 514–15, 517–18, 522–4 Girsanov’s theorem 67–8, 102 Greeks 233, 246, 248, 253, 255, 258, 260 Heston model 753, 754, 760 Hull–White model 743, 744 immediate-touch options 339, 345–7 in-out parity 389, 408 independent increment property 453, 517 knock out/knock in options 424–5 knock-out equity accumulator 428 Levy approximation 457, 459 local volatility 650, 685 lookback options 604, 613, 618, 623, 627–8, 634–5, 640, 642–4 market price of risk 124 martingale property 105 Merton model 128, 131, 430 non-dividend-paying asset price as num´eraire 163–4, 166 one-touch options 331, 336, 338 Ornstein–Uhlenbeck process 654 power options 534, 535 probabilistic approach 353, 357–8, 364, 366, 372, 374, 380, 382 put-call parity 442, 496, 497, 522, 523–4 rachet options 588, 589 rainbow options 559, 565, 569, 570 reflection principle 386, 421, 422 similarity reduction 482, 483, 485 spread options 547–9, 552, 554, 555 stationary increment property 510 stochastic volatility 651, 710–13, 715, 723, 726, 736, 743–4, 753–4, 760 stop-loss options 609 symmetry of arithmetic average rate/strike 525, 526, 527, 528 time-dependent options 305, 331, 336, 338, 339, 345 time-independent options 292, 301, 302, 304 timer options 783 two-dimensional Black–Scholes equation 537 up-and-out/in options 357–8, 372, 374, 390, 397, 402, 410, 415 variance swaps 769, 774, 777 volatility derivatives 769, 774, 777, 780, 783

842 stationary increment property 510 stochastic differential equations (SDEs) 155–6 arithmetic average options 487, 491, 496 arithmetic Brownian motion 151 asset-or-nothing options 139, 141 backward Kolmogorov equation 687, 713 Black equation 422 Black–Scholes equation 89, 91, 93, 421, 480, 692 Black–Scholes model with transaction costs 125 capped options 532 chooser options 600, 601 cliquet options 588 corridor options 533 cross-currency options 572, 575, 580, 584 digital options 134, 137, 138 down-and-out/in options 364, 380, 394, 400, 405, 412, 418 Dupire equation 699 European option price under stochastic interest rate 174 European option valuation 95, 101 forward Kolmogorov equation 694, 726 forward start options 586 geometric average options 500, 504, 508, 514, 522 geometric mean-reversion process 659 Greeks 233, 246 in-out parity 389 knock out/knock in options 424–5 knock-out equity accumulator 428 local volatility 649 lookback options 627, 634, 640, 643 market price of risk 124 martingale pricing theory 68–9 martingale property 105 Merton model 129, 158, 429 perpetual American options 301, 303 power options 534–6 put-call parity 442, 496, 522 rachet options 588 reflection principle 386, 421, 422 similarity reduction 482, 483, 485 symmetry of arithmetic average rate/strike 525, 528 time-dependent options 331, 336, 339, 345 up-and-out/in options 357, 372, 397, 402, 410, 415 stochastic interest rates 167–90 stochastic processes, delta hedging 64 stochastic volatility 651, 710–69 backward Kolmogorov equation 713–25, 727, 729 Black–Scholes equation 723–6, 756–60, 763

Index forward Kolmogorov equation 726–36, 737, 738 generalised model 710–13 Heston model 753–69 Hull–White model 743–53 local volatility model 743 timer options 783 variance swaps 778 stock dividend effect 6 stock index futures 2, 10–11 stock index options, definition 1–2 stock split effect 6 stop-loss options 609–13 straddles 8, 44–6, 56–8 strangles 8, 47–9, 56–8 straps 8, 52–4 strike price, definition 2 strips 8, 49–52, 262–4 Subrahmanyam see Brenner–Subrahmanyam approximation sum of squares 103, 143 swaps 1, 769–79 symmetry of arithmetic average rate/strike 525–9 synthetic forwards 7 T-forward measure 177–80 Taylor’s expansion/series 156 backward Kolmogorov equation 688, 691, 713–14, 721 binomial tree models 209 Black–Scholes equation 480 Bos–Vandermark model 236 Brenner–Subrahmanyam approximation 662 continuous limit of binomial model 308 continuous-time limit of binomial model 70, 202, 203, 308 generalised perpetual American options 292 self-financing trading strategy 65 time-dependent options 306, 308 timer options 784 variance swaps 775 Taylor’s formula 659 Taylor’s theorem arithmetic Brownian motion 152 backward Kolmogorov equation 688 Black model 144 Black–Scholes equation 90, 92, 94, 538 Black–Scholes model with transaction costs 126 Chambers–Nawalkha approximation 684 Cox–Ross–Rubinstein method 195 cross-currency options 573 European option price under stochastic interest rate 183, 184

Index exchange options 546 Greeks 250, 259, 261 knock out/knock in options 425 Li non-ATM volatility approximation 670 Manaster–Koehler method 682 market price of risk 124 Merton model 129 power options 535 spread options 552 time-dependent options 309, 318 two-dimensional Black–Scholes equation 538 term structure of volatility 649, 705 term-structure variance 751 terminal payoffs corridor options 533 cross-currency options 584, 586 in-out parity 409 theoretical Black-Scholes formula 647, 649 theta 72, 228–30 3-period binomial tree models 211–17, 310–14, 328–31, 432–5 time value, definition 1 time-dependent continuous dividend yield 72 time-dependent options 305–49 American options 305–49 asymptotic optimal exercise boundaries 316–21 Barone-Adesi and Whaley formula 324–7 binomial tree models 308–14, 327–31 Black approximation 322–4 continuous limit of binomial model 308–9 immediate-touch options 339–49 linear complementarity 307–8 one-touch options 331–9 smooth pasting condition 314–16, 325 upper bound of American option price 321–2 time-dependent variance 751 time-dependent volatility 705–6 time-independent options 292–305 see also perpetual American options timer cash contracts 786 timer options 783–6 timer share contracts 785, 786 tower property Curran approximation 469 Heston model 754, 755 Hull–White model 745 trading strategies delta hedging 237, 239 hedging options 310–14 self-financing 64–9, 191–3

843 transaction costs 73, 125–8 tree-based methods 190–217 binomial tree models 204–17 Boyle method 197–9 Cox–Ross–Rubinstein method 193–6 Jarrow–Rudd method 196–7 Kamrad–Ritchken method 199–202 risk-neutral approach 190–1, 192–3 self-financing strategy 191–3 trinomial models 197, 199, 201 two-dimensional Black–Scholes equation exchange options 540 path-independent options 537–9, 540 stochastic volatility 724 two-dimensional Girsanov’s theorem 582, 585 backward Kolmogorov equation 713 European option price under stochastic interest rate 168, 171, 177 forward Kolmogorov equation 726 generalised stochastic volatility model 711, 712 Heston model 753 Hull–White model 744 two-dimensional PDEs 573, 575–9 Uhlenbeck see Ornstein–Uhlenbeck up-and-out/in barrier options 351, 352 European “partial barrier” option price 435–7 with immediate rebates 415–18 PDE approach 355, 356 probabilistic approach 357–64, 372–80 rebates at expiry 410–12 reflection principle 390–3, 397–400, 402–5, 410–11 upper bound of American option price 321–2 valuation European options 95–105, 131–4 Merton model 131–4 PDE approach 95–101 probabilistic approach 101–5 see also risk-neutral valuation Vandermark see Bos–Vandermark model vanilla options capped options 532 down-and-out/in options 365, 381 knock out/knock in options 427 lookback options 628, 635 up-and-out/in options 357–8, 373 see also American options; European options variance swaps 769–79 variation margin 6

844 Vasicek process see Ornstein–Uhlenbeck process vega 72, 224–6, 244, 252, 678 vol-of-vol 651 volatility Greeks 262 see also spot price volatility volatility derivatives 769–86 timer options 783–6 variance swaps 769–79 volatility models 647–786 historical volatility 647–9, 652–84 implied volatility 648–9, 652–84 local volatility 649–51, 685–710 stochastic volatility 651, 710–69 volatility derivatives 769–86

Index volatility skews 649 volatility smiles 649 vomma 226–8 Whaley see Barone-Adesi and Whaley formula White see Hull–White model Wiener processes Heston model 756 probabilistic approach 353–4 reflection principle 304 see also standard Wiener processes written collars 42–4 zero-coupon bonds 168–9, 173, 176, 180–2, 184, 187