Exotic Derivatives and Risk: Theory, Extensions and Applications 9812797475, 9789812797476

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Theory, Extensions and Applications

Mondher Bellalah Université de Cergy-Pontoise, France and

Dubai Group, UAE

World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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SHANGHAI

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HONG KONG

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TA I P E I

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CHENNAI

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

EXOTIC DERIVATIVES AND RISK: THEORY, EXTENSIONS AND APPLICATIONS Copyright © 2009 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-279-747-6 ISBN-10 981-279-747-5

Typeset by Stallion Press Email: [email protected]

Printed in Singapore.

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Foreword by Harry M. Markowitz (University of California, San Diego)

Herein follows a remarkable volume, suitable as both a textbook and a reference book. Mondher Bellalah starts with an introduction to options and basic hedges built from specific options. He then presents an accessible account of the formulae used in valuing options. This account includes historically important formulae as well as the currently most used results of Black–Scholes, Merton and others. Bellalah then proceeds to the main task of the volume, to show how to value an endless assortment of exotic options. Mondher Bellalah is to be congratulated for this tour de force of the field.

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Foreword by James J. Heckman (University of Chicago and University College Dublin)

Mondher Bellalah offers a lucid and comprehensive introduction to the important field of modern asset pricing. This field has witnessed a remarkable growth over the past 50 years. It is an example of economic science at its best where theory meets data, and shapes and improves on reality. Economic theory has suggested a variety of new and “exotic” financial instruments to spread risk. Created from the minds of theorists and traders guided by theory, these instruments are traded in large volume and now define modern capital markets. Bellalah offers a step-by-step introduction to this evolving theory starting from its classical foundations. He takes the reader to the frontier by systematically building up the theory. His examples and intuition are splendid and the formal proofs are clearly stated and build on each other. I strongly recommend this book to anyone seeking to gain a deep understanding of the intricacies of asset pricing.

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Foreword by George M. Constantinides (University of Chicago)

Both the trading of options and the theory of option pricing have long histories. The first use of option contracts took place during the Dutch tulip mania in the 17th century. Organized trading in calls and puts began in London during the 18th century, but such trading was banned on several occasions. The creation of the Chicago Board Options Exchange (CBOE) in 1973 greatly encouraged the trading of options. Initially, trading took place at the CBOE only in calls of 16 common stocks, but soon expanded to many more stocks, and in 1977, put options were also listed. The great success of option trading at the CBOE contributed to their trading in other exchanges, such as the American, Philadelphia and Pacific Stock Exchanges. Currently, daily option trading is a multibillion-dollar global industry. The theory of option pricing has had a similar history that dates to Bachelier (1900). Sixty-five years after Bachelier’s remarkable study, Samuelson (1965) revisited the question of pricing a call. Samuelson recognized that Bachelier’s assumption that the price of the underlying asset follows a continuous random walk leads to negative asset prices, and thus makes a correction by assuming a geometric continuous random walk. Samuelson obtained a formula very similar to the Black–Scholes–Merton formula, but discounted the cash flows of the call at the expected rate of return of the underlying asset. The seminal papers of Black and Scholes (1973) and Merton (1973) ushered in the modern era of derivatives. Exotic Derivatives and Risk is a lucid textbook treatment of the principles of derivatives pricing and hedging. At the same time, it is an exhaustively comprehensive encyclopedia of the vast array of exotic options, fixedincome options, corporate claims, credit derivatives and real options. Written by an expert in the field, Mondher Bellalah’s comprehensive and rigorous book is an indispensable reference on any professional’s desk. ix

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About the Author After obtaining his Ph.D. in Finance in 1990 at France’s leading University Paris-Dauphine, Mondher Bellalah began his career both as a Professor of Finance (HEC, INSEAD, University of Maine, and University of CergyPontoise) and as an international consultant and portfolio manager. He started out as a market maker on the Paris Bourse, before being put in charge of BNP’s financial engineering research team as Head of Derivatives and Structured Products. Dr. Mondher has acted as an advisor to various leading financial institutions, including BNP, Rothschild Bank, Euronext, Houlihan Lokey Howard & Zukin, Associés en Finance, the NatWest, Central Bank of Tunisia, Dubai Holding, etc., and has been Chief Risk Officer, Managing Director in Alternative Investments, Head of Capital Markets and Head of Trading. Dr. Mondher has also enjoyed a distinguished academic career as a tenured Professor of Finance at the University of Cergy-Pontoise in Paris for about 20 years. During this time, he has authored more than 14 books and 150 articles in leading academic and professional journals, and was awarded the Turgot Prize for the best French-language book on risk management in 2005. English-language books co-written/co-edited by Dr. Mondher include Options, Futures and Exotic Derivatives: Theory, Application and Practice published by John Wiley in 1998, and Risk Management and Value: Valuation and Asset Pricing published by World Scientific in 2008. His French-language books include Quantitative Portfolio Management and New Financial Markets; Options, Futures and Risk Management; and Risk Management and Classical and Exotic Derivatives. Dr. Mondher is an associate editor of the International Journal of Finance, Journal of Finance and Banking and International Journal of Business, and has been published in leading academic journals including Financial Review, Journal of Futures Markets, International Journal of Finance, International Journal of Theoretical and Applied Finance as well as in the Harvard Business Review. xi

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About Dubai Group Dubai Group is the leading diversified financial services company of Dubai Holding focusing on Banking, Investments and Insurance, both regionally and globally. Dubai Group was established in the year 2000, when it was known as The Investment Office. It was renamed to Dubai Investment Group in the earlier part of 2004 when a decision was taken to raise the company’s profile and expand its investment activities. In January 2007, Dubai Investment Group was renamed to Dubai Group following Dubai Holding’s restructuring creating a financial conglomerate with seven investment companies, each focusing on a specific sector and geographical focus: • • • • • • •

Dubai Ventures Group Dubai Financial Group Dubai Capital Group Dubai Insurance Group Dubai Banking Group Dubai Investment Group Noor Investment Group

Dubai Group is headquartered in Dubai with offices in Pittsburgh, New York, London, Istanbul, Hong Kong and Kuala Lumpur. The areas of operations of Dubai Group span the Middle East & North Africa Region (MENA), the European Union (EU), North America, Asia and CIS countries. Through its subsidiaries and affiliates, Dubai Group has business interests in 26 countries employing 13,450 individuals, serving over 4 million customers with 748 branches worldwide. Log on to www.dubaigroup.com for additional information.

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Contents

Foreword by Harry M. Markowitz

v

Foreword by James J. Heckman

vii

Foreword by George M. Constantinides

ix

About the Author

xi

About Dubai Group 1. 2. 3.

xiii

Derivatives and Asset Pricing in a Discrete-Time Setting: Basic Concepts and Strategies Option Pricing in Continuous-Time: The Black–Scholes–Merton Theory and Its Extensions

1 74

Exchange, Forward Start, Chooser Options and Their Applications

131

4.

Rainbow Options and Their Applications

175

5.

Extendible Options and Their Applications

202

6.

Currency Translated Options, Hybrid Securities and Their Applications

226

7.

Binaries, Barriers and Their Applications

267

8.

Lookback Options, Double Lookback Options and Their Applications

317

9. Asian and Flexible Asian Options and Their Applications xv

348

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Exotic Derivatives and Risk: Theory, Extensions and Applications

Steps, Parisian and Static Hedging of Exotic Options

372

11. Value at Risk: Basic Concepts and Applications in Risk Management

403

12.

Credit Risk and Credit Valuation: The Basic Concepts

446

13.

Credit Derivatives: The Basic Concepts

484

14.

Default Risk and the Pricing of Corporate Bonds, Swaps and Options

511

Contingent Claims Analysis and Its Applications in Corporate Finance: The Case of Real Options

541

Extended Discounted Cash Flow Techniques and Real Options Analysis within Information Uncertainty

550

Option Pricing When the Underlying Asset is Nonobservable

584

15. 16. 17.

Index

597

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Chapter 1

Derivatives and Asset Pricing in a Discrete-Time Setting: Basic Concepts and Strategies This chapter is organized as follows: 1. Section 2 develops the basic strategies using calls and puts. It presents the main concepts in option strategies. 2. Section 3 illustrates several combined strategies. These strategies can be used in different markets for different underlying assets. 3. Section 4 presents the main framework for asset pricing in a discrete-time context. It develops the mean–variance framework. 4. Section 5 presents the Capital Asset Pricing Model (CAPM) in its simplest version in the lines of Markowitz (1952) and Sharpe (1964). It also develops the general techniques for the derivation of the efficient frontier and presets Merton’s (1987) simple model of capital market equilibrium with incomplete information. Incomplete information refers to the fact that there are some costs in gathering data and transmitting information from one agent to another. All these models are presented in a discretetime context. 5. Section 6 presents the discrete-time approach for option pricing. It develops the Cox, Ross and Rubinstein (1979) model for the valuation of standard equity options. 6. Section 7 extends the standard discrete-time binomial model of Cox, Ross and Rubinstein (1979) to account for the effects of distributions to the underlying asset.

1. Introduction

T

his chapter studies the basic concept of options and their uses. It allows the reader to understand the main risks and return patterns 1

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associated with investments in financial markets and in particular in derivative assets. Derivatives correspond to futures, forward contracts, swaps, standard options and more complex options. An option gives the right to its holder to buy (for a call) or sell (for a put) a specified asset at a given strike price for a specified period of time. This is a standard definition of a standard option. Futures or forward contracts have similar definitions as options except that the buyer or the seller of the contract has no option: he has an obligation. Simple CAPMs allow the reader to understand the main concept of asset pricing in a standard context and in the presence of incomplete information. Simple CAPMs allow the reader to understand the concepts of risk and return in finance. These models can also be applied to the valuation of options with and without incomplete information using the standard analysis in Black and Scholes (1973), Black (1976) and Bellalah (1999, 2000). Asset pricing theory includes the valuation of a wide range of financial assets and derivative securities. Modern financial theory is based on some standard assumptions regarding markets and investors. It has had an impact on the development of financial markets. Over the last three decades, the financial market has lived through a wave of financial innovations and structural changes in the securities industry. What is a derivative? A derivative is a generic term to encompass all financial transactions which are not directly traded in the primary physical market. It refers to a financial instrument that helps to manage a given risk. It includes forwards, futures, options, commodity contracts, etc. What is a forward contract? A forward contract is the simplest and most basic hedging instrument. It is an agreement between two parties to set the price today for a transaction that will not be completed until a specified date in the future. The only way for the buyer or the seller to cancel the contract at a later date is to enter into a reverse forward contract with the same bank or another institution. However, a reverse contract implies a gain or a loss because the forward rate is likely to change as time passes. Forward rate contracts are flexible and allow for customized hedges since all the terms can be negotiated with the counterparty. However, each side of the contract bears what is called counterparty risk, that is, the risk that the other side defaults on the future

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commitments. That is why futures contracts are often preferred to forward contracts. What is a futures contract? A future is an exchange-traded contract between a buyer and seller and the clearinghouse of a futures exchange to buy or sell a standard quantity and quality of a commodity at a specified future date and price. The clearinghouse acts as a counterparty in all transactions and is responsible for holding traders’ surety bonds to guarantee that transactions are completed. Like forward contracts, futures contracts are used to lock in the interest rate, exchange rate or commodity price. But, futures contracts are organized in such a way that the counterparty risk of default is always completely eliminated because the clearinghouse steps in between a buyer and a seller, each time a deal is struck in the pit. The clearinghouse adopts the position of the buyer to every seller, and of the seller to every buyer, i.e. the clearinghouse keeps a zero net position. This means that every trader in the futures markets has obligations only to the clearinghouse, and has strong expectations that the clearinghouse will maintain its side of the bargain as well. The credibility of the system is maintained through the requirements of margin and daily settlements. The margin is a deposit in the form of cash, goverment securities, stock in the clearing corporation or letters of credit issued by an approved bank. The main purpose of the margin is to provide a safeguard to ensure that traders will honor their obligations. It is usually set to the maximum loss a trader can experience in a normal trading day. Daily settlements, called making to market, involve debiting the cash accounts of those whose positions lost money for the day and crediting the cash accounts of those whose positions earned money. However, the elimination of default risk has a cost. Futures contracts are standardized with respect to quantities and delivery dates and limited to frequently traded financial assets. Therefore, available futures contracts may not correspond perfectly to the risk to be hedged, thereby leaving hedgers with basis risk and correlation risk, which cannot be fully eliminated. What are standard options? Options are more flexible than forwards and futures because they protect the buyer against unfavorable outcomes, but allow him to enjoy the benefits associated with favorable outcomes. This price to be paid for this win–win position is called the option premium. A standard or a vanilla option is a security that gives its holder the right to buy or sell the underlying asset

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within a specified period of time, at a given price, called the strike price, striking price or exercise price. The right to buy is a call and the right to sell is a put. A call is in-the-money when the underlying asset price is higher than the strike price. It is out-of-the-money, if the underlying asset price is lower than the strike price. The call is at-the-money, if the underlying asset price is equal to the strike price. A put is in-the-money when the underlying asset price is lower than the strike price. It is out-of-the-money, if the underlying asset price is higher than the strike price. The put is at-the-money, if the underlying asset price is equal to the strike price. These definitions apply at maturity and at each instant before expiration. A European style option can be exercised only on the last day of the contract, called the maturity date or the expiration date. An American style option can be exercised at any time during the contract’s life. This chapter deals with the main strategies of derivatives markets and the pricing of assets and options in a discrete-time setting. Using the definition of a standard or a plain vanilla option, it is evident that the higher the underlying asset price, the greater the call’s value. When the underlying asset price is much greater than the strike price, the current option value is nearly equal to the difference between the underlying asset price and the discounted value of the strike price. The discounted value of the strike price is given by the price of a pure discount bond, maturing at the same time as the option, with a face or nominal value equal to the strike price. Hence, if the maturity date is very near, the call’s value (put’s value) is nearly equal to the difference between the underlying asset price and the strike price or zero. If the maturity date is very far, then the call’s value is nearly equal to that of the underlying asset since the bond’s price will be very low. The call’s value cannot be negative and cannot exceed the underlying asset price. In the first part of this chapter, we develop the basic strategies and synthetic option positions: long or short the underlying asset, long a call, long a put and short a put. Then we present some combinations and more elaborated strategies as: long a straddle, short a straddle, long or short a strangle, long a tunnel, short a tunnel, long a call or put bull spread, long a call or a put bear spread, long or short a butterfly, long or short a condor, etc. This first part allows the reader to understand the main strategies and risk–return trade-offs in option markets. In the second part of this chapter, we introduce the main concepts regarding asset pricing in a discrete-time setting. These models are proposed because they can be used in the pricing of options.

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Portfolio theory refers to the work of Markowitz (1952) on portfolio selection. Investors prefer to increase their wealth and to minimize the risk linked with the potential gain. It is not possible to obtain the maximum expected return and the minimum variance. According to Markowitz (1952), “The portfolio with maximum expected return is not necessarily the one with minimum variance. There is a rate at which the investor can gain expected return by taking on variance, or reduce variance by giving up expected return”. The limitations of the standard portfolio theory and mainly its restrictive assumptions, lead to the extensions of the theory in several direction. The most notable extension is the introduction of information uncertainty and its effects on the pricing of assets. In fact, the acquisition of information and its transmission to other agents are central activities in all areas of finance. Recognition of the different speeds of information diffusion is important in empirical research also. The perfect market model can provide a good description of the financial system in the long run. The analysis in Merton (1987) shows that a reconciling of finance theory with empirical violations of the complete-information, perfect market model need not imply a departure from the standard paradigm. However, as it appears in Merton (1987), “It does, however suggest that researchers be cognizant of the insensitivity of this model to institutional complexities and…. I believe that even a modest recognition of institutional structures and information costs can go a long way toward explaining financial behavior that is otherwise seen anomalous to the standard friction-less-market model”. For these reasons, we present also Merton’s (1987) simple model of capital market equilibrium with incomplete information. This model can be easily applied for the valuation of options and futures contracts as in Bellalah and Jacquillat (1995). In the third part of this chapter, we deal with the valuation of derivative assets in a discrete-time setting using the binomial approach. The valuation of options in a discrete time setting is more pedagogical than in a continuous time setting. Ironically enough, however, the more complex approach, namely the Black–Scholes (1973) one, was discovered before the simple binomial approach. Even if the discrete-time approach is not always computationally efficient, option valuation with the lattice approach is very flexible. It can handle many situations where no analytical solutions are possible. We are interested in the lattice approach pioneered by Cox, Ross and Rubinstein (1979). These authors proposed a binomial model in a discrete-time setting for the valuation of options.

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2. Basic Strategies and Synthetic Positions Using Standard Options This section develops the main standard option strategies and synthetic option positions.

2.1. Options and Synthetic Positions Synthetic positions can be constructed by options on spot assets, options on futures contracts and their underlying assets. If we use 0 to denote a horizontal line, −1 for the slope under 0 and 1 for the slope above 0, then it is possible to represent the diagram pay-offs of a long call by (0, 1), a short call by (0, −1), a long put by (−1, 0) and a short put by (1, 0). Adopting this notation for the basic option pay-offs, it is possible to construct all the synthetic positions as well as most elaborated diagram strategies using this representation. We denote by S, (F ): price of the underlying asset, which may be a spot asset, S (or a futures contract F ), K: strike price, C: call price, P: put price. We use the following symbols —: 0, /: 1, \: −1. The results of the basic strategies can be represented as follows: Long a call: (0, 1): Short a call: (0, −1): Long a put: (−1, 0): Short a put: (1, 0): Long the underlying asset: (1, 1) Short the underlying asset: (−1, −1) The symbols (−1, 0, 1) refer to a downward movement, (−1), a flat position (0) or an upward movement (1). The risk-return trade-off of the basic strategies can be represented using the different symbols. Using the above notations, it is possible to construct the risk-reward trade-off of any option

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strategy. For example, long a call (0, 1) and short a put (1, 0) are equivalent to long the underlying asset (1, 1). Also, short a call (0, −1) and long a put (−1, 0) are equivalent to a short position in the underlying asset (−1, −1). We give the basic synthetic positions when the options have the same strike prices and maturity dates. Long a synthetic underlying asset = long a call + short a put (1, 1) = (0, 1) + (1, 0) Short a synthetic underlying asset = short a call + long a put (−1, −1) = (0, −1) + (−1, 0) Long a synthetic call = long the underlying asset + long a put (0, 1) = (1, 1) + (−1, 0) Short a synthetic call = short the underlying asset + short a put (0, −1) = (−1, −1) + (1, 0) Long a synthetic put = short the underlying asset + long a call (−1, 0) = (−1, 1) + (0, 1) Short a synthetic put = long the underlying asset + short a call (1, 0) = (1, 1) + (0, −1) The knowledge of synthetic positions is necessary for market participants since it allows the implementation of hedged positions. When managing an option position, buying a call and a put with the same strike price are two equivalent strategies since when buying a call, the trader or the market maker hedges his transaction by the sale of the underlying asset and when buying a put, he hedges his transaction by purchasing the underlying asset. Buying the call and selling the put are equivalent to a long put with the same strike price. This transaction enables the trader or market maker to make a direct sale of the put since a position in a long call and a short put is equivalent to a long position in the underlying asset.

2.2. Long or Short the Underlying Asset The risk-return profile for a position which is long or short the underlying asset (for example, a futures contract) shows unlimited profit or loss. If we represent the underlying asset price with a horizontal line and the profit or loss with a vertical line, the pay-off to a long or a short position in the undelying asset can be easily represented. If the asset price rises or falls by one point, the profit or loss will be of the same amount.

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2.3. Long a Call Expectations: The trader expects a rising market and (or) a high volatility until the maturity date. Definition: Buy a call c with a strike price K. Specific features: The potential gain is not limited and the potential loss is limited to the option premium. Buying the call at 1.9 reveals the risk-reward profile given in Figure 1 at expiration. Table 1: Long a call: S = 102, r = 5%, volatility = 20%, T = 100 days. Type

Point

Break-even point Maximal loss Maximal gain

Options

A

Value K+c c Not limited if S > K

Q: 10

Long a call: 1.9

Strike price = Prime = Cost = Break-even point =

110 1.9 19 111.9

Q: quantity Underlying asset price S 90.00 95.00 100.00 105.00 110.00 115.00 120.00 125.00 130.00

Variation in (%)

Call

Performance in (%)

−12 −7 −2 3 8 13 18 23 27

−1.9 −1.9 −1.9 −1.9 −1.9 3.1 8.1 13.1 18.1

−100 −100 −100 −100 −100 165 430 696 961

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20 18

18,1

16 14

13,1

profit

12 10 8

8,1

6 4

3,1

2 0 -2

-1,9

-1,9

-1,9

-1,9

-1,9

-4 90

95

100

105

110

115

120

125

130

S

Figure 1:

Long a call.

If S = 111.9 at maturity; (110 + 1.9), the profit is zero. This is the break-even point of the position. The profit is not limited beyond this level. The maximum loss or performance corresponds to 1.9 or 100%. In Figure 1, the break-even point is given by the sum of the strike price and the option premium.

2.4. Short a Call Expectations: The trader expects a falling market and (or) a lower volatility until the maturity date. Definition: Sell a call c with a strike price K. Specific features: The potential gain is limited to the perceived premium and the potential loss is not limited. The risk-reward trade-off is inverted when selling calls. The results of the strategy are given in Figure 2. In Figure 2, the break-even point is given by the sum of the strike price and the option premium.

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Table 2: Short a call: S = 102, r = 5%, volatility = 20%, T = 100 days. Type Break-even point Maximal loss Maximal gain

Options

Point

Value

A

K+c Not limited Premium

Q: 10

Short a call: 1.9

Strike price = Premium = Profit = Break-even point =

110 1.9 19 111.9

S

Variation (%)

90.00 95.00 100.00 105.00 110.00 115.00 120.00 125.00 130.00

−12 −7 −2 3 8 13 18 23 27

Call 1.9 1.9 1.9 1.9 1.9 −3.1 −8.1 −13.1 −18.1

Performance (%) 100 100 100 100 100 −165 −430 −696 −961

2.5. Long a Put Expectations: The trader expects a falling market and (or) a higher volatility until the maturity date. Definition: Buy a put p with a strike price K. Specific features: The potential gain is not limited and the potential loss is limited to the option premium. In Figure 3, the break-even point is given by the algebraic sum of the strike price and the option premium.

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4 2

1,9

1,9

1,9

1,9

1,9

0 -2 -3,1

profit

-4 -6 -8

-8,1

-10 -12 -13,1

-14 -16 -18

-18,1

-20 90

95

100

105

110

115

120

125

130

S

Figure 2:

Short a call.

2.6. Short a Put Expectations: The trader expects a stable and (or) a rising market. Definition: Sell a put p with a strike price K. Specific features: The potential gain is limited to the option premium and the potential loss is unlimited. Figure 3 represents in a certain way the opposite of the risk-reward profile in Figure 2. The profit is limited when the underlying asset price increases and the risk is unlimited when the underlying asset price is decreases.

3. Combined Strategies This section illustrates several combined strategies involving call and put options. The main features of each strategy are provided.

3.1. Long a Straddle Expectations: The trader expects a high volatility until the maturity date.

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Table 3: Long a put: S = 102, r = 5%, volatility = 20%, T = 100 days. Type

Point

Break-even point Maximal loss Maximal gain

Options

S 60.00 70.00 80.00 90.00 100.00 110.00 120.00 130.00 140.00

A

Value K−p Premium Not limited

Q: 10

Long a put: 2.7

Strike price = Prime = Cost = Break-even point =

100 2.7 27 97.28

Variation (%)

Put

Performance (%)

−41 −31 −22 −12 −2 8 18 27 37

37.3 27.3 17.3 7.3 −2.7 −2.7 −2.7 −2.7 −2.7

1371 1003 635 268 −100 −100 −100 −100 −100

Definition: Buy a call, c and simultaneously buy a put, p on the same underlying for the same maturity date and the same strike price. Specific features: • The initial investment is important since the investor buys simultaneously the call and the put. • The loss is limited to the initial cost (c and p). • The maximum potential gain is not limited when the market goes up or down. Buying a straddle needs a simultaneous purchase of call and a put with the same strike price for the same maturity. When the put is worthless, the call is deep-in-the-money. When the call is worthless, the put is in-the-money.

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40 37,3 35 30 27,3

profit

25 20 17,3 15 10 7,3 5 0 -2,7 -2,7 -2,7 -2,7 -2,7 -5 60

70

80

90

100

110

120

130

140

S

Figure 3:

Long a put.

Notes: • The strike price is chosen according to the trader expectations about the future market direction. • Simulation Underlying asset S = 102 Interest rate r(%) = 5 Volatility (%) = 20 Maturity (in days) = 50

3.2. Short a Straddle Expectations: The trader expects a low volatility until the maturity date. Definition: Sell a call, c and simultaneously sell a put, p on the same underlying for the same maturity date and the same strike price. Specific features: • The initial revenue is limited to the option premiums. • The loss is not limited when the market goes up or down. • The maximum potential gain is limited to the initial premium (c and p).

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Table 4: Short a put: S = 102, r = 5%, volatility = 20%, T = 100 days. Type Break-even point Maximal loss Maximal gain

Options

S

Point

Value

A

K−p Not limited Premium

Q: 10

Short a put: 2.7

Strike price = Premium = Profit = Break-even point =

100 2.7 27 97.28

Variation (%)

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

−22 −17 −12 −7 −2 3 8 13 18

Put −17.3 −12.3 −7.3 −2.3 2.7 2.7 2.7 2.7 2.7

Performance (%) −635 −451 −268 −84 100 100 100 100 100

When the underlying asset price is expected to be in a specified interval at maturity, the trader can sell simultaneously a call and a put. The profit is limited to the premium received and the risk may be unlimited. If the underlying asset is not expected to move much either side, the investor can sell the put and the call. The maximum profit at expiration is obtained when S is in a given interval.

3.3. Long a Strangle Expectations: The trader expects a high volatility during the options’ life. Definition: • Buy a call with a strike price Kc . • Buy a put with a strike price Kp . Where the Kp < Kc .

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5 2,7

2, 7

2, 7

2, 7

2, 7

0 -2 ,3 -5 -7 ,3

profit -10 -1 2 , 3 -15 -1 7 , 3 -20 80

85

90

95

100

105

110

115

120

S Figure 4:

Short a put.

Specific features: • This strategy costs less than the straddle. • The maximum loss is limited to the initial cost of (c + p). • The net result is a profit only when the market movement is important. In this example, the market must increase by 5%, (107.49 − 102)/102 or decrease by 9%, (92.41 − 102)/102. Notes: The trader buys the 105 call and the 95 put. The theoretical prices of these options are, respectively, 2.04 and 0.55, or a total of 2.58. The quantity is 10 and the total cost of the strategy is 25.8. The two break-even points are computed as follows: • 105 + (2.04 + 0.55) = 107.59 or a variation of 5.38%. • 95 − (2.04 + 0.55) = 92.41 or a variation of −9.40%. If the underlying asset price is between the two strike prices at expiration, the maximum loss is reduced to the initial cost 25.8. The net result is a loss if the underlying asset price is between the two break-even points, 92.41

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Table 5: Type

Point

Break-even point Maximal loss Maximal gain

Options

Long a straddle. Value S = K − (c + p) S = K + (c + p) (c + p) if S = K K − (c + p) if S tends toward 0 Limited if S is beyond the limits

A B C

Q

1

1

10

Long a call

Long a put

Strategy

100 4.5 45 104.50

100 1.8 18 98.18

6.3 63 5

Strike price = Prime = Cost = Break-even point =

S

Variation (%)

Call

Put

Straddle

Performance (%)

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

−22 −17 −12 −7 −2 3 8 13 18

−4.5 −4.5 −4.5 −4.5 −4.5 0.5 5.5 10.5 15.5

18.2 13.2 8.2 3.2 −1.8 −1.8 −1.8 −1.8 −1.8

13.7 8.7 3.7 −1.3 −6.3 −1.3 3.7 8.7 13.7

216 137 58 −21 −100 −21 58 137 216

and 107.59. This loss is less than the initial cost. However, if the underlying asset price is above the break-even points, either side, the trader benefits from the leverage effect. For example, if the underlying asset price is 90 at expiration, or a variation of 12%, the net result is 93%. If the underlying asset goes up by 18% to attain a level of 120, the net profit of 12.4, compared to 2.58, represents a performance of 480%. Simulation The parameters used in the simulation are: S = 102, r = 5%, volatility = 20%, maturity = 50 days.

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20

15

13.7

10 profit

October 23, 2008

13.7

8.7

8.7

5

3.7

3.7

0

-1.3

-1.3

Call Put

-5

Straddle

-6.3 -10 80

85

90

95

100

105

110

115

120

S

Figure 5:

Buying a straddle.

Table 6:

Shorting a straddle.

S

Variation (%)

Call

Put

Straddle

Performance (%)

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

−22 −17 −12 −7 −2 3 8 13 18

4.5 4.5 4.5 4.5 4.5 −0.5 −5.5 −10.5 −15.5

−18.2 −13.2 −8.2 −3.2 1.8 1.8 1.8 1.8 1.8

−13.7 −8.7 −3.7 1.3 6.3 1.3 −3.7 −8.7 −13.7

−216 −137 −58 21 100 21 −58 −137 −216

17

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10 Call

6.3

Put

5

Straddle

1.3

0

profit

18

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1.3

-3.7

-5

-3.7

-8.7

-10

-8.7

-13.7

-15

-13.7

-20 80

85

90

95

100

105

110

115

120

S

Type Break-even point Maximal loss Maximal gain

Figure 6:

Short a straddle.

Table 7:

Long a strangle.

Point

Value

A B

S = Kp − (c + p) S = Kc + (c + p) (c + p) Kp − (c + p) Limited

A B

if Kp < S < Kc if S tends toward 0 if S is higher

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Strike price Premium Cost Break-even point

Table 8:

Long a call

Long a put

105 2.04 20.4 107.59

95 0.55 5.5 92.41

19

Strategy

25.8

Profit (per unit) of a long strangle strategy.

S

Variation (%)

Call

Put

Strangle

Performance (%)

85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00 125.00

−17 −12 −7 −2 3 8 13 18 23

−2.0 −2.0 −2.0 −2.0 −2.0 3.0 8.0 13.0 18.0

9.5 4.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5 −0.5

7.4 2.4 −2.6 −2.6 −2.6 2.4 7.4 12.4 17.4

287 93 −100 −100 −100 93 287 480 674

3.4. Short a Strangle Expectations: The trader expects a low volatility during the options’ life. Definition: Sell a call with a strike price Kc and sell a put with a strike price Kp where the Kp < Kc . Specific features: The maximum gain is limited to the initial premium of (c + p) and the strategy can show a loss. The reader can make the specific comments by comparing this strategy with the long strangle.

3.5. Long a Tunnel Expectations: The trader expects a high volatility during the options’ life. Definition: Buy an out-of-the money call and sell an out-of-the money put as in Table 10.

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20 Call

17.4

Put Strangle

15

12.4

profit

10 7.4

7.4

5 2.4

2.4

0

-2.6

-5 85

90

95

100

105

110

115

120

125

S

Figure 7: Profit (per unit) of a long strangle strategy.

3.6. Short a Tunnel This is the opposite of the previous strategy.

3.7. Long a Call Bull Spread A strategy can be implemented by buying a call with a lower strike price and selling a call with a higher strike price. If the underlying asset price is below the lower strike price at expiration, the maximum loss is limited to the difference between the two option premiums. If the underlying asset price is above the higher strike price at expiration, the lower strike price call is worth the intrinsic value. This strategy shows a limited profit (a loss).

3.8. Long a Put Bull Spread Expectations: Buying a put spread is equivalent to buying the higher strike price put and selling the lower strike price put. If the underlying asset is

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Table 9:

Short a strangle.

S

Variation (%)

Call

Put

Strangle

Performance (%)

85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00 125.00

−17 −12 −7 −2 3 8 13 18 23

2.0 2.0 2.0 2.0 2.0 −3.0 −8.0 −13.0 −18.0

−9.5 −4.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

−7.4 −2.4 2.6 2.6 2.6 −2.4 −7.4 −12.4 −17.4

−287 −93 100 100 100 −93 −287 −480 −674

5

21

2.6

0 -2.4

-2.4

profit

-5 -7.4

-7.4

-10 -12.4 -15

Call Put

-17.4

Strangle

-20 85

90

95

100

105

110

115

120

125

S

Figure 8:

Short a strangle.

around the lower strike price at maturity, the higher strike price put is worth the intrinsic value and the lower strike price is worthless. The maximum profit is given by the difference between the two option premiums. The strategy is done with a debit. The trader can sell the put spread by selling the

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Table 10: Options

Underlying asset S

10

Long a call

Short a put

Strike price = Premium = Cost = Break-even point =

570 22.3 223 592.32

550 15.3 153 534.68

Variation (%)

Call out-of-the money

Put out-of-the money

−5 −4 −2 0 2 4 5 7 9

−22.3 −22.3 −22.3 −22.3 −22.3 −12.3 −2.3 7.7 17.7

−4.7 5.3 15.3 15.3 15.3 15.3 15.3 15.3 15.3

530.00 540.00 550.00 560.00 570.00 580.00 590.00 600.00 610.00

Tunnel

7.0 −70

Performance (%)

−27.0 −17.0 −7.0 −7.0 −7.0 3.0 13.0 23.0 33.0

−386 −243 −100 −100 −100 43 186 329 471

40 33,0

30 23, 0

20

profit

13, 0

10 3, 0

0 -7 ,0

-7 ,0

-10

-7 ,0 -17, 0

-20 -2 7, 0

Call OUT Put OUT Tunnel

-30 -40 530

540

550

560

57 0

580

590

600

610

S

Figure 9: Long a tunnel (buy an out-of-money call and sell an out-of-the money put).

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23

Table 11: Q

1

1

10

Short a call

Long a put

Strike price = Premium = Cost = Break-even Point =

570 22.3 223 592.32

550 15.3 153 534.68

Options

Underlying asset

Variation (%)

Call OUT

Put OUT

−5 −4 −2 0 2 4 5 7 9

22.3 22.3 22.3 22.3 22.3 12.3 2.3 −7.7 −17.7

4.7 −5.3 −15.3 −15.3 −15.3 −15.3 −15.3 −15.3 −15.3

530.00 540.00 550.00 560.00 570.00 580.00 590.00 600.00 610.00

Tunnel

7.0 −70 10

Performance (%)

27.0 17.0 7.0 7.0 7.0 −3.0 −13.0 −23.0 −33.0

386 243 100 100 100 −43 −186 −329 −471

40 30 2 7 ,0

20 1 7 ,0

profit

10 7 ,0

7 ,0 7 ,0

0 -3 ,0

-1 0 -1 3 ,0

-2 0 -2 3 ,0

-3 0

C a ll O U T -3 3 ,0

Put O U T -4 0

Tunnel 530

540

550

560

570

580

590

600

610

S

Figure 10: Short a tunnel (sell an out-of-the money call and buy an out-of-the money put).

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Table 12: days. S 490.00 510.00 530.00 550.00 570.00 590.00 610.00 630.00 650.00

Bull spread with calls S = 102, r = 5%, volatility = 20%, T = 100

Variation (%)

Call IN

Call OUT

Spread

Performance (%)

−14 −11 −7 −4 0 3 7 10 14

−19.9 −19.9 −19.9 −19.9 −19.9 0.1 20.1 40.1 60.1

11.0 11.0 11.0 11.0 11.0 11.0 −9.0 −29.0 −49.0

−8.9 −8.9 −8.9 −8.9 −8.9 11.1 11.1 11.1 11.1

−100 −100 −100 −100 −100 125 125 125 125

80

60

40

20 profit

24

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11.1

0

-20

-8.9

Call IN

-40

Call OUT Spread

-60 490

510

530

550

570

590

610

S

Figure 11: Buying a bull spread with calls.

630

650

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Table 13: Buying a bull spread with puts. S

Variation (%) −14 −11 −7 −4 0 3 7 10 14

490.00 510.00 530.00 550.00 570.00 590.00 610.00 630.00 650.00

Put OUT

Put IN

Spread

Performance (%)

66.0 46.0 26.0 6.0 −14.0 −14.0 −14.0 −14.0 −14.0

−75.0 −55.0 −35.0 −15.0 5.0 25.0 25.0 25.0 25.0

−9.0 −9.0 −9.0 −9.0 −9.0 11.0 11.0 11.0 11.0

82 82 82 82 82 −100 −100 −100 −100

80 60 40 11.0

profit

20 0 -20

-9.0

-40 Put OUT

-60

Put IN Spread

-80 -100 490

510

530

550

570

590

610

630

650

S

Figure 12: Buying a bull spread with puts.

higher strike price put and buying the lower strike price put. The strategy is done with a credit.

3.9. Short a Call Bear Spread The strategy is illustrated in Figure 13. The investor buys a call with a strike K1 and sells a call with a strike K2 with K1 < K2 .

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Table 14: Selling a call bear spread. S

Variation (%)

Call IN

Call OUT

−14 −11 −7 −4 0 4 7 11 14

18.8 18.8 18.8 18.8 18.8 −1.2 −21.2 −41.2 −61.2

−10.3 −10.3 −10.3 −10.3 −10.3 −10.3 9.7 29.7 49.7

490.00 510.00 530.00 550.00 570.00 590.00 610.00 630.00 650.00

Spread

Performance (%)

8.5 8.5 8.5 8.5 8.5 −11.5 −11.5 −11.5 −11.5

100 100 100 100 100 −134 −134 −134 −134

60 40 20

profit

26

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8.5

0 -11.5

-20 -40 Call IN Call

-60

Spread

-80 490

510

530

550

570

590

610

S

Figure 13: Selling a call bear spread.

630

650

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3.10. Shorting a Put Bear Spread Table 15: Selling a put bear spread. S

Variation (%)

Put OUT

Put IN

Spread

Performance (%)

490.00 510.00 530.00 550.00 570.00 590.00 610.00 630.00 650.00

−14 −11 −7 −4 0 4 7 11 14

−65.1 −45.1 −25.1 −5.1 14.9 14.9 14.9 14.9 14.9

73.8 53.8 33.8 13.8 −6.2 −26.2 −26.2 −26.2 −26.2

8.7 8.7 8.7 8.7 8.7 −11.3 −11.3 −11.3 −11.3

−77 −77 −77 −77 −77 100 100 100 100

100 Put OUT

80

Put IN Spread

60

profit

40 20

8.7

0 -11.3

-20 -40 -60 -80 490

510

530

550

570

590

610

630

650

S

Figure 14: Selling a put bear spread.

3.11. Long a Butterfly Anticipation: The strategy consists in buying a call with a strike K1 , selling two calls with strikes K2 and buying a call with strike K3 with K1 < K2 < K3 .

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Table 16:

Long a butterfly.

S

Variation (%)

Call OUT

Call AT

Call IN

Butterfly

Performance (%)

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

−22 −17 −12 −7 −2 3 8 13 18

−12.7 −12.7 −12.7 −7.7 −2.7 2.3 7.3 12.3 17.3

9.0 9.0 9.0 9.0 9.0 −1.0 −11.0 −21.0 −31.0

−0.7 −0.7 −0.7 −0.7 −0.7 −0.7 −0.7 4.3 9.3

−4.5 −4.5 −4.5 0.5 5.5 0.5 −4.5 −4.5 −4.5

−20 −20 −20 2 25 2 −20 −20 −20

20

10 5.5 0.5

profit

0

0.5

-4.5

-10

-4.5

-20 Call Call AT Call IN

-30

Butterfly

-40 80

85

90

95

100

105

S

Figure 15:

Long a butterfly.

110

115

120

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Table 17:

29

Short a butterfly.

S

Variation (%)

Call OUT

Call AT

Call IN

Butterfly

Performance (%)

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

−22 −17 −12 −7 −2 3 8 13 18

12.7 12.7 12.7 7.7 2.7 −2.3 −7.3 −12.3 −17.3

−9.0 −9.0 −9.0 −9.0 −9.0 1.0 11.0 21.0 31.0

0.7 0.7 0.7 0.7 0.7 0.7 0.7 −4.3 −9.3

4.5 4.5 4.5 −0.5 −5.5 −0.5 4.5 4.5 4.5

20 20 20 −2 −25 −2 20 20 20

3.12. Short a Butterfly Anticipation: The strategy consists in selling two calls with a strike K1 and a strike K3 and buying the two calls with a strike K2 .

4. Asset Pricing in a Discrete-Time Setting: The Mean–Variance Framework Asset pricing in a discrete time setting is often analyzed with respect to simple models of capital market equilibrium. These models are based on the concepts of risk and return. They can also be applied for the valuation of derivatives.

4.1. Risk and Return: Some Definitions The mean–variance framework refers to a risk-return trade-off. Table 18 shows an uncertain return and its corresponding probability. The sum of the probabilities equals one. m


Pi = P1 + P2 + P3 + P4 + P5 + P6

i=1

=

1 2 4 3 1 1 + + + + + = 1. 12 12 12 12 12 12

The expected return from the investment can be computed as a weighted average of each uncertain return by its corresponding probability.

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Table 18: The probable results of the investment according to the state of nature. Rate of return R1 R2 R3 R4 R5 R6

= 6% = 8% = 10% = 12% = 14% = 16%

Probability Pi = 1/12 = 2/12 = 4/12 = 3/12 = 1/12 = 1/12

P1 P2 P3 P4 P5 P6 

i=1 Pi = 1

Total

E(R) =

n


Probability in %, Pi



8.33 16.66 33.33 25 8.33 8.33

i=1 Pi = 100%

Pi Ri = 10.67%.

i=1

The variance is computed as the deviations around the mean: σ = 2

n


Pi [[Ri − E(R)]2 ].

i=1

Table 19 shows the procedure for the computation of the variance of returns in this context. Table 19: Computing the variance. Pi 1/12 2/12 4/12 3/12 1/12 1/12

Ri

E(R)

(Ri − E(R))2

Pi × (Ri − E(R))2

6.00 8.00 10.00 12.00 14.00 16.00

10.67 10.67 10.67 10.67 10.67 10.67

21.78 7.11 0.44 1.78 11.11 28.44

1.81 1.19 0.15 0.44 0.93 2.37

Hence σ 2 = 6.89 (%). Risk is often defined as the deviations of the expected return with respect to the mean return. The following example illustrates the procedure for the computation of risk and return. Portfolio selection consists in the selection of a portfolio with respect to the mean variance framework. The investor prefers a higher expected return for a given variance or a lower variance

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31

for a given expected return. The computation of the variance and expected return for a portfolio with two or N assets is simple.

4.2. Portfolio Selection Portfolio theory is based on the concepts of risk and return. Consider a portfolio with a proportion XA of asset A and XB of asset B, with: EA < EB σA < σB , XA + XB = 1. The expected return is EP = XA EA + XB EB .

(1)

σp2 = XA2 σA2 + XB2 σB2 + 2XA XB ρAB σA σB

(2)

The variance is

where ρAB is the correlation coefficient between A and B. When the two assets are perfectly correlated, ρAB = 1, and the variance of the portfolio is: σp2 = XA2 σA2 + XB2 σB2 + 2XA XB σA σB or: σp2 = (XA σA + XB σB )2 . The standard deviation is: σp = XA σA + XB σB . Using this system for different values of ρAB in the interval (−1, 1), it is possible to generate point by point all the curves reflecting the relationships between the pairs (Ep , σp ) of a portfolio. Consider the pairs (Ep , σp ). There is a rate at which the investor can gain expected return by taking on variance, or reduce variance by giving up expected return. Investors choose the assets to be included in their portfolios using the mean variance framework. Figure 16 shows that for each level of risk, it is possible to determine the portfolio with the highest expected return. The points corresponding to this situation allow the definition of the efficient frontier. For the case of a portfolio with two assets A and B where A is the risk-less asset, the expected return and the variance: EP = XA EA + XB EB σp2 = XA2 σA2 + XB2 σB2 + 2XA XB ρAB σA σB

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Figure 16:

Portfolio choice.

Ep

C

RF B σp

Figure 17: Introduction of a risk-free asset.

become EP = XA EA + XB EB σp2 = XB2 σB2 . When a riskless asset is used, it is possible to determine the “best” combination between the riskless asset and point C on the efficient frontier. Point C dominates all the other portfolios on the efficient frontier. It is often referred to as the market portfolio, M.

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Risk

Non systematic risk (diversifiable)

Systematic risk Number of stocks

Figure 18: Diversification principal.

4.3. Systematic Risk and Diversification: An Introduction The diversification principle is based on a relationship between risk and the number of stocks to be included in a portfolio. The idea is illustrated in Figure 18. When investors hold the market portfolio, the contribution of each asset to the risk of a portfolio can be easily determined as in Table 20. Table 20: Variance–covariance matrix of the market portfolio. 1

2

.. .

N

1

X12 σ12

X1 X2 Cov(R1 , R2 )

.. .

X1 XN Cov(R1 , RN )

2

X2 X1 Cov(R2 , R1 )

X22 σ22

Stock

3

X3 X1 Cov(R3 , R1 )

X3 X2 Cov(R3 , R2 )

· ·

· ·

· ·

N

XN X1 Cov(RN , R1 )

XN X2 Cov(RN , R2 )

X2 XN Cov(R2 , RN ) .. . · · · · ·

X3 XN Cov(R3 , RN ) · · 2 σ2 XN N

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For example, the contribution of asset 3 in line 3 shows its covariance with the other assets. Line 3 can be written as: X3 X1 cov(R1 , R3 ) + X3 X2 cov(R3 , R2 ) + X32 cov(R3 , R3 ) + X3 X4 cov(R3 , R4 ) + · · · + X3 XN cov(R3 , RN ) = X3 [X1 cov(R3 , R1 ) + X2 cov(R3 , R3 ) + X4 cov(R3 , R4 ) + · · · + X3 XN cov(R3 , RN ). This is the contribution of asset 3 to the global risk of the portfolio weighted by the fraction of this asset in the value of the portfolio. The contribution of an asset to the risk of the market portfolio is measured by its covariance cov(Ri , RM ) with the market portfolio and its beta. The beta is given by: βi = [Cov(Ri , RM )]/[σ 2 (RM )] or βi = ρiM (σi /σM ).

5. Asset Pricing Models in Discrete Time: The Capital Asset Pricing Model, CAPM and the CAPMI of Merton (1987) At equilibrium, there is a simple relationship between the expected return and risk, given by the beta of an asset.

5.1. The Capital Asset Pricing Model, CAPM When short sales are allowed and the investor can borrow and lend at the riskless rate, the composition of the optimal portfolio requires the maximization of the slope between a given portfolio and the risk free rate, or when the partial derivative with respect to the assets in the portfolio is set to zero, we obtain the following system of simultaneous equations: ¯ k − RF ) = x1 σ1k + x2 σ2k +  + xk σk2 + xN−1 σN−1,k + xN σN,k . (R Since investors have homogeneous expectations regarding the optimal portfolio, the right-hand side of this equality can be written as: ¯ k − RF ) = γ cov(Rk , Rm ). (R

(3)

We can check that the following quantity ¯ k − RF ) = γ(x1 σ1k + x2 σ2k +  + xk σk2 + xN−1 σN−1,k + xN σN,k ) (4) (R

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is equivalent to: ¯ m − RF ) = γ cov(Rm , Rm ). (R Recall that the return on the market portfolio is given by: N


X∗i Ri

i=1

where X∗i correspond to the weights invested in the market portfolio. The covariance between the return an asset k and the market portfolio can be written as: ¯ k )(Rm − R ¯ m )} cov(Rk , Rm ) = E{(Rk − R or:  ¯ k) cov(Rk , Rm ) = E (Rk − R

 N


X∗i Ri −

i=1

N


 ¯i X∗i R

.

i=1

N

X∗i gives:  N  
¯ k) ¯ i) cov(Rk , Rm ) = E (Rk − R X∗i (Ri − R .

Factoring by the quantity

i=1

i=1

Developing the terms within the expectation operator gives: ¯ k )X∗1 (R1 − R ¯ 1 ) + (Rk − R ¯ k )X∗2 (R2 − R ¯ 2) cov(Rk , Rm ) = E{(Rk − R ¯ k )X∗k (Rk − R ¯ k) + K + (Rk − R ¯ k )X∗N (RN − R ¯ N )}. + K + (Rk − R Factoring by X∗i and applying the expectation operator provides: ¯ k )(R1 − R ¯ 1 )} cov(Rk , Rm ) = X∗1 E{(Rk − R ¯ k )(R2 − R ¯ 2 )} + X∗2 E{(Rk − R ¯ k )(R3 − R ¯ 3 )} + X∗3 E{(Rk − R ¯ k )(Rk − R ¯ k )} + · · · + X∗k E{(Rk − R ¯ k )(RN − R ¯ N )}. + · · · + X∗N E{(Rk − R

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When these terms are compared with the right-hand side of Eq. (4), we see that they are the same. Hence, Eq. (4) can be written as: γ cov(Rk , Rm ) = ¯ k − RF . R The equality is verified for each asset and portfolio and in particular for ¯ m − RF ) = γ cov(Rm , Rm ) or: (R ¯ m − RF ) = γσm2 . the market portfolio: (R ¯ m − RF )/σm2 . Replacing this value of γ in Eq. (3), we obtain: Hence, γ = (R ¯ k = RF + R



 ¯ m − RF ) (R cov[Rk , Rm ]. σm2

This is the standard version of the capital asset pricing model.

5.2. The Efficient Frontier when Investors can Borrow and Lend in the Presence of Short Selling Restrictions In this case, we face the same problem, except the fact that the weights ¯ p − RF )/σp must be positive. The maximization problem becomes: F = (R N under the constraint: i=1 xi = 1, xi ≥ 0. Since the variance has terms in x2 and products xi xj , the solution can be found using the Kuhn–Tucker conditions.

5.3. The Efficient Frontier when Investors are not Allowed to Borrow and Lend at the Risk Free Rate in the Presence of Short Selling Restrictions In this case, we face the following maximization problem: F=




xi2 σi2

+

N N



xi xj σij

i=1 j=1 i=j

under the constraint: N
i=1

for i = 1, . . . , N.

xi = 1,

N
i=1

¯i = R ¯ p, xi R

xi ≥ 0

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The efficient frontier can be determined by varying Rp between the return on the minimum variance portfolio and the return on the portfolio with the maximum variance.

5.4. Capital Market Equilibrium with Incomplete Information Merton’s (1987) model is based on the standard assumptions of frictionless markets, no transaction costs and no taxes, and borrowing and short selling without restrictions. There are n firms in the economy and N investors. Investors pay information costs λ before they include assets in their portfolios. It is important to regard information costs as: the cost of gathering and processing data, and the cost of information transmission from one party to another. In the literature of the principal agent and signalling models, the cost of transmitting information can be considerable. Investors pay information costs before they can process detailed information released from time to time about the firm. Information comes from the firm, stock market advisory services, brokerage houses, professional portfolio managers, etc. The background of information costs fits well with the theory of “generic” or “neglected” stocks which are not followed by large numbers of professional analysts. The relationship between the equilibrium market value Vk on firm k if all investors were informed about firm k and its value in the context of incomplete information is given by:     λk λk Vk∗ = Vk 1 + , hence Vk∗ = Vk∗ 1 + . R R The term λk reflects information costs on the asset k. It has dimensions of incremental expected rate of return and R refers to one plus the riskless rate. This equation shows that “the effect of incomplete information on equilibrium price is similar to applying an additional discount rate”. When information is complete, the model reduces to the standard capital asset pricing model of Sharpe (1964). In fact, if we define in a standard fashion: ˜ k, R ˜ M )/[var(R ˜ M )], then the equilibrium expected return on βk = cov(R security k can be written as ¯ M − R − λ m ) + λk . ¯ k = R + βk ( R R

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6. Option Pricing in a Discrete-Time Setting: The Cox, Ross and Rubinstein Model for Equity Options Cox, Ross and Rubinstein (1979) propose the first discrete-time model for the pricing of stock options. Rendleman and Barter (1980) develop a similar model for the pricing of interest rate sensitive instruments.

6.1. The Monoperiodic Model To illustrate the foundations of the binomial model, consider the following data: — — — —

Underlying asset price: S = 40, Strike price: K or E = 40, Riskless interest rate: r = 10% or R = 1 + r = 1.1, Time of maturity: 1 year.

At the end of the year, the underlying stock can increase by 20%, from 40 to (40 × 1.2), or 48, as it can decrease by the same amount from 40 to (40 × 0.8), or 32 as in Figure 19. The dynamics of the option is nearly similar to that of the underlying asset. The call option price at the maturity date is given by the greater of zero and the intrinsic value. As in Figure 20, the option price can go up to Cu or down to Cd . uS = 48

S = 40

dS = 32

or : u = (1 + 0.2) = 1.2

and

d = (1 − 0.2) = 0.8

Figure 19: One period binomial model. max (0, uS − E) = (0, 48 − 40) = 8 = Cu C= max (0, dS − E) = (0, 32 − 40) = 0 = Cd

Figure 20:

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uS − HC = 32 S – HC

dS − HC = 32

Figure 21: Dynamics of the hedge portfolio.

The strike price is often denoted by K or E. It is possible to construct an initial hedge portfolio using the underlying asset S and a certain number H of options as (S − HC). If this portfolio hedges the investor against risk, it must lead to the same result at the maturity date as in Figure 21. We can compute the number H as follows: H=

S(u − d) 40(1.2 − 0.8) = = 16/8 = 2. (Cu − Cd ) (8 − 0)

When the stock price increases, the value of the hedge portfolio is: uS − HCu = 1.2(40) − 2(8) = 48 − 16 = 32. When the stock price decreases, the value of the hedge portfolio is: dS − HCd = 0.8(40) − 2(0) = 32. What is the option price at time 0 in this simple binomial model? Since the initial portfolio value is (S − HC), its final value must be multiplied by the riskless rate since it is a hedge portfolio. The value of a hedged portfolio at the maturity date becomes R(S − HC). In order to avoid risk-less arbitrage, we must have: R(S − HC) = (uS − HCu ) which gives C = (S(R − u) + Hcu )/HR. Since the value of H is given by: H = (S(u − d))/(Cu − Cd ). The call price is given by 

 (R − d) (u − R) + Cd R. C = Cu (u − d) (u − d) This is the option price in a mono-periodic binomial model. Example. Using the following data: Cu = 8, Cd = 0, u = 1.2, d = 0.8, R = 1.1, the option price is: 

 (1.2 − 1.1) (1.1 − 0.8) +0 1.1 = (6 + 0)/1.1 = 5.4545. C= 8 (1.2 − 0.8) (1.2 − 0.8)

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The call price can also be written as: C = [pCu + (1 − p)Cd ]/R with p = (R − d)/(u − d), (1 − p) = (u − R)/(u − d) where p refers to the probability associated to an increase in the underlying asset price.

6.2. The Multiperiodic Model This simple mono-periodic model can be repeated N times to construct the multi-periodic binomial option pricing model. Time to maturity T is divided into N intervals of length t where the underlying asset price increases from S to uS or decreases from S down to dS with a probability p and (1 − p). In a risk-neutral world, the expected value of S is Sert . The expected value can also be calculated as follows: pSu + (1 − p)Sd. The equality between the two expected values gives: Sert = pSu + (1 − p)Sd. Simplifying by S gives: ert = pu + (1 − p)d. (5) √ The variance of S over the same time interval t is σ 2 S 2 t since the variance of a random variable X is given by E(X2 ) − E(X)2 . Calculating the variance and simplifying gives: σ 2 t = pu2 + (1 − p)d 2 − [pu+(1 − p)d]2 .

(6)

Using Eqs. (5) and (6), and u = 1/d, it is possible to show that the following relationships are verified: u = eσ

√ t

,

1 d= , 4

m = ert ,

p=

m−d . u−d

(7)

At each node, the underlying asset value can be written as Suj d i−j for j varying from 0 to i. The first index i correspond to the period and the second index j indicate the position. For example, when the option’s maturity date is in one period, i = 1 and j varies from 0 to i, i.e., 0 to 1. Using 0 for the lowest position at each period, when the underlying asset value decreases, we have: Su0 d 1−0 = Sd. When it increases, we have: Su1 d 1−1 = Su.

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The value of a European or an American option at each pair (i, j) is denoted by Fi,j . The option price at time 0 can be computed by a recursive starting from the maturity date T . The option price is given by its expected future value discounted to the present at the appropriate riskless rate. At maturity, the payoff from a European call is: FN,j = max[0, Suj d N−j − K]. The payoff from a European put: FN,j = max[0, K − Suj d N−j ]. The option value at each node can be computed using the two immediate successive nodes. The expected value must be discounted using the riskless rate as follows: Fi,j = e−rt · [pFi+1,j+1 + (1 − p)Fi+1,j ] for 0 ≤ i ≤ M − 1 and 0 ≤ j ≤ i. Since the value of an American call option must be at least equal to its intrinsic value, the following condition must be satisfied: Fi,j = max[Suj d i−j − K, e−rt (pFi+1,j+1 + (1 − p)Fi+1,j )]. The value of an American put option must satisfy the following condition: Fi,j = max[K − Suj d i−j , e−rt (pFi+1,j+1 + (1 − p)Fi+1,j )]. This model appears in CRR (1979), Cox and Rubinstein (1985), Hull (2000), etc.

6.3. Examples 6.3.1. Examples with Two Periods Consider the following data for the pricing of a European call: S = 100,

K = 100,

T = 1 year,

N = 2,

σ = 0.2,

r = 0.1.

In a first step, the values of the model parameters must be computed: u=e

√ σ t



=e

0.2

m = ert

1 2



1 1 −0.2 2 = 0.8681, = 1.1519, d = = e 4  1 m−d 0.1 2 = 1.0732, = 0.7227. =e p= u−d

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Suu = 132.68 Su =115.19 S = 100

Sud = 100 Sd = 86.81 Sdd = 75.36

Figure 22: Dynamics of the underlying asset price. S2,2 = 132.68 S1,1 = 115.19 S0,0 = 100

S2,1 = 100 S1,0 = 86.81 S2,0 = 75.36

Figure 23: Dynamics of the underlying asset price. C2,2 = 132.68 − 100 = 32.68 C1,1 = ? C0,0 = ?

C2,1 = 100 −100 = 0 C1,0 = ? C2,0 = 75.36 − 100 = 0

Figure 24: Dynamics of the option price.

For an initial value of S = 100, the two possible values in the next period are: Su = 100(1.1519) = 115.19,

Sd = 100(0.8681) = 86.81.

These two values of the underlying asset price lead to three possible values as: Suu = 115.19(1.1519) = 132.68, Sud = 115.19(0.8681) = 100, Sdd = 86.81(0.8681) = 75.36. Using the index representation, (i, j), we have Figure 23. In the above representation, S0,0 refers to the initial time 0 and the lowest position 0. S1,0 corresponds to period 1 and the lowest position 0. S1,1 refers to period 1 and the first position after 0, i.e., 1. The dynamics of the option price are given in Figure 24. The option value can be computed by simple application of the following formula: Fi,j = e−rt · [pFi+1,j+1 + (1 − p)Fi+1,j ].

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The option value at node (1, 1), i.e., C1,1 is given by: C1,1 = e−rt · [pC2,2 + (1 − p)C2,1 ] or: C1,1 = e−0.1(1/2) · [0.7227(32.68) + (1 − 0.7227)0] = 22.465. The option value at node (1, 0), i.e., C1,0 is: C1,0 = e−rt · [pC2,1 + (1 − p)C2,0 ] or: C1,0 = e−0.1(1/2) · [0.7227(22.465) + (1 − 0.7227)0] = 0. Using the possible values in one period, what is the option price at time 0 ? Using the same formula, C0,0 is given by: C0,0 = e−rt · [pC1,1 + (1 − p)C1,0 ] or : C0,0 = e−0.1(1/2) · [0.7227(0) + (1 − 0.7227)0] = 15.443. The option value at time 0 is 15.443.

6.3.2. Other Applications of the Binomial Model of Cox,Ross and Rubinstein for Two Periods Consider the following data for the valuation of European and American call and put options: S = 100, K = 100, r = 5%, σ = 30%, N = 2, T = 1 year. European call prices: Using the above date, the dynamics of the underlying asset are given by: 152.84

u²S u 123.63 u

uS

d

S

S 00

u

100

d

dS 80.88

d

d ²S 65.41

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In this case, option values are given by: C2,2 = 52.84,

C1,1 = 0,

C2,0 = 0,

C1,1 =

0.5064 ∗ 52.84 + 0 ∗ (1 − 0.5064) = 26.105, 1.025

C0,0 =

0.5064 ∗ 26.105 + 0 = 12.897. 1.025

C1,0 = 0

Hence, the option price is 12.897. European put prices: Option prices are given by: P2,2 = 0,

P2,1 = 0,

P2,0 = 100 − 65.41 = 34.59,

P1,1 =

0.5064 ∗ 0 + 0 = 0, 1.025

P1,0 =

0.5064 ∗ 0 + 0.4936 ∗ 34.59 = 16.657, 1.025

P0,0 =

0 + 0.4936 ∗ 16.657 = 8.021. 1.025

Hence, the option price is 8.021. We can check that the put–call parity theorem is verified: P = C + Ke−rt − S = 12.897 + 100e−0.05∗1 − 100 = 8.02. American call prices: Option prices are computed as:

0.5064 ∗ 52.84 + 0.4936 ∗ 0 C1,1 = max ; 23.63 = 26.105 1.025

0.5064 ∗ 0 + 0 C1,0 = max ;0 = 0 1.025 C0,0 =

0.5064 ∗ 26.105 = 12.88. 1.025

American put prices: Option prices are computed as:

0.5064 ∗ 0 + 0.4936 ∗ 0 P1,1 = max ; 100 − 80.88 = 0 1.025

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P1,0 = max P0,0 =

45

0.5064 ∗ 0 + 0.4936 ∗ 34.59 ; 100 − 80.88 = 19.12 1.025

0.5064 ∗ 0 + 0.4936 ∗ 19.12 = 9.2. 1.025

6.3.3. Applications of the Binomial Model of Cox, Ross and Rubinstein for Three Periods Consider the following data for the valuation of European and American options: S = 100, K = 100, r = 5%, σ = 30%, N = 3, T = 1 year. European call prices: Using the above data, we have √ t

1 = 0.33, 3 √ 1 1 u = e0.30 0.33 = 1.1881, d = = = 0.8417, u 1.1881 ert − d e0.05∗0.33 − 0.8417 p= = = 0.5050. u−d 1.1881 − 0.8417 u = eσ

with Nt = T ⇔ t =

167.71 3 uS

141.16

u²S 118.81 118.81

uS

uS S dS

S 00

84.17

dS 84.17

d ²S 70.85

3 dS

59.63

The option pay-off at maturity is: C3,3 = 167.71 − 100 = 67.71, C3,2 = 118.81 − 100 = 18.81, C3,1 = 84.17 − 100 = 0, C3,0 = 59.63 − 100 = 0.

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Before maturity, option prices are computed as: C2,2 =

0.5050 ∗ 67.71 + 0.495 ∗ 18.81 = 42.78, 1.017

C2,1 =

0.5050 ∗ 18.81 + 0.495 ∗ 0 = 9.34, 1.017

C2,0 =

0.5050 ∗ 0 + 0.495 ∗ 0 = 0, 1.017

C1,1 =

0.5050 ∗ 42.78 + 0.495 ∗ 9.34 = 25.79, 1.017

C1,0 =

0.5050 ∗ 9.34 + 0.495 ∗ 0 = 4.64, 1.017

0.5050 ∗ 25.79 + 0.495 ∗ 4.64 = 15.06. 1.017 The option price is C0,0 = 15.06. C0,0 =

European put prices: The option pay-off at maturity is given by: P3,3 P3,2 P3,1 P3,0

= 100 − 167.71 = 0, = 100 − 118.81 = 0, = 100 − 84.17 = 15.83, = 100 − 59.63 = 40.37.

Before maturity, option prices are given by: P2,2 =

0.5050 ∗ 0 + 0.495 ∗ 0 = 0, 1.017

P2,1 =

0.5050 ∗ 0 + 0.495 ∗ 15.83 = 7.70, 1.017

P2,0 =

0.5050 ∗ 15.83 + 0.495 ∗ 40.37 = 27.51, 1.017

P1,1 =

0.5050 ∗ 0 + 0.495 ∗ 7.70 = 3.75, 1.017

P1,0 =

0.5050 ∗ 7.70 + 0.495 ∗ 27.51 = 17.21, 1.017

P0,0 =

0.5050 ∗ 3.75 + 0.495 ∗ 17.21 = 10.24. 1.017

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We can check the put call parity relationship in this context: P = C + Ke−rt − S = 15.06 + 100e−0.05∗1 − 100 = 10.2. American call prices: American call option prices are computed as follows:

0.5050 ∗ 27.71 + 0.495 ∗ 18.81 C2,2 = max ; 141.16 − 100 = 42.78 1.017

0.5050 ∗ 18.81 + 0.495 ∗ 0 C2,1 = max ; 100 − 100 = 9.34 1.017

0.5050 ∗ 0 + 0.495 ∗ 0 C2,0 = max ;0 = 0 1.017

0.5050 ∗ 42.78 + 0.495 ∗ 9.34 C1,1 = max ; 118.8 − 100 = 25.79 1.017

0.5050 ∗ 9.34 + 0.495 ∗ 0 C1,0 = max ; 84.17 − 100 = 4.64 1.017 0.5050 ∗ 25.79 + 0.495 ∗ 4.64 = 15.06. 1.017 American put prices: American put prices are computed as follows:

0.5050 ∗ 0 + 0.495 ∗ 0 P2,2 = max ; 100 − 141.16 = 0 1.017

0.5050 ∗ 0 + 0.495 ∗ 15.83 P2,1 = max ; 100 − 100 = 7.70 1.017

0.5050 ∗ 15.83 + 0.495 ∗ 40.37 P2,0 = max ; 100 − 70.85 = 29.15 1.017

0.5050 ∗ 0 + 0.495 ∗ 7.70 P1,1 = max ; 100 − 118.8 = 3.75 1.017

0.5050 ∗ 7.70 + 0.495 ∗ 29.15 P1,0 = max ; 100 − 84.17 = 18.01 1.017 C0,0 =

P0,0 =

0.5050 ∗ 3.75 + 0.495 ∗ 18.01 = 10.63. 1.017

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6.3.4. Applications of the Binomial Model of Cox, Ross and Rubinstein for Four Periods Consider the following data for the valuation of European and American call and put options: S = 100,

r = 5%, σ = 30%, T = 1 year. √ √ 1 N = 4, N ∗ t = T ⇔ t = = 0.25u = eσ t = e0.30∗ 0.25 = 1.1618 4 1 1 d= = = 0.8607, u 1.1618 p=

E = 100,

ert − d e0.05∗0.25 − 0.8607 = = 0.5044. u−d 1.1618 − 0.8607

European call prices: Using the above data, the dynamics of the underlying asset are given by: 182.19 4 uS

156.82

3 uS

134.98

2 uS

134.98 2 uS

116.18

uS

116.18

uS S

100

S

100

S 00

dS dS

86.07

86.07 d ²S

74 .08

74.08

d ²S

3 dS

63.76 54.88

4 d S

Option prices at maturity are given by: C4,4 = 182.19 − 100 = 82.19,

C4,3 = 134.98 − 100 = 34.98,

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C4,2 = 100 − 100 = 0, C4,1 = 74.08 − 100 = 0, C4,0 = 54.88 − 100 = 0. Before maturity, option prices are given by: C3,3 =

0.5044 ∗ 82.19 + 0.4956 ∗ 34.98 = 58.04, 1.013

C3,2 =

0.5044 ∗ 34.98 + 0.4956 ∗ 0 = 17.42, 1.013

C3,1 =

0.5044 ∗ 0 + 0.4956 ∗ 0 = 0, 1.013

C3,0 =

0.5044 ∗ 0 + 0.4956 ∗ 0 = 0, 1.013

C2,2 =

0.5044 ∗ 58.04 + 0.4956 ∗ 17.42 = 37.42, 1.013

C2,1 =

0.5044 ∗ 17.42 + 0.4956 ∗ 0 = 8.67, 1.013

C2,0 =

0.5044 ∗ 0 + 0.4956 ∗ 0 = 0, 1.013

C1,1 =

0.5044 ∗ 37.42 + 0.4956 ∗ 8.67 = 22.87, 1.013

C1,0 =

0.5044 ∗ 8.67 + 0.4956 ∗ 0 = 4.32, 1.013

C0,0 =

0.5044 ∗ 22.87 + 0.4956 ∗ 4.32 = 13.50. 1.013

The European call price is 13.50. European put prices: At maturity, the option pay-off is given by: P4,4 = 100 − 182.19 = 0, P4,3 = 100 − 134.98 = 0, P4,2 = 100 − 100 = 0, P4,1 = 100 − 74.08 = 25.92, P4,0 = 100 − 54.88 = 45.12,

P3,3 =

0.5044 ∗ 0 + 0.4956 ∗ 0 = 0, 1.013

49

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0.5044 ∗ 0 + 0 = 0, 1.013 0.5044 ∗ 0 + 0.4956 ∗ 25.92 P3,1 = = 12.68, 1.013 0.5044 ∗ 25.92 + 0.4956 ∗ 45.12 P3,0 = = 34.98. 1.013 P3,2 =

Option prices are given by: P2,2 =

0.5044 ∗ 0 + 0.4956 ∗ 0 = 0, 1.013

P2,1 =

0.5044 ∗ 0 + 0.4956 ∗ 12.68 = 6.2, 1.013

P2,0 =

0.5044 ∗ 12.68 + 0.4956 ∗ 34.98 = 23.43, 1.013

P1,1 =

0.5044 ∗ 0 + 0.4956 ∗ 6.2 = 3.03, 1.013

P1,0 =

0.5044 ∗ 6.2 + 0.4956 ∗ 23.43 = 14.55, 1.013

P0,0 =

0.5044 ∗ 22.87 + 0.4956 ∗ 4.32 = 8.63. 1.013

We can check that the put–call relationship is verified in this context: P = C + Ke−rt − S = 13.50 + 100e−0.05∗1 − 100 = 8.62. American call prices: American call option prices are computed as follows:

C3,3 C3,2

0.5044 ∗ 82.19 + 0.4956 ∗ 34.98 = max ; 156.82 − 100 = 58.04, 1.013

0.5044 ∗ 34.98 + 0.4956 ∗ 0 = max ; 116.18 − 100 = 17.42, 1.013

C3,1 = max(0; 0) = 0,

C3,0 = max(0; 0) = 0.

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C2,2 = max(37.42; 134.98 − 100) = 37.42, C2,1 = max(8.67; 0) = 8.67,

C2,0 = max(0; 0) = 0,

C1,1 = max(22.87; 16.18) = 22.87, C1,0 = max(4.32; 0) = 4.32, C0,0 =

0.5044 ∗ 22.87 + 0.4956 ∗ 4.32 = 13.50. 1.013

American put prices: American put prices are computed as follows: P3,3 = max(0; 100 − 156.82) = 0, P3,2 = max(0; 100 − 116.18) = 0,

0.5044 ∗ 0 + 0.4956 ∗ 25.92 P3,1 = max ; 100 − 86.07 = 13.93, 1.013

0.5044 ∗ 25.92 + 0.4956 ∗ 45.12 P3,0 = max ; 100 − 63.76 = 36.24, 1.013

0.5044 ∗ 0 + 0.4956 ∗ 0 P2,2 = max ; 0 = 0, 1.013

0.5044 ∗ 0 + 0.4956 ∗ 13.93 P2,1 = max ; 0 = 6.81, 1.013

0.5044 ∗ 13.93 + 0.4956 ∗ 36.24 P2,0 = max ; 25.92 = 25.92, 1.013

0.5044 ∗ 0 + 0.4956 ∗ 6.81 P1,1 = max ; 0 = 3.33, 1.013

0.5044 ∗ 6.81 + 0.4956 ∗ 25.92 P1,0 = max ; 13.93 = 16.07, 1.013 P0,0 =

0.5044 ∗ 3.33 + 0.4956 ∗ 16.07 = 9.52. 1.013

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6.3.5. Other Applications of the Cox, Ross and Rubinstein for Four Periods Consider the valuation of European and American call and put options in the following context: S = 100, K = 100, σ = 20%, r = 14%, N = 4, T = 1. Since T = 1, we have ⇒ T =

1 T = = 0.25, N 4

u = e0.2

√ 1/4

= 1.105170916,

1 = 0.904837418, p = erT − d = 0.6528228716, u 1 − p = 0.3471771284, m = erT = 1.035619709. d=

The price dynamics of the underlying asset are given in the following figure: Dynamics of the underlying asset 149.1824698 134.9858808

S 44

S 33 122.1402758 110.5170918

S 22

S 11

110.5170918 100

S 21 S 00 100

S 43 S 32 90.4837418

S 31 S 10 90.4837418

S 42 100 81.67307531

S 41

S 20 81.67307531

122.1402758

S 30 74.08182207

S 40 67.0320046

Using the initial underlying asset value S00 = 100, we can generate all the other values using: Sij = S · uj · d i−j with i ∈ {0, 1, 2, 3, 4} and j ∈ {0, 1, 2, 3, 4}. For example, S10 = 100∗1.1051709180 ∗0.904837481 = 904837418. The European call price can be computed as follows:

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Computation of the European call price 49.18246976 38.42533913

C 44

C 33

22.14027582

28.90089383 21.16757008

C 11

15.20255587

C 43 C 22

C 32

8.797780777

0

C 21

C 00 5.545850916

C 42

13.95655018

C 31

0

C 10

0

C 41

C 20 0

C 30 C 40

0

0

At maturity, option prices are computed as: C4j = Max{0; Sij − K} with j ∈ {0, 1, 2, 3, 4},

C40 = 0.

The other values are computed as: Ci,j =

Ci+1,j+1 ∗ p + Ci+1,j ∗ (1 − p) for i < 4, m

where p is the probability. European put prices: The same underlying asset prices can be used in the computation of the European put price.

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0 0

P 44

P 33

0

P 43 0

0

P 22

0

0.6829305332

P 32

P 42

P11

P21

2.138379407

P 31

2.037162768

P 00

6.076799822

P 41 18.12692469

P10 5.094561371

P 20 P 30

32.9679954

22.47871956

P 40

11.36630668

At maturity, European put prices are given by: P4j = Max{0; K − S4j } with j ∈ {0, 1, 2, 3, 4}. Before maturity, put prices are computed as: Pij =

Pi+1,j+1 ∗ p + Pi+1,j ∗ (1 − p) m

for i < 4.

American call prices: 49.18246976 38.42533913

C 44

C 33 C 43 28.90089383 21.16757008

15.20255587

C 11

C 00

22.14027582

C 22

C 32 13.95655018

C 21

0

C 31

8.797780777

0

C 10 5.545850916

C 42

C 41 0

C 20 C 30 0 0

C 40 0

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The following condition must be satisfied at each node in the computation of the American call price:   Ci+1,j+1 ∗ p + Ci+1,j ∗ (1 − p) Cij = Max Sij − E; for i < 4. m Option values are computed as follows: C33 = Max{13498 − 100; 38.42539913} = 38.42539913, C32 = Max{110.5170918 − 100; 13.95655018}, C31 = Max{90.483741 − 100; 0}, C30 = Max{74.08182207 − 100; 0}, C22 = Max{122.1402758 − 100; 28.90089383}, C21 = Max{100 − 100; 8.797780777}, C20 = Max{81.87307531 − 100; 0}, C11 = Max{110.5170918 − 100; 21.16757008}, C00 = 15.20255587. The American call price is 15.20255587. American put prices: The same procedure is used for the computation of option prices. 0

P 44

0 0

1.069468054

P11

3.864353262

P 00

P 33

0

P 22 3.190193433

P 32

9.5162582 18.12692469

P 20

P 42

0

P 21 P10

P 43

P 31

18.12692469 9.5162582

P 41 25.91817793

P 30

32.9679954

P 40

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The following condition must be satisfied before maturity:   Pi+1,j+1 ∗ p + Ci+1,j ∗ (1 − p) for i < 4, Pij = Max K − Sij ; m or: P33 = Max{100 − 134.98, 58.808 − 100; 0}, P32 = Max{100 − 110.5170918; 0}, P31 = Max{100 − 90.4837418; 6.0767}, P30 = Max{100 − 74.08182207; 22.47}, P22 = Max{100 − 122.14; 0}, P21 = Max{100 − 100; 0},  P20 = Max 100 − 81.87307531; 9.5162582 ∗ p + 25.918177930(1 − p) m P11 P10

 = 14.68746632,

  3.190193433(1 − P) = Max 100 − 110.5170918; = 1.069468054, m   1.069468054 ∗ P + 9.5162582(1 − P) = Max 100 − 90.4837418; m = 3.864353262.

Hence, the option price is 3.86435.

6.3.6. Examples with Five Periods Example: Consider the valuation of European and American options in the following context: Underlying asset, S = 100, strike price K = 100, interest rate = 0.1, volatility = 0.4, T = 5 months, N = 5. In this case, we have : p = 0.5073, d = 0.8909, u = 1.1224.

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Dynamics of the underlying asset price

178.1312 158.7055

141.3982 125.9784 112.2401 100

141.3982 125.9784

112.2401 100.0000

89.0947

112.2401 100.0000

89.0947 79.3787

89.0947 79.3787

70.7222

70.7222 63.0098 56.1384

The valuation of European put options 0.0000 0.0000 0.0000 1.2720 4.2282 8.6380

0.0000 0.0000

2.6033 7.3442

13.3256

0.0000 5.3282

12.3506 19.7110

10.9053 19.7914

27.6249

29.2778 36.1603 43.8616

The valuation of American put options 0.0000 0.0000 0.0000 1.2720 4.3250 8.9769

0.0000 0.0000

2.6033 7.5423

13.9195

0.0000 5.3282

12.7561 20.7226

10.9053 20.6213

29.2778

29.2778 36.9902 43.8616

57

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The valuation of European call options 78.1312 59.5354 43.0511 29.7194 19.7467 12.7191

41.3982 26.8082

16.4963 9.8132

5.6987

12.2401 6.1581

3.0982 1.5587

0.0000 0.0000

0.0000

0.0000 0.0000 0.0000

The valuation of American call options 78.1312 59.5354 43.0511 29.7194 19.7467 12.7191

41.3982 26.8082

16.4963 9.8132

5.6987

12.2401 6.1581

3.0982 1.5587

0.0000 0.0000

0.0000

0.0000 0.0000 0.0000

7. The Binomial Model and the Distributions to the Underlying Assets We denote by: C(·), c(·): prices of American and European call options, P(·), p(·): prices of American and European put options.

7.1. The Model The model is a simple extension of the basic lattice approach which has the flexibility to account for the magnitude and the timing of dividends, and

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other cash payments. The basic lattice approach suggested by CRR (1979) considers the situation where there is only one state variable: the price of a non-dividend paying stock. The time to maturity of the option is divided into N equal intervals of length t during which the stock price moves from its initial value S to one of two new values Su and Sd with probabilities p and (1 √− p). When √ u = 1/d, it can be shown that: p = (a − d)/(u − d), u = eσ t , d = e−σ t , a = , ert . The nature of the lattice of stock prices is completely specified and the nodes correspond to Suj d i−j for j = 0, 1, . . . , i. The option is evaluated by starting at time T and working backward. We denote by Fi,j , the option value at time t + it when the stock price is Suj d i−j . At time t + it, the option holder can choose to exercise the option and receives the amount by which K (or S) exceeds the current stock price (or K) or wait. The American call is given by: Fi,j = max[Suj d i−j − K, e−rt (pFi+1,j+1 + (1 − p)Fi+1,j )]. The American put is given by: Fi,j = max[K − Suj d i−j , e−rt (pFi+1,j+1 + (1 − p)Fi+1,j )]. The extension of the lattice approach to the valuation of American options on stocks paying a known cash income is as follows. When there is only one cash income at date, τ, between kt and (k + 1)t, it is possible to design trees where the number of nodes at time t is always (i + 1). The analysis which parallels that in Hull (2000) and Briys et al. (1998) can be simplified by assuming that the implicit spot stock price has two components: a part which is stochastic and a part which is the present value of all future cash payments during the option’s life. When there is just one ex-cash income date τ, during the option’s life and kt ≤ τ ≤ (k + 1)t, then at time x, the value of the stochastic component S is given by: S ∗ (x) = S(x), when x > τ S ∗ (x) = S(x) − (Di + Ri )e−r(τ−x) ,

when x ≤ τ.

Assume σ ∗ is the constant volatility of S ∗ . Using the parameters p, u, and d, at time t + it, the nodes on the tree define the stock prices: If it < τ: S ∗ (t)uj d i−j + (Di + Ri )e−r(τ−it) , If it ≥ τ: S ∗ (t)uj d i−j , j = 0, 1, . . . , i.

j = 0, 1, . . . , i.

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7.2. Simulations for a Small Number of Periods in the Presence of Dividends Example 1. Applications of the Cox, Ross and Rubinstein model for five periods. Consider the following data for the valuation of European and American call and put options: S = 110, K = 115, r = 10%, σ = 40%, N = 5, t = 5 months, t = 1 month. Date of dividend: 105 days, Dividend amount: D = 10. The European call price: Using the above data, the dynamics of the underlying asset are computed using: u = eσ

√ t

,

u = e0.4

√ 1/2

,

or u = 1.1224,

d=

ert − d = 0.5073, u−d S ∗ = S − D = 110 − 10 = 100, S0,0 = S ∗ u0 d 0 + De−r(105/365) = 109.7164, S1,1 = S ∗ u1 d 0 + De−r((105/365)−(1/12)) = 100(1.1224), + 10De−r((105/365)−(1/12)) = 122.0377, S1,0 = S ∗ u0 d 1 + De−r((105/365)−(1/12)) = 98.8925, S2,2 = S ∗ u2 d 0 + De−r((105/365)−(2/12)) = 135.8579, S2,1 = S ∗ u1 d 1 + De−r((105/365)−(2/12)) = 109.8797, S2,0 = S ∗ u0 d 2 + De−r((105/365)−(2/12)) = 89.2586, S3,3 = S ∗ u3 d 0 + De−r((105/365)−(3/12)) = 151.3603, S3,2 = S ∗ u2 d 1 + De−r((105/365)−(3/12)) = 122.2024, S3,1 = S ∗ u1 d 2 + De−r((105/365)−(3/12)) = 99.0572, S3,0 = S ∗ u0 d 3 + De−r((105/365)−(3/12)) = 80.6848, S4,4 = S ∗ u4 d 0 = 158.7050, S4,3 = S ∗ u3 d 1 = 125.782, S4,2 = S ∗ u2 d 2 = 100.00, S4,1 = S ∗ u1 d 3 = 79.3788, S4,0 = S ∗ u0 d 4 = 63.0100, S5,5 = S ∗ u5 d 0 = 178.1305, S5,4 = S ∗ u4 d 1 = 141.3979, S5,3 = S ∗ u3 d 2 = 112.2400, p=

1 = 0.8909, u

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S5,2 = S ∗ u2 d 3 = 89.0948, S5,1 = S ∗ u1 d 4 = 70.7224, S5,0 = S ∗ u0 d 5 = 56.1386. The dynamics of the underlying asset are represented in the following way: S 5,5

S 4,4 S 3,3 S 2,2

S 4,3 S 5,3

S 3,2

S 1,1

S 4,2

S 2,1

S 00

S 5,4

S 5,2

S 3,1 S 1,0

S 4,1 S 5,1

S 2,0 S 3,0 S 4,0

S 5,0

The option’s maturity value is computed as: C5,5 C5,4 C5,3 C5,2 C5,1 C5,0

= max[0; S5,5 − K] = 63.1305, = max[0; S5,4 − K] = 26.3979, = max[0; S5,3 − K] = 0, = max[0; S5,2 − K] = 0, = max[0; S5,1 − K] = 0, = max[0; S5,0 − K] = 0.

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The American call option price is computed as:  C4,4 = max

 p × C5,5 + q × C5,4 ; IV = max[44.6586; S4,4 − K] ert

= max[44.6586; 43.7050] = 44.6586 where IV stands for the option intrinsic value. 

C4,3

C4,2 C4,1 C4,0 C3,3

C3,2

C3,1 C3,0

p × C5,4 + q × C5,3 = max ; S4,3 − K ert



= max[13.2805; 10.9782] = 13.2805,   p × C5,3 + q × C5,2 ; max[0; S4,2 − K] = 0, = max ert   p × C5,2 + q × C5,1 = max ; max[0; S4,1 − E] = 0, ert   p × C5,1 + q × C5,0 = max ; max[0; S4,0 − K] = 0, ert   p × C4,4 + q × C4,3 = max ; S3,3 − K ert = max[28.9563; 36.3603] = 36.3603,   p × C4,3 + q × C4,2 = max ; S3,2 − K ert = max[6.6813; 7.2024] = 7.2024,   p × C4,2 + q × C4,1 = max ; S3,1 − K = 0, ert   p × C4,1 + q × C4,0 = max ; S3,0 − K = 0. ert

American call option prices are computed as follows:

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63.105 44.6586 36.3603 21.8117 12.7437

26.3979 13.2805

7.2024 0

7.3019

3.6235 0 1.8229

0

0

0

0 0

0 0 0

 C2,2 = max

C2,1

C2,0 C1,1

C1,0

p × C3,3 + q × C3,2 ; S2,2 − K ert



= max[21.8117; 20.8579] = 21.8117,   p × C3,2 + q × C3,1 = max ; max[0; S2,1 − K] ert = max[3.6235; 0] = 3.6235,   p × C3,1 + q × C3,0 = max ; max[0; S2,0 − K] = 0, ert   p × C2,2 + q × C2,1 = max ; max[0; S1,1 − K] ert = max[12.7437; 7.0377] = 12.7437,   p × C2,1 + q × C2,0 = max ; max[0; S1,0 − K] ert

63

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C0,0

= max[1.8229; 0] = 1.8229,   p × C1,1 + q × C1,0 = max ; max[0; S0,0 − K] ert = max[7.3019; 0] = 7.3019.

The American put price: Using the above data, the American put price is computed as follows at different nodes: P5,5 P5,4 P5,3 P5,2 P5,1 P5,0

= max[0; K − S5,5 ] = 0, = max[0; K − S5,4 ] = 0, = max[0; K − S5,3 ] = 2.76, = max[0; K − S5,2 ] = 25.9052, = max[0; K − S5,1 ] = 44.2776, = max[0; K − S5,0 ] = 58.8614.

Before maturity, the option price is computed as:   p × P5,5 + q × P5,4 P4,4 = max ; max[0; K − S4,4 ] = 0, ert   p × P5,4 + q × P5,3 P4,3 = max ; max[0; K − S4,3 ] ert

P4,2

P4,1

P4,0

= max[1.3486; 0] = 1.3486,   p × P5,3 + q × P5,2 = max ; max[0; K − S ] 4,2 ert = max[14.0461; 0] = 15,   p × P5,2 + q × P5,1 = max ; max[0; K − S4,1 ] ert = max[34.6672; 35.6212] = 35.6212,   p × P5,1 + q × P5,0 = max ; max[0; K − S4,0 ] ert = max[51.0360; 51.9900] = 51.9900.

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Option prices are reported in the following figure: 0

0 0

0.6589 1.3486

2.76

4.2441 8.0076

15

10.0605

25.9052

16.2201 24.9513

17.0991

35.6212

24.6367

44.2776

33.7211 43.3236

51.9900 58.8614



P3,3

P3,2

P3,1



p × P4,4 + q × P4,3 ; max[0; K − S3,3 ] = max ert

= max[0.6589; 0] = 0.6589,   p × P4,3 + q × P4,2 = max ; max[0; K − S3,2 ] ert = max[8.0076; 0] = 8.0076,   p × P4,2 + q × P4,1 = max ; max[0; K − S ] 3,1 ert = max[24.9513; 15.9428] = 24.9513,

65

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 P3,0 = max

P2,2

P2,1

P2,0

P1,1

P1,0

P0,0



p × P4,1 + q × P4,0 ; max[0; K − S3,0 ] ert

= max[43.3236; 34.3152] = 43.3236,   p × P3,3 + q × P3,2 = max ; max[0; K − S ] 2,2 ert = max[4.2441; 0] = 4.2441,   p × P3,2 + q × P3,1 = max ; max[0; K − S2,1 ] ert = max[16.2201; 5.1203] = 16.2201,   p × P3,1 + q × P3,0 = max ; max[0; K − S2,0 ] ert = max[33.7211; 25.7414] = 33.7211,   p × P2,2 + q × P2,1 = max ; max[0; K − S1,1 ] ert = max[10.0605; 0] = 10.0605,   p × P2,1 + q × P2,0 = max ; max[0; K − S1,0 ] ert = max[24.6367; 16.1075] = 24.6367,   p × P1,1 + q × P1,0 ; max[0; K − S0,0 ] = max ert = max[17.0991; 5.2836] = 17.0991.

Example 2. Consider the valuation of European and American options in the following context: Underlying asset, S = 100, strike price K = 100, interest rate = 0.1, volatility = 0.4, T = 5 months, N = 5, dividend = 10, dividend date = 105. In this case, we have : p = 0.5073, d = 0.8909, and u = 1.1224.

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67

178.1312 158.7055

151.3606 135.8581 122.0378 110

141.3982 125.9784

122.2025 109.8797

98.8925

112.2401 100.0000

99.0571 89.2584

89.0947 79.3787

80.6846

70.7222 63.0098 56.1384

The valuation of European put options 0.0000 0.0000 0.0000 1.2720 4.2282 8.6380

0.0000 0.0000

2.6033 7.3442

13.3256

0.0000 5.3282

12.3506 19.7110

10.9053 19.7914

27.6249

29.2778 36.1603 43.8616

The valuation of American put options 0.0000 0.0000 0.0000 1.2720 4.3250 8.8801

0.0000 0.0000

2.6033 7.5423

13.7214

0.0000 5.3282

12.7561 20.3171

10.9053 20.6213

28.4479

29.2778 36.9902 43.8616

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The valuation of European call options 78.1312 59.5354 43.0511 29.7194 19.7467 12.7191

41.3982 26.8082

16.4963 9.8132

5.6987

12.2401 6.1581

3.0982 1.5587

0.0000 0.0000

0.0000

0.0000 0.0000 0.0000

The valuation of American call options 78.1312 59.5354 51.3606 36.6880 24.6554 15.8944

41.3982 26.8082

22.2025 12.6840

7.1430

12.2401 6.1581

3.0982 1.5587

0.0000 0.0000

0.0000

0.0000 0.0000 0.0000

Example 3. Consider the valuation of European and American options in the following context: Underlying asset, S = 80, strike price K = 100, interest rate = 0.1, volatility = 0.4, T = 5 months, N = 5, dividend = 10, dividend date = 105. In this case, we have: p = 0.5073, d = 0.8909, u = 1.1224.

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Dynamics of the underlying for five periods

69

142.5050 126.9644 123.0810

110.6624 99.5898 90

113.1186 100.7827

99.7545 89.8797

81.0735

89.7921 80.0000

81.2382 73.3827

71.2758 63.5030

66.5402

56.5778 50.4078 44.9107

Valuation of European put options 0.0000 0.0000 2.4369 7.0283 12.9178 19.3423

0.0000 4.9875

11.8756 19.2016

26.2863

10.2079 19.1701

27.0714 34.0280

28.7242 35.6672

41.7694

43.4222 48.7623 55.0893

Valuation of American put options 0.0000 0.0000 2.4369 7.2265 13.3136 19.8853

0.0000 4.9875

12.2811 19.8077

26.9900

10.2079 20.0000

27.8943 34.8442

28.7242 36.4970

42.5923

43.4222 49.5922 55.0893

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Valuation of European call options 42.5050 27.7943 17.2083 10.2801 5.9882 3.4234

13.1186 6.6001

3.3206 1.6706

0.8405

0.0000 0.0000

0.0000 0.0000

0.0000 0.0000

0.0000

0.0000 0.0000 0.0000

Valuation of American call options 42.5050 27.7943 23.0810 13.2347 7.4747 4.1713

13.1186 6.6001

3.3206 1.6706

0.8405

0.0000 0.0000

0.0000 0.0000

0.0000 0.0000

0.0000

0.0000 0.0000 0.0000

Summary This chapter develops the main concepts regarding the pricing of assets and derivatives in a simple discrete-time context. The analysis in a discrete-time setting is more intuitive and provides the foundations about the convergence to continuous-time models. This chapter provides the reader with the main concepts and tools used in this book regarding risk, return, uncertainty, incomplete information and asset pricing in an intuitive manner. The most well known strategies in portfolio management involve combinations of options. They include vertical spreads, calendar spreads, diagonal spreads, ratio spreads, volatility spreads and synthetic contracts.

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A vertical spread involves the purchase of an option and the sale of an other with the same time to maturity and a different strike price. When the strategy produces a cash-out flow, we say that the investor is long the spread. When the strategy generates a cash-inflow, the investor is said short the spread. The strategy can be implemented by calls or puts using different strike prices. A vertical bull spread is implemented when an at-the-money option is bought and an out-of-the-money option is sold. A calendar spread strategy represents a position where the investor is long an option with a longer term and short an option with a short term for the same strike price. A diagonal spread involves the purchase of an option with a longer term and the sale of an other with a short term where both options have different strike prices. A bullish diagonal spread is implemented when the purchased option is at parity and the short option is out-of-the-money. The option price depends on the underlying asset price S, the strike price K, the interest rate r, the time to maturity, T , the volatility σ and dividend payouts. The option maturity corresponds to the number of days until expiration. It is often given in a fraction of a year or in days. The dividends must be known or estimated before using an option pricing model. Asset pricing and option pricing are based on the concepts of risk and return. Portfolio theory provides the basis for asset valuation. Sharpe (1963), among others, shows how an investor should divide wealth between risky assets and a riskless asset. He shows that “ the proportionate composition of the non-cash assets is independent of their aggregate share of the investment balance. This fact makes it possible to describe the investor’s decisions as if there was a single non-cash asset, a composite formed by combining the multitude of actual non-cash assets in fixed proportions”. Sharpe (1963) develops a simplified model for portfolio analysis by observing that stocks are likely to co-move with the market. The contributions of Markowitz and Sharpe were honored by the first Nobel Prize in financial economics in 1991. The analysis in Merton’s (1987) provides a simple model of capital market equilibrium with incomplete information. He shows that a reconciling of finance theory with empirical violations of the complete-information, perfect market model need not imply a departure from the standard paradigm. As Merton (1987) asserts “Financial markets dominated by rational agents may nevertheless produce anomalous behavior relative to the perfect

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market model. Institutional complexities and information costs may cause considerable variations in the time scales over which different types of anomalies are expected to be eliminated in the market place”. These models can be applied for the valuation of derivative assets. We present also a discrete-time approach for the valuation of options. The analysis is based on the standard model of Cox, Ross and Rubinstein (1979). This binomial model can be used for the valuation of options on different underlying assets.

Questions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

Define the strategy of buying (selling) calls. Define the strategy of buying (selling) puts. Define the strategy of buying (selling) spreads. Define the strategy of buying (selling) combinations. What are the determinants of an option price? Define the specific features of long (or short) a straddle. Define the specific features of long (or short) a strangle. Define the specific features of long (or short) a tunnel. Define the specific features of long (or short) spreads. Define the specific features of long (or short) butterfly. What is risk? How can we compute expected return? How is the efficient frontier derived? How is the Capital Asset Pricing Model derived? Describe Merton’s (1987) simple model of capital market equilibrium with incomplete information. Describe the Cox, Ross and Rubinstein model for equity options for one period. Describe the Cox, Ross and Rubinstein model for equity options for two periods (N periods). What are the valuation parameters in the lattice approach for stock prices? How is an option priced in the lattice approach for stock prices? What modifications are necessary to the standard lattice approach to apply it to American options? What are the effects of cash distributions on the stock price? What are the main definitions of risk and return? What is portfolio selection and how can it be implemented?

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24. What is portfolio diversification in a standard context? 25. How is the standard CAPM derived? Can it be applied to the pricing of assets and options? 26. Does information affect the pricing of assets? 27. How is Merton’s (1987) simple model of capital market equilibrium with incomplete information obtained? 28. What is the definition of information? 29. How can options be priced? 30. What are the specific features of a general binomial model? 31. What are the effects of dividends on option pricing? 32. Analyze the simulation results provided in the tables of this chapter. What are your main comments and remarks regarding results of the simulations?

Bibliography Bellalah, M (1999). The valuation of futures and commodity options with information costs. Journal of Futures Markets, September. Bellalah, M (2000). A risk management approach to options trading on the Paris bourse. Derivatives Strategy, 5(6), 31–33. Bellalah, M and B Jacquillat (1995). Option valuation with information costs: theory and tests. The Financial Review, 30(3), 617–635. Black, F (1976). The pricing of commodity contracts. Journal of Financial Economics, 79(3), 167–179. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659. Briys, E, M Bellalah, F de Varenne and H Mai (1998). Options, Futures and Other Exotics, John Wiley and Sons. Cox, J, S Ross and M Rubinstein (1979). Option pricing: a simplified approach. Journal of Financial Economics, 7, 229–263. Cox, JC and M Rubinstein (1985). Options Markets, Prentice-Hall. Hull, J (2000). Options, Futures, and Other Derivative Securities, Prentice Hall International Editions. Markowitz, HM (1952). Portfolio selection. Journal of Finance, 7(1), 77–91. Merton, R (1987). An equilibrium market model with incomplete information. Journal of Finance, 483–511. Rendleman, RJ and BJ Barter (1980). The pricing of options on debts securities. Journal of Financial and Quantitative Analysis, 15 (March), 11–24. Sharpe, W (1963). A simplified model for portfolio analysis. Management Science, 10, 277–293. Sharpe, WF (1964). Capital asset prices: a theory of market equilibrium under conditions of risk. Journal of Finance, 19, 425–442.

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Chapter 2

Option Pricing in Continuous-Time: The Black–Scholes–Merton Theory and Its Extensions This chapter is organized as follows: 1. Section 2 provides an overview of the option pricing theory in the preBlack–Scholes period. 2. Section 3 develops the foundations of the Black–Scholes–Merton Theory. 3. Section 4 reviews the main results in Black’s (1976) model for the pricing of derivative assets when the underlying asset is traded on a forward or a futures market. 4. Section 5 develops the main results in Garman and Kohlhagen’s (1983) model for the pricing of currency options. 5. Section 6 presents the main results in the models of Merton (1973) and Barone-Adesi and Whaley (1987) model for the pricing of European commodity and commodity futures options. 6. Section 7 develops option price sensitivities. 7. Section 8 presents Ito’s lemma and some of its applications. 8. Section 9 develops Taylor series, Ito’s theorem and the replication argument. 9. Section 10 derives the differential equation for a derivative security on a spot asset in the presence of a continuous dividend yield and information costs. 10. Section 11 develops a general context for the valuation of securities dependent on several variables in the presence of incomplete information. 11. Section 12 presents the general differential equation for the pricing of derivatives.

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12. Section 13 extends the risk-neutral argument in the presence of information costs. 13. Section 14 extends the analysis to commodity futures prices within incomplete information. 14. Appendix 1 provides the risk measures in analytical models. 15. Appendix 2 gives the relationship between hedging parameters. 16. Appendix 3 introduces the valuation of options within information uncertainty. 17. Appendix 4 develops a general equation for the pricing of derivative securities. 18. Appendix 5 extends the risk-neutral valuation argument in the valuation of derivatives. 19. Appendix 6 provides an approximation of the cumulative normal distribution function. 20. Appendix 7 gives an approximation of the bivariate normal density function.

1. Introduction

T

he previous chapter presents the main concepts regarding option strategies, asset pricing and derivatives in a discrete-time framework. This chapter extends the valuation of derivatives to a continuous-time setting. Numerous researchers have worked on building a theory of rational option pricing and a general theory of contingent claims valuation. The story began in 1900, when the French Mathematician, Louis Bachelier, obtained an option pricing formula. His model is based on the assumption that stock prices follow a Brownian motion. Since then, numerous studies on option valuation have blossomed. The proposed formulas involve one or more arbitrary parameters. They were developed by Sprenkle (1961), Ayres (1963), Boness (1964), Samuelson (1965), Thorp and Kassouf (1967), Samuelson and Merton (1969) and Chen (1970) among others. The Black and Scholes (1973) formulation, hereafter B–S, solved a problem which has occupied economists for at least three-quarters of a century. This formulation represented a significant breakthrough in attacking the option pricing problem. In fact, the Black–Scholes theory is attractive since it delivers a closed-form solution to the pricing of European options. Assuming that the option is a function of a single source of uncertainty, namely the underlying asset price, and using a portfolio which combines options and the underlying asset, Black–Scholes constructed a riskless

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hedge which allowed them to derive an analytical formula. This model provides a no arbitrage value for European options on shares. It is a function of the share price S, the strike price K, the time to maturity T , the risk free interest rate r and the volatility of the stock price, σ. This model involves only observable variables to the exception of volatility and it has become the benchmark for traders and market makers. It also contributed to the rapid growth of the options markets by making a brand new pricing technology available to market players. About the same time, the necessary conditions to be satisfied by any rational option pricing theory were summarized in Merton’s (1973) theorems. The post-Black–Scholes period has seen many theoretical developments. The contributions of many financial economists to the extensions and generalizations of Black–Scholes type models has enriched our understanding of derivative assets and their seemingly endless applications. The first specific option pricing model for the valuation of futures options is introduced by Black (1976). Black (1976) derived the formula for futures and forward contracts and options under the assumption that investors create riskless hedges between options and the futures or forward contracts. The formula relies implicitly on the CAPM. Futures markets are not different in principal from the market for any other asset. The returns on any risky asset are governed by the asset contribution to the risk of a well diversified portfolio. The classic CAPM is applied by Dusak (1973) in the analysis of the risk premium and the valuation of futures contracts. Black (1976) model is used in Barone-Adesi and Whaley (1987) for the valuation of American commodity options. This model is referred to as the BAW (1987) model. It is helpful, as in Smithson (1991), to consider the Black–Scholes model within a family tree of option pricing models. This allows the identification of three major tribes within the family of option pricing models: analytical models, analytic approximations and numerical models. Each analytical tribe can be divided into three distinct lineages, precursors to the Black–Scholes model, extensions of the Black–Scholes model and generalisations of the Black–Scholes model.This chapter presents in detail the basic theory of rational option pricing of European options in different contexts. Starting with the analysis of the option pricing theory in the pre-Black– Scholes period, we develop the basic theory and its extensions in a continuous time framework. We develop option price sensitivities, Ito’s lemma, Taylor series and Ito’s theorem and the replication argument. We also derive the differential equation for a derivative security on a spot asset in the presence

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of a continuous dividend yield and information costs, a general context for the valuation of securities dependent on several variables in the presence of incomplete information. Finally, we present the general differential equation for the pricing of derivatives, extend the risk-neutral argument and the analysis to commodity futures prices within incomplete information. For the analysis of information costs and valuation, we can refer to Bellalah (2001), Bellalah, Prigent and Villa (2001), Bellalah M, Bellalah Ma and Portait (2001), Bellalah and Prigent (2001), Bellalah and Selmi (2001), etc.

2. Precursors to the Black–Scholes Model The story begins in 1900 with a doctoral dissertation at the Sorbonne in Paris, France, in which Louis Bachelier gave an analytical valuation formula for options.

2.1. Bachelier Formula Using an arithmetic Brownian motion for the dynamics of share prices and a normal distribution for share returns, he obtained the following formula for the valuation of a European call option on a non-dividend paying stock:

 
√ S−K S−K K−S c(S, T ) = SN − KN + σ Tn √ √ √ σ T σ T σ T where: S: underlying common stock price, K: option’s strike price, T : option’s time to maturity, σ: instantaneous standard deviation of return, N(.): cumulative normal density function, and n(.): density function of the normal distribution. As pointed out by Merton (1973) and Smith (1976), this formulation allows for both negative security and option prices and does not account for the time value of money. Sprenkle (1961) reformulated the option pricing problem by assuming that the dynamics of stock prices are log-normally distributed. By introducing a drift in the random walk, he ruled out negative security prices and allowed risk aversion. By letting asset prices have multiplicative, rather than additive fluctuations, the distribution of the option’s underlying asset at maturity is log-normal rather than normal.

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2.2. Sprenkle Formula Sprenkle (1961) derived the following formula: c(S, T ) = SeρTN(d1 ) − (1 − Z)KN(d2 )     2 ln KS + ρ + σ2 T d1 = √ σ T     2 ln KS + ρ − σ2 T d2 = √ σ T where ρ is the average rate of growth of the share price and Z corresponds to the degree of risk aversion. As it appears in this formula, the parameters corresponding to the average rate of growth of the share price and the degree of risk aversion must be estimated. This reduces considerably the use of this formula. Sprenkle (1961) tries to estimate the values of these parameters, but he was unable to do that.

2.3. Boness Formula Boness (1964) presented an option pricing formula accounting for the time value of money through the discounting of the terminal stock price using the expected rate of return to the stock. The option pricing formula proposed is c(S, T ) = SN(d1 ) − e−ρTKN(d2 )     2 ln KS + ρ + σ2 T d1 = √ σ T     2 ln KS + ρ − σ2 T d2 = √ σ T where ρ is the expected rate of return to the stock.

2.4. Samuelson Formula Samuelson (1965) allowed the option to have a different level of risk from the stock. Defining ρ as the average rate of growth of the share price and w

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as the average rate of growth of the call’s value, he proposed the following formula: c(S, T ) = Se(ρ−w)TN(d1 ) − e−wTKN(d2 )     2 ln KS + ρ + σ2 T d1 = √ σ T     2 ln KS + ρ − σ2 T d2 = . √ σ T Note that all the proposed formulas show one or more arbitrary parameters, depending on the investors preferences toward risk or the rate of return on the stock. Samuelson and Merton (1969) proposed a theory of option valuation by treating the option price as a function of the stock price. They advanced the theory by realizing that the discount rate must be determined in part by the requirement that investors hold all the amounts of stocks and the option. Their final formula depends on the utility function assumed for a “typical” investor.

2.5. The Black–Scholes–Merton Theory In this theory, the main intuition behind the risk-free hedge is simple. Consider an at-the-money European call giving the right to its holder to buy one unit of the underlying asset in one month at a strike price of $100. Assume that the final asset price is either 105 or 95. An investor selling a call on the unit of the asset will receive either 5 or 0. In this context, selling two calls against each unit of the asset will create a terminal portfolio value of 95. The certain terminal value of this portfolio must be equal today to the discounted value of 95 at the riskless interest rate. If this rate is 1%, the present value is (95/1.01) . The current option value is (100 − (95/1.01))/2. If the observed market price is above (or below) the theoretical price, it is possible to implement anarbitrage strategy by selling the call and buying (selling) a portfolio comprising a long position in a half unit of the asset and a short position in risk-free bonds. The Black–Scholes–Merton model is the continuous-time version of this example. The theory assumes that the underlying asset follows a geometric Brownian motion and is based on the construction of a

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risk-free hedge between the option and its underlying asset. This implies that the call pay-out can be duplicated by a portfolio consisting of the asset and risk-free bonds. In this theory, the option value is the same for a riskneutral investor and a risk-averse investor. Hence, options can be valued in a risk-neutral world, i.e. expected returns on all assets are equal to the risk-free rate of interest.

3. The Black–Scholes Model Under the following assumptions, the value of the option will depend only on the price of the underlying asset S, time t and on other variables assumed constants. These assumptions or “ideal conditions” as expressed by Black– Scholes are the following: — The option is European. — The short term interest rate is known. — The underlying asset follows a random walk with a variance rate proportional to the stock price. It pays no dividends or other distributions. — There is no transaction costs and short selling is allowed, i.e. an investor can sell a security that he does not own. — Trading takes place continuously and the standard form of the capital market model holds at each instant. The main attractions of the Black–Scholes model are that their formula is a function of “observable” variables and that the model can be extended to the pricing of any type of option.

3.1. The Black–Scholes Model and the Capital Asset Pricing Model, CAPM The capital asset pricing model of Sharpe (1964) can be stated as follows: ¯ S − r = βS [R ¯ m − r] R where: ¯ S : equilibrium expected return on security S, — R ¯ m : equilibrium expected return on the market portfolio, — R — r: 1 + the riskless rate of interest,

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˜ S /R ˜ m) (R — βS = cov var(R˜ m ) : the beta of security S, that is the covariance of the return on that security with the return on the market portfolio, divided by the variance of market return.

The model gives a general method for discounting future cash flows under uncertainty. The model is referred to the previous chapter for more details about this model. Denote by C(S, t) the value of the option as a function of the underlying asset and time. To derive their valuation formula, B–S assumed that the hedged position was continuously rebalanced in order to remain riskless. They found that the price of a European call or put must verify a certain differential equation, which is based on the assumption that the price of the underlying asset follows a geometric Wiener process S = α dt + σz S where α and σ refer to the instantaneous rate of return and the standard deviation of the underlying asset, respectively, and z is a Brownian motion. The relationship between an option’s beta and its underlying security’s beta is
 CS βC = S βS C where: — — — —

βc : the option’s beta, βS : the stock’s beta, C: the option value, CS : the first derivative of the option with respect to its underlying asset. It is also the hedge ratio or the option’s delta in a covered position. According to the CAPM, the expected return on a security should be: ¯ m − r] ¯ S − r = βS [R R

¯ m is the expected return ¯ S is the expected return on the asset S and R where R on the market portfolio. This equation may also be written as:
 S ¯ m − r)]t. E = [r + βS (R S

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Using the CAPM, the expected return on a call option should be:
 C ¯ m − r)]t. E = [r + βC (R C Multiplying the previous two equations by S and C gives ¯ m − r)]t E(S) = [rS + SβS (R ¯ m − r)]t. E(C) = [rC + CβC (R When substituting for the option’s elasticity βc , the above equation becomes after transformation: ¯ m − r)]t. E(C) = [rC + SCS βS (R Assuming a hedged position is constructed and “continuously” rebalanced, and since C is a continuous and differentiable function of two variables, it is possible to use Taylor series expansion to expand C: C =

1 CSS (S)2 + CS S + Ct t. 2

This is just an extension of simple results to obtain Ito’s lemma. Taking expectations of both sides of this equation and replacing S, we obtain E(C) =

1 2 2 σ S CSS t + CS E(S) + Ct t. 2

Replacing the expected value of S from E(S/S) gives: E(C) =

1 2 2 ¯ m − r)]t + Ct t. σ S CSS t + CS [rS + SβS (R 2

Combining E(C) and this last equation and rearranging gives: 1 2 2 σ S CSS + rSCS − rC + Ct = 0. 2 This partial differential equation corresponds to the Black–Scholes valuation equation. Let T be the maturity date of the call and E be its strike price. The last equation subject to the following boundary condition at maturity C(S, T ) = S − K, C(S, T ) = 0,

if S ≥ K if S < K

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is solved using standard methods for the price of a European call, which is found to be equal to C(S, T ) = SN(d1 ) − Ke−rTN(d2 ) √ √ with d1 = [ln( KS ) + (r + 21 σ 2 )T ]/σ T , d2 = d1 − σ T and where N(.) is the cumulative normal density function.

3.2. An Alternative Derivation of the Black–Scholes Model Assuming that the option price is a function of the stock price and time to maturity, c(S, t) and that over “short” time intervals, t, a hedged portfolio consisting of the option, the underlying asset and a riskless security can be formed, where portfolio weights are chosen to eliminate “market risk”, Black–Scholes expressed the expected return on the option in terms of the option price function and its partial derivatives. In fact, following Black– Scholes, it is possible to create a hedged position consisting of a sale of 1 options against one share of stock long. If the stock price changes [∂c(S,t)/∂S] by a small amount S, the option changes by an amount [∂c(S, t)/∂S]S. Hence, the change in value in the long position (the stock) is approximately 1 offset by the change in [∂c(S,t)/∂S] options. This hedge can be maintained continuously so that the return on the hedged position becomes completely independent of the change in the underlying asset value, i.e. the return on the hedged position becomes certain. The value of equity in a hedged position,containing a stock purchase  1 and a sale of [∂c(S,t)/∂S] options is S − C(S, t)/ ∂c(S,t) . Over a short interval ∂S t, the change in this position is c(S, t)

S − ∂c(S, t) ∂S

(1)

where c(S, t) is given by c(S + S, t + t) − c(S, t). Using stochastic calculus for c(S, t) gives c(S, t) =

∂c(S, t) 1 ∂2 c(S, t) 2 2 ∂c(S, t) S + t. σ S t + 2 ∂S 2 ∂S ∂t

(2)

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The change in the value of equity in the hedged position is found by substituting c(S, t) from Eq. (2) into Eq. (1):   ∂c(S,t) 1 2 2 ∂2 c(S,t) σ S + t ∂t 2 ∂S 2 − . ∂c(S,t) ∂S

Since the return to the equity in the hedged position is certain, it must be equal to rt, where r stands for the short term interest rate. Hence, the change in the equity must be equal to the value of the equity times rt, or    ∂c(S,t) 1 2 2 ∂2 c(S,t) σ S + t 2 2 ∂t c(S, t) ∂S − = S − ∂c(S,t) rt. ∂c(S,t) ∂S

∂S

Dropping the time and rearranging gives the Black–Scholes partial differential equation ∂c(S, t) ∂c(S, t) 1 2 2 ∂2 c(S, t) σ S + rS = 0. − rc(S, t) + 2 ∂S 2 ∂t ∂S This partial differential equation must be solved under the boundary condition expressing the call’s value at maturity date: c(S, t ∗ ) = max[0, St ∗ −K], where K is the option’s strike price. For the European put, the equation must be solved under the following maturity date condition: P(S, t ∗ ) = max[0, K − St ∗ ]. To solve this differential equation, under the call boundary condition, Black– Scholes make the following substitution: 


  S 1 2 2 σ2 r(t−t ∗ ) ∗ ln − r − σ (t − t) , y 2 r− c(S, t) = e σ 2 K 2
 2(t ∗ − t) 1 2 2 − r− σ . (3) σ2 2 Using this substitution, the differential equation becomes ∂2 y ∂y = 2. ∂t ∂S

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This differential equation is the heat transfer equation of physics. The boundary condition is rewritten as y(u, 0) = 0, if u < 0 otherwise, 
1 uσ 2  y(u, 0) = K e

2 r− 21 σ 2

−1 .

The solution to this problem is the solution to the heat transfer equation given in Churchill (1963):
2

∞ 
21 (u+q√2s)σ 2  − q2 1 1 2 y(u, s) = √ −1 e dq. K e r− 2 σ 2 √−u2s Substituting (4) gives the following solution for the European call price with T = t ∗ − t: c(S, T ) = SN(d1 ) − Ke−rT N(d2 )     2 ln KS + r + σ2 T √ d1 = , d2 = d1 − σ T √ σ T where N(.) is the cumulative normal density function given by N(d) =  d (− x2 ) √1 e 2 dx. 2 −∞

3.3. The Put–Call Parity Relationship The put–call parity relationship can be derived as follows. Consider a portfolio A which comprises a call option with a maturity date t ∗ and a discount bond that pays K dollars at the option’s maturity date. Consider also a portfolio B, with a put option and one share. The value of portfolio A at maturity is max[0, St ∗ − K] + K = max[K, St ∗ ]. The value of portfolio B at maturity is max[0, K − St ∗ ] + St ∗ = max[K, St ∗ ]. Since both portfolios have the same value at maturity, they must have the same initial value at time t, otherwise arbitrage will be profitable. Therefore, ∗ the following put–call relationship must hold ct − pt = St − Ke−r(t −t) , with t ∗ − t = T .

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If this relationship does not hold, then arbitrage would be profitable. In fact, suppose for example, that ct − pt > St − Ke−r(t

∗ −t)

.

At time t, the investor can construct a portfolio by buying the put and the underlying asset and selling the call. This strategy yields a result equal to ct − pt − St . If this amount is positive, it can be invested at the riskless rate until the maturity date t ∗ , otherwise it can be borrowed at the same rate for the same period. At the option maturity date, the options will be in-the-money or out-ofthe-money according to the position of the underlying asset St ∗ with respect to the strike price K. If St ∗ > K, the call is worth its intrinsic value. Since the investor sold the call, he is assigned on that call. He will receive the strike price, delivers the stock and closes his position in the cash account. The put is worthless. Hence, the position is worth K + er(t

∗ −t)

[ct − pt − St ] > 0.

If ST < K, the put is worth its intrinsic value. Since the investor is long the put, he exercises his option. He will receive upon exercise the strike price, delivers the stock and closes his position in the cash account. The call is worthless. Hence, the position is worth K + er(t

∗ −t)

[ct − pt − St ] > 0.

In both cases, the investor makes a profit without initial cashoutlay. This is a riskless arbitrage which must not exist in efficient markets. Therefore, the above put–call parity relationship must hold. Using this relationship, the European put option value is given by p(S, T ) = −SN(−d1 ) + Ke−rT N(−d2 )     2 ln KS + r + σ2 T √ d1 = , d2 = d1 − σ T √ σ T where N(.) is the cumulative normal density function given by N(d) =  x2  − d √1 e 2 dx. 2 −∞ We illustrate by the following examples the application of the Black– Scholes (1973) model for the determination of call and put prices.

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Table 1: S 80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

87

Simulations of Black and Scholes put prices.

Price

Delta

Gamma

Vega

Theta

19.44832 15.56924 12.17306 9.29821 6.94392 5.07582 3.63657 2.55742 1.76806

−0.81955 −0.72938 −0.62762 −0.52216 −0.42055 −0.32847 −0.24939 −0.18449 −0.13331

0.01676 0.01967 0.02105 0.02086 0.01933 0.01692 0.01412 0.01130 0.00871

0.21155 0.28240 0.34124 0.37895 0.39156 0.38031 0.35019 0.30789 0.26002

−0.00575 −0.00769 −0.00931 −0.01035 −0.01069 −0.01038 −0.00955 −0.00838 −0.00707

S = 100, K = 100, t = 22/12/2002, T = 22/12/2003, r = 2%, σ = 20%.

3.4. Examples Tables 1–4 provide simulation results for European call and put prices using the Black–Scholes model. The tables provide also Greek-letters. The delta is given by the option’s first partial derivative with respect to the underlying asset price. It represents the hedge ratio in the context of the Black– Scholes model. The option’s gamma corresponds to the option second partial derivative with respect to the underlying asset or to the delta partial derivative with respect to the asset price. The option’s theta is given by the option’s first partial derivative with respect to the time remaining to maturity. The option’s vega is given by the option price derivative with respect to the volatility parameter. The derivation of these parameters appears in Appendix 1.

4. The Black Model for Commodity Contracts Using some assumptions similar to those used in deriving the original B–S option formula, Black (1976) presented a model for the pricing of commodity options and forward contracts. In this model, the spot price S(t) of an asset or a commodity is the price at which an investor can buy or sell it for an immediate delivery at current time, time t. This price may rise steadily, fall and fluctuate randomly. The futures price F(t, t ∗ ) of a commodity can be defined as the price at which an investor agrees to buy or sell at a given time in the future, t ∗ , without putting up any money immediately.

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Table 2: S 80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

Simulations of Black Scholes call prices.

Price

Delta

Gamma

Vega

Theta

1.43382 2.55474 4.15856 6.28371 8.92943 12.06132 15.62208 19.54292 23.75356

0.18045 0.27062 0.37238 0.47784 0.57945 0.67153 0.75061 0.81551 0.86669

0.01676 0.01967 0.02105 0.02086 0.01933 0.01692 0.01412 0.01130 0.00871

0.21155 0.28240 0.34124 0.37895 0.39156 0.38031 0.35019 0.30789 0.26002

−0.00575 −0.00769 −0.00931 −0.01035 −0.01069 −0.01038 −0.00955 −0.00838 −0.00707

S = 100, K = 100, t = 22/12/2002, T = 22/12/2003, r = 2%, σ = 20%.

Table 3: S 80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

Simulations of Black and Scholes call prices.

Price

Delta

Gamma

Vega

Theta

0.36459 0.93156 1.99540 3.70489 6.12966 9.24535 12.95409 17.12243 21.61673

0.07582 0.15729 0.27358 0.41284 0.55640 0.68669 0.79246 0.87053 0.92356

0.01332 0.02070 0.02656 0.02900 0.02763 0.02332 0.01781 0.01245 0.00807

0.08172 0.14559 0.21252 0.26201 0.27966 0.26377 0.22359 0.17279 0.12322

−0.00442 −0.00791 −0.01158 −0.01429 −0.01526 −0.01439 −0.01218 −0.00939 −0.00668

S = 100, K = 100, t = 22/12/2002, T = 22/06/2003, r = 2%, σ = 20%.

When t = t ∗ , the futures price is equal to the spot price. A forward contract is a contract to buy or sell at a price that stays fixed until the maturity date, whereas the futures contract is settled every day and rewritten at the new futures price. Following Black (1976), let v be the value of the forward contract, u the value of the futures contract and c the value of an option contract. Each of these contracts, is a function of the futures price F(t, t ∗ ) as well as other variables. So, we can write at instant t, the values of these contracts, respectively, as V(F, t), u(F, t) and c(F, t). The value of the forward contract depends also on the price of the underlying asset, K at time t ∗ and can be written V(F, t, K, t ∗ ). The futures price is the price at which a forward contract presents a zero current value. It is written as V(F, t, F, t ∗ ) = 0.

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Table 4: S 80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

89

Simulations of Black and Scholes put prices.

Price

Delta

Gamma

Vega

Theta

19.36686 14.93383 10.99767 7.70716 5.13193 3.24762 1.95636 1.12471 0.61900

−0.92418 −0.84271 −0.72642 −0.58716 −0.44360 −0.31331 −0.20754 −0.12947 −0.07644

0.01332 0.02070 0.02656 0.02900 0.02763 0.02332 0.01781 0.01245 0.00807

0.08172 0.14559 0.21252 0.26201 0.27966 0.26377 0.22359 0.17279 0.12322

−0.00442 −0.00791 −0.01158 −0.01429 −0.01526 −0.01439 −0.01218 −0.00939 −0.00668

S = 100, K = 100, t = 22/12/2002, T = 22/06/2003, r = 2%, σ = 20%.

This equation says that the forward contract’s value is zero when the contract is initiated and the contract price, K, is always equal to the current futures price F(t, t ∗ ). The main difference between a futures contract and a forward contract is that a futures contract may be assimilated to a series of forward contracts. This is because the futures contract is rewritten every day with a new contract price equal to the corresponding futures price. Hence when F rises, i.e. F > K, the forward contract has a positive value and when F falls, F < K, the forward contract has a negative value. When the transaction takes place, the futures price equals the spot price and the value of the forward contract equals the spot price minus the contract price or the spot price V(F, t ∗ , K, t ∗ ) = F − K. At maturity, the value of a commodity option is given by the maximum of zero and the difference between the spot price and the contract price. Since at that date, the futures price equals the spot price, it follows that if F ≥ K, then c(F, t ∗ ) = F − K, otherwise c(F, t ∗ ) = 0. In order to value commodity contracts and commodity options, Black (1976) assumes that: — The futures price changes are distributed log-normally with a constant variance rate σ 2 . — All the parameters of the capital asset pricing model are constant through time. — There are no transaction costs and no taxes.

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Under these assumptions, it is possible to create a riskless hedge by taking a long position in the option and a short position in the futures contract. Let [∂c(F, t)/∂F ] be the weight affected to the short position in the futures contract, which is the derivative of c(F, t) with respect to F . The change in the hedged position may be written as

∂c(F, t) c(F, t) − F. ∂F Using the fact that the return to a hedged portfolio must be equal to the risk-free interest rate and expanding c(F, t) gives the following partial differential equation

∂c(F, t) 1 2 2 ∂2 c(F, t) = rc(F, t) − σ F ∂t 2 ∂F 2 or

1 2 2 ∂2 c(F, t) ∂c(F, t) = 0. (4) − rc(F, t) + σ F ∂F 2 ∂t 2 Denoting T = t ∗ − t, using the call’s payoff and Eq. (4), the value of a commodity option is c(F, T ) = e−rT [FN(d1 ) − KN(d2 )]  F  σ2 √ ln K + T d1 = √ 2 , d2 = d1 − σ T σ T where N(.) is the cumulative normal density function. It is convenient to note that the commodity option’s value is the same as the value of an option on a security paying a continuous dividend. The rate of distribution is equal to the stock price times the interest rate. If F e−rT is substituted in the original formula derived by Black–Scholes, the result is exactly the above formula. In the same context, the formula for the European put is p(F, T ) = e−rT [−FN(d1 ) + KN(−d2 )]  F  σ2 √ ln K + T d1 = √ 2 , d2 = d1 − σ T σ T where N(.) is the cumulative normal density function. The value of the put option can be obtained directly from the put–call parity. The put–call parity relationship for futures options is p − c = e−rT (K − F ).

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5. The Extension to Foreign Currencies: The Garman and Kohlhagen Model Foreign currency options are priced along the lines of Black–Scholes (1973), Merton (1973) and Garman and Kohlhagen (1983). Using the same assumptions as in the Black–Scholes (1973) model, Garman and Kohlhagen (1983) presented the following formula for a European currency call: ∗

c(S, T ) = Se−r T N(d1 ) + Ke−rT N(d2 )     2 ln KS + r − r ∗ + σ2 T d1 = √ σ T √ d2 = d1 − σ T where S, the spot rate; K, the strike price; r, the domestic interest rate; r ∗ , the foreign interest rate; σ, the volatility of spot rates; and T , the option’s time to maturity. The formula for a European currency put is ∗

p(S, T ) = −Se−r T N(−d1 ) − Ke−rT N(−d2 )     2 ln KS + r − r ∗ + σ2 T √ , d2 = d1 − σ T . d1 = √ σ T Note that the main difference between these formulae and those of B–S for the pricing of equity options is that the foreign risk-free rate is used in the adjustment of the spot rate. The spot rate is adjusted by the known “dividend”, i.e. the foreign interest earnings, whereas the domestic risk-free rate enters the calculation of the present value of the strike price since the domestic currency is paid over on exercise. Examples Tables 5–8 provide simulation results for option prices using the Garman– Kohlhagen model. The tables give also the Greek-letters. The reader can make comments about the values of the Greek-letters.

6. The Extension to Other Commodities: The Merton, Barone-Adesi and Whaley Model and Its Applications The model presented in Barone-Adesi and Whaley (1987) is a direct extension of models presented by Black–Scholes (1973), Merton (1973) and

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Table 5:

Simulations of Garman–Kohlhagen call prices.

S

Price

Delta

Gamma

Vega

Theta

0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04

0.05384 0.05815 0.06266 0.06737 0.07227 0.07736 0.08264 0.08810 0.09375

0.43109 0.45081 0.47042 0.48984 0.50904 0.52798 0.54661 0.56492 0.58286

0.52931 0.50961 0.49003 0.47063 0.45145 0.43253 0.41391 0.39562 0.37769

0.00364 0.00370 0.00376 0.00380 0.00383 0.00386 0.00387 0.00388 0.00387

0.00008 0.00008 0.00009 0.00009 0.00009 0.00008 0.00008 0.00008 0.00008

S = 1, K = 1, t = 07/02/2002, T = 07/02/2003, r = 3%, r ∗ = 4%, σ = 20%.

Table 6:

Simulations of Garman–Kohlhagen call prices.

S

Price

Delta

Gamma

Vega

Theta

1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14

0.10315 0.10988 0.11680 0.12392 0.13123 0.13872 0.14639 0.15423 0.16224

0.61071 0.62921 0.64714 0.66449 0.68123 0.69736 0.71287 0.72774 0.74198

0.34986 0.33137 0.31345 0.29611 0.27937 0.26325 0.24775 0.23289 0.21865

0.00385 0.00382 0.00378 0.00373 0.00368 0.00362 0.00355 0.00347 0.00339

0.00008 0.00008 0.00007 0.00007 0.00007 0.00007 0.00006 0.00006 0.00006

S = 1.1, K = 1, t = 07/02/2002, T = 07/02/2003, r = 3%, r ∗ = 4%, σ = 20%.

Black (1976). The absence of riskless arbitrage opportunities imply that the following relationship exists between the futures contract, F , and the price of its underlying spot commodity, S: F = SebT ; where T is the time to expiration and b is the cost of carrying the commodity. When the underlying commodity dynamics are given by: dS = α dt + σ dW S

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Table 7:

93

Simulations of Garman–Kohlhagen put prices.

S

Price

Delta

Gamma

Vega

Theta

0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04

0.10195 0.09666 0.09156 0.08666 0.08195 0.07744 0.07311 0.06896 0.06500

−0.52960 −0.50987 −0.49027 −0.47084 −0.45164 −0.43271 −0.41407 −0.39577 −0.37783

0.52931 0.50961 0.49003 0.47063 0.45145 0.43253 0.41391 0.39562 0.37769

0.00364 0.00370 0.00376 0.00380 0.00383 0.00386 0.00387 0.00388 0.00387

0.00011 0.00011 0.00011 0.00011 0.00011 0.00011 0.00011 0.00011 0.00011

S = 1, K = 1, t = 07/02/2002, T = 07/02/2003, r = 3%, r ∗ = 4%, σ = 20%.

Table 8: Simulations of Garman–Kohlhagen put prices and the Greek-letters. S

Price

Delta

Gamma

Vega

Theta

1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14

0.05904 0.05519 0.05155 0.04810 0.04484 0.04177 0.03887 0.03615 0.03358

−0.34997 −0.33148 −0.31354 −0.29620 −0.27945 −0.26332 −0.24781 −0.23294 −0.21870

0.34986 0.33137 0.31345 0.29611 0.27937 0.26325 0.24775 0.23289 0.21865

0.00385 0.00382 0.00378 0.00373 0.00368 0.00362 0.00355 0.00347 0.00339

0.00011 0.00011 0.00011 0.00011 0.00011 0.00010 0.00010 0.00010 0.00010

S = 1.1, K = 1, t = 07/02/2002, T = 07/02/2003, r = 3%, r∗ = 4%, σ = 20%.

where α is the expected instantaneous relative price change of the commodity and σ is its standard deviation, then the dynamics of the futures price are given by the following differential equation: dF = (α − b)dt + σ dW. F Assuming that a hedged portfolio containing the option and the underlying commodity can be constructed and adjusted continuously, the partial

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differential equation that must be satisfied by the option price, c, is





∂c(S, t) ∂c(S, t) 1 2 2 ∂2 c(S, t) + = 0. − rc(S, t) + bS σ S ∂S 2 ∂S ∂t 2 This equation appeared first indirectly in Merton (1973). When the cost of carry b is equal to the riskless interest rate, this equation reduces to that of B–S (1973). When the cost of carry is zero, this equation reduces to that of Black (1976). When the cost of carry is equal to the difference between the domestic and the foreign interest rate, this equation reduces to that in Garman and Kohlhagen (1983). The short term interest rate r, and the cost of carrying the commodity, b, are assumed to be constant and proportional rates. Using the terminal boundary condition c(S, T ) = max[0, ST − K]; Merton (1973) shows indirectly that the European call price is: c(S, T ) = Se(b−r)T N(d1 ) − Ke−rT N(d2 )   2 √ ln KS + (b + σ2 )T , d2 = d1 − σ T . d1 = √ σ T Using the boundary condition for the put p(S, T ) = max[0, K − ST ] the European put price is given by p(S, T ) = −Se(b−r)T N(−d1 ) + Ke−rT N(−d2 )   2 √ ln KS + (b + σ2 )T , d2 = d1 − σ T . d1 = √ σ T The call formula provides the composition of the asset-bond portfolio that mimics exactly the call’s payoff. A long position in a call can be replicated by buying e(b−r)T N(d1 ) units of the underlying asset and selling N(d2 ) units of risk-free bonds, each unit with strike price Ke−rT . When the asset price varies, the units invested in the underlying asset and risk-free bonds will change. Using a continuous rebalancing of the portfolio, the pay-outs will be identical to those of the call. The same strategy can be used to duplicate the put’s payoff.

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7. Option Price Sensitivities: Some Specific Examples 7.1. The Delta 7.1.1. The Call’s Delta The call’s delta is given by c = N(d1 ). The use of this formula requires the computation of d1 given by d1 =

ln KS + (r + 21 σ 2 )T . √ σ T

Appendix 2 provides the detailed derivations of these parameters. Example Let the underlying asset price S = 18, the strike price K = 15, the short term interest rate r = 10%, the maturity date T = 0.25 and the volatility σ = 15%, the option’s delta is given by c = N(d1 ). Applying this formula needs the calculation of d1 :



18 1 1 2 ln d1 = + 0.1 + × 0.5 0.25 = 2.8017. √ 15 2 0.15 0.25 Hence, the delta is c = N(2.8017) = 0.997. This delta value means that the hedge of the purchase of a call needs the sale of 0.997 units of the underlying asset. When the underlying asset price rises by 1 unit, from 18 to 19, the option price rises from 3.3659 to approximately (3.3659+0.997), or 4.3629. When the asset price falls by one unit, the option price changes from 3.3659 to approximately (3.3659 − 0.997), or 2.3689.

7.1.2. The Put’s Delta The put’s delta has the same meaning as the call’s delta. It is also given by the option’s first derivative with respect to the underlying asset price. When selling (buying ) a put option, the hedge needs selling (buying) delta units of the underlying asset. The put’s delta is given by p = c − 1 = 0.0997 − 1 = −0.003. The hedge ratio is −0.003. When the underlying asset price rises from 18 to 19, the put price falls from 0.0045 to approximately (0.0045 − 0.003),

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or 0.0015. When it falls from 18 to 17, the put price rises from 0.0045 to approximately (0.0045 + 0.003), or 0.0075. Appendix 1 provides the derivation of the Greek letters in the context of analytical models.

7.2. The Gamma 7.2.1. The Call’s Gamma c = In the Black–Scholes model, the call’s gamma is given by c = ∂ ∂S 1 2 1√ 1 − 2 d1 √ n(d1 ) with n(d1 ) = 2π e . Using the same data as in the example Sσ T 1

1 n(d1 ) = √6.2831 e− 2 (2.8017) = 0.09826 and c = 18(0.15)1 √0.25 0.09826 = 0.0727. When the underlying asset price is 18 and its delta is 0.997, a fall in the asset price by one unit yields a change in the delta from 0.997 to approximately (0.997 − 0.0727), or 0.9243. Also, a rise in the asset price from 18 to 19, yields a change in the delta from 0.997 to (0.997+0.0727), or 1. This means that the option is deeply-in-the-money, and its value is given by its intrisic value (S − K). The same arguments apply to put options. The call and the put have the same gamma. 2

7.2.2. The Put’s Gamma The put’s gamma is given by p = 1 2 √1 e− 2 d1 2π

∂p ∂S

=

1√ Sσ T

n(d1 ) with n(d1 ) =

or p = = 0.0727. When the asset price changes by one unit, the put price changes by the delta amount and the delta changes by an amount equals to the gamma. 1√ 0.09826 18(0.15) 0.25

7.3. The Theta 7.3.1. The Call’s Theta In the B-S model, the theta is given by

c =

∂c −Sσn(d1 ) − rKe−rT N(d2 ). = √ ∂T 2 T

Using the same data as in the example above, we obtain:

c = −0.2653 − 1.4571 = −1.1918.

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When the time to maturity is shortened by 1% year, the call’s price decreases by 0.01 (1.1918), or 0.011918 and its price changes from 3.3659 to approximately (3.3659 − 0.01918), or 3.3467.

7.3.2. The Put’s Theta In the B–S model, the put’s theta is given by

p =

Sσn(d1 ) ∂p + rKe−rTN(d2 ) =− √ ∂T 2 T

or p = −0.2653 + 0.0058 = −0.2594. Using the same reasoning, the put price changes from 0.0045 to approximately (0.0045 − 0.0025), or 0.002.

7.4. The Vega 7.4.1. The Call’s Vega

√ ∂c In the B–S model, the call’s vega is given by vc = ∂σ = S T n(d1 ) or using √ the above data vc = 18 0.25(0.09826) = 0.88434. Hence, when the volatility rises by 1 point, the call price increases by 0.88434. The increase in volatility by 1% changes the option price from 3.3659 to (3.3659 + 1% (0.88434)), or 3.37474. In the same context, the put’s vega is equal to the call’s vega. The put price changes from 0.0045 to (0.0045 + 1% (0.88434)), or 0.0133434. When the volatility falls by 1% the call’s price changes from 3.3659 to (3.3659 − 1% (0.88434)), or 3.36156. In the same way, the put price is modified from 0.0045 to approximately (0.0045 − 1% (0.88434)), or zero since option prices cannot be negative.

7.4.2. The Put’s Vega ∂p = In the Black–Scholes model, the put’s vega is given by vp = ∂σ √ √ S T n(d1 ) or vp = 18 0.25(0.09826) = 0.88434 and it has the same meaning as the call’s vega. Appendix 2 provides the relationships between hedging parameters.

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8. Ito’s Lemma and Its Applications Financial models are rarely described by a function that depends on a single variable. In general, a function which is itself a function of more than one variable is used. Ito’s lemma, which is the fundamental instrument in stochastic calculus, allows such functions to be differentiated. We first derive Ito’s lemma with reference to simple results using Taylor series approximations. We then give a more rigorous definition of Ito’s theorem. Let f be a continuous and differentiable function of a variable x. If x is a small change in x, then using Taylor series, the resulting change in f is given by:


  df 1 d2 f 1 d3f 2 f ∼ x + x x3 + · · · . + dx 2 dx2 6 dx3 If f depends on two variables x and y, then Taylor series expansion of f is


 ∂f 1 ∂2f ∂f x + y + x2 f ∼ ∂x ∂y 2 ∂x2
2  
∂ f 1 ∂2 f 2 xy + · · · . y + + 2 ∂y∂y 2 ∂y In the limit case, when x and y are close to zero, Eq. (9) becomes

 ∂f ∂f f ∼ dx + dy. ∂x ∂y Now, if f depends on two variables x and t in lieu of x and y, the analogous to Eq. (9) is


 
 ∂f 1 ∂2 f 1 ∂2 f ∂f 2 x + t + x + t 2 f ∼ ∂x ∂t 2 ∂x2 2 ∂t 2
2  ∂ f xt + · · · . (5) + ∂x∂t Consider a derivative security, f(x, t), which value depends on time and on the asset price x. Assuming that x follows the general Ito process, dx = a(x, t)dt + b(x, t)dW or

√ x = a(x, t)t + bξ t.

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In the limit, when x and t are close to zero, we cannot ignore as before the term in x2 since it is equal to x2 = b2 ξ 2 t + terms in higher order in t. In this case, the term in t cannot be neglected. Since the term ξ is normally distributed with a zero mean, E(ξ) = 0 and a unit variance, E(ξ 2 ) − E(ξ)2 = 1, then E(ξ 2 ) = 1 and E(ξ)2 t is t . The variance of ξ 2 t is of order t 2 and consequently, as t approaches zero, ξ 2 t becomes certain and equals its expected value, t. In the limit, Eq. (5) becomes


 ∂f ∂f 1 ∂2f df = x + t + b2 dt. ∂x ∂t 2 ∂x2 This is exactly Ito’s lemma. Substituting a(x, t)dt + b(x, t)dW for dx gives





 ∂f ∂f 1 ∂2f ∂f 2 df = a+ + b dW. b dt + 2 ∂x ∂t 2 ∂x ∂x Example Apply Ito’s lemma to derive the process of f = ln(S). First calculate the derivatives


2  ∂f ∂f 1 ∂ f 1 = ; = 0. = − 2; 2 ∂S S ∂S S ∂t Then from Ito’s lemma, one obtains





 ∂f ∂f 1 ∂2 f ∂f 2 2 df = µS + + σS dW σ S dt + 2 ∂x ∂t 2 ∂S ∂S or



1 df = µ − σ 2 dt + σ dW. 2

This last equation shows that, f follows a generalized Wiener process with a constant drift of (µ − 21 σ 2 ) and a variance rate of σ 2 . The generalization of Ito’s lemma is useful for a function that depends on n stochastic variables xi , where i varies from 1 to n. Consider the following dynamics for the variables xi : dxi = ai dt + bi dzi .

(6)

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Using a Taylor series expansion of f gives f =

 ∂f 1   ∂2f ∂f xi + t + xi xj ∂t 2 i j ∂xi ∂xj ∂xi i +

∂2 f xi t + · · · . ∂xi ∂t

(7)

Equation (6) can be discretized as follows: √ xi = ai t + bi i  zi where the term i corresponds to a random sample from a standardized normal distribution. The terms i and j reflecting the Wiener processes present a correlation coefficient ρi,j . It is possible to show that when the time interval tends to zero, in the limit, the term xi2 = bi2 dt and the product xi xj = bi bj ρi,j dt. Hence, in the limit, when the time interval becomes close to zero, Eq. (7) can be written as df =

 ∂f 1   ∂2 f ∂f dxi + dt + bi bj ρij dt. ∂xi ∂t 2 i j ∂xi ∂xj i

This gives the generalized version of Ito’s lemma. Substituting Eq. (6) in the above equation gives:   2   ∂f   ∂f 1 ∂ f ∂f + ai + bi bj ρij  + bi dzi . df =  ∂xi ∂t 2 i j ∂xi ∂xj ∂xi i Example Use Ito’s Lemma to show that:

t 1 m n n − 1 t m n−2 m n−1 τ X (τ)dX(τ) = t X (t) − t X (τ)dτ n 2 0 0

m t m−1 n − t X (τ)dτ. n 0 Solution When F = Xn (t),

nm ∈ N ∗

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then

t

dF = t m Xn (t) − t m Xn (0) = tmXn (t) 0

since X(0) = 0. Using Ito’s Lemma gives: 1 dF = nt m Xn−1 dX + n(n − 1)Xn−2 dt + mt m−1 Xn (t)dt. 2 Hence, we have:

t

t

t 1 dF = n τ m Xn−1 (τ)dX(τ) + n(n − 1) τ m Xn−2 (τ)dτ 2 0 0 0

t +m τ m−1 Xn (τ)dτ = t m Xn (t)

0 t

1 τ m Xn−1 (τ)dX(τ) + (n − 1)τ m Xn−2 (τ) dτ 2 0

t m 1 + τ m−1 Xn (τ)dτ = t m Xn (t) n 0 n

t

t 1 m n n − 1 m n−2 m n−1 ⇔ τ X (τ)dX(τ) = t X (t) − τ X (τ) dτ n 2 0 0

m t m−1 n − τ X (τ) dτ. n 0 ⇔

9. Taylor Series,Ito’s Theorem and the Replication Argument We denote by c(S, t) the option value at time t as a function of the underlying asset price S and time t. Assume that the underlying asset price follows a geometric Brownian motion: dS = µ dt + σ dW(t) S where µ and σ 2 correspond, respectively, to the instantaneous mean and the variance of the rate of return of the stock.

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9.1. The Relationship Between Taylor Series and Ito’s Differential Using Taylor series differential, it is possible to express the price change of the option over a small interval of time [t, t + dt] as:
dc =

∂c ∂S




dS +


 ∂c 1 ∂2 c dt + (dS)2 , ∂t 2 ∂2 S

(8)

where the last term appears because (dS)2 is of order dt. The last term in Eq. (8) appears because the term (dS)2 is of order dt. Omberg (1991) makes a decomposition of the last term in Eq. (8) into its expected value and an error term.This allows one to establish a link between Taylor series (dc) and Ito’s differential dcI as
dc =

∂c ∂S




dS +


 ∂c 1 ∂2 c dt + σ 2 S 2 dW 2 + de(t), ∂t 2 ∂2 S

which can be written as the sum of two components corresponding to the Ito’s differential dcI and an error term de(t) dc = dcI + de(t), where 

 ∂c 1 ∂2 c ∂c dS + dt + σ 2 S 2 dt ∂S ∂t 2 ∂2 S
   1 ∂2 c σ 2 S 2 dW 2 − dt . de(t) = 2 2 ∂ S


dcI =

and

9.2. Ito’s Differential and the Replication Portfolio 9.2.1. The Standard Case in Frictionless Markets The pay-off of a derivative asset can be created using the discount bond, some options and the underlying asset. The portfolio which duplicates the pay-off of the asset is called the replicating portfolio. When using Ito’s lemma, the error term de(t) is often neglected and, the equation for the option is approximated only by the term dcI . The quantity dcI is replicated by QS ∂c units of the underlying asset and an amount of cash Qc with QS = ∂S

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    2  ∂c + 21 ∂∂2 Sc σ 2 S 2 where r stands for the risk-free rate of and Qc = 1r ∂S return. Hence, the dynamics of the replicating portfolio are given by
 ∂c dR = (9) dS + rQc dt ∂S where R refers to the replicating portfolio.

9.2.2. An Extension to Account for Information Costs in the Valuation of Derivatives Information costs are defined in the spirit of Merton (1987) as in the previous chapter. These costs appear in option pricing models in the analysis conducted by Bellalah and Jacquillat (1995) and Bellalah (1999, 2000a,b, 2001). The trading of financial derivatives on organized exchanges has exploded since the beginning of 1970s. The trading on “over-the-counter” or OTC market has exploded since the mid-1980s. Since the publication of the pionnering papers by Black–Scholes (1973) and Merton (1973), three industries have blossomed: an exchange industry in derivatives, an OTC industry in structured products and an academic industry in derivative research. Each industry needs a specific knowledge regarding the pricing and the production costs of the products offered to the clients. As it appears in Scholes (1998), derivative instruments provide (and will provide) lower-cost solutions to investor problems than will competing alternatives. These solutions will involve the repackaging of coarse financial products into their constituent parts to serve the investor demands. The “commoditisation” of instruments and the increased competition in the over-the-counter (OTC) market reduce profit margins for different players. The inevitable result is that products become more and more complex requiring more and more expenses in information acquisition. The problems of information, liquidity, transparency, commissions and charges are specific features of these markets. Differences in information are important in financial and real markets. They are used in several contexts to explain some puzzling phenomena like the “smile effect”,1 etc. Since Merton’s CAPMI can explain several anomalies in financial markets, its application in the valuation of derivative securities can be useful in explaining some anomalies in the option markets as the smile effect. 1 See the models in Bellalah and Jacquillat (1995) and Bellalah (1999).

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As it appears in the work of Black (1989), Scholes (1998) and as Merton (1998) asserts: “Fisher Black always maintained with me that the CAPM-version of the option model derivation was more robust because continuous trading is not feasible and there are transaction costs.”

This approach will be used here by applying the CAPMI of Merton (1987). As it is well-known, in all standard asset pricing models, assets that show only diversifiable risk or nonsystematic risk are valued to yield an expected return equal to the riskless rate. In Merton’s context, the expected return is equal to the riskless rate plus the shadow costs of incomplete information. The derivation of an option pricing model is based on an arbitrage strategy which consists in hedging the underlying asset and rebalancing continuously until expiration. This strategy is only possible in a frictionless market. Investors spend time and money to gather information about the financial instruments and financial markets. Consider for example a financial institution using a given market. If the costs of portfolio selection, models conception, etc. are computed, then it can require at least a return of say, for example λ = 3%, before acting in this market. This cost is in some sense the minimal return required before implementing a given strategy. If you consider the above replicating strategy, then the returns from the replicating portfolio must be at least:
dR =

 ∂c dS + (r + λ)Qc dt ∂S

     1 ∂c 1 ∂2 c 2 2 with Qc = r+λ S + σ , where R refers to the replicating ∂S 2 ∂2 S portfolio. This shows that the required return must cover at least the costs necessary for constructing the replicating portfolio plus the risk-free rate. In fact, when constructing a portfolio, some money is spent and a return for that must be required. Hence, there must be a minimal cost and a minimal return required for investing in information at the aggregate market level. For this reason, the required return must be at least λ plus the riskless rate. For an introduction to information costs and their use in asset pricing, the reader can refer to Appendix 3.

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9.3. Ito’s Differential and the Arbitrage Portfolio If one uses arbitrage arguments, then the option value must be equal to the value of its replicating portfolio.

9.3.1. The Standard Analysis Using arbitrage arguments, we must have




 1 ∂c 1 ∂2 c ∂c 2 2 S+ + σ S c = Qs S + Qc or c = ∂S r ∂t 2 ∂2 S and ∂c 1 ∂2 c 2 2 σ S + rc − r S + 2 2∂ S ∂S




∂c ∂t

 = 0.

This equation is often referred to in financial economics as the Black– Scholes–Merton partial differential  ∂c  equation. Note that the value of the S + Qc . It is possible to implement a replicating portfolio is R = ∂S hedged position by buying the derivative asset and selling delta units of the underlying asset:
 ∂c S = Qc H = c − ∂S where the subscript H refers to the hedged portfolio. A hedged position or portfolio is a portfolio whose return at equilibrium must be equal in theory to the short-term risk-free rate of interest. This is the main contribution of Black–Scholes (1973) to the pricing of derivative assets. Merton (1973) uses the same argument as Black–Scholes (1973) by implementing the concept of self-financing portfolio. This portfolio is also constructed by buying the option and selling the replicating portfolio or vice versa. The condition on the self-financing portfolio is
 ∂c A = c − S − Qc = 0 ∂S where S refers to the self-financing portfolio. The omitted error term in the above analysis, de(t), can reflect a replication error, a hedging error or an arbitrage error. It can have different interpretations. The term de(t) is neglected or omitted when the revision of the portfolio is done to allow for the replicating portfolio to be self-financing.

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When this term is positive, this may refer to additional cash that must be put in the portfolio. When it is negative, a withdrawal of cash from the portfolio is possible.

9.3.2. An Extension to Account for Information Costs in Option Pricing Theory In the same way, the previous analysis can be extended to account for information costs. In this context, we must have:




 1 ∂c 1 ∂2 c ∂c 2 2 S+ + σ S c = QS S + Qc or c = ∂S (r + λ) ∂t 2 ∂2 S and 1 2





 
 ∂c ∂c ∂2 c 2 2 S σ S + = 0. + (r + λ)c − (r + λ) ∂2 S ∂S ∂t

This equation corresponds to an extended version of the well-known Black–Scholes–Merton partial differential equation accounting for the effect of information costs. For the sake of simplicity, we assume that information costs are equal in both markets: the option market and the underlying asset market. Or in practice, institutions and investors support these costs on both markets. Therefore, a more suitable analysis must account for two costs: an information cost λc on the option market and an information cost λS on the underlying asset market. In this case, we obtain the following more general equation as in Bellalah and Jacquillat (1995) and Bellalah (1999): ∂c ∂c 1 2 2 ∂2 c σ S 2 + (r + λc )c − (r + λS ) S + = 0. 2 ∂ S ∂S ∂t

10. Differential Equation for a Derivative Security on a Spot Asset in the Presence of a Continuous Dividend Yield and Information Costs We denote by V the price of a derivative security on a stock with a continuous dividend yield q. The dynamics of the underlying asset are given by: dS = µS dt + σS dz, where the drift term µ and the volatility σ are constants.

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Using Ito’s lemma for the function V(S, t) gives
 ∂V ∂V dV 1 ∂2 V 2 2 dV = σ S dt + µS + + σS dz. 2 dS dt 2 ∂S ∂S It is possible to construct a portfolio  by holding a position in the derivative security and a certain number of units of the underlying asset  = −V +(∂V /dS)S. Over a short time interval, the change in the portfolio value can be written as
 ∂V 1 ∂2 V 2 2  = − − σ S t. ∂t 2 ∂S 2 Over the same time interval, dividends are given by qS(∂V /∂S)t. We denote by W the change in the wealth of the portfolio holder. In this case, we have
 ∂V 1 ∂2 V 2 2 ∂V W = − − t. σ S + qS ∂t 2 ∂S 2 dS Since this change is independent of the Wiener process, the portfolio is instantaneously risk-less and must earn the risk-free rate plus information costs or
 ∂V ∂V ∂V 1 ∂2 V 2 2 − σ S + qS − t = −(r + λV )Vt + (r + λS )S t. 2 ∂t 2 ∂S ∂S ∂S This gives ∂V 1 ∂2 V 2 2 ∂V + (r + λS − q)S + σ S = (r + λV )V. ∂t ∂S 2 ∂S 2 This equation must be satisfied by the derivative security in the presence of information costs and a continuous dividend yield.

11. The Valuation of Securities Dependent on Several Variables in the Presence of Incomplete Information: A General Method When a variable does not indicate the price of a traded security, the pricing of derivatives must account for the market price of risk. The market price of risk γ for a traded security is given by γ=

µ−r−λ σ

(10)

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where µ indicates the expected return from the security. This equation can also be written as µ − r − λ = γσ.

(11)

The excess return over the risk-free rate in the presence of shadow costs on a security corresponds to its market price of risk multiplied by its volatility. When γ > 0, the expected return on an asset is higher than the risk-free rate plus information costs. When γ = 0, the expected return on an asset is exactly the risk-free rate plus information costs. When γ < 0, the expected return on an asset is less than the risk-free rate plus information costs. When a variable does not indicate the price of a traded security, its market price of risk corresponds to the market price of risk of a traded security whose price is a function only on the value of the variable and time. The value of the market price of risk of the variable is the same at each instant of time. In fact, we can show as in Hull (2000) that two traded securities depending on the same asset must have the same price of risk, i.e. that Eq. (10) must be verified. Consider the following dynamics for a variable θ which is not a tradable asset: dθ = µ(θ, t)dt + s(θ, t)dz. θ We denote by V1 and V2 , respectively, the prices of two derivative securities as a function of θ and t. The dynamics of these derivatives can be written as dV1 = µ1 dt + σ1 dz V1 dV2 = µ1 dt + σ2 dz. V2 These two processes can be written in discrete time as V1 = µ1 V1 t + σ1 V1 z V2 = µ2 V2 t + σ2 V2 z.

(12) (13)

It is possible to construct a portfolio  which is risk-free using σ2 V2 of the first derivative security and −σ1 V1 of the second derivative security:  = σ2 V2 V1 − σ1 V1 V2 .

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The change in the value of this portfolio can be written as  = σ2 V2 V1 − σ1 V1 V2 . Using Eqs. (12) and (13), the change in the portfolio value can be written as  = µ1 σ2 V1 V2 − µ2 σ1 V1 V2 t. Since the portfolio  is instantaneously risk-less, it must earn the riskfree rate plus information costs on both markets. Hence, we must have µ1 σ2 − µ2 σ1 = (r + λ1 )σ2 − (r + λ2 )σ1 or µ2 − (r + λ2 ) µ1 − (r + λ1 ) = . σ1 σ2

(14)

The term (µ − (r + λ))/σ must be the same for all securities that depend on time and the variable θ. It is also possible to show that V1 and V2 must depend positively on θ. Since the volatility of V1 is σ1 , it is possible to use Ito’s lemma for σ1 to obtain: σ1 V1 = sθ(∂V1 /∂θ). Hence, when V1 is positively related to the variable θ, the σ1 is positive and corresponds to the volatility of V1 . But, when f1 is negatively related to the variable θ, σ1 is negative and the equation for V1 can be written as dV1 = µ1 dt + (−σ1 )(−dz). V1 This indicates that the volatility is −σ1 rather than σ1 . The result in Eq. (14) can be generalized to n state variables. Consider n variables which are assumed to follow Ito diffusion processes where for each state variable i between 1 and n, we have dθi = mi θi dt + si θi dzi where dzi are Wiener processes. The terms mi and si correspond to the expected growth rate and the volatility of the θi with i = 1, . . . , n. The price process for a derivative security that depends on the variables θi can be written as:  dV = µ dt + σi dzi V i=1 n

where µ corresponds to the expected return from the security and σi is its volatility. The volatility of V is σi when all the underlying variables

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except θi are kept fixed. This result is obtained directly using an extension of the generalized version of Ito’s lemma in its discrete form. We show in Appendix 4 that µ − r − λV =

n 

γi σi

(15)

i=1

where γi indicates the market price of risk for the variable θi . Equation (15) shows that the expected excess return on the security (option) in the presence of shadow costs depends on γi and σi . When γi σi > 0, a higher return is required by investors to get compensated for the risk arising from the variables θi . When γi σi < 0, a lower return is required by investors to get compensated for the risk arising from θi .

12. The General Differential Equation for the Pricing of Derivatives We denote by: — — — — — —

θi : value of ith state variable, mi : expected growth in ith state variable, γi : market price of risk of ith state variable, si : volatility of ith state variable, r: instantaneous risk-free rate, λi : shadow cost of incomplete information of ith state variable,

where i takes the values from 1 to n. Garman (1976) and Cox, Ingersoll and Ross (1985) have shown that the price of any contingent claim must satisfy the following partial differential equation: ∂2 f 1 ∂V  ∂V + θi (mi − γi si ) + ρi,k si sk θi θk = rf ∂t ∂θi 2 i,k ∂θi ∂θk i where ρi,j stands for the correlation coefficient between the variables θi and θk . We show in Appendix 4 how to obtain a similar equation in the presence of incomplete information. In this context, the equation becomes ∂V  ∂V ∂2 V 1 + θi (mi − γi si ) + ρi,k si sk θi θk = (r + λ)V. (16) ∂t ∂θi 2 i,k ∂θi ∂θk i

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In the presence of a single state variable, θ, the equation becomes ∂V 1 ∂2 V ∂V + θ (m − γs) + s2 θ 2 2 = (r + λ)V. ∂θ 2 ∂ θ ∂t

(17)

*/ For a non-dividend paying security, the expected return and volatility must satisfy m − r − λ = γs m − γs = r + λ. In this case, Eq. (17) becomes the extended Black–Scholes equation in the presence of information costs. */ For a dividend-paying security at a rate q, we have q + m − r − λ = γs or m − γs = r + λ − q. In this case, Eq. (17) becomes ∂V 1 ∂2 V ∂V + (r + λs − q)S + s2 S 2 2 = (r + λV )V. ∂t ∂S 2 ∂ S

13. Extension of the Risk-Neutral Argument in the Presence of Information Costs We know that the market price of risk is given by γ=

µ−r−λ , σ

or µ − r − λ = γσ. Appendix 5 shows how to price a derivative as if the world were risk neutral. This is possible when the expected growth rate of each state variable is (mi − γi si ) rather than mi . For the case of a non-dividend paying traded asset, we have mi − r − λi = γi si

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or mi − γi si = r + λi . This result shows that a change in the expected growth rate of the state variable from θi to (mi − γi si ) is equivalent to using an expected return from the security equal to the risk-less rate plus shadow costs of incomplete information. For the case of a dividend paying traded asset, we have qi + mi − r − λi = γi si or mi − γi si = r + λi − qi . This result shows that a change in the expected growth rate of the state variable from θi to (mi − γi si ) is equivalent to using an expected return (including continuous dividends at a rate q) from the security equal to the risk-less rate plus shadow costs of incomplete information. This analysis allows the pricing of any derivative security as the value of its expected payoff discounted to the present at the risk-free rate plus the information cost on that security or ˆ T] f = e−(r+λV )(T −t) E[f ˆ to the where VT corresponds to the security’s payoff at maturity T and E expectation operator in a risk-neutral economy. This refers to an economy where the drift rate in θi corresponds to (mi − γi si ). When the interest rate r is stochastic, it is considered as the other underlying state variables. In this case, the drift rate in r becomes γr sr , where γr refers to the market price of risk related to r. The term sr indicates its volatility. In this case, the pricing of a derivative is given by the discounting of its terminal payoff at the average value of r as: ˆ −(¯r+λV )(T −t) VT ] V = E[e where r¯ corresponds to the average risk-free rate between current time t and maturity T .

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14. Extension to Commodity Futures Prices within Incomplete Information Consider the pricing of a long position in a commodity forward contract with delivery price K and maturity T .

14.1. Commodity Futures Prices We denote by S the spot price of the commodity, F the futures price, µ the growth rate of the commodity price, σ its volatility and  its market price of risk. Using the extension of the risk-neutral valuation principle, the price is given by ˆ T − K] V = e−(r+λV )(T −t) E[S or ˆ T ] − K) f = e−(r+λf )(T −t) (E[S ˆ where E(.) refers to the expected value in a risk-neutral economy. The forward or futures price F corresponds to the value of K that makes the value of the contract f equal to zero in this last equation. So, we have ˆ T ]. F = E[S

(18)

This equation shows that the futures price corresponds to the expected spot price in a risk-neutral world. If (γσ) is constant and the drift µ is a function of time, then ˆ T ] = E[ST ]e−γσ(T −t) E[S where E corresponds to the real expectations or the expectations in the real world. Using Eq. (18), we have F = E[ST ]e−γσ(T −t) .

(19)

If  = 0, the futures price is an unbiased estimate of the expected spot price. However, if γ > 0, the futures price corresponds to a downward-biased estimate of the expected spot price. If γ < 0, the futures price is a downward-biased estimate of the expected spot price.

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14.2. Convenience Yields When the drift or the expected growth in the commodity price is constant, we have E(ST ) = Seµ(T −t) . Using Eq. (19), we have F = Se(µ−γσ)(T −t) . This last equation is consistent with the cost of carry model, when the convenience yield y satisfies the following relationship: µ − γσ = r + λ + g − y. This equation shows that the commodity can be assimilated to a traded security paying a continuous dividend yield equal to the convenience yield y less the storage costs u. Hence, the convenience yield must satisfy the following relationship: y = g + r + λ − µ + γσ. When the convenience yield is zero, we have µ − γσ = r + λ + g. This result shows that some commodities can be assimilated to traded securities paying negative dividend yields which are equal to storage costs.

Summary “Because options are specialised and relatively unimportant financial securities, the amount of time and space devoted to the development of a pricing theory might be questioned”, Professor Robert Merton (1973), Bell Journal of Economics and Management Science. Thirty years ago, no one could have imagined the changes that were about to occur in finance theory and the financial industry. The seeds of change were contained in option theory being conducted by the Nobel Laureates Fisher Black, Myron Scholes and Robert Merton. Valuing claims to future income streams is one of the central problems in finance. The first known attempt to value options appears in Bachelier (1900) doctoral dissertation using an arithmetic Brownian motion. This process amounts to negative asset prices. Sprenkle (1961) and Samuelson (1965) used a geometric Brownian motion that eliminates the occurrence of negative asset prices. Samuelson and Merton (1969) proposed a theory of option valuation by treating the option price as a function of the stock price. They advanced the

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theory by realizing that the discount rate must be determined in part by the requirement that investors hold all the amounts of stocks and the option. Their final formula depends on the utility function assumed for a “typical” investor. Several discussions are done with Robert Merton (1973) who was also working on option valuation. Merton (1973) pointed out that assuming continuous trading in the option or its underlying asset can preserve a hedged portfolio between the option and its underlying asset. Merton was able to prove that in the presence of a non-constant interest rate, a discount bond maturing at the option expiration date must be used. Black–Scholes (1973) and Merton (1973) showed that the construction of a riskless hedge between the option and its underlying asset, allows the derivation of an option pricing formula regardless of investors risk preferences. The main attractions of the Black–Scholes model are that their formula is a function of “observable” variables and that the model can be extended to the pricing of any type of option. Using some assumptions similar to those used in deriving the original B–S option formula, Black (1976) presented a model for the pricing of commodity options and forward contracts. Black (1976) showed that in the absence of interest rate uncertainty, a European commodity option on a futures (or a forward) contract can be priced using a minor modification of the Black and Scholes (1973) option pricing formula. In deriving expressions for the behavior of the futures price, they assumed that both taxes and transaction costs are zero and that the CAPM applies at each instant of time. This analysis was extended by several authors to account for other observable variables. This chapter presented in detail the basic concepts and techniques underlying rational derivative asset pricing in the context of analytical European models along the lines of Black–Scholes (1973), Black (1976) and Merton (1973). First, an overview of the analytical models proposed by the precursors is given. Second, the simple model of Black–Scholes (1973) is derived in detail for the valuation of options on spot assets and some of its applications are presented. Third, the Black model, which is an extension of the Black–Scholes model for the valuation of futures contracts and commodity options is analysed. Also, applications of the model are proposed.

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Fourth, the basic limitations of the Black–Scholes–Merton theory are studied and the models are applied to the valuation of several financial contracts. The Black–Scholes hedge works in the real, discrete, frictionful world when the hedger uses the correct volatility of the prices at which they actually trade and when the asset prices do not jump too much. The Black–Scholes formula gives a rough approximation to the formula investors would use if they knew how to account for the above factors. Modifications of the Black–Scholes formula can move it to the hypothetical perfect formula. Fifth, we derive the differential equation for a derivative security on a spot asset in the presence of a continuous dividend yield and information costs. Sixth, we provide the valuation of securities dependent on several variables in the presence of incomplete information. Seventh, we propose the general differential equation in the same context. Finally, we show how to extend the risk-neutral argument and the theory to the valuation of commodity futures contracts within incomplete information.

Questions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

What is wrong in Bachelier’s formula? What is wrong in Sprenkle’s formula? What is wrong in Boness’s formula? What is wrong in Samuelson’s formula? What are the main differences between the Black and Scholes model and the precursors models? How can we obtain the put–call parity relationship for options on spot assets? How can we obtain the put–call parity relationship for options on futures contracts? Is the Black model appropriate for the valuation of derivative assets whose values depend on interest rates? Justify your answer. Is there any difference between a futures price and the value of a futures contract? What are the holes in the Black–Scholes–Merton theory? What are information costs? What are the main results in the models of Merton (1973) and BaroneAdesi and Whaley (1987) model for the pricing of European commodity and commodity futures options?

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13. 14. 15. 16.

17.

18. 19. 20.

117

What are option price sensitivities? Describe Ito’s lemma and some of its applications. Explain Taylor series, Ito’s theorem and the replication argument. How can we derive the differential equation for a derivative security on a spot asset in the presence of a continuous dividend yield and information costs? How can we develop a general context for the valuation of securities dependent on several variables in the presence of incomplete information? How can we develop the general differential equation for the pricing of derivatives? How can one extend the risk-neutral argument in the presence of information costs? How can one extend the analysis to commodity futures prices within incomplete information?

Appendix 1: Greek Letter Risk Measures in Analytical Models A.1 Black–Scholes Model Call sensitivity parameters 1 ∂c = √ n(d1 ), ∂S Sσ T Sσn(d1 ) ∂c

c = − rKe−rTN(d2 ), = √ ∂T 2 T √ ∂c ∂c vc = = S T n(d1 ), ρc = = KT e−rTN(d2 ). ∂σ ∂r c = N(d1 ),

c =

Put sensitivity parameters ∂p 1 = √ n(d1 ), ∂S Sσ T Sσn(d1 ) ∂p

p = + rKe−rTN(−d2 ), =− √ ∂T 2 T √ ∂p ∂p vp = = S T n(d1 ), ρp = = −KT e−rTN(−d2 ). ∂σ ∂r p = c − 1,

p =

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A.2 Black’s Model The option sensitivity parameters in the Black’s Model are presented as follows: Call sensitivity parameters c = e−rTN(d1 ),

c =

e−rT ∂c = √ n(d1 ), ∂S Sσ T

Se−rT σn(d1 ) ∂c + rSe−rTN(d1 ) − rKe−rTN(d2 ), =− √ ∂T 2 T √ ∂c vc = = Se−rt T n(d1 ). ∂σ

c =

Put sensitivity parameters p = c − e−rT ,

p = −

p =

∂p 1 = √ n(d1 ), ∂S Sσ T

Sσe−rT n(d1 ) ∂p − rSe−rTN(−d1 ) + rKe−rTN(−d2 ), = √ ∂T 2 T √ ∂p vp = = S T n(d1 ). ∂σ

A.3 Garman and Kohlhagen’s Model Call sensitivity parameters ∗

e−r T ∂c c = e = N(d1 ), c = √ n(d1 ), ∂S Sσ T ∗ ∂c Se−r σn(d1 ) ∗ −r ∗ T −rT ,

c = N(d1 ) − rKe N(d2 ) − = r Se √ ∂T 2 T ∂c ∗ ρc = ∗ = −TSe−r TN(d1 ), ∂r ∂c ∗ √ vc = = Se−r T T n(d1 ). ∂σ −r ∗ T

Put sensitivity parameters p = e

−r ∗ T

∗

[N(d1 − 1],

∂p e−r T = p = √ n(d1 ), ∂S Sσ T

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∗

Sσe−r T n(d1 ) ∂p ∗ , = −r ∗ Se−r TN(−d1 ) + rKe−rTN(−d2 ) −

p = √ ∂T 2 T ∂p ∂p ∗ √ ∗ vp = = Se−r T T n(d1 ), ρ = ∗ = TSe−r TN(−d1 ). ∂σ ∂r

A.4 Merton’s and Barone-Adesi and Whaley’s Model Call sensitivity parameters e(b−r) ∂c = √ n(d1 ), ∂S Sσ T ∂p Se(b−r)T n(d1 ) ,

c = = (r − b)Se(b−r)TN(d1 ) − rKe−rTN(d2 ) − √ ∂T 2 T ∂c ∂c ∗ √ vc = = TSe(b−r)TN(d1 ). = Se−r T T n(d1 ), ρc = ∂b ∂σ  = e(b−r)N(d1 ),

c =

Put sensitivity parameters ∂p e(b−r)T = √ n(d1 ), ∂S Sσ T ∂p Se(b−r) σN(d2 )

p = , = Se(b−r)N(−d1 ) + rKerTN(d2 ) √ ∂T 2 T √ ∂p ∂p vp = = Se(b−r)T T n(d1 ), ρ = = −TSe(b−r)TN(−d2 ). ∂σ ∂b p = −e(b−r)T [N(−d1 ) + 1],

p =

Appendix 2: The Relationship Between Hedging Parameters Using the definitions of the delta, gamma and theta, the B–S equation can be written as



∂c(S, t) ∂c(S, t) 1 2 2 ∂2 c(S, t) = rc(S, t) − rS − σ S ∂t ∂S 2 ∂S 2 or 1 − = −rc(S, t) + rS + σ 2 S 2  2

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or 1 rc(S, t) = + rS + σ 2 S 2 . 2 For a delta-neutral position, the following relationship applies 1 rc(S, t) = + σ 2 S 2 . 2 Using the definitions of the hedging parameters, the Black equation can be written as

∂c(F, t) 1 2 2 ∂2 c(F, t) = rc(F, t) − σ F ∂t 2 ∂S 2 or 1

= rc(F, t) − σ 2 F 2 . 2 Using the definitions of the delta, gamma and theta, the Merton and BAW (1987) equation can be written as



1 2 2 ∂2 c(S, t) ∂c(S, t) σ S − rc(S, t) + = 0 + bS 2 ∂S 2 ∂S or 1 rc(S, t) = + bS + S 2 σ 2 . 2 For a delta-neutral position, the following relationship applies: 1 rc(S, T ) = S 2 σ 2  + . 2

Appendix 3: On the Valuation of Options and Information Costs An important question in financial economics is how frictions affect equilibrium in capital markets since in a world of costly information, some investors will have incomplete information. Merton (1987) advanced the investor recognition hypothesis in a mean–variance model.

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A.1 Incomplete Information, Costly Arbitrage and Asset Pricing Shapiro (2000) examines equilibrium in a dynamic pure-exchange economy under a generalization of Merton’s (1987) investor recognition hypothesis IRH). The premise in Merton’s (1987) model and Shapiro’s (2000) extension is that the costs of gathering and processing data lead some investors to focus on stocks with high visibility and also to entrust a portion of their wealth to money managers employed by pension plans. In this context, a trading strategy shaped by real-world information costs should incorporate an investment in well-known, visible stocks, and an investment delegated to professional money managers. In this theory, an investor considers only the stocks visible to him, i.e. those about which he has sufficient information to implement optimal portfolio rebalancing. In general, information about larger firms is likely to be available at a lower cost. The claim that large firms are more widely known is consistent with the empirical evidence that large firms have more shareholders as in Merton (1987). As documented by Falkenstein (1996), large firms present in general longer listing histories. Falkenstein (1996) documents that both size and age of a firm are positively correlated with the number of new stories in major newspapers about the firm. For these reasons, it is important to account for information costs in the pricing of assets and derivatives. Most of the option pricing models are based on an arbitrage argument. While most traders are aware of the Black– Scholes (1973) theory, the arbitrage mechanism assumed cannot work in a real options market in the same way that it does in a supposed frictionless market. From a theoretical standpoint, it is possible to account for some imperfections in the derivation of option pricing models. However, the mathematical problems raised by treating real market conditions are often too complex to be tractable in theory. From an empirically standpoint, it is possible to simulate trading strategies using historical option prices to see how much options arbitrage is affected by market imperfections. In this case, the transaction cost structure varies considerably among investors, traders and market makers which complicates the empirical tests. In practice, an arbitrageur follows a strategy that limits both trading costs (transaction costs) and risk. The first main point here is that hedging in not implemented in a continuous time framework. The second point is that transaction costs are different from the costs of collecting information or information costs.

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The third point, is that in less liquid markets, it is not always possible to implement an arbitrage strategy as described in the Black–Scholes theory. The fourth point is that the appropriate hedge must account for some of the costs of arbitrage. In a standard Black–Scholes approach in which the hedge is implemented instantaneously, the force of arbitrage drives the option price to its theoretical value.

A.2 The Intuition of the Derivation of Option Pricing Models with Costly Arbitrage As it appears in the work of Scholes (1998), the option pricing technology was adopted because it reduces transaction costs. The wave of financial innovation might be explained by the reduction in the cost of computer and communications technology. This lower-cost technology plays a significant role in the globalization of products and financial markets. Advances in financial theory allowed financial services firms to meet the complex needs of the clients around the world at lower cost than was previously possible.2 The arbitrage argument in Modigliani and Miller (see Miller, 1988) provided a general model of corporate finance by showing that the value of the firm is independent of how it financed its activities.3 Our analysis is based mainly on the use of capital asset pricing models for the valuation of derivatives. The work of Markowitz (1952), Sharpe (1964) and Lintner (1965) on the capital asset pricing model provided the general equilibrium model of asset prices under uncertainty. This model represents a fundamental tool in measuring the risk of a security under uncertainty. The first work of Black and Scholes was to test the standard CAPM by developing the concept of a zero-beta portfolio. A zero-beta-minimum variance portfolio can be implemented by buying low beta stocks and selling high beta stocks. If the realized returns on this portfolio are different from the interest rate, this would be a violation of the predictions of the original 2 Financial service firms are the leaders in using derivatives in their own risk-management programs.

In fact, using the available information and the option pricing technology, financial services firms can value their commitments and decide what risks to transfer and what risks to retain. As it appears in the tables reported in Scholes (1998), the OTC market in derivatives has grown much faster than the exchange market during the last 10 years. Therefore, it is expected that clients would find it cheaper to execute a program through their financial service than to execute it themselves in the exchange market. 3 The generalization of the Modigliani and Miller analysis in the presence of shadow costs of incomplete information is done in our papers (Bellalah, 2000a,b).

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CAPM.4 We justify the use of Merton’s (1987) model by the increasing empirical support for the implications of that simple model of capital market equilibrium with incomplete information. As it appears in Merton’s (1998) paper, his main contribution to the Black–Scholes option pricing theory was to demontrate the following result: in the limit of continuous trading, the Black–Scholes dynamic trading strategy designed to offset the risk exposure of an option would provide a perfect hedge. Hence, when trading is done without cost, the Black–Scholes dynamic strategy using the option’s underlying asset and a risk-free bond would exactly replicate the option’s payoff.5 This approach is used in the original Black–Scholes model who derived their formula using the standard CAPM.

Appendix 4: A General Equation for Derivative Securities Consider a derivative security whose price depends on n state variables and time t. The security can be priced under the standard Black–Scholes assumptions. The state variables are assumed to follow Ito diffusion processes where for each state variable i between 1 and n, we have dθi = mi θi dt + si θi dzi where the growth rate mi and the volatility si can be functions of any of the n variables and time. Let us denote, respectively, by — Vj : price of the jth traded security for j between 1 and n + 1, — r: risk-free rate, 4As it appears in Scholes (1998), his first work on option valuation was to apply the capital asset pricing model to value the warrants. The expected return of the warrant could not be constant for each time period if the beta of the stock was constant each period. This leads to the use of the CAPM to establish a zero-beta portfolio of common stocks and warrants. The portfolio is implemented by selling enough shares of common stock per each warrant held each period in order to create a zero-beta portfolio. In the context of the CAPM, the expected return on the net investment in the zero-beta portfolio would be equal to the riskless rate of interest. 5 In the absence of a continuous trading, which represents an idealized prospect, replication with discrete trading intervals is at best only approximate. Merton (1976) study a mixture of jump and diffusion processes to capture the prospect of nonlocal movements in the underlying asset’s return process. Replication is not possible when the sample path of the underlying asset is not continuous. In this case, the derivation of an option pricing model is completed by using an equilibrium asset pricing model.

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— λj : information cost for the jth traded security for j between 1 and n+1, — ρi,k : correlation coefficient between dzi and dzk . Since the (n + 1) traded securities depend on θi , then using Ito’s lemma, we have  dfj = µj Vj dt + σi,j Vj dzi (A.1) i

where µj Vj =

∂Vj  ∂Vj 1 ∂2 f + mi θi + ρi,k si sj θi θj ∂θk ∂t ∂θi 2 i,k ∂θi i ∂fj si θi . ∂θi

σij fj =

(A.2)

(A.3)

In this context, it is possible to construct a portfolio using the (n + 1) traded securities. We denote by aj the amount of the jth security in the portfolio  so that  aj Vj = j

where the aj are chosen in a way to eliminate the stochastic components of the returns. Using Eq. (A.1), we have  aj σij fj = 0 (A.4) j

for i between 1 and n. The instantaneous return from this portfolio can be written as  d = aj µj Vj dt j

 where the cost of constructing this portfolio is j aj Vj . If this portfolio is riskless, it must earn the riskless rate plus information costs corresponding to each asset in the portfolio   aj µj fj = aj fj (r + λj ) (A.5) j

j

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which is equivalent to 

aj fj (µj − r − λj ) = 0.

(A.6)

j

Equations (A.4) and (A.6) are consistent, only if  fj (µj − r + λj ) = γi σij fj

(A.7)

i

or µj − r − λ j =



γi σij

(A.8)

i

where for γi , i is between 1 and n. Using Eqs. (A.2) and (A.3) and replacing into Eq. (A.7) gives the following equation:  ∂Vj ∂Vj  ∂Vj ∂2 Vj 1 mi θi + ρik si sk θi θk −(r +λj )Vj = γi si θi . + ∂t ∂θi 2 i,k ∂θi ∂θk ∂θi i i This last equation reduces to ∂Vj  ∂Vj ∂2 Vj 1 θi (mi − γi si ) + ρik si sk θi θk = (r + λj )Vj . + ∂θi 2 i,k ∂θi ∂θk ∂t i Hence, any security f contingent on the state variables θi and time must satisfy the following second order differential equation: ∂V  ∂Vj ∂2 V 1 + θi (mi − γi si ) + ρik si sk θi θk = (r + λ)V. ∂t ∂θi 2 i,k ∂θi ∂θk i (A.9)

Appendix 5: Extension to the Risk-Neutral Valuation Argument When the variable θi is not a traded asset, it is possible to assume the existence of a traded asset θ˜ i paying a continuous dividend qˆ where qˆ = r + λi − mi + γi si .

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It is important to note that this term corresponding to the dividend yield allows the conversion of θi to a tradeable security. The values of θˆi and θi must be equal and the following differential equation must be verified: ∂V  ∂V ∂2 V 1 + θˆi (r + λi − qˆ i ) + ρi,k si sk θˆi θˆk = (r + λ)V. ˆi ˆi ∂θˆk ∂t 2 ∂ θ ∂ θ i i,k This equation is independent of risk preferences. Since (r + λi − q˜ ) = mi − γi si , the derivative security can be valued in a risk-neutral economy if the drift term in θi is modified from mi to mi − γi si .

Appendix 6: The Cumulative Normal Distribution Function The following approximation of the cumulative normal distribution function N(x) produces values to within four decimal place accuracy:

x 1 N(x) = √ exp(−z2 /2)dz 2π −∞ 

N(x) = 1 − n(x)(a1 k + a2 k2 + a3 k3 ), when 1 − n(−x), when

x0 x 1, the call (put) will start (α − 1) percent out-of-the-money (in-the-money). In the presence of a cost of carry b, the forward start call and put options are given by: c = Se−(b−r)t [e−(b−r)(T −t) N(d1 ) − αe−r(T −t) N(d2 )]

 1   ln α + b + 21 σ 2 (T − t) , d2 = d1 − σ (T − t). d1 = √ σ (T − t) The put formula is p = Se−(b−r)t [e−(b−r)(T −t) N(d1 ) − αe−r(T −t) N(−d2 )]. Tables 3 and 4 provide some simulations of forward start options values.

5. Ratchet Options A cliquet, moving strike or Ratchet option corresponds to a series of forward starting options. In this case, the strike price for the next exercise date is set equal to a given constant times the asset price as of the previous date. In general, the strike price of the first option is set equal to the current asset

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Table 3: S 96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

Simulations of forward start call values.

Price

Delta

Gamma

Vega

4.29928 4.34406 4.38885 4.43363 4.47841 4.52320 4.56798 4.61277 4.65755

0.04478 0.04478 0.04478 0.04478 0.04478 0.04478 0.04478 0.04478 0.04478

0.00000 −0.00000 −0.00000 −0.00000 0.00000 −0.00000 0.00000 0.00000 0.00000

0.18910 0.19107 0.19304 0.19501 0.19698 0.19895 0.20092 0.20289 0.20486

Theta 0.02625 0.02653 0.02680 0.02707 0.02735 0.02762 0.02789 0.02817 0.02844

S = 100, t = 11/01/2003, T = 05/06/2003, r = 4%, σ = 20%, forward start date = 06/03/2003.

Table 4: S 96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

Simulations of standard forward start puts.

Price

Delta

Gamma

Vega

Theta

3.34667 3.38153 3.41639 3.45125 3.48611 3.52097 3.55583 3.59069 3.62555

0.03486 0.03486 0.03486 0.03486 0.03486 0.03486 0.03486 0.03486 0.03486

0.00000 0.00000 −0.00000 0.00000 0.00000 0.00000 −0.00000 0.00000 −0.00000

0.18910 0.19107 0.19304 0.19501 0.19698 0.19895 0.20092 0.20289 0.20486

0.01584 0.01600 0.01617 0.01633 0.01650 0.01666 0.01683 0.01699 0.01716

S = 100, t = 11/01/2003, T = 05/06/2003, r = 4%, σ = 20%, forward start date = 06/03/2003.

price. Hence, a cliquet option can be valued as a sum of forward starting options as: c=

n 

Se−(b−r)ti [e−(b−r)(Ti −ti ) N(d1 ) − αe−r(Ti −ti ) N(d2 )]

i=1

where n corresponds to the number of settlements, ti time of the forward start.

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6. Pay Later Options Pay-later options provide a certain insurance against large one-way price movements and are traded on stock indices, foreign currencies and other commodities. The buyer of pay-later options has the obligation to exercise his option when it is in-the-money and to pay the premium. The exercise takes place regardless of the importance of the difference between the underlying asset price and the strike price, i.e., the amount by which the option is in-the-money.

6.1. Standard Pricing Following Turnbull (1989), we use the following notation: ST : F: cT : r∗ :

price of the underlying asset at the option’s maturity date, current forward rate, option premium paid at the option’s maturity date, foreign interest rate.

At the option’s maturity date, the pay later European call’s pay-off is cpayl (ST , 0, K) = (ST − K − cT )1ST >K . The option pays out ST − K − cT when ST > K, otherwise it has a zero pay-off. Applying standard arbitrage arguments, the value of the pay-later European call option is given by: ∗

cpayl (ST , T, K) = Se−r TN(d1 ) − (K + cT )e−rTN(d2 ) d1 =

[ln(S/K) + (r − r ∗ + 21 σ 2 )T ] , √ σ T

√ d2 = d1 − σ T .

When the option contract is initiated, the premium cT must be fixed in a way such that its current value is zero, i.e. Cpayl (ST , T, K) = 0. This implies that 

N(d1 ) N(d1 ) ∗ cT = Se(r−r )T −K =F − K. N(d2 ) N(d2 ) The pay-later call option formula can also be written as ∗

cpayl (S, T, K) = Se−r TN(d1 ) − Ke−rTN(d2 ) − cT e−rTN(d2 ).

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This price corresponds also to the value of a foreign exchange option less cT digital options. This decomposition offers a natural way for the hedging of the pay-later option. In fact, it shows that the purchase of cT digital calls and the sale of standard calls with the same time to maturity and strike price give a perfect hedge. At the option’s maturity date, the pay-later European put option’s payoff is ppayl (ST , 0, K) = (K − ST − pT )1ST K . Hence, the option pays out ST −K−cT when ST > K, otherwise it has a zero pay-off. Applying standard arbitrage arguments, the value of the pay-later

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European call option is given by: cpayl (S, T, K) = Se−(r d1 =

∗ +λ −λ )T o S

N1 (d1 ) − (K + cT )e−(r+λo )TN1 (d2 )

[ln(S/K) + (r − r ∗ + λS + 21 σ 2 )T ] , √ σ T

√ d2 = d1 − σ T .

When the option contract is initiated, the premium cT must be fixed in a way such that its current value is zero, i.e. cpayl (ST , T, K) = 0. This implies that  

(r−r ∗ +λS )T N1 (d1 ) λS T N1 (d1 ) cT = Se − K = Fe − K. N1 (d2 ) N1 (d2 ) The pay-later call option formula can also be written as: cpayl (S, T, K) = Se−(r

∗ +λ −λ )T o S

N1 (d1 ) − Ke−(r+λo )TN1 (d2 )

− cT e−(r+λo )TN1 (d2 ). At the option’s maturity date, the pay-later European put option’s pay-off is ppayl (ST , 0, K) = (K − ST − pT )1ST T , R: one plus the riskless interest rate, d: one plus the payout rate, η: a binary variable taking the value 1 (−1) when the underlying option is a call (put), φ: a binary variable taking the value 1 (−1) for a call on a call and a call on a put (a put on a call and a put on a put). The pay-off of a compound option at time T can be written as: Ccom = max[0, φPVt (max[0, ηSt ∗ − ηK1 | t ∗ ]) − φK2 ] where PVt (.) is the present value at time t of (.).

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The current value of the compound option is given by its expected value, under the appropriate probability measure Ccom = R−t E[max[0, φc(St , t) − φK2 ]] where E(.) is the mathematical expectation operator and c(St , t) is the call formula. This can be written in an integral form as  ∞ −t Ccom = R [max[0, φc(Seu , t) − φK2 ]]f(u)du −∞

√ √ 2 with u = ln(St /S), f(v) = (1/σ 2πt)e−v /2 , v = (u − µt)/σ 2t, µ = ln(R/d) − σ 2 /2. The above integral can be valued using a decomposition to the three pay-off variables, S, K1 and K2 , denoted, respectively, by I(1), I(2) and I(3) where  φ∞ −t ∗ u −t I1 = φSR e N(zt )f(u)du = φηSd N(φηx, ηy, φρ) ln

Scr S

and −t



φ∞

I2 = φK1 R

ln

Scr S

N(zt

√ − σ t ∗ − t)f(u)du

√ √ = φηK1 R−tN(φηx − φησ t, ηy − ησ t ∗ , φρ) and I3 = φK2 R−t



φ∞

ln

where

Scr S

√ f(u)du = φK2 R−tN(φηx − φησ t)



t , t∗  −t   √ Sd x = ln σ t + Scr R−t  −t ∗   √ Sd y = ln σ t + KR−t ∗

ρ=

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and Scr is solution to the following equation: ηScr d −(t

∗ −t)

N(zt ) − ηK1 R−(t

√ N(zt − σ t ∗ − t) − K2 = 0

∗ −t)

with ∗

z = ln

Scr d −t −t ∗ K1 R−t −t √∗ σ t −t

1 √ + σ t ∗ − t. 2

This equation gives the value of the critical underlying asset price. It can be computed using an iterative procedure. The current value of a compound option is given by ∗

Ccom = φηSd −t N(φηx, ηy, φρ) − φηK1 R−tN(φηx √ √ − φησ t, ηy − ησ t ∗ , φρ) √ − φK2 R−tN(φηx − φησ t). ∗

∗

For continuous compounding, the terms d −t , d −(t −t) , R−t can be writ∗ ∗ ten as e−dt , e−d(t −t) , and e−rt . Tables 33–36 provide simulations results for compound options and the corresponding Greek letters.

Table 33:

Compound FX-call/call.

S

Price

Delta

Gamma

Vega

Theta

0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04

0.02740 0.03007 0.03290 0.03590 0.03907 0.04243 0.04595 0.04966 0.05355

0.26587 0.28279 0.30000 0.31747 0.33515 0.35299 0.37096 0.38901 0.40710

0.69341 0.67661 0.65950 0.64213 0.62454 0.60677 0.58887 0.57089 0.55287

0.00305 0.00317 0.00330 0.00341 0.00352 0.00363 0.00372 0.00381 0.00389

0.02740 0.03007 0.03290 0.03590 0.03907 0.04243 0.04595 0.04966 0.05355

S = 1, K = 1, t = 07/02/2003, T = 07/02/2004, r = 3%, r∗ = 4%, σ = 20%, compound date = 06/02/2004, compound strike = 0.1.

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Table 34: Compound FX-call/put. S

Price

Delta

Gamma

Vega

Theta

0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04

0.02685 0.02296 0.01931 0.01590 0.01272 0.00976 0.00702 0.00449 0.00216

−0.38911 −0.36509 −0.34150 −0.31841 −0.29589 −0.27398 −0.25275 −0.23222 −0.20814

0.38911 0.36509 0.34150 0.31841 0.29589 0.27398 0.25275 0.23222 0.20814

0.00447 0.00450 0.00451 0.00451 0.00449 0.00445 0.00440 0.00434 0.00426

0.02685 0.02296 0.01931 0.01590 0.01272 0.00976 0.00702 0.00449 0.00216

S = 1, K = 1, t = 07/02/2003, T = 07/02/2004, r = 3%, r ∗ = 4%, σ = 20%, compound date = 06/02/2004, compound strike = 0.1.

Table 35: Compound FX-put/call. S 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04

Price

Delta

Gamma

Vega

Theta

0.04056 0.03865 0.03678 0.03496 0.03319 0.03147 0.02980 0.02818 0.02662

−0.19084 −0.18668 −0.18217 −0.17735 −0.17227 −0.16697 −0.16149 −0.15586 −0.15013

0.19076 0.18661 0.18211 0.17730 0.17222 0.16692 0.16144 0.15582 0.15010

0.00069 0.00078 0.00086 0.00093 0.00100 0.00107 0.00113 0.00118 0.00123

0.04056 0.03865 0.03678 0.03496 0.03319 0.03147 0.02980 0.02818 0.02662

S = 1, K = 1, t = 07/02/2004, T = 07/02/2005, r = 3%, r ∗ = 4%, σ = 20%, compound date = 06/02/2005, compound strike = 0.1.

Table 36: S 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04

Compound FX-put/put.

Price

Delta

Gamma

Vega

Theta

0.04313 0.04505 0.04696 0.04884 0.05070 0.05254 0.05434 0.05611 0.05784

0.19185 0.19041 0.18851 0.18617 0.18343 0.18032 0.17686 0.17311 0.16907

−0.19161 −0.19019 −0.18830 −0.18598 −0.18326 −0.18016 −0.17672 −0.17298 −0.16896

−0.00019 −0.00028 −0.00037 −0.00047 −0.00055 −0.00064 −0.00072 −0.00080 −0.00087

0.04313 0.04505 0.04696 0.04884 0.05070 0.05254 0.05434 0.05611 0.05784

S = 1, K = 1, t = 07/02/2003, T = 07/02/2004, r = 3%, r ∗ = 4%, σ = 20%,

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Summary The option to exchange one risky asset for another is analyzed and valued. The identification of this option allows the pricing of several financial contracts. Examples include the performance incentive fee, the margin account, the exchange offer, and the standby commitment. This concept is useful in the valuation of complex options such as options on the minimum or the maximum of several assets. Margrabe provided valuation formulas for exchange options giving the right to exchange one risky asset for another. The Black–Scholes–Merton formula appears as a particular case of the Margrabe general formula. Options with an uncertain strike price are also analyzed and valued. Their prices are simulated and the effects of variations in parameter values are analyzed. The concept of an option with an uncertain strike price is useful for the pricing of bonds and bond options when inflation is accounted for. Forward start options provide an answer to the following question: how much can one pay for the opportunity to decide after a known time in the future, known as the ‘grant date’, to obtain an at-the-money call with a given time to matutity with no additional cost? A Ratchet option corresponds to a series of forward starting options. In this case, the strike price for the next exercise date is set equal to a given constant times the asset price as of the previous date. In general, the strike price of the first option is set equal to the current asset price. Pay-later options provide a certain insurance against large one-way price movements and are traded on stock indices, foreign currencies and other commodities. The buyer of pay-later options has the obligation to exercise his option when it is in the money and to pay the premium. A chooser allows its holder at the “choice date” to trade this claim for either a call or a put. The claim is a regular chooser when the call and the put have identical strike prices and time to maturity. The claim is a “complex chooser” when the call and the put have different strike prices or time to maturities. Hence, the chooser is neither a call nor a put. A complex chooser option is defined in the same way as the simple chooser except that either the strike prices or (and) the time to maturities for the call and the put are different. A compound option is an option whose underlying asset is an option. Since an option may be a call or a put, we may find four types of compound options: a call on a call, a call on a put, a put on a put and a put on a call.

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173

This chapter also presents a framework for the analysis and valuation of forward start options, pay-later options, simple chooser and complex chooser options and several other forms of compound options with and without information costs. Forward start, pay-later options and simple chooser options are analyzed and valued. Then compound options are applied and valued in different contexts with and without the presence of information costs.

Questions 1. 2. 3. 4. 5.

What is an exchange option? What do are the main applications of this concept? How do information costs affect the pricing procedure? What is an option with an uncertain strike price? What are the definitions and applications of the following options: forward start options, Ratchet options and pay-later options? 6. What are the definitions and applications of simple chooser options and complex choosers? 7. What are the payoffs of compound options? 8. How are compound options valued with and without information costs?

Bibliography Bellalah, M (1999). The valuation of futures and commodity options with information costs. Journal of Futures Markets, September. Bellalah, M (2000). A risk management approach to options trading on the Paris bourse. Derivatives Strategy, 5(6), 31–33. Bellalah, M (2001). Market imperfections; information costs and the valuation of derivatives: some general results. International Journal of Finance, 13(3), 1895–1927. Bellalah, M, Ma Bellalah and R Portait (2001). The cost of capital in international finance. International Journal of Finance, 13(3), 1958–1973. Bellalah, M and JL Prigent (2001). Pricing standard and exotic options in the presence of a finite mixture of Gaussian distributions. International Journal of Finance, 13(3), 1975–2000. Bellalah, M, JL Prigent and C Villa (2001). Skew without skewness: asymmetric smiles; information costs and stochastic volatilities. International Journal of Finance, 13(2), 1826–1837. Bellalah, M and F Selmi (2001). On the quadratic criteria for hedging under transaction costs. International Journal of Finance, 13(3), 2001–2020. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659.

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Fisher, S (1978). Call option pricing when the exercise price is uncertain and the valuation of index bonds. Journal of Finance, 33, 169–176. Geske, R (1979). A note on an analytical valuation formula for unprotected American call options with known dividends. Journal of Financial Economics, 7, 375–380. Margrabe, W (1978). The value of an option to exchange one asset for another. Journal of Finance, 33, 177–186. Merton, R (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183. Modigliani, F and GA Pogue (1975). Alternative investment performance fee arrangements and implications for SEC regulatory policy. Bell Journal of Economics, 6 (Spring), 127–160. Rubinstein, M (1991). Options for the undecided. Risk, 4, 70–73. Turnbull, S (1989). The price is right. Risk, 5, 54–55. Turnbull, S and L Wakeman (1991). A quick algorithm for pricing European average options. Journal of Financial and Quantitative Analysis, 28, 1–20. Willmott, P (1998). Derivatives, John Wiley and Sons.

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Chapter 4

Rainbow Options and Their Applications

This chapter is organized as follows: 1. In Section 2, rainbow options are analyzed and valued using a continuous-time framework. In particular, analytic formulas are given for options on the minimum and the maximum of two assets. 2. In Section 3, an extension of the results in Stulz (1982) by Rubinstein (1991) to the valuation of an option delivering the best of two assets and cash is proposed. 3. In Section 4, an extension to the valuation of an option delivering the best of two assets and cash is proposed by accounting for information costs. 4. In Section 5, options on several assets and non-path dependent options on assets with dividends are studied and valued in the presence of information costs. 5. In Section 6, some applications of rainbow options are presented. In particular, applications to currency bonds, multi-currency bonds and corporate option bonds are proposed. 6. In Section 7, spread options are valued with and without information uncertainty. For the analysis of information costs and valuation, we can refer to Bellalah (2001), Bellalah, Prigent and Villa (2001), M. Bellalah, Ma. Bellalah and R. Portait (2001), Bellalah and Prigent (2001), Bellalah and Selmi (2001), etc. 7. In Section 8, other similar options such as portfolio options and dual strike options are analyzed. 8. In Section 9, wildcard options are analyzed and valued in different contexts.

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1. Introduction ainbow options, and in particular, options on the minimum or the maximum of two or more risky assets, have proved to be useful in the pricing of a wide variety of contingent claims, traded assets and financial instruments whose values depend on extreme values. For example, in the Eurobond market, the option bond gives the right to the bearer to choose among two or more currencies in which the payment is to be made. A discount option bond gives the right to the bearer to choose at maturity between two currencies at a predetermined exchange rate. Compensation plans, risk-sharing contracts, collateralized loans, and growth opportunities among other contracts, may present a pay-off function corresponding to that of an option on the minimum or the maximum of two or more risky assets. More generally, two-color rainbow options refer not only to options on the maximum (minimum) of two assets, but also to all options whose payoff depends on two underlying assets: options delivering the best of two assets and cash, spread options, portfolio options, dual strike options, etc. Stulz (1982) and Johnson (1987) derived prices for options on the minimum or the maximum of two risky assets. As for ordinary options, it is possible to obtain some parity relationships between options on the minimum, the maximum, the underlying asset and the interest rate. Options on baskets or basket options can be priced and hedged in a straight forward manner. In fact, the pricing of these options is based on the Black–Scholes hedging argument in higher dimensions. Some of the applications of rainbow options concern foreign financial instruments and currencey bonds. A currency bond is a bond denominated in a different currency from that prevailing in the country in which the common stock of the firm is traded. A standard spread option entitles its holder the right to call or put the spread value against a predetermined strike price and corresponds in general to the difference between two prices. Spread options include spreads in futures markets, bond markets and energy markets. Spread options in futures markets correspond, for example, to the spread or the basis between Nebraska’s corn and Chicago’s corn. This spread may be due to location or grade among other things. Spread options in bond markets may result from the differences in yield spreads between bonds issued in different countries. Spread options in energy markets may be due to the differences between, for example, refined and unrefined products such as crack spreads.

R

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Wildcard options, which are implicit delivery options, exist in several financial contracts and have been extensively studied in the literature. The wildcard option feature has been investigated for U.S. listed options and futures contracts.1 This implicit option arises when the exercise value of a derivative asset is determined before the final exercise date and when exercise closes the underlying asset position.

2. Valuation of Options on the Maximum (Minimum) Stultz (1982) and Johnson (1987) derived valuation formulas for options on the maximum and the minimum of two or more risky assets. It is important to note that several exchange-traded futures contracts can be valued using the formula for the option on the minimum.

2.1. The Call on the Minimum of Two Assets Following, Stulz (1982), let S1 and S2 stand, respectively, for the prices of two risky assets. At maturity, the pay-off of a European call on the minimum of these two assets is cmin = max[min(S1 , S2 ) − K, 0].

(1)

Assume that the price dynamics of the two assets are given by dS1 = µ1 dt + σ1 dW1 , S1

dS2 = µ2 dt + σ2 dW2 S2

where µ1 , µ2 , σ1 and σ2 are constants. The price of a European call on the minimum of S1 and S2 , with a maturity date T and a strike price K, denoted by cmin (S1 , S2 , K, T − t) is equal to the value of a self-financing portfolio which has the same value as the option at date T . Denoting the value of that portfolio by P = P(S1 , S2 , τ) with τ = T − t, and using Ito’s lemma gives the dynamics for P as: dP =

∂P ∂P ∂P dt dS1 + dS2 − ∂S1 ∂S2 ∂τ
 ∂2P ∂2P 1 ∂2P 2 2 2 2 S σ + (S2 ) σ2 + 2 S1 S2 ρ12 σ1 σH dt. + 2 ∂2 S1 1 1 ∂2 S2 ∂S1 ∂S2

1 See for example Fleming and Whaley (1994), French and Maberly (1992), Valerio (1993) and Cohen

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If the portfolio comprises S1 , S2 , a riskless asset and is self-financing, then its value must satisfy the following differential equation:   ∂P ∂P S1 − r S2 −Pτ = rP − r ∂S1 ∂S2 
∂2P ∂2P 1 ∂2P 2 2 2 2 + 2 S S ρ σ σ (S ) σ (S ) σ + + 1 2 12 1 2 . (2) 1 2 1 2 ∂2 S2 ∂S1 ∂S2 2 ∂2 S1 Using Eq. (2) and the following boundary conditions P(S1 , S2 , 0) = max[min(S1 , S2 ) − K, 0] P(0, S2 , τ) = 0, P(S1 , 0, τ) = 0, Stulz (1982) provides the appropriate formulas for the valuation of these options. In the presence of a cost of carry b1 for asset 1 and b2 for asset 2, the formula for the pricing of a call on the minimum of two assets is: cmin (S1 , S2 , K, τ) = S1 e(b1 −r)τN(y1 , −d, −ρ1 ) √ + S2 e(b2 −r)τN(y2 , d − σ τ, −ρ2 ) √ √ − Ke−rτN(y1 − σ1 τ, y2 − σ2 τ, ρ) with

(3)

 

      + b1 − b2 + 21 σ22 τ ln SK1 + b1 + 21 σ12 τ , y1 = d= √ √ σ τ σ1 τ  S2  ln K + (b2 + 21 σ22 )τ y2 = , σ 2 = σ12 + σ22 − 2ρ12 σ1 σ2 √ σ2 τ ρ1 σ = σ1 − ρσ2 , ρ2 σ = σ2 − ρσ1 ln

S1 S2

and where N(α, β, ρ) is the bivariate cumulative normal distribution where α and β are the upper limits of integration, and ρ is the correlation coefficient. We can write in a compact form the price of a European call on the minimum of two assets as cmin (S1 , S2 , K, τ) = S1 e(b1 −r)τN(β1 , β2 , ρc ) + S2 e(b2 −r)τN(α1 , α2 , ρc ) − Ke−rτN(γ1 , γ2 , ρ12 ). If the strike is zero and there is no cost of carry, Eq. (3) reduces to cmin (S1 , S2 , 0, τ) = S1 − cE (S1 , S2 , 1, τ) = S1 − N(d11 ) + S2 N(d22 )

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√ √ with d11 = [ln(S1 /S2 ) + (1/2)σ 2 τ]/σ τ, d22 = d11 − σ τ, where cE (S1 , S2 , 1, τ) stands for the price of an option to exchange one unit of asset S2 for one unit of asset S1 . This formula is also given in Margrabe (1978).

2.2. The Call on the Maximum of Two Assets As for ordinary options, it is possible to obtain some parity relationships between options on the minimum, the maximum, the underlying asset and the interest rate. At maturity, the pay-off of a European call on the maximum of two assets is cmax (S1 , S2 , K, τ) = max[max(S1 , S2 ) − K, 0]. The value of this option is given by cmax (S1 , S2 , K, τ) = C(S1 , K, τ) − cmin (S1 , S2 , 0, τ) + C(S2 , K, τ) where C(S2 , K, τ) stands for the price of a European call on asset S1 , with a strike price K and a maturity date τ. This result can be easily verified. In fact, if S1 ≥ S2 , S1 is the maximum and the option on the maximum pays at maturity, max[0, S1 − K]. In this context, a portfolio comprising a call on S2 , a call on S1 and a sale of a call on the minimum of S2 and S1 also pays max[0, S1 − K] as cmin (S1 , S2 , 0, τ) = C(S2 , K, 0). The same analysis applies when S2 is the maximum of S1 and S2 . The value of the call on the maximum of two assets can also be written as: √ cmax (S1 , S2 , K, τ) = S1 e(b−r)τN(y1 , d, ρ1 ) + S2 e(b−r)τN(y2 , −d + σ τ, ρ2 ) √ √ − Ke−rτ [1 − N(−y1 + σ1 τ, −y2 + σ2 τ, ρ)].

2.3. The Put on the Minimum (Maximum) of Two Assets Let pmin (S1 , S2 , K, τ) be the price of a European put on the minimum of two assets S1 and S2 . Its price must satisfy the following relationship: pmin (S1 , S2 , K, τ) = e−rτK − cmin (S1 , S2 , 0, τ) + cmin (S1 , S2 , K, τ). This result can be seen by the construction of two portfolios A and B.

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Portfolio A comprises a put on the minimum of S1 and S2 . Portfolio B comprises a call on the minimum of S1 and S2 , a discount bond paying K at maturity and a sale of a call on the minimum with a zero strike price. Consider the pay-offs at maturity. If the minimum of S1 and S2 is greater than or equal to K, then portfolio A is worthless and portfolio B pays K − min[S1 , S2 ] + min[S1 , S2 ] − K = 0. If the minimum of S1 and S2 is S1 and is less than K, then portfolios A and B are worth K − S1 . If the minimum of S1 and S2 is S2 and is less than K, then portfolios A and B are worth K − S2 . Since these portfolios are worth the same at maturity, they must have the same initial value. The same proof applies for a put on the maximum, pmax (S1 , S2 , K, τ) and its value is given by Stulz (1982) as pmax (S1 , S2 , K, τ) = e−rτK − cmax (S1 , S2 , 0, τ) + cmax (S1 , S2 , K, τ). In the presence of a cost of carry, the value of the put on the minimum of two assets is given by: pmin (S1 , S2 , K, τ) = e−rτK − cmin (S1 , S2 , 0, τ) + cmin (S1 , S2 , K, τ) where √ cmin (S1 , S2 , 0, τ) = S1 e(b1 −r)τ − S1 e(b1 −r)τN(d) + S2 e(b2 −r)τN(d − σ τ). In the same context, the value of the put on the maximum of two assets is given by: pmax (S1 , S2 , K, τ) = e−rτK − cmax (S1 , S2 , 0, τ) + cmax (S1 , S2 , K, τ) where √ cmax (S1 , S2 , 0, τ) = S2 e(b2 −r)τ + S1 e(b1 −r)τN(d) − S2 e(b2 −r)τN(d − σ τ).

3. Options Delivering the Best of Two Assets and Cash: Standard Pricing Following the analysis in Rubinstein (1991), we denote by: Si : initial underlying price for asset 1 or 2, STi : terminal price for asset i: 1, 2,

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1 plus the riskless rate, 1 plus the pay-out rate for asset i: 1, 2, a fixed amount of cash potentially received at expiration, time to maturity, volatility of the underlying asset i: 1, 2, and correlation coefficient of the natural logarithms of 1 plus the rates of return of the two underlying assets.

R: di : K: t: σi : ρ:

The pay-off of this option is given by Cbac = max[ST1 , ST2 , K].

(4)

The value of this option is given by its expected pay-off discounted at the riskless interest rate under the appropriate probability Cbac = R−t E[max[ST1 , ST2 , K]]. This may be also written as: +∞ +∞ −t max[S1 ex , S2 ey , K]f(x, y)dx dy Cbac = R −∞

−∞

with x = ln(ST1 /S1 ), y = ln(ST2 /S2 ), where f(x, y) is the bivariate density function given by u

e(− 2 ) f(x, y) =

2πtσ1 σ2 1 − ρ2 with u=


1 (x − µ1 t)2 (x − µ1 t)(y − µ2 t) (y − µ2 t)2 + − 2ρ 1 − ρ2 σ1 σ2 (1 − ρ2 )t σ12 t σ22 t   R 1 µ1 = ln − σ12 , d1 2   1 R µ2 = ln − σ22 . d2 2

Since the option value depends on the positions of ST1 ,ST2 and K, it is convenient to break the pay-off into three components I1 , I2 and I3 : I1 = R−t times the expected final value of ST1 under the condition that ST1 > K and ST1 > ST2 . I2 = R−t times the expected final value of ST2 under the condition that ST2 > K and ST2 > ST1 . I3 = R−t times K times the probability that K > ST1 and K > ST2 .

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This decomposition allows to write the option value as the sum of the three components I1 + I2 + I3 with the following densities: 

− 21 v21





− 1 v2



(x − µ1 t) e 2 2 (y − µ2 t) , v2 = , v1 = f1 (x) = √ √ , f2 (y) = √ √ σ1 t σ2 t σ1 2πt σ1 2πt e

and the following conditional densities: 

f1 (x | y) =

e ,

σ1 2π(1 − ρ2 )t 

f2 (x | y) =





− 21 w1

− 21 w2

w1 =

(x − µ1 t) −

e ,

σ1 2π(1 − ρ2 )t

w2 =

2 − µ2 t)

σ12 (1 − ρ2 )t





σ1 ρ(y σ2

(y − µ2 t) −

σ2 ρ(x σ1

2 − µ1 t)

σ22 (1 − ρ2 )t

Using these densities, I1 can be written as:     +∞ x−ln S2 S1 I1 = S1 R−t    f2 (x | y)dy ex f1 (x) dx −∞

K S1

ln

or I1 = S1 d1−t [N(y1 ) − N(−x1 , y1 , ρ1 )]. I2 is given by: I2 = S2 R−t



+∞



  ln SK

  S y−ln S1 2

−∞

2

 f1 (y | x) dx ey f2 (y) dy

or I2 = S2 d2−t [N(y2 ) − N(−x2 , y2 , ρ2 )]. I3 is given by: I3 = KR−t



  ln SK

−∞

1



  ln SK

−∞

2

 f2 (x | y) dy f1 (x) dx

or √ √ I3 = KR−tN(−x1 + σ1 t, −x2 + σ2 t, ρ)

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with x1 = y1 =

 ln

ln

S1 d1−t KR−t

√

σ1 t  −t  S1 d1 S2 d2−t

√ t



 1 √ + σ1 t, 2 1 √ + t, 2

2 = σ12 + σ22 − 2ρσ1 σ2 ,

x2 =

ln

ln

S2 d2−t KR−t

183



√

σ2 t  −t 

1 √ + σ2 t, 2

S2 d2 S1 d1−t

1 √ + t. √ 2 t (ρσ2 − σ1 ) (ρσ1 − σ2 ) , ρ2 = . ρ1 = y2 =

The value of the option delivering the best of two assets and cash is Cbac = I1 + I2 + I3 or Cbac = S1 d1−t [N(y1 ) − N(−x1 , y1 , ρ1 )] + S2 d2−t [N(y2 ) − N(−x2 , y2 , ρ2 )] √ √ + KR−tN(−x1 + σ1 t, −x2 + σ2 t, ρ). For the case of continuous compounding, the terms d1−t , d2−t and R−t become e−d1 t , e−d2 t and e−rt , respectively.

4. Valuation of Options Delivering the Best of Two Assets and Cash with Shadow Costs of Incomplete Information We denote by λ1 , λ2 and λc the associated information costs. The pay-off of this option is Cbac = max[ST1 , ST2 , K]. The value of this option is given by its expected pay-off discounted at the riskless interest rate under the appropriate probability Cbac = e−(r+λc )t E[max[ST1 , ST2 , K]]. This may be also written as +∞ −(r+λc )t Cbac = e −∞

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+∞

−∞

max[S1 ex , S2 ey , K]f(x, y) dx dy

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with x = ln(ST1 /S1 ), y = ln(ST2 /S2 ) and where f(x, y) is the bivariate density function given by u

e(− 2 ) f(x, y) =

2πtσ1 σ2 1 − ρ2 with


1 (x − µ1 t)2 (x − µ1 t)(y − µ2 t) (y − µ2 t)2 u= + − 2ρ 1 − ρ2 σ1 σ2 (1 − ρ2 )t σ12 t σ22 t

1 µi = r + λi + di − σi2 . 2 Since the option value depends on the positions of ST1 , ST2 and K, it is convenient to break the pay-off into three components I1 , I2 and I3 : I1 is computed as I1 = e−(r+λc )t times the expected final value of ST1 under the condition that ST1 > K and ST1 > ST2 . I2 is computed as I2 = e−(r+λc )t times the expected final value of ST2 under the condition that ST2 > K and ST2 > ST1 . I3 is computed as I3 = e−(r+λc )t times K times the probability that K > ST1 and K > ST2 . This decomposition allows us to write the option value as the sum of the three components I1 + I2 + I3 with the following densities: 

f1 (x) =

e

− 21 v21

√

σ1 2πt 

f2 (y) =



e

− 21 v22

√

,

v1 =

(x − µ1 t) √ , σ1 t

,

v2 =

(y − µ2 t) √ . σ2 t



σ1 2πt

The following conditional densities are used:   1 [(x − µ1 t) − σσ21 ρ(y − µ2 t)]2 e − 2 w1 , , w1 = f1 (x | y) =

σ12 (1 − ρ2 )t σ1 2π(1 − ρ2 )t   − 21 w2 [(y − µ2 t) − σσ21 ρ(x − µ1 t)]2 e f2 (x | y) =

. (5) , w2 = σ22 (1 − ρ2 )t σ1 2π(1 − ρ2 )t

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Using these densities, I1 can be written as    S2 I1 = S1 e

+∞

−(r+λc )t

 ln

x−ln



K S1

S1



f2 (x | y) dy ex f1 (x) dx

−∞

or I1 = S1 e−d1 t e(λ1 −λc )t [N(y1 ) − N(−x1 , y1 , ρ1 )]. I2 can be written as I2 = S2 e

−(r+λc )t





+∞ 

ln



K S2

 y−ln

S1 S2





f1 (y | x) dx ey f2 (y) dy

−∞

or I2 = S2 e−d2 t e(λ2 −λc )t [N(y2 ) − N(−x2 , y2 , ρ2 )]. I3 can be written as I3 = Ke or

−(r+λc )t





ln −∞

K S1





 ln −∞

K S2





f2 (x | y) dy f1 (x) dx

√ √ I3 = Ke−(r+λc )tN(−x1 + σ1 t, −x2 + σ2 t, ρ)

with x1 = x2 = y1 = y2 =

ln

 S1 

ln

 S2 

ln

 S1 

ln

 S2 

K

K

K

K

+ (r + λ1 − d1 )t 1 √ + σ1 t, √ 2 σ1 t + (r + λ2 − d2 )t 1 √ + σ2 t √ 2 σ2 t + (d2 − d1 )t 1 √ + t, √ 2 t + (d1 − d2 )t 1 √ + t √ 2 t

and 2 = σ12 + σ22 − 2ρσ1 σ2 ,

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ρ1 =

(ρσ2 − σ1 ) ,

ρ2 =

(ρσ1 − σ2 ) .

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The value of the option delivering the best of two assets and cash is: Cbac = I1 + I2 + I3 or Cbac = S1 e−d1 t e(λ1 −λc )t [N(y1 ) − N(−x1 , y1 , ρ1 )] + S2 e−d2 t e(λ2 −λc )t [N(y2 ) − N(−x2 , y2 , ρ2 )] √ √ + Ke−(r+λc )tN(−x1 + σ1 t, −x2 + σ2 t, ρ).

5. Options on Several Assets: The Case of Non-Path Dependent Options on Assets with Dividends and Information Costs Options on baskets or basket options can be priced and hedged in a straight forward manner. In fact, the pricing of these options is based on the Black– Scholes hedging argument in higher dimensions.

5.1. The Valuation of Options on Several Assets Following the analysis in Willmott (1998), a portfolio consisting of a long position in a basket option and a short position in a number i of each asset Si can be constructed as follows: = V(S1 , . . . , Sm , t) −

m 

i Si .

i=1

Over each small interval of time, the change in this portfolio value can be written as    m m  m  2   ∂V 1 ∂V ∂ V   d = + σi σj ρij Si Sj − i dSi . dt + ∂t 2 i=1 j=1 ∂Si ∂Sj ∂Si i=1 ∂V When the proportion invested in the option i is equal to i = ∂S then the i portfolio becomes risk-free. Therefore, each asset must yield the riskless rate plus the information cost coresponding to that asset, as in the original modified Black–Scholes equation. This gives

 ∂V ∂2 V ∂V 1  σi σj ρij Si Sj + (r + λi )Si − (r + λV )V = 0 + 2 i=1 j=1 ∂Si ∂Sj ∂Si ∂t i=1 m

m

m

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187

where λi corresponds to an information cost supported by investors in a given market. This is the multidimensional equivalent version to the modified Black–Scholes in the presence of information costs. In the presence of a dividend yield bi on the ith asset, this equation can be modified as follows: m m m  ∂V ∂2 V ∂V 1   + (r+λi −bi )Si −(r+λV )V = 0. ρij σi σj Si Sj + ∂Si ∂Si ∂Sj i=1 ∂t 2 i=1 j=1

Note that there is an information cost for each asset.

5.2. The Valuation of Non-Path Dependent Options on Assets with Dividends The value of a European non-path-dependent option with a payoff given by (S1 , S2 , . . . , Sm ) at time T , in the presence of a dividend yield di and shadow costs of information is given by V = e−(r+λV )(T −t) (2π(T − t))−m/2 (Det )−1/2 (σ1 , . . . , σm )−1 0

∞

··· 0

∞

  Payoff(S1 , . . . , Sm ) 1 T −1 exp − α α dS1 , . . . , dSm S1 · · · Sm 2

(6)

with 1 αi = σi (T − t)1/2





Si log Si





σ2 + r + λ i − bi − i 2



 (T − t) .

This result is obtained using the Green function. The Green’s function is given by

V  (S, t) =

−r(T −t)

σS 

e √

2π(T − t)

e

 2 
2 − log S r− σ2 (T −t) S 2σ 2 (T −t)

for any S  . Tables 1–4 provide simulation results for the option price as well as the delta and the gamma with respect to the first asset and the second asset.

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Table 1:

Price

Delta 1

Gamma 1

Delta 2

Gamma 2

2.58960 2.75440 2.92299 3.09510 3.27041 3.44862 3.62939 3.81242 3.99737

0.16282 0.16677 0.17041 0.17378 0.17683 0.17956 0.18196 0.18405 0.18583

0.00395 0.00365 0.00336 0.00305 0.00273 0.00241 0.00209 0.00177 0.00146

0.11297 0.11297 0.11297 0.11297 0.11297 0.11297 0.11297 0.11297 0.11297

0.00033 0.00033 0.00033 0.00033 0.00033 0.00033 0.00033 0.00033 0.00033

S1 96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

Simulations of the values of a call on the minimum of two assets.

S1 = 100, t = 11/01/2002, T = 11/01/2003, r = 4%, σ1 = 20%, S2 = 100, σ2 = 30%, ρ = 0.8%.

Table 2:

Price

Delta 1

Gamma 1

Delta 2

Gamma 2

12.14544 11.67764 11.22122 10.77603 10.34191 9.91871 9.50625 9.10437 8.71290

−0.47349 −0.46203 −0.45072 −0.43957 −0.42858 −0.41775 −0.40709 −0.39660 −0.38628

0.01145 0.01131 0.01115 0.01099 0.01083 0.01066 0.01049 0.01032 0.01014

−0.42858 −0.42858 −0.42858 −0.42858 −0.42858 −0.42858 −0.42858 −0.42858 −0.42858

0.01083 0.01083 0.01083 0.01083 0.01083 0.01083 0.01083 0.01083 0.01083

S1 96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

Simulations of the values of a put on the minimum of two assets.

S1 = 100, t = 11/01/2002, T = 11/01/2003, r = 4%, σ1 = 20%, S2 = 100, σ2 = 30%, ρ = 0.8%.

Table 3: S1 96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

Simulations of the values of a call on the maximum of two assets. Price

Delta 1

Gamma 1

Delta 2

Gamma 2

18.81055 19.19449 19.59494 20.01179 20.44495 20.89426 21.35954 21.84061 22.33724

0.37575 0.39228 0.40875 0.42511 0.44134 0.45741 0.47329 0.48896 0.50440

0.01653 0.01646 0.01636 0.01623 0.01607 0.01588 0.01567 0.01544 0.01519

0.49878 0.49878 0.49878 0.49878 0.49878 0.49878 0.49878 0.49878 0.49878

0.01230 0.01230 0.01230 0.01230 0.01230 0.01230 0.01230 0.01230 0.01230

S1 = 100, t = 11/01/2002, T = 11/01/2003, r = 4%, σ1 = 20%, S2 = 100, σ2 = 30%, ρ = 0.8%.

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Table 4: S1 96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

189

Simulations of the values of a put on the maximum of two assets.

Price

Delta 1

Gamma 1

Delta 2

Gamma 2

2.80196 2.65370 2.51056 2.37261 2.23988 2.11239 1.99014 1.87308 1.76118

−0.15077 −0.14569 −0.14053 −0.13532 −0.13008 −0.12484 −0.11962 −0.11444 −0.10932

0.00508 0.00516 0.00521 0.00524 0.00524 0.00522 0.00518 0.00512 0.00505

−0.07264 −0.07264 −0.07264 −0.07264 −0.07264 −0.07264 −0.07264 −0.07264 −0.07264

0.00147 0.00147 0.00147 0.00147 0.00147 0.00147 0.00147 0.00147 0.00147

S1 = 100, t = 11/01/2002, T = 11/01/2003, r = 4%, σ1 = 20%, S2 = 100, σ2 = 30%, ρ = 0.8%.

6. Applications of Rainbow Options This section provides some applications of rainbow options.

6.1. Pricing Currency Bonds A currency bond is a bond denominated in a different currency from that prevailing in the country in which the common stock of the firm is traded. To price this simple form of currency bonds, we use the following notations: S1 : K∗ : r∗ : S:

value of the firm in domestic currency, face value of the bond in foreign currency, riskless rate of interest in foreign currency, spot exchange rate.

At maturity, the bond’s pay-off is given by min[S1 , SK∗ ] where SK∗ is ∗ uncertain. Let S2 (t) = S(t)e−r τ K∗ be the price of a zero-coupon bond at ∗ ∗ S1 t paying SK at T . Let B ( S , K∗ , τ) be the price of the foreign currency bond (in foreign currency). We can show that the bond price is also equal to the product of a standard European call and the inverse of the exchange rate, i.e.   1 ∗ S1 ∗ B , K , τ = cmin (S1 , S2 , 0, τ). S S In fact, Stulz (1982) observed that cmin (S1 , S2 , 0, τ) corresponds to a standard call with K = 1, r = 0 and underlying asset price S1 /S2 . Since

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cmin (S1 , S2 , 0, τ) is a decreasing function of σ 2 , the bond’s price is also a decreasing function of the volatility of the firm’s value and the exchange rate value. Using the interest rate parity theorem f(t, T ) = S(t)e(r−r

∗ )τ

where f(t, T ) stands for the forward exchange rate at t for a delivery at T , it is possible to show that S2 depends on f(t, T). Using this relationship, S2 can also be written as S2 = f(t, T )erτ K∗ . It follows that the higher the forward exchange rate and the expected future spot exchange rate, the higher the price of the bond. This result is immediate when changes in exchange rate are certain. In fact, in this context, the bond is assimilated to a bond in a domestic currency with a fixed strike price f(t, T)K∗ . Hence, an increase in the forward rate increases the bond’s face value in domestic currency.

6.2. Multi-Currency Bonds Let SA , (SB ) be the present price of one unit of currency of country A(B) and rA (rB∗ ) be the riskless interest rate in country A(B). Let B(SA KA∗ , SB KB∗ , K, τ) stand for the discount bond price, giving the option to the bearer, at maturity, to get either K units of domestic currency, or KA∗ units of currency of country A, or KB∗ units of currency of country B. A portfolio comprising a domestic discount bond, two options on SA KA∗ and SB KB∗ , with the same strike price K, and an option on the minimum of SA KA∗ and SB KB∗ , gives exactly the same pay-off as this option bond. If we define the values of two assets H and V as ∗

S2 = e−rA τ SA KA∗ ,

∗

S1 = e−rB τ SB KB∗

then the value of the bond is B(SA KA∗ , SB KB∗ , K, τ) = cmax (S1 , S2 , K, τ) + Ke−rτ where cmax (S1 , S2 , K, τ) stands for the price of a European option on the maximum of S1 and S2 , with a strike price K and a maturity date τ.

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6.3. Corporate Option Bonds Let B(S1 , S2 , K, τ) stand for the price of an option bond issued by a corporation. The issuer gives the bearer the option at maturity to choose between payments of K∗ units of the foreign currency and K units of the domestic currency. The bond’s pay-off is equivalent to that of a portfolio comprising a discount bond with face value K, an option on the minimum of S1 and ∗ S2 = e−r τ SK∗ , with a strike price K and a sale of a standard European put on S1 , with the same strike price. In this context, the price of the bond is given by: B(S1 , K, SK∗ , τ) = Ke−rτ − P(S1 , K, τ) + cmin (S1 , S2 , K, τ). At maturity, the currency option is worthless when the value of the firm is smaller than K. When S1 > SK∗ > K, the option on the minimum pays SK∗ − K, and when S1 > K > SK∗ it pays nothing. If SK∗ > S1 > K, the bond value is V and the portfolio value is K + (min[S1 , S2 ] − K) = S1 .

7. Spread Options and Basket Options A standard spread option entitles its holder the right to call or put the spread value against a predetermined strike price and corresponds in general to the difference between two prices. Examples of spread options include spreads in futures markets, bond markets and energy markets. Spread options in futures markets refer, for example, to the spread or the basis between Nebraska’s corn and Chicago’s corn. This spread may be due to location or grade among other things. Spread options in bond markets may be due to the differences in yield spreads between bonds issued in different countries. Spread options in energy markets result from the differences between, for example, refined and unrefined products such as crack spreads.

7.1. Definitions Spread options can be defined using two underlying assets, contracts or commodities which are closely related. This correlation between the

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assets or commodities results from demand substitution or the potential for transformation. The pay-off of spread options is cspr = max[0, (ST2 − ST1 ) − K] where the binary variable takes the value 1 for a call and −1 for a put. Even though the models proposed above allow the valuation of spread options, there are pros and cons to these approaches, since as shown by Garman (1992), the valuation of spread options is rather complex. Besides, they present the paradox of negative vegas.

7.2. Spread Option Valuation within Information Uncertainty Ravindran (1993) proposed a Black–Scholes approach for the pricing of European options on the spread between two indexes. Consider the following dynamics for the case of two underlying assets (i = 1, 2):   σ2 d(ln Si ) = µi − i dt + σi dzi (for i = 1, 2). (7) 2 The two Wiener processes are correlated by a coefficient ρ. When n = 1, the solution for a European call in the presence of information uncertainty is: call = e−rd T [S0 erd T e(λS −λc )TN(d1 ) − Ke−λc TN(d2 )] where



d1 =

ln

S0 K

   2 + rd +λS + σ2 T √ σ T

N(y) =

y

−∞

,

(8)

√ d2 = d1 − σ T 2

z √1 e− 2 2π

dz,

where rd is the domestic risk free rate of interest and λS and λC are information costs on the asset S and the option C. The European call price can also be written as: e−(rd +λc )T EST [max(ST − K, 0)]. Consider the following payout function at time T for the two correlated assets: [max(aS1,T + bS2,T + c, 0)], where a is positive, and b and c are negative. At initial time, this security value is: e−rd T ES1,T S2,T [max(aS1,T + bS2,T + c, 0)].

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Using conditional expectations, we have: ES1,T S2,T [max(aS1,T + bS2,T + c, 0)]   = ES2,T ES1,T [max(aS1,T + bS2,T + c, 0) | S2,T ] .

(9)

The conditional distribution of ln S1,T , given ln S2,T is: ln S1,T | ln S2,T ∼ N(µ∗ , σ∗2 )

(10)

1 (ln S2,T − µ2 ) and σ∗2 = σ12 (1 − ρ2 ). where µ∗ = µ1 + ρσ σ2 The conditional expectation in Eq. (9) can be computed using Eq. (10). When ln ST has a normal distribution, with mean α and variance β2 , then:     2 α − ln w α − ln w α+ β2 N + β − wN EST [max(ST − w, 0)] = e β β (11)

where w is an arbitrary constant. √ 2 When w = K, α = ln S0 + rd + λS − σ2 and β = σ T , this gives Eq. (8) without the discount factor. Using Eqs. (10) and (11), the conditional expectation in Eq. (9) is computed as: ES1,T [max(aS1,T + bS2,T + c, 0) | S2,T ]  
    b c , 0 S2,T = aES1,T max S1,T − − S2,T − a a
    σ2 µ∗ − ln K∗ µ∗ − ln K∗ µ∗ + 2∗ =a e N + σ∗ − K ∗ N σ∗ σ∗

(12)

where K∗ represents: − ba S2,T − ac while µ∗ and σ∗ are given in Eq. (10). The expectation in Eq. (12) can be computed under the assumption that ln S2,T has a√normal distribution with mean ln S2,0 + (µ2 − 1/2σ22 )T and volatility σ2 T . Example: Ravindran (1993) considers the example of a European call spread option on the forward prices for the following parameters: a = 1, b = −1, −c = K = 5, F1,0 = 10, F2,0 = 7, µ = 0, rs = 0.07, ρ = 0.5, T = 0.5, σi = 0.2. Using the Gaussian quadrature method of a function for an interval [0, 1], we have: e−rd T EF1,T F2,T [max(F1,T − F2,T − K, 0)] = 0.03969.

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Ravindran (1995) shows that this method is efficient in the computation of option prices.

8. Other Similar Options 8.1. Options on the Worst of Two Assets A call on the worst of two assets have the following pay-off: cwta = max[0, min[S1 , S2 ] − K]. Following the analysis in Stulz (1982), the value of this option is given by cwta (S1 , S2 , K, τ) = S1 N(d1 (S1 , σ1 ), d2 (S1 , S2 ), d3 (σ1 , σ2 ))N + S2 N(d1 (S2 , σ2 ), d2 (S2 , S1 ), d3 (σ2 , σ1 )) √ √ − Ke−rτN(d1 (S1 , σ1 ) − σ1 τ, d1 (S2 , σ2 ) − σ2 τ, ρ) with  d1 (Si , σi ) = for i = 1, 2 and d2 (Si , Sj ) =

ln



ln

 Sj   Si

Si Ki e−rτ



+ 21 σi2 τ

√ σi τ 

− 21 σ 2 τ √ σ τ

for i, j = 1, 2 and d3 (σi , σj ) =

(ρσj − σi )/σ, where σ = σ12 + σ22 − 2ρσ1 σ2 . Where N(., ., .) is the bivariate cumulative normal distribution. A put on the worst of two assets has the following pay-off: pwta = max[0, K − min[S1 , S2 ]]. Table 5 provides simulations of options values as well as the Greek letters. The reader can comment on the results.

8.2. Options on the Worst of Two Assets and Cash An option on the worst of two assets and cash has the following pay-off: min[S1 , S2 , K]. This can be written as: min[S1 , S2 , K] = K − max[0, K − min[S1 , S2 ]]. This option is equivalent to holding a cash position K and selling a put on the worst of two assets with a strike price K.

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Table 5: S1 96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

195

Simulations of the values of an option on the worst of two assets. Price

Delta 1

Gamma 1

Delta 2

Gamma 2

83.92299 84.39079 84.84721 85.29240 85.72652 86.14972 86.56218 86.96406 87.35553

0.47349 0.46203 0.45072 0.43957 0.42858 0.41775 0.40709 0.39660 0.38628

−0.01145 −0.01131 −0.01115 −0.01099 −0.01083 −0.01066 −0.01049 −0.01032 −0.01014

0.42858 0.42858 0.42858 0.42858 0.42858 0.42858 0.42858 0.42858 0.42858

−0.01083 −0.01083 −0.01083 −0.01083 −0.01083 −0.01083 −0.01083 −0.01083 −0.01083

S1 = 100, t = 11/01/2002, T = 11/01/2003, r = 4%, σ1 = 20%, S2 = 100, σ2 = 30%, ρ = 0.8%.

8.3. Portfolio Options Portfolio options have the following pay-off: cport = max[0, (n1 ST1 + n2 ST2 − K)] where n1 and n2 correspond to the number of units of the two assets in the portfolio.

8.4. Dual Strike Options Dual strike options have the following pay-off: cdso = max[0, 1 (ST1 − K1 ), 2 (ST2 − K2 )] where the binary variables 1 and 2 takes the values 1 or −1. For the valuation of these options, see the numerical techniques in Boyle et al. (1989).

8.5. Options on the Best (Better) of Two Assets Options on the best (better) of two assets have the following pay-off: cbta = max[0, max[S1 , S2 ] − K]. The payoff of a call on the best of two assets can be written as: max[0, S1 − K] + max[0, S2 − K] − max[0, min[S1 , S2 ] − K].

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The term max[0, S1 − K] is equivalent to buying a standard call on the first asset. The term max[0, S2 − K] is equivalent to buying a standard call on the second asset. The term max[0, min[S1 , S2 ] − K] corresponds to the sale of a call on the worst of two assets. Puts on the best (better) of two assets have the following pay-off: pbta = max[0, K − max[S1 , S2 ]]. The payoff of a European put on the best of two assets can be written as: max[0, K − S1 ] + max[S1 , S2 ] = K − max[S1 , S2 ] + max[0, max[S1 , S2 ] − K]. Tables 6 and 7 provide simulations of options values as well as the Greek letters. The reader can comment on the results. Table 6: S1 96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

Simulations of the values of an option on the best of two assets.

Price

Delta 1

Gamma 1

Delta 2

Gamma 2

112.07701 112.60921 113.15279 113.70760 114.27348 114.85028 115.43782 116.03594 116.64447

0.52651 0.53797 0.54928 0.56043 0.57142 0.58225 0.59291 0.60340 0.61372

0.01145 0.01131 0.01115 0.01099 0.01083 0.01066 0.01049 0.01032 0.01014

0.57142 0.57142 0.57142 0.57142 0.57142 0.57142 0.57142 0.57142 0.57142

0.01083 0.01083 0.01083 0.01083 0.01083 0.01083 0.01083 0.01083 0.01083

S1 = 100, t = 11/01/2002, T = 11/01/2003, r = 4%, σ1 = 20%, S2 = 100, σ2 = 30%, ρ = 0.8%.

Table 7: S1 96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

Simulations of the values of an option on the best of two assets and cash. Price

Delta 1

Gamma 1

Delta 2

Gamma 2

114.87897 115.26291 115.66335 116.08021 116.51336 116.96267 117.42796 117.90903 118.40565

0.37575 0.39228 0.40875 0.42511 0.44134 0.45741 0.47329 0.48896 0.50440

0.01653 0.01646 0.01636 0.01623 0.01607 0.01588 0.01567 0.01544 0.01519

0.49878 0.49878 0.49878 0.49878 0.49878 0.49878 0.49878 0.49878 0.49878

0.01230 0.01230 0.01230 0.01230 0.01230 0.01230 0.01230 0.01230 0.01230

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8.6. Options on the Best of Two Assets and Cash An option on the best of two assets and cash has the following pay-off: max[S1 , S2 , K]. This can be written as: max[S1 , S2 , K] = K + max[0, max[S1 , S2 ] − K]. This option is equivalent to investing K in cash and buying a call on the best of two assets with a strike price K.

8.7. Options on the Product of Two Assets Options on the product of two assets have the following pay-off: cpta = S1 · S2 . Tables 8–11 provide simulations of options values as well as the Greek letters. The reader can comment on the results.

Table 8: Simulations of the values of options on the product of two assets (call). S1 96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

Price

Delta 1

Gamma 1

Delta 2

Gamma 2

8900.22070 8993.93205 9087.64339 9181.35474 9275.06608 9368.77743 9462.48877 9556.20012 9649.91146

93.71134 93.71134 93.71134 93.71134 93.71134 93.71134 93.71134 93.71134 93.71134

0.00000 −0.00000 0.00000 0.00000 −0.00000 0.00000 −0.00000 0.00000 0.00000

104.12372 104.12372 104.12372 104.12372 104.12372 104.12372 104.12372 104.12372 104.12372

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

S1 = 100, t = 11/01/2002, T = 11/01/2003, r = 4%, σ1 = 20%, S2 = 90, σ2 = 30%, ρ = 0.5%.

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Table 9: S1 96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

Simulations of the values of a quotient of two assets (call).

Price

Delta 1

Gamma 1

Delta 2

Gamma 2

0.13523 0.13972 0.14388 0.14772 0.15125 0.15448 0.15742 0.16007 0.16244

0.00465 0.00432 0.00400 0.00368 0.00338 0.00308 0.00279 0.00251 0.00224

−0.00033 −0.00032 −0.00032 −0.00031 −0.00030 −0.00029 −0.00028 −0.00027 −0.00026

−0.00338 −0.00338 −0.00338 −0.00338 −0.00338 −0.00338 −0.00338 −0.00338 −0.00338

−0.00023 −0.00023 −0.00023 −0.00023 −0.00023 −0.00023 −0.00023 −0.00023 −0.00023

S1 = 100, t = 11/01/2002, T = 11/01/2003, r = 4%, σ1 = 30%, S2 = 100, σ2 = 20%, ρ = 0.8%.

Table 10: Simulations of the values of a quotient of two assets (put). S1 96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

Price

Delta 1

Gamma 1

Delta 2

Gamma 2

0.13648 0.13097 0.12514 0.11899 0.11252 0.10576 0.09870 0.09136 0.08374

−0.00534 −0.00567 −0.00600 −0.00631 −0.00662 −0.00691 −0.00720 −0.00748 −0.00775

−0.00033 −0.00032 −0.00032 −0.00031 −0.00030 −0.00029 −0.00028 −0.00027 −0.00026

0.00661 0.00661 0.00661 0.00661 0.00661 0.00661 0.00661 0.00661 0.00661

−0.00043 −0.00043 −0.00043 −0.00043 −0.00043 −0.00043 −0.00043 −0.00043 −0.00043

S1 = 100, t = 11/01/2002, T = 11/01/2003, r = 4%, σ1 = 30%, S2 = 100, σ2 = 20%, ρ = 0.8%.

Table 11: Simulations of the values of relative exchange options. S1 96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

Price

Delta 1

Gamma 1

Delta 2

Gamma 2

1.54215 1.53232 1.52251 1.51271 1.50292 1.49315 1.48338 1.47364 1.46390

−0.00983 −0.00982 −0.00981 −0.00979 −0.00978 −0.00977 −0.00976 −0.00974 −0.00973

0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001

0.02481 0.02481 0.02481 0.02481 0.02481 0.02481 0.02481 0.02481 0.02481

0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001

S1 = 100, t = 11/01/2002, T = 11/01/2003, r = 4%, σ1 = 40%, S2 = 100, σ2 = 10%, ρ = 0.9%.

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9. Wildcard Options Several models are proposed in the literature for the valuation of wildcard options in different markets. Fleming and Whaley (1994) analyzed and priced wildcard options embedded in the S&P 100 index option contract, OEX. The OEX wildcard options arise because the settlement price is given with reference to the S&P 100 index level at 3:00 pm Central Standard Time (CST), at the close of the New York Stock Exchange, while the option holder may postpone the exercise decision until 3:20 pm. If the market rises (falls) during this wildcard period of 20 minutes, the holder of a slightly in-the-money put option (or call) may find it optimal to exercise his option. Since the wildcard option is linked to the exercise procedure, it appears each day between 3:00 and 3:20 pm until the option’s maturity date. Hence, there is a wildcard option sequence which must be taken into account when pricing S&P 100 index option contracts. Fleming and Whaley (1994) provided a methodology that builds upon the standard CRR (1979) binomial model for valuing American-style options to price the embedded options. They applied their model to the wildcard early exercise premium, which is defined as the difference between an American option value with embedded wildcard options and an American option without this privilege. The results by Fleming and Whaley (1994) show that the wildcard premium is an important component of an OEX option value. It accounts, for example, for about 12 cents of the value of slightly in-the-money call and put options.

Summary In this chapter, several forms of rainbow options and options on the minimum or the maximum of many assets are analyzed and valued in a continuous and a discrete time setting. Some parity relationships between options on the minimum, the maximum, the underlying asset and the interest rate are proposed. Options on baskets or basket options can be priced in a straight forward manner. The valuation of these options is based on the Black–Scholes (1973) hedging argument in higher dimensions. A currency bond is a bond denominated in a different currency from that prevailing in the country in which the common stock of the firm is traded. It can be priced using the concept of rainbow options. 08:35:53.

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A standard spread option entitles its holder the right to call or put the spread value against a predetermined strike price. It corresponds in general to the difference between two prices. Examples of spread options include spreads in futures markets, bond markets and energy markets. Spread options in futures markets refer, for example, to the spread or the basis between Nebraska’s corn and Chicago’s corn. This spread may be due to location or grade among other things. Spread options in bond markets may be due to the differences in yield spreads between bonds issued in different countries. Spread options in energy markets result from the differences between, for example, refined and unrefined products such as crack spreads. First, using the approach in Stulz (1982), options on the minimum or the maximum of two assets are studied and valued. Then, options delivering the best of two assets and cash are analyzed and valued. Also, options on the maximum or the minimum of several assets are analyzed. Second, we propose the Ravindran’s (1993) approach for the pricing of European options on the spread between two indexes. Third, some applications to currency bonds, multi-currency bonds, corporate option bonds, spread options, portfolio options. The chapter provides also the appropriate models for the valuation of the index options and the wildcard features embedded in these contracts.

Questions 1. What are the specific features of a call (a put) on the minimum of two assets? 2. What are the specific features of a call (a put) on the maximum of two assets? 3. What are the specific features of options delivering the best of two assets and cash? 4. How can these options be valued with and without information uncertainty? 5. What are the main applications of rainbow options? 6. What is the definition of currency bonds? 7. What is the definition of multi-currency bonds? 8. What is the definition of corporate option bonds? 9. What are the specific features of spread options? 10. What are the specific features of wildcard options?

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Bibliography Bellalah, M (2001). Market imperfections; information costs and the valuation of derivatives: some general results. International Journal of Finance, 13(3), 1895–1927. Bellalah, M, Ma Bellalah and R Portait (2001). The cost of capital in international finance. International Journal of Finance, 13(3), 1958–1973. Bellalah, M and JL Prigent (2001). Pricing standard and exotic options in the presence of a finite mixture of Gaussian distributions. International Journal of Finance, 13(3), 1975–2000. Bellalah, M, JL Prigent and C Villa (2001). Skew without skewness: asymmetric smiles; information costs and stochastic volatilities. International Journal of Finance, 13(2), 1826–1837. Bellalah, M and F Selmi (2001). On the quadratic criteria for hedging under transaction costs. International Journal of Finance, 13(3), 2001–2020. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659. Boyle, PP, J Evnine and S Gibbs (1989). Numerical evaluation of multivariate contingent claims. Review of Financial Studies, 2(2), 241–250. Cohen, H (1995). Isolating the wildcard option. Mathematical Finance, 5(2), 155–165. Cox, JC, SA Ross and M Rubinstein (1979). Option pricing: a simplified approach. Journal of Financial Economics, 7(3), 229–263. Fleming, J and R Whaley (1994). The value of wildcard options. Journal of Finance, XLIX(1), March, 215–236. French, D and E Maberly (1992). Early exercise of American index options. Journal of Financial Research, Summer, 2, 127–137. Garman, M (1992). Spread the load. Risk, 5(11), 68–84. Johnson, H (1987). Options on the minimum or maximum of several assets. Journal of Financial and Quantitative Analysis, 22, 227–283. Margrabe, W (1978). The value of an option to exchange one asset for another. Journal of Finance, 33, 177–186. Ravindran, K (1993). Low-fat spreads. Risk, 6, 66–67. Rubinstein, M (1991). Options for the undecided. Risk, 4, 70–73. Stulz, RM (1982). Options on the minimum or maximum of two risky assets. Journal of Financial Economics, 10, 161–185. Valerio, N (1993). The valuation of cash-settlement options containing the wildcard feature. Journal of Financial Engineering, 2(4), 335–364. Willmott, P (1998). Derivatives, John Wiley and Sons.

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Chapter 5

Extendible Options and Their Applications

This chapter is organized as follows: 1. In Section 2, extendible options are identified, analyzed and valued with and without market frictions. 2. In Section 3, simple writer extendible options are analyzed and priced with and without the presence of information costs. 3. In Section 4, extendible options are applied in the analysis of extendible bonds and extendible warrants. Simulation of option values are provided. 4. In Section 5, pay-on-exercise options are defined, analyzed and valued with and without information costs. 5. In Section 6, outperformance options are presented and valued. 6. In Section 7, guaranteed exchange-rate contracts in foreign stock investments are analyzed and valued. 7. In Section 8, passport options are valued in the presence of information costs.

1. Introduction

T

here are many types of financial contracts or contingent claims allowing the issuer or the contract’s holder to extend the maturity date of the initial contract. For example, corporate warrants give the issuer the right to extend the maturity date of the contract unilaterally. Also, bonds with embedded options give the right to the issuer to extend the initial maturity date. More generally, in any financial contract with provisions concerning a rescheduling of payments, a renegotiation of terms can imply the existence

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of options with extendible maturities. These options can be extended by either the option holder or the option writer. When this option allows the holder to extend the initial maturity date to another date, he must pay an additional premium A to the option writer. Note that when pricing these options, the strike price is often adjusted. A pay-on-exercise option is defined with respect to its strike price and a given payment. When compared to a standard option, the option contract requires that the option holder pays the option writer a specified amount when the option expires in the money. At maturity, the buyer of a pay-onexercise call receives the payoff of a standard call when the underlying asset price is above the strike price and pays the option writer the specified amount. Since the holder of such options, must pay the specified amount, this option is always worth less than a standard call. Hence, the buyer of a pay-on-exercise call is entitled to a positive net pay-off only when the underlying asset price is greater than the sum of the strike price and the specified amount. However, if the underlying asset price lies between the strike price and the sum of the strike price and the specified amount, the buyer has to make a net payment to the option seller. An outperformance option on an asset one versus an asset two with a given maturity date and a face value of one dollar is a contract that gives the positive difference between the values of the two assets. The holder of this option receives at the maturity date any positive excess return of the asset one over the asset two. Guaranteed exchange-rate contracts have a dollar pay-off which is independent of the exchange rate prevailing at the maturity date. These contracts protect against exchange risk investors who hold foreign stocks or indexes. A guaranteed exchange-rate forward contract on a foreign stock is an agreement to receive on a certain date the stock’s prevailing price in exchange for a specified foreign-currency delivery price. The prices are converted to dollars at a pre-specified exchange rate. Consider at trader following a given asset using its view of the market direction. The amount of money accumulated from trading in this asset corresponds to the trading account. This amount can be increased or lost. A passport option or a perfect trader option is a call on the trading account. It gives its holder the positive amount in the account at the horizon date or zero.

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2. Pricing Extendible Options 2.1. The Valuation Context The valuation of these options is realized in the Black–Scholes context. Following Longstaff (1990), the dynamics of the underlying asset are described by dS = α dt + σ dW S where α and σ are constants. The valuation equation that must be satisfied by the option price, V(S, t) is 1 2 2 ∂2 V ∂V ∂V σ S 2 + rS − rV + = 0. 2 ∂ S ∂S ∂t Let CE (S, K1 , T1 , K2 , T2 , A) be the current value of an extendible call as a function of the two strike prices K1 and K2 , times to maturity T1 and T2 , the underlying asset S and the premium A to be paid in the event of extension. At date T1 , the call’s pay-off is CE (S, K1 , T1 , K2 , T2 , A) = max(0, C(S, K2 , T2 − T1 ) − A, S − K1 ) i.e., the option holder can choose between three pay-offs: the intrinsic value (S − K1 ), zero, or the difference between the premium A and a standard European call with a strike price K2 and a maturity date (T2 − T1 ). This may also be written as CE = max{max[0, C(S, K2 , T2 − T1 ) − A], max[0, S − K1 ]}. This pay-off function corresponds to a maximum of two risky pay-offs: the pay-off of a standard call and that of a call on a call. The pay-off looks like that of an option on the maximum of two assets. When A is positive, there is some critical value of S at T1 denoted by I1 below which the option is not extended, and another critical value I2 above which the option is again not extended. Hence, extension occurs when S is in the interval [I1 , I2 ]. At T1 , the value of I1 is solution to the equation C(I1 , K2 , T2 − T1 ) = A and I1 must lay between A and A + K2 e−r(T2 −T1 ) . When A = 0, I1 = 0 and when I1 ≥ K1 , the extension privilege is worthless. A sufficient condition for I1 to be less than K1 is A < K2 − K2 e−r(T2 −T1 ) .

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At T1 , the value of I2 is given by the solution to the equation C(I2 , K2 , T2 − T1 ) = I2 − K1 + A.

2.2. Extendible Calls 2.2.1. The Extendible Call Without a Cost of Carry The value of the extendible call given by Longstaff is: CE (S, K1 , T1 , K2 , T2 , A) = C(S, K1 , T1 ) + SN2 (γ1 , γ2 , −∞, γ3 , ρ)
    −K2 e−rT2N γ1 − σ 2 T1 , γ2 − σ 2 T1 , −∞, γ3 − σ 2 T2 , ρ
   − SN(γ1 , γ4 )+, K1 e−rT1N γ1 − σ 2 T1 , γ4 − σ 2 T1
   − Ae−rT1N γ1 − σ 2 T1 , γ2 − σ 2 T1 where

       σ2 S γ1 = ln T1 + r+ σ 2 T1 I2 2        σ2 S σ 2 T1 T1 + r+ γ2 = ln I1 2        S σ2 γ3 = ln σ 2 T2 T2 + r+ K2 2        σ2 S σ 2 T1 , T1 + r+ γ4 = ln K1 2

 ρ=

T1 T2

with N2 (a, b, c, d, ρ): the cumulative probability of the standard bivariate normal density with correlation coefficient ρ for the region [a, b]x[c, d]. N(a, b): the cumulative probability of the standard normal density in the region [a, b] C(S, K1 , T1 ): the value of a standard call option in a Black and Scholes context. The extendible call has some interesting properties. It is worth at least as much as an equivalent standard call without the extension privilege. It is 08:36:01.

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greater than or equal to the maximum of a standard call, C(S, K1 , T1 ) and a compound option on C(S, K2 , T2 − T1 ). Note that the call is worthless when S = 0. When I1 is nil and I2 is infinite, the extendible call is worth a standard call, C(S, K2 , T2 ). The extendible call price is an increasing function of S, r, σ and a decreasing function of K1 .

2.2.2. The Extendible Call in the Presence of a Cost of Carry In the presence of a cost of carry b, the formula for the extendible call is given by: CE (S, K1 , T1 , K2 , T2 , A) = C(S, K1 , T1 ) + Se(b−r)T2N2 (γ1 , γ2 , −∞, γ3 , ρ)
    − K2 e−rT2N γ1 − σ 2 T1 , γ2 − σ 2 T1 , −∞, γ3 − σ 2 T2 , ρ 
  − Se(b−r)T1N(γ1 , γ4 ) + K1 e−rT1N γ1 − σ 2 T1 , γ4 − σ 2 T1
   − Ae−rT1N γ1 − σ 2 T1 , γ2 − σ 2 T1 where the interest rate r is replaced by the cost of carry b in the following formulas:        S σ2 γ1 = ln T1 σ 2 T1 + b+ I2 2        S σ2 γ2 = ln σ 2 T1 T1 + b+ I1 2        S σ2 T2 γ3 = ln σ 2 T2 + b+ K2 2         σ2 T1 S T1 + b+ γ4 = ln σ 2 T1 , ρ = . K1 2 T2 Table 1 provides simulation results for the values of options and the Greek letters. Tables 2–4 provide simulations of standard extendible option values for different parameters.

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Table 1:

207

Simulation results for the values of extendible calls.

S

Price

Delta

Gamma

Vega

Theta

96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

8.22621 8.77647 9.34548 9.93290 10.53836 11.16147 11.80182 12.45896 13.13242

0.54988 0.56873 0.58724 0.60537 0.62311 0.64043 0.65731 0.67370 0.68963

0.01885 0.01851 0.01841 0.01774 0.01732 0.01687 0.01640 0.01592 0.01543

0.50571 0.50771 0.50854 0.50823 0.50681 0.50434 0.50083 0.49638 0.49102

0.01001 0.01024 0.01046 0.01066 0.01086 0.01104 0.01120 0.01136 0.01150

S = 100, K = 100, t = 11/01/2002, T = 11/01/2003, r = 4%, σ = 20%, extendible maturity date = 11/06/2004, K2 = 110, additional premium = 4.

Table 2:

Simulation results for the values of extendible calls.

S

Price

Delta

Gamma

Vega

Theta

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

1.49102 2.63070 4.24971 6.38402 9.03184 12.15924 15.71041 19.61847 23.81495

0.18434 0.27411 0.37487 0.47896 0.57920 0.67006 0.74831 0.81275 0.86383

0.01674 0.01951 0.02080 0.02056 0.01905 0.01673 0.01400 0.01125 0.00873

0.24246 0.32142 0.38670 0.42850 0.44261 0.43046 0.39742 0.35076 0.29766

−0.00473 −0.00628 −0.00757 −0.00839 −0.00867 −0.00844 −0.00779 −0.00687 −0.00583

S = 110, K1 = 100, t = 28/12/2001, T = 27/12/2002, r = 2%, σ = 20%, extendible date = 27/12/2003, K2 = 105, A = 5.

Table 3:

Simulation results for the values of extendible calls.

S

Price

Delta

Gamma

Vega

Theta

88.00 93.00 99.00 104.00 110.00 115.50 121.00 126.50 132.00

7.61027 10.19395 13.17356 16.52056 20.20043 24.17575 28.40893 32.86391 37.50725

0.43251 0.50656 0.57618 0.63997 0.69713 0.74740 0.79094 0.82812 0.85953

0.01370 0.01304 0.01206 0.01090 0.00965 0.00841 0.00722 0.00612 0.00514

0.43886 0.47311 0.49186 0.49591 0.48709 0.46784 0.44074 0.40832 0.37282

−0.00755 −0.00810 −0.00841 −0.00849 −0.00837 −0.00808 −0.00767 −0.00716 −0.00660

S = 110, K1 = 100, t = 28/12/2001, T = 27/12/2002, r = 2%, σ = 30%, extendible date = 08:36:01.

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Table 4:

Simulation results for the values of extendible calls.

S

Price

Delta

Gamma

Vega

Theta

88.00 93.00 99.00 104.00 110.00 115.50 121.00 126.50 132.00

5.48272 7.87505 10.75702 14.09610 17.84521 21.95004 26.35477 31.00624 35.85654

0.38948 0.48026 0.56687 0.64603 0.71575 0.77533 0.82488 0.86520 0.89737

0.01662 0.01617 0.01502 0.01341 0.01160 0.00974 0.00799 0.00642 0.00507

0.41118 0.45367 0.47404 0.47303 0.45394 0.42123 0.37981 0.33403 0.28748

−0.00748 −0.00824 −0.00862 −0.00861 −0.00829 −0.00772 −0.00699 −0.00617 −0.00534

S = 110, K1 = 100, t = 28/12/2001, T = 27/12/2002, r = 2%, σ = 25%, extendible date = 27/12/2003, K2 = 105, A = 5.

2.3. Extendible Calls with Shadow Costs of Incomplete Information By introducing shadow costs of incomplete information, λi for each security, simple computations allows us to give the following formula for the value of the extendible call: CE (S, K1 , T1 , K2 , T2 , A)  = e(λCE −λS )(T2 −T1 ) C(S, K1 , T1 ) + SN2 (γ1 , γ2 , −∞, γ3 , ρ)
    − K2 e−(r+λS )(T2 −T1 )N γ1 − σ 2 T1 , γ2 − σ 2 T1 , −∞, γ3 σ 2 T2 , ρ
   − SN(γ1 , γ4 ) + K1 e−(r+λS )T1N γ1 − σ 2 T1 , γ4 − σ 2 T1
   − Ae−(r+λS )T1N γ1 − σ 2 T1 , γ2 − σ 2 T1 where        S σ2 γ1 = ln T1 + r + λS + σ 2 T1 , I2 2        S σ2 σ 2 T1 , T1 + r + λS + γ2 = ln I1 2        S σ2 T2 σ 2 T2 , + r + λS + γ3 = ln K2 2 08:36:01.

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       σ2 S T1 σ 2 T1 , + r + λS + γ4 = ln K1 2

209

 ρ=

T1 . T2

We can derive a similar formula in the presence of a cost of carry b. In this case, we can use the previous formula and only change the discounting factor.

2.4. Extendible Puts Let PE (S, K1 , T1 , K2 , T2 , A) be the value of an extendible put and P(S, K, T ) be the price of a Black-Scholes put. At maturity date, T1 , the pay-off is: PE (S, K1 , T1 , K2 , T2 , A) = max(0, P(S, K2 , T2 − T1 ) − A, K1 − S) which can also be written as PE = max{max[0, P(S, K2 , T2 − T1 ) − A], max[0, K1 − S]}.

2.4.1. The Extendible Put in the Absence of a Cost of Carry This analysis is very similar to that of an extendible call and implies the existence of two values of the underlying asset in the interval [I1 , I2 ] that must be determined from the following equations: P(I1 , K2 , T2 − T1 ) = K1 − I1 + A and P(I2 , K2 , T2 − T1 ) = A. The solution given by Longstaff for the extendible put price is PE (S, K1 , T1 , K2 , T2 , A) = P(S, K1 , T1 ) − SN2 (γ1 , γ2 , −∞, −γ3 , ρ)
    + K2 e−rT2N γ1 − σ 2 T1 , γ2 − σ 2 T1 , −∞, −γ3 + σ 2 T2 , ρ
   + SN(γ4 , γ2 ) − K1 e−rT1N γ4 − σ 2 T1 , γ2 − σ 2 T1
   − Ae−rT1N γ1 − σ 2 T1 , γ2 − σ 2 T1 . Note that when I1 = K1 , the option is not extendible and its value reduces to P(S, K1 , T1 ). When A = 0 and I1 = 0, the extendible put option value is P(S, K2 , T2 ). When A is positive and I1 = 0, the extendible put value is equal to that of a call on P(S, K2 , T2 − T1 ). The option value is an increasing function of K1 , T1 , K2 , T2 , σ and a decreasing function of S, r and A.

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2.4.2. The Extendible Put in the Presence of a Cost of Carry In the presence of a cost of carry b, the solution for the extendible put price is PE (S, K1 , T1 , K2 , T2 , A) = P(S, K1 , T1 ) − Se(b−r)T2 N2 (γ1 , γ2 , −∞, −γ3 , ρ)
    + K2 e−rT2N γ1 − σ 2 T1 , γ2 − σ 2 T1 , −∞, −γ3 + σ 2 T2 , ρ
   + Se(b−r)T1N(γ4 , γ2 ) − K1 e−rT1N γ4 − σ 2 T1 , γ2 − σ 2 T1
   − Ae−rT1N γ1 − σ 2 T1 , γ2 − σ 2 T1 where the different parameters have the same meaning as before.

2.5. Extendible Puts with Shadow Costs of Incomplete Information The solution given by Longstaff for the extendible put price is PE (S, K1 , T1 , K2 , T2 , A)  = e(λPE −λS )(T2 −T1 ) P(S, K1 , T1 ) − SN2 (γ1 , γ2 , −∞, −γ3 , ρ)
    + K2 e−(r+λS )T2N γ1 − σ 2 T1 , γ2 − σ 2 T1 , −∞, −γ3 + σ 2 T2 , ρ
   − SN(γ4 , γ2 ) − K1 e−(r+λS )T1N γ4 − σ 2 T1 , γ2 − σ 2 T1
   − Ae−(r+λS )T1N γ1 − σ 2 T1 , γ2 − σ 2 T1 . It is straightforward to obtain a similar formula in the presence of a cost of carry and information costs. Where λi refers to information costs on asset i.

3. Simple Writer Extendible Options 3.1. Simple Writer Extendible Calls Let CW (X, K1 , T1 , K2 , T2 ) to be the value of a simple writer extendible call for which T1 is the initial maturity date, and T2 is the extended maturity date. If the call is extended by the writer, its strike price is adjusted from K1 to K2 , and A is zero. The CW pay-off at T1 is CW (S, K2 , T2 − T1 ), S K1 , if S 08:36:01.

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The use of the extensible call formula under the restrictions A = 0, I1 = 0 and I2 = K1 gives the following price: CW (S, K1 , T1 , K2 , T2 ) = C(S, K1 , T1 ) + SN2 (γ3 , −γ4 , −ρ) 
  − K2 e−rT2N γ3 − σ 2 T1 , −γ4 + σ 2 T1 , −ρ . A similar formula can be derived in the presence of a cost of carry b. In this case, we have: CW (S, K1 , T1 , K2 , T2 ) = C(S, K1 , T1 ) + Se(b−r)T2 e(b−r)T2N2 (γ3 , −γ4 , −ρ)
   − K2 e−rT2N γ3 − σ 2 T1 , −γ4 + σ 2 T1 , −ρ .

3.2. Simple Writer Extendible Calls with Shadow Costs Let CW (X, K1 , T1 , K2 , T2 ) to be the value of a simple writer extendible call. The CW pay-off at T1 is CW (S, K2 , T2 − T1 ), if S < K1 at T1 S − K1 , if S ≥ K1 at T1 . The value of this option within a context of incomplete information is CW (S, K1 , T1 , K2 , T2 )  = e(λCW −λS )(T2 −T1 ) C(S, K1 , T1 ) + SN2 (γ3 , −γ4 , −ρ)
   − K2 e−(r+λS )(T2 −T1 )N γ3 − σ 2 T1 , −γ4 + σ 2 T1 , −ρ . A formula for the pricing of this option can be derived in the presence of shadow costs of incomplete and a cost of carry b.

3.3. Simple Writer Extendible Puts Let PW (S, K1 , T1 , K2 , T2 ) be the value of a simple writer extendible put. At the maturity date, its pay-off is given by PW (S, K2 , T2 − T1 ), if S ≥ K1 at T1 K1 − S, if S < K1 at T1 .

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This pay-off corresponds to two components: a standard put and the value of the extension feature. Substituting A = 0, I1 = K1 and I2 = ∞ in the formula for the extensible put gives the PW value: PW (S, K1 , T1 , K2 , T2 ) = P(S, K2 , T2 − T1 ) − Se(b−r)T2N2 (−γ3 , γ4 , −ρ) 
  + K2 e−rT2N −γ3 + σ 2 T2 , γ4 − σ 2 T1 , −ρ .

3.4. SimpleWriter Extendible Puts with Shadow Costs The value of a simple writer extendible put in the presence of shadow costs of incomplete information is given by  PW (S, K1 , T1 , K2 , T2 ) = e(λPW −λS )(T2 −T1 ) P(S, K2 , T2 − T1 ) − SN2 (−γ3 , γ4 , −ρ) + K2 e−(r+λS )T2
   × N −γ3 + σ 2 T2 , γ4 − σ 2 T1 , −ρ .

4. Applications of Extendible Options 4.1. Extendible Bonds As shown in the analysis and valuation of compound options, the stockholder’s claim on the assets of a levered firm, may be assimilated to a call on the value of the firm with a strike price equal to the debt’s face or nominal value and a maturity date equals to that of the outstanding debt. Many corporate bonds have embedded options allowing the issuing firm to extend the maturity date of the original debt. By analogy, extending the life of the debt is equivalent to extending the time to maturity of the call. Hence, stockholders have an extendible call on the firm’s value, since extending the debt’s time to maturity gives them additional time to turn the firm around. Hence, this privilege is particularily valuable. Other applications of extendible option analysis are possible in pricing the firm’s capital structure.

4.2. Extendible Warrants Warrants are often issued by corporations with simple or complex provisions. Some long-term warrants stipulate a change in the strike price and can consequently be modelled as extendible calls. In fact, if you denote by T1 the date when the strike price changes from K1 to K2 , and by T2 the

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maturity date of the warrant, then at T1 the warrant holder must decide to exercise or not his warrant at the strike price K1 . If the warrant is not exercised at T1 , it is extended to T2 (with A = 0). This type of warrant can be valued using the formula for extendible calls.

5. Pay-On-Exercise Options A pay-on-exercise option is defined with respect to its strike price K and payment B. When compared to a standard option, the option contract requires that the option holder pays the option writer an amount B when the option expires in the money. At maturity, the buyer of a pay-on-exercise call receives the pay-off of a standard call when the underlying asset price is above K and pays the option writer an amount B. Since the holder of such options, must pay an amount B, this option is always worth less than a standard call. Hence, the buyer of a pay-on-exercise call is entitled to a positive net pay-off only when the underlying asset price is greater than K + B. However, if the underlying asset price lies between K and (K + B), the buyer has to make a net payment to the option seller.

5.1. The Pricing of Pay-On-Exercise Options: Theoretical Value We denote by: S: the current underlying asset price, d: a continuous dividend yield (in years), t: time to maturity, B: the payment that must be made upon exercise, r: the annual riskless rate, and σ: the annual volatility. This option can valued in a Black and Scholes economy in the presence of a forward payment. We consider a forward on the underlying asset with a strike price K and a delivery date t. The value of this forward at time 0 is Se−dt − Ke−rt . The forward price is written as SF = Se(r−d)t . Recall that the value of a standard call in a Black–Scholes world is 
2 ln(SF /K) + v2 T ca = Se−rtSF N(x) − KN(x − v), x = v 08:36:01.

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√ with v = σ t. In this context, the value of a pay-on-exercise call is given by the same formula where K is replaced by K + B: ca = Se−rt SF N(x) − (K + B)N(x − v) 
2 ln(SF /(K + B)) + v2 T x= v √ with v = σ t. It is possible to determine the value of B
for which  the SF N(x0 ) contract initial value is zero. In this case, B is given by B = N(x0 −v0 ) − K where the index 0 refers to the time of issue. This quantity corresponds to the fixed payment in the contract. The value of a pay-on-exercise put can be determined by a direct application of the put-call parity relationship p − c = (K + B)e−rt − Se−dt .

5.2. The Valuation of Pay-On-Exercise Options with Shadow Costs The value of the forward contract at time 0 is Se−dt − Ke−(r+λS )T , where d stands for a continuous dividend yield and λS is the information cost on S. The forward price is written as SF = Se(r+λS −d)T . The value of a pay-on-exercise call is given by: ca = e−(r+λc )T [SF N(d1 ) − (K + B)N(d2 )] √ √ d1 = [ln ln(SF /K) + (σ 2 T/2)]/σ T , d2 = d1 − σ T . The value of B for which the contract initial value is zero is given by B = e−(r+λc )T

SF N(d1,0 ) −K N(d2,0 )

where the index 0 refers to the time of issue. This quantity corresponds to the fixed payment in the contract. The value of a pay-on-exercise put can be determined by a direct application of the put-call parity relationship p − c = (K + B)e−(r+λc )T − Se−dT .

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6. Valuing and Hedging Outperformance Options An outperformance option on an asset A versus an asset B with a maturity date T and a face value of one dollar is a contract with the following pay-off in dollars: c = max[A(T ) − B(T ), 0]. The holder of this option receives at the maturity date any positive excess return of the asset A over the asset B. It is possible to analyze and value outperformance options by assimilating them as options issued in an imaginary country on a foreign underlying asset whose value is denominated in a foreign currency.

6.1. Analysis and Valuation Following Derman (1992), we denote by T : time to maturity, t: any time between 0 and T , Si (0), (Si (t)): value in currency i at time t = 0 (at time t) of one dollar worth of an asset S (a stock), σ(Si ): volatility of stock S in currency i, dS : continuous dividend yield for the asset S, ciAB (t): the value in currency i at time t of an outperformance option with the above pay-off, BS(S, K, r, σ, T − t): Black and Scholes formula for a stock with price S, a strike price K, a riskless rate r, a volatility σ and a time to maturity T − t. The pay-off of an outperformance call in dollar can be written as c$AB (T ) = max[A$ (T ) − B$ (T ), 0].

6.2. Outperformance Options in B-Share Currency Units Following the analysis in Derman (1992), it can be shown that the value of an outperformance option in B-shares is similar to that of a standard call. Consider an investor who lives in a country where the currency is the B-share whose value at t = 0 is one dollar. The value of one share of stock A in this country can be written as AB (t) = A$ (t)/B$ (t), since the value of one share of stock B, denoted BB is one and the riskless rate is the B

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dividend rate, dB , the pay-off of the outperformance call can be expressed in B-shares as cBAB (T ) = max[AB (T ) − 1, 0]. It is convenient to note that this pay-off corresponds to a standard option on AB (t) with a unit strike price. The volatilty of the asset AB is equivalent to the volatility of the asset A$ expressed in B-shares. Since we know that in the presence of a correlation ρAB between the returns of the two assets A$ (t) and B$ (t), the volatility of the asset AB is given by  σ(AB ) = σ 2 (A$ ) + σ 2 (B$ ) − 2ρAB σ(A$ σ(B$ ) then, the value of the outperformance call is given by the Black and Scholes formula with ctAB = BS(AB , 1, dB , σ(AB ), T − t). Now, the value of the outperformance option in dollars at time t can be obtained from the last formula. To do this, we take the value of this option in B-shares and convert it to dollars using the cross-rate, B$ (t), c$AB (t) = cBAB (t)B$ (t) or c$AB (t) = B$ (t)BS(AB , 1, dB , σ(AB ), T − t). Tables 5–10 provide simulations of performance option values for different parameters. Table 5: Standard performance call. S

Price

Delta

Gamma

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

0.01105 0.01981 0.03221 0.04832 0.06786 0.09026 0.11479 0.14067 0.16719

0.00141 0.00211 0.00286 0.00358 0.00422 0.00472 0.00507 0.00526 0.00533

0.00013 0.00015 0.00015 0.00014 0.00011 0.00008 0.00005 0.00002 −0.00000

Vega 0.00143 0.00186 0.00213 0.00217 0.00198 0.00159 0.00105 0.00046 −0.00014

S = 100, K = 100, t = 27/12/2001, T = 27/12/2002, r = 2%, σ = 20%.

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Table 6: Standard performance call. S

Price

Delta

Gamma

Vega

80.80 85.85 90.90 95.95 101.00 106.05 111.10 116.15 121.20

0.01222 0.02166 0.03484 0.05178 0.07214 0.09526 0.12039 0.14674 0.17358

0.00152 0.00223 0.00299 0.00371 0.00433 0.00480 0.00512 0.00529 0.00532

0.00013 0.00015 0.00015 0.00013 0.00011 0.00008 0.00004 0.00002 −0.00001

0.00151 0.00192 0.00215 0.00215 0.00192 0.00148 0.00092 0.00032 −0.00028

S = 101, K = 100, t = 27/12/2001, T = 27/12/2002, r = 2%, σ = 20%.

Table 7:

Standard performance put.

S

Price

Delta

Gamma

Vega

Theta

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

0.28091 0.21614 0.16318 0.12081 0.08772 0.06250 0.04374 0.03009 0.02038

−0.01421 −0.01173 −0.00949 −0.00750 −0.00578 −0.00435 −0.00320 −0.00230 −0.00162

0.00051 0.00047 0.00042 0.00037 0.00031 0.00025 0.00020 0.00015 0.00011

0.00648 0.00660 0.00661 0.00642 0.00601 0.00543 0.00472 0.00396 0.00322

−0.00018 −0.00018 −0.00018 −0.00017 −0.00016 −0.00015 −0.00013 −0.00011 −0.00009

S = 100, K = 100, t = 27/12/2001, T = 27/12/2002, r = 2%, σ = 20%.

Table 8:

Standard performance put.

S

Price

Delta

Gamma

Vega

Theta

80.80 85.85 90.90 95.95 101.00 106.05 111.10 116.15 121.20

0.26970 0.20633 0.15480 0.11385 0.08209 0.05807 0.04034 0.02755 0.01852

−0.01380 −0.01133 −0.00911 −0.00715 −0.00547 −0.00409 −0.00298 −0.00212 −0.00148

0.00051 0.00046 0.00041 0.00036 0.00030 0.00024 0.00019 0.00014 0.00010

0.00650 0.00661 0.00659 0.00636 0.00591 0.00529 0.00455 0.00379 0.00305

−0.00018 −0.00018 −0.00018 −0.00017 −0.00016 −0.00014 −0.00012 −0.00010 −0.00008

= 101, Kand = Risk: 100,Theory, t = 27/12/2001, T Applications, = 27/12/2002, r = 2%,Publishing σ = 20%. Bellalah, Mondher. ExoticSDerivatives Extensions and World Scientific Co 08:36:01.

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Table 9:

Performance FX-standard performance option call.

S

Price

Delta

Gamma

Vega

Theta

0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04

0.04154 0.04479 0.04818 0.05170 0.05534 0.05910 0.06298 0.06697 0.07107

0.32534 0.33857 0.35149 0.36405 0.37620 0.38792 0.39918 0.40996 0.42023

−0.06266 −0.07853 −0.09405 −0.10917 −0.12384 −0.13805 −0.15176 −0.16496 −0.17763

0.00216 0.00216 0.00215 0.00213 0.00210 0.00206 0.00202 0.00196 0.00190

0.00005 0.00005 0.00005 0.00004 0.00004 0.00004 0.00004 0.00004 0.00003

S = 1, K = 1, t = 07/02/2002, T = 07/02/2003, r = 3%, r∗ = 4%, σ = 20%.

Table 10:

Performance FX-standard performance option.

S

Price

Delta

Gamma

Vega

Theta

0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04

0.13394 0.12624 0.11889 0.11189 0.10523 0.09889 0.09286 0.08714 0.08172

−0.77075 −0.73503 −0.70031 −0.66662 −0.63396 −0.60233 −0.57175 −0.54221 −0.51371

0.77048 0.73479 0.70009 0.66642 0.63378 0.60217 0.57160 0.54208 0.51359

0.00645 0.00640 0.00635 0.00629 0.00622 0.00614 0.00605 0.00596 0.00586

0.00019 0.00018 0.00018 0.00018 0.00018 0.00018 0.00017 0.00017 0.00017

S = 1, K = 1, t = 07/02/2002, T = 07/02/2003, r = 3%, r∗ = 4%, σ = 20%.

7. Guaranteed Exchange-Rate Contracts in Foreign Stock Investments Guaranteed exchange-rate contracts are derivative assets having a dollar pay-off which is independent of the exchange rate prevailing at the maturity date. They protect against exchange risk investors who hold foreign stocks or indexes. However, a change in the expected covariance between the exchange rate and the foreign asset price may produce a change in the contract’s value. A guaranteed exchange-rate forward contract on a foreign stock is an agreement to receive on a certain date the stock’s prevailing price in

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exchange for a specified foreign-currency delivery price. The prices are converted to dollars at a pre-specified exchange rate.

7.1. Valuing a Guaranteed Exchange-Rate Forward Contract Following Derman, Karasinski and Wecker (1990), we use the following notations: T : time to delivery, S(0) (S(t)): stock price in German euro at time T = 0 (t), d: continuous dividend yield for the stock S, X(T ): spot dollar value of the German euro at time T , K: stock’s delivery price in German mark, X0 : value of the euro in dollars applied to convert the GER pay-off to dollars, F(0): value of the GER forward in dollars at time t = 0, SF (T ): stock’s GER forward price in marks, r$ : U.S. riskless interest rate, rg : German riskless interest rate, σs : volatility of stock S in euro, σx : volatility of the euro’s value in dollars, σxs : covariance between returns of the euro in dollars and the stock price in euro, ρxs : correlation coefficient σσxxsσs . At the delivery date, the dollar value of the forward contract is (S(T )−K)X0 . The value of the GER forward contract at time 0 is given by the discounted value of this pay-off where all investments earn the U.S. riskless rate. The expected value of this pay-off needs the knowledge of the probability distributions of the dollar value of the euro and the dollar value of the German stock in a way, such that expected returns on these investments are the U.S. riskless rate. In a Black and Scholes economy, the distribution of X(T ) is log-normal with a mean growth rate rx and a volatility σx . The distribution of S(T ) is log-normal with a mean growth rate rs and a volatility σs . In this context, the fair dollar value of a GER contract can be obtained from the discounting of its expected dollar-valued pay-off at the U.S. riskless rate. The dollar-based investment in the euro and the stock must have a mean growth rate r$ . If we fix rx and calculate the expected growth rate in the German euro, it is possible to show that r$ = rx + rg . 08:36:01.

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If we fix rs and calculate the expected growth rate of an investment in the German stock, then the following strategy can be used. Consider an investor who buys one share of the stock at S(0)X(0) dollars for S(0) euro, converted at X(0). If the stock pays d, he can reinvest this and own edt shares at delivery. The dollar value of his position is edTS(T )X(T ). The expected value of this position is E[edTS(T )X(T )] = e(d+rs +rx +σxs )TS(0)X(0). It is possible to show that r$ = d + rs + rx + σxs . Since r$ = rx + rg or rg = r$ + rx , and since rs = r$ − rx − d − σxs , it follows that the expected growth rate for the stock value in euro, rs = rg − d − σxs . The analysis shows that the log-normal distribution for the euro has a mean E[X(t)] = X(0)e(r$ −rg )T and a volatilty σx . The log-normal distribution for the stock has a mean E[S(t)] = S(0)e(rg −d−σxs )T and a volatilty σs . We denote by d  = d + σxs . Since, the dollar value of the forward contract at the delivery date is (S(T )−K)X0 , then its initial value can be calculated using the last equation 

F(0) = e−r$ TE[(S(T ) − K)X0 ] = e−r$ T [S(0)X0 e(rg −d )T − KX0 ]. Let SF (t) (the forward price for delivery at t) be the value in euro, of the GER delivery price K that makes the forward contract’s value F(0) = 0,  i.e., SF (t) = S(0)e(rg −d )t . By comparing this last equation and E[S(t)] = S(0)e(rg −d−σxs )t with  d = d + σxs , it is clear that the forward price is simply the expected value of the stock in this economy.

7.2. Valuing a Guaranteed Exchange-Rate Option Consider the pricing of a GER option with a strike price of K German euro. The pay-off of a European GER put struck at X0 K dollars is max[X0 K − X0 S(t), 0], where the underlying imaginary stock has a volatility σs and a mean growth rate (rg − d  ). The put value in dollars at time 0 is P0 = e−r$ TE[max[X0 K − X0 S(T ), 0]]. The value at time 0 is 

P0 = X0 e−r$ T [KN(−d2 ) − S(0)e(rg −d )T N(−d1 )] 08:36:01.

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with



 S0  ln K + rg − d  + 21 σs2 T , d1 = √ σs T

221

√ d2 = d1 − σs T .

It is possible to check that the value of a GER forward is equivalent to a portfolio with a long call and a short put with the same strike price K in euro, with a guaranteed exchange rate X0 .

7.3. Guaranteed Exchange-Rate Contracts in Foreign Stock Investments with Shadow Costs 7.3.1. Valuing a Guaranteed Exchange-Rate Forward Contract Consider an investor who buys one share of the stock at S(0)X(0) dollars for S(0) euro, converted at X(0). If the stock pays d, he can reinvest this and own edT shares at delivery. The dollar value of his position is edT S(T )X(T ). The expected value of this position is E[edT S(T )X(T )] = e(d+rs +rx +σxs +λS )T S(0)X(0) where λS refers to the information costs on the underlying stock S. It is possible to show that r$ = d + rs + rx + σxs − λS where λS is the information cost on the stock market. Since r$ = rx + rg or rg = r$ − rx , and since rs = r$ − rx − d − σxs + λS , it follows that the expected growth rate for the stock value in euros is rs = rg − d − σxs + λS . The above analysis shows that the log-normal distribution for the euro has a mean: E[X(T )] = X(0)e(r$ −rg )T and a volatilty σx . The log-normal distribution for the stock has a mean E[S(T )] = S(0)e(rg −d−σxs +λS )T and a volatilty σs . We denote by d  = d +σxs . Since, the dollar value of the forward contract at the delivery date is (S(T ) − K)X0 , then its initial value can be calculated using the last equation: F(0) = e−(r$ +λc )TE[(S(T ) − K)X0 ]  = e−(r$ +λc )T [S(0)X0 e(rg −d +λS )T − KX0 ].

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Let SF (T ) (the forward price for delivery at T ) be the value in euros of the GER delivery price K that makes the forward contract’s value F(0) = 0, i.e. 

SF (T ) = S(0)e(rg −d +λS )T . By comparing this last equation and E[S(T )] = S(0)e(rg −d−σxs +λS )T with d  = d + σxs , it is clear that the forward price is simply the expected value of the stock in this economy.

7.3.2. Valuing a Guaranteed Exchange-Rate Option with Shadow Costs The put value in dollars at time 0 is p0 = e−(r$ +λc )TE[max[X0 K − X0 S(T ), 0]]. The value at time 0 is 

P0 = X0 e−r$ T [Ke−λc TN(−d2 ) − S(0)e(rg −d −λc +λS )T N(−d1 )] with

      √ S0 1 + rg − d  + λS + σs2 T σs T , d1 = ln K 2

√ d2 = d1 − σ T .

8. Passport Options and Information Uncertainty Consider a trader following a given asset using its view of the market direction. The amount of money accumulated from trading in this asset is referred to as the trading account. This amount can be increased or lost. In this context, a passport option referred to also as a perfect trader option corresponds to a call on the trading account and gives its holder the positive amount in the account at the horizon date or zero. Following the analysis in Willmott (1998), we denote by A the value of the trading account comprising stocks and accumulated cash. This variable satisfies the following stochastic differential equation: dA = r(A − qS)dt + q dS where the strategy q corresponds to the amount of stock S held at time t as a function of S, A and t. The term (A − qS) indicates the growth in cash

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holding resulting from the addition of interest. The pay-off at time T of the passport option is max(A, 0). This option can be valued by constructing a portfolio comprising the option and the sale of delta units of the underlying asset S:  = V − S. The change in the value of this portfolio can be written as   2 ∂V 1 2 2 2 ∂2 V 1 2 2 ∂2 V 2 2 ∂ V d = + q σ S + qσ S + σ S ∂S∂A 2 ∂A2 ∂S 2 ∂t 2 ∂V ∂V dS + dA − dS ∂S ∂A where the absolute value of q is less than one. If we choose the following hedge ratio  = (∂V /∂S) + q(∂V /∂A) then, using the hedging argument, the option price must satify the following equation: +

∂V ∂2 V ∂2 V ∂2 V ∂V 1 1 + σ 2 S 2 2 + qσ 2 S 2 + q2 σ 2 S 2 2 + (r + λS )S ∂t 2 ∂S ∂S∂A 2 ∂A ∂S ∂V − (r + λV )V = 0. ∂A The highest value of this contract appears when q maximizes the terms in this previous equation   2 1 2 2 2 ∂2 V 2 2 ∂ V max qσ S + q σ S . |q|≤1 ∂S∂A 2 ∂A2 + rA

Since the option pay-off is given by V(S, A, T ) = max(A, 0) the solution can be of the form V(S, A, t) = SH(ξ, t),

ξ=

A . S

Hence, the general equation becomes ∂H 1 ∂2 H + σ 2 (ξ − q)2 2 = 0, ∂t 2 ∂ξ and the pay-off is H(ξ, T ) = max(ξ, 0).

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2 The optimal strategy is therefore, max|q|≤1 (ξ − q)2 ∂∂ξH2 . In this case, the optimal strategy q is given by  −1, when ξ > 0 q= 1, when ξ < 0 and the option value satisfies the following equation: ∂H ∂2 H 1 + σ 2 (| ξ | +1)2 2 = 0. ∂ξ ∂t 2

Summary The stockholder’s claim on the assets of a levered firm may be assimilated to a call on the value of the firm with a strike price equal to the debt’s face or nominal value and a maturity date equals to that of the outstanding debt. Many corporate contracts have embedded options allowing the issuing firm to extend the maturity date of the original debt. By analogy, extending the life of the debt is equivalent to extending the time to maturity of the call. Hence, stockholders have an extendible call on the firm’s value, since extending the debt’s time to maturity gives them additional time to turn the firm around. A pay-on-exercise option can be introduced with respect to its strike price and a specified payment. When compared to a standard option, the option holder pays the option writer a specified amount when the option expires in the money. At the option’s maturity date, the buyer of a pay-onexercise call receives the pay-off of a standard call when the underlying asset price is above the strike price and pays the option writer the specified amount. Since the holder of such options, must pay the specified amount, this option is always worth less than a standard call. An outperformance option on an asset A versus an asset B with a given maturity date and a face value of one dollar is a contract that gives the positive difference between the values of the two assets. The holder of this option receives at the maturity date any positive excess return of the asset A over the asset B. It is possible to analyze and value outperformance options by assimilating them as options issued in an imaginary country on a foreign underlying asset whose value is denominated in a foreign currency. Guaranteed exchange-rate contracts have a dollar pay-off which is independent of the exchange rate prevailing at the maturity date. They protect against exchange risk investors who hold foreign stocks or indexes. 08:36:01.

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A guaranteed exchange-rate forward contract on a foreign stock is an agreement to receive on a certain date the stock’s prevailing price in exchange for a specified foreign-currency delivery price. The prices are converted to dollars at a prespecified exchange rate. The value at delivery of a guaranteed exchange-rate forward contract on a given stock is given by the difference between the stock’s price and the delivery price in a foreign currency. This difference is converted to dollars at a fixed exchange rate. In the same context, a GER call (put) entitles its holder the right to receive (deliver) on a certain date the stock’s prevailing price in exchange for a specified foreign-currency delivery price. The prices are converted to dollars at a prespecified exchange rate. In this chapter, options with extendible maturities are identified, analyzed and valued. First, the general context is provided. Second, options are priced and their values are simulated. Third, some applications are presented. In particular, extendible bonds and extendible warrants are analyzed and identified to options with extendible maturities.

Questions 1. 2. 3. 4. 5. 6.

What are the specific features of extendible calls (puts)? What are the specific features of simple writer extendible calls (puts)? What are the main applications of extendible options? What are the specific features of pay-on-exercise options? What are the specific features of outperformance options? What are the specific features of guaranteed exchange-rate contracts in foreign stock investments? 7. What are the specific features of passport options?

Bibliography Derman, E (1992). Valuing and hedging outperformance options, Goldman Sachs. Quantitative Strategies Research Notes. Derman, E, P Karasinski and J Wecker (1990). Understanding guaranteed exchange-rate contracts in foreign stock investments, Goldman Sachs. Quantitative Strategies Research Notes. Longstaff, FA (1990). Time varying term premiums and traditional hypothesis about the term structure. Journal of Finance, 45, 1307–1314. Willmott, P (1998). Derivatives, John Wiley and Sons.

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Chapter 6

Currency Translated Options, Hybrid Securities and Their Applications

This chapter is organized as follows: 1. In Section 2, equity-linked foreign exchange options and quantos are analyzed and valued. 2. In Section 3, equity-linked foreign exchange options and quantos are studied in the presence of information costs. 3. In Section 4, several forms of capped contracts like the range forward contract, collars, indexed notes, etc., are studied. 4. In Section 5, some hybrid instruments are analyzed and valued. Also, simulation results are provided. 5. In Section 6, some applications are provided for the pricing of swaps and bonds within information uncertainty. 6. In Section 7, equity swaps and swaptions are analyzed and valued. In particular, the following swaps are studied: equity swaps with a fixed notional principal, pay floating, receive equity return on fixed notional principal, the two-way equity swap, the cross-currency two-way equity swap, the equity swap with variable notional principal, the capped equity swap and equity swaptions.

1. Introduction or more than 15 years, foreign currency markets have been characF terized by wide price changes. The high volatility of exchange rates has exposed treasurers and international investors to a high level of currency risk. Currency options markets have been developed to provide new Bellalah, Mondher. Exotic Derivatives and Risk: Theory, Extensions and226 Applications, World Scientific Publishing Co 08:36:08.

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means of dealing with this growing risk. Currency options are traded on several security exchanges throughout the world. A sizeable overthe-counter market has also developed, offering a variety of specialized currency options. Among such specialized options are the so-called quantos, hybrids, ratios, capped options, and so on. It is often interesting for investors to link a strategy in a foreign equity and a currency to create pay-offs corresponding to a foreign equity option struck in foreign curency, a foreign equity option struck in domestic curency, an equitylinked foreign exchange option, a fixed exchange rate foreign equity option. Ratio options, capped options and hybrid securities are also traded in different forms in OTC markets. Some financial contracts impose a cap or a floor on the possible pay-offs at maturity. By limiting the possible pay-offs, the issuer’s risk is reduced. A capped option can be analyzed as a combination of a call and a put option. Also, several firms have issued some financial contracts that resemble commodity futures and debt. Hybrid foreign currency derivatives are based on some use of the put–call parity relation. Currency risk management has thus become a delicate compromise between flexibility, protection and cost. The achievement of such a tradeoff amounts to tailoring an instrument that perfectly matches the needs of the investor, conditional on his anticipations. Banks worldwide have been quite successful in marketing the hybrid securities, although theoretically they can be replicated by combinations of instruments traded on organized markets. A typical hybrid like the cylinder option from the Citibank or the range forward contract from Salomon Brothers allows its holder to customize the hedge by selecting the appropriate strike and maturity that might not be available on an organized exchange. Besides, hybrid foreign currency options are mostly European options, which make them less expensive than their American counterparts. The potential for a wide variety of specifications has attracted many customers. In that respect, the French hybrid currency market is interesting, since French banks have been actively competing among themselves to market several hybrid contracts. These contracts are designed in such a way that the buyer does not have to pay any upfront cost and they are referred to as zero-premium hybrid foreign currency options.

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2. Equity-Linked Foreign Exchange Options and Quantos As shown in Garman and Kohlhagen (1983), the Black and Scholes (1973) formula for stock options applies to the valuation of options on currencies where the foreign interest rate replaces the dividend yield. When an investor wants to link a strategy in a foreign stock and a currency, he can use at least four different type of options: a foreign equity option struck in foreign currency, a foreign equity option struck in domestic currency, fixed exchange rate foreign equity options also known as quanto options or an equity-linked foreign exchange option. These different types of options are analyzed and valued in this section. For the analysis of information costs and valuation, we can refer to Bellalah (2001), Bellalah, Prigent and Villa (2001), M. Bellalah, Ma. Bellalah and R. Portait (2001), Bellalah and Prigent (2001), Bellalah and Selmi (2001), etc.

2.1. The Foreign Equity Call Struck in Foreign Currency The pay-off of a foreign equity call struck in foreign currency is C1∗ = X∗ max[S ∗ − K , 0], where S ∗ is the equity price in the currency of the investor’s country and K is a foreign currency amount. The spot exchange rate expressed in domestic currency of a unit of foreign currency, X∗ , stands in front of the pay-off to show that the latter must be converted into domestic currency. The domestic currency value of this call option is: √ 
C1 = S  X(d)−TN(d1 ) − K X(r ∗ )−TN d1 − σS  T 



S  (d)−T d1 = log K (r ∗ )−T

 

√ 1 √ σS  T + σS  T 2

where σS  is the volatility of S  . For the continuous compounding of interest rates, the term (d)−T ∗ must be replaced by e−dT , the term (r ∗ )−T by e−r T and the term (r)−T by e−rT .

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2.2. The Foreign Equity Call Struck in Domestic Currency The pay-off of a foreign equity option struck in domestic currency is C2∗ = max[S ∗ X∗ − K, 0], where K is the domestic currency amount. For the foreign option writer, the pay-off is C2∗ = max[S ∗ − KX∗ , 0], where X = 1/X. X corresponds to the exchange rate quoted at the price of a unit of domestic currency in terms of the foreign currency. This payoff corresponds to that of an option to exchange one asset (K units of our currency) for another asset (a share of stock). Its value is √ 
C2∗ = S  (d)−TN(d2 ) − KX (r)−TN d2 − σS  X T    −T   √ √ S (d) 1 σS  X T + σS  X T d2 = log −T  KX (r) 2  σ(S  X ) = σS2 + σX2  − 2ρS  X σS  σX where ρS  X is the correlation coefficient between the rates of return on S  and X . The domestic value of this option in the same context is √ 
C2 = S X(d)−TN(d2 ) − K(r)−TN d2 − σS X T      √ √ S X(d)−T 1 X X T + T σ σ d2 = log S S K(r)−T 2  σ(S  X) = σS2 + σX2 − 2ρS  X σS  σX = σ(S  X ) . As before, for the continuous compounding of interest rates, the term ∗ (d)−T must be replaced by e−dT , the term (r ∗ )−T by e−r T and the term (r)−T by e−rT .

2.3. Fixed Exchange Rate Foreign Equity Call The pay-off of a fixed exchange rate foreign equity call, known as a Quanto is C3∗ = X¯ max[S ∗ − K , 0] = max[S ∗ X¯ − K, 0] where X¯ is the rate at which the conversion will be made. It can be written in reciprocal units as ¯ ∗ max[S ∗ − K , 0]. C3∗ = XX

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Rubinstein and Reiner (1991a,b) give the value of this option in foreign currency as

  √ rd −T −(ρ  σ  σX )T     −T ¯ C3 = XX e SX S N(d3 ) − K (r) N(d3 − σS  T ) S r∗ 



S  (d)−T d3 = log K (r ∗ )−T



 − ρS  X σS  σX T

√ 1 √ σS  T + σS  T . 2

The domestic value of this option is  

−T √
 rd e−(ρS X σS σX )TN(d3 ) − K (r)−TN d3 − σS  T C3 = X¯ S  ∗ r 



S  (d)−T d3 = log K (r ∗ )−T



 − ρS  X σS  σX T

√ 1 √ σS  T + σS  T . 2

For the continuous compounding of interest rates, the term (d)−T must be ∗ replaced by e−dT , the term (r ∗ )−T by e−r T and the term (r)−T by e−rT .

2.4. An Equity Linked Foreign Exchange Call The pay-off of an equity linked foreign exchange call is C4∗ = S ∗ max[X∗ − K, 0]. It can also be written as C4∗ = S ∗ max[1 − KX∗ − K, 0] = KS ∗ max



 1 − X∗ , 0 . K

The foreign value of this call option is given by  C4∗ = S  (d)−TN(d4 ) − KS  X 



X(r ∗ )−T d4 = log K(r)−T

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rd r∗

−T

√ 
 e−(ρS X σS σX )TN d4 − σX T

+ ρS  X σS  σX T



√ 1 √ σX T + σX T . 2

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The domestic value of this call option is 



−T

C4 = S X(d) N(d4 ) − K

rd S ∗ r

−T

√ e−(ρS X σS σX )TN(d4 − σX T ).

Table 1 gives the results for the different models: Black–Scholes (B–S), Garman–Kohlhagen (G–K), foreign-equity/foreign-strike (FE/FS), foreignequity/domestic-strike (FE/DS), fixed-rate-foreign-equity (FR/FE), equitylinked-foreign-exchange (FL/FE), and equity-linked-foreign-exchange (EL/FE). Table 1:

Summary of the main currency formulas.

Type

Asset

Strike

Rate

Distribution

σ

B–S G–K FE/FS FE/DS FL/FE EL/FE

S X S X S X S X¯ S X

K K KX K K X¯ KS 

r r r∗ r r

d r∗ d d

σS σX σS  σS X σS  σX

rd eρS X σS  σX r∗

rd eρS X σS  σX r∗

d

Tables 2–8 provide simulations results for the values of foreign currency options in different contexts. We provide also the Greek letters. This allows the reader to make some comments. Table 2:

Simulations of foreign equity call struck in foreign currency.

S

Price

Delta

Gamma

Vega

Theta

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

4.31208 6.15817 8.41102 11.06549 14.10374 17.49871 21.21759 25.22464 29.48371

0.32902 0.40989 0.49121 0.57012 0.64441 0.71261 0.77387 0.82786 0.87471

0.01598 0.01633 0.01606 0.01528 0.01416 0.01282 0.01138 0.00993 0.00854

0.30619 0.35417 0.39151 0.41644 0.42862 0.42886 0.41875 0.40031 0.37569

−0.01255 −0.01453 −0.01607 −0.01710 −0.01760 −0.01761 −0.01719 −0.01642 −0.01540

S = 100, K = 100, t = 28/12/2003, T = 27/12/2004, r = 2%, σ = 30%, (domestic/foreign) = 1.1.

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Table 3:

Simulations of foreign equity call struck in foreign currency.

S

Price

Delta

Gamma

Vega

Theta

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

3.92008 5.59834 7.64638 10.05954 12.82158 15.90791 19.28872 22.93149 26.80338

0.29911 0.37263 0.44655 0.51829 0.58583 0.64783 0.70352 0.75260 0.79519

0.01453 0.01484 0.01460 0.01389 0.01287 0.01166 0.01035 0.00903 0.00777

0.27835 0.32197 0.35591 0.37858 0.38965 0.38987 0.38068 0.36392 0.34154

−0.01141 −0.01321 −0.01461 −0.01555 −0.01600 −0.01601 −0.01563 −0.01493 −0.01400

S = 100, K = 100, t = 28/12/2003, T = 27/12/2004, r = 2%, σ = 30%, S ∗ = 1.

Table 4:

Simulations of foreign equity call struck in foreign currency.

S

Price

Delta

Gamma

Vega

Theta

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

3.72407 5.31842 7.26406 9.55656 12.18051 15.11252 18.32428 21.78492 25.46321

0.28415 0.35400 0.42422 0.49238 0.55654 0.61544 0.66834 0.71497 0.75543

0.01380 0.01410 0.01387 0.01320 0.01223 0.01108 0.00983 0.00858 0.00738

0.26444 0.30587 0.33812 0.35965 0.37017 0.37038 0.36165 0.34573 0.32446

−0.01084 −0.01255 −0.01388 −0.01477 −0.01520 −0.01521 −0.01484 −0.01418 −0.01330

S = 0.95, K = 100, t = 28/12/2003, T = 27/12/2004, r = 2%, σ = 30%.

Table 5:

Simulations of foreign equity put struck in foreign currency.

S

Price

Delta

Gamma

Vega

Theta

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

20.84295 17.68730 14.88293 12.42544 10.29938 8.48139 6.94315 5.65379 4.58208

−0.66585 −0.59600 −0.52578 −0.45762 −0.39346 −0.33456 −0.28166 −0.23503 −0.19457

0.01380 0.01410 0.01387 0.01320 0.01223 0.01108 0.00983 0.00858 0.00738

0.26444 0.30587 0.33812 0.35965 0.37017 0.37038 0.36165 0.34573 0.32446

−0.01084 −0.01255 −0.01388 −0.01477 −0.01520 −0.01521 −0.01484 −0.01418 −0.01330

S ∗ = 0.95, K = 100, t = 28/12/2003, T = 27/12/2004, r = 2%, σ = 30%.

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Table 6:

233

Simulations of foreign equity put struck in foreign currency.

S

Price

Delta

Gamma

Vega

Theta

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

21.93994 18.61821 15.66625 13.07941 10.84145 8.92778 7.30858 5.95136 4.82324

−0.70089 −0.62737 −0.55345 −0.48171 −0.41417 −0.35217 −0.29648 −0.24740 −0.20481

0.01453 0.01484 0.01460 0.01389 0.01287 0.01166 0.01035 0.00903 0.00777

0.27835 0.32197 0.35591 0.37858 0.38965 0.38987 0.38068 0.36392 0.34154

−0.01141 −0.01321 −0.01461 −0.01555 −0.01600 −0.01601 −0.01563 −0.01493 −0.01400

K = 100, t = 28/12/2003, T = 27/12/2004, r ∗ = 1%, σ = 30%.

Table 7:

Simulations of foreign equity put struck in foreign currency.

S

Price

Delta

Gamma

Vega

Theta

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

24.13394 20.48003 17.23287 14.38735 11.92560 9.82056 8.03944 6.54650 5.30557

−0.77098 −0.69011 −0.60879 −0.52988 −0.45559 −0.38739 −0.32613 −0.27214 −0.22529

0.01598 0.01633 0.01606 0.01528 0.01416 0.01282 0.01138 0.00993 0.00854

0.30619 0.35417 0.39151 0.41644 0.42862 0.42886 0.41875 0.40031 0.37569

−0.01255 −0.01453 −0.01607 −0.01710 −0.01760 −0.01761 −0.01719 −0.01642 −0.01540

K = 100, t = 28/12/2003, T = 27/12/2004, r ∗ = 1.1%, σ = 30%.

Table 8:

Simulations of foreign equity call struck in domestic currency.

S

Price

Delta

Gamma

Vega

Theta

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

7.06210 9.63947 12.64854 16.05871 19.83119 23.92307 28.29092 32.89315 37.69156

0.47095 0.55956 0.64319 0.71972 0.78796 0.84746 0.89840 0.94132 0.97700

0.01798 0.01720 0.01593 0.01434 0.01260 0.01086 0.00920 0.00769 0.00634

0.31745 0.34418 0.35823 0.36022 0.35176 0.33498 0.31221 0.28562 0.25712

−0.01460 −0.01584 −0.01648 −0.01657 −0.01618 −0.01541 −0.01435 −0.01312 −0.01181

S = 100, K = 100, t = 28/12/2003, T = 27/12/2004, r = 2%, σ = 20%, exchange rate (domestic/foreign) = 1.1, r ∗ = 2%, σs = exchange rate volatility 15%, ρ = 50%.

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3. Equity-Linked Foreign Exchange Options and Quantos in the Presence of Shadow Costs of Incomplete Information 3.1. Pricing Foreign Equity Call Struck in Foreign Currency The domestic currency value of this call is

√ C1 = S  X(d + λc − λS )−TN(d1 ) − K X(r ∗ + λc )−TN(d1 − σS  T ) √ √ where d1 = [log(S  (d + λc − λS )−T /K (r ∗ + λc )−T )]/σS  T + 21 σS  T , in which σS  is the volatility of S  . The λi corresponds to information costs. This option can be easily hedged by an amount S in stocks and B units of foreign cash with S = (d + λc − λS )−TN(d1 ) √ 
B = −K (r ∗ + λc )−TN d1 − σS  T . In the presence of continuous compounding, the domestic currency value of this call option is given by: √ ∗ C1 = S Xe−(d+λc −λS )TN(d1 ) − K Xe−(r +λc )TN(d1 − σS  T ) √ √ ∗ where d1 = [ln(S  e−(d+λc −λS )T /K e−(r +λc )T )]/σS  T + 21 σS  T , in which σS  is the volatility of S  and r ∗ is the foreign risk-free rate. This option can be hedged by an amount S in stocks and B units of foreign cash with: S = e−(d+λc −λS )TN(d1 ) √ 
∗ B = −K e−(r +λc )TN d1 − σS  T .

3.2. Pricing Foreign Equity Call Struck in Domestic Currency The value of this option in the presence of information costs is: √ 
C2∗ = S  (d + λc − λS )−TN(d2 ) − KX (r + λc )−TN d2 − σS  X T with





  √ √ S  (d + λc − λS )−T 1  X  X T + T, σ σ d2 = log S S KX (r + λc )−T 2    σ(S X ) = σS2 + σX2  − 2ρS  X σS  σX , 08:36:08.

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where ρS  X is the correlation co-efficient between the rates of return on S  and X . If we multiply this formula by the exchange rate and substitute 1/X for X , we obtain the domestic value of this option in the same context, i.e., √ C2 = S X(d + λc − λS )−TN(d2 ) − K(r + λc )−TN(d2 − σS  X T ) with





  √ √ S X(d + λc − λS )−T 1 X X T + T, σ σ d2 = log S S K(r + λc )−T 2  σ(S  X) = σS2 + σX2 − 2ρS  X σS  σX = σ(S  X ) .

This option can again be easily hedged by an amount S  in stocks and B units of foreign cash with S  = (d + λc − λS )−TN(d2 ) √ 
B = −K(r + λc )−TN d2 − σS  X T . This formula is equivalent to that of Black and Scholes in the presence of information costs with S  X replacing S and σS  X replacing σ. It is as if the Black and Scholes risk-neutral pricing approach were applied to the underlying asset S  X. In the presence of continuous compounding, the value of this option with information costs is: √ 
C2∗ = S  e−(d+λc −λS )TN(d2 ) − KX e−(r+λc )TN d2 − σS  X T with

   −(d+λc −λS )T   √ √ Se 1 σS  X T + σS  X T , d2 = ln  −(r+λ )T c KX e 2  σ(S  X ) = σS2 + σX2  − 2ρS  X σS  σX ,

where ρS  X is the correlation co-efficient between the rates of return on S  and X . If we multiply this formula by the exchange rate and substitute 1/X for X , we get the domestic value of this option in the same context, i.e., √ C2 = S  Xe−(d+λc −λS )TN(d2 ) − Ke−(r+λc )TN(d2 − σS  X T )

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   −(d+λc −λS )T   √ √ S Xe 1 X X d2 = ln T + T, σ σ S S Ke−(r+λc )T 2  σ(S  X) = σS2 + σX2 − 2ρS  X σS  σX = σ(S  X ) .

This option can again be easily hedged by an amount S  in stocks and B units of foreign cash with: S  = e−(d+λc −λS )TN(d2 )

√ B = −Ke−(r+λc )TN(d2 − σS  X T ).

3.3. Pricing Fixed Exchange Rate Foreign Equity Call ¯ ∗ max[S ∗ − K , 0]. The pay-off in reciprocal units is C3∗ = XX It can be expressed as: ¯  ev max[S  eu − K , 0] C3∗ = XX where u and v stand for the natural logarithm of one plus the returns of S  and X . Following the methodology in Rubinstein and Reiner (1991a,b) and the joint distribution for u and v, the value of this option in foreign currency in the presence of information costs is  −T    (r + λc )(d + λc − λS ) ¯ C3 = XX S e−(ρS X σS σX )TN(d3 ) (r ∗ + λc )

√ 
 −T − K (r + λc ) N d3 − σS  T with

     √ S (d + λc − λS )−T 1 √    σS  T . σ σ T σ T + − ρ d3 = log S X S X S K (r ∗ + λc )−T 2

The domestic value of this option is  −T  (r + λc )(d + λc − λS ) ¯ C3 = X S e−(ρS X σS σX )TN(d3 ) (r ∗ + λc )

√ 
 −T − K (r + λc ) N d3 − σS  T

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with





S  (d + λc − λS )−T d3 = log K (r ∗ + λc )−T



 − ρS  X σS  σX T

237

√ 1 √ σS  T + σS  T . 2

This option can be hedged in an unusual form by an amount S in stocks, B in foreign cash and B in domestic currency where S =

X¯ (r + λc )(d + λc − λS )−T −ρS X σS σX T e N(d3 ) X (r ∗ + λc ) B = −S  S  .

In the presence of continuous compounding, the value of this option in foreign currency is:   −(r+λc )(d+λc −λS )T  e   ¯ C3 = XX S e−(ρS X σS σX )TN(d3 ) e−(r∗ +λc )T  √ 
 −(r+λc )T −K e N d3 − σS  T with:

   −(d+λc −λS )T   √ Se 1 √  X σS  σX T  d3 = ln σS  T . − ρ σ T + S S ∗ K e−(r +λc )T 2

The domestic value of the option is:   −(r+λc )(d+λc −λS )T  e ¯ C3 = X S  e−(ρS X σS σX )TN(d3 ) e−(r∗ +λc )T  √ 
 −(r+λc )T −K e N d3 − σS  T with:

    −(d+λc −λS )T  √ 1 √ Se    d3 = ln σ σ T σ T + σS  T . − ρ S X S X S ∗ K e−(r +λc )T 2

This option can be hedged in an unusual form by an amount S in stocks, B in foreign cash and B in domestic currency where: X¯ e−(r+λc )(d+λc −λS )T −ρS X σS σX T e N(d3 ) X e−(r∗ +λc )T B = −S  S  .

S =

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3.4. Pricing an Equity Linked Foreign Exchange Call An equity linked foreign exchange call is C4∗ = S ∗ max[X∗ − K, 0]. Note that this contract is the complement of the previous one. This pay-off can be written also as   1 − X∗ , 0 . C4∗ = S ∗ max[1 − KX∗ − K, 0] = KS ∗ max K The foreign value of this call option is given by: C4∗

−T





= S (d + λc − λS ) N(d4 ) − KS X √ 
× e−(ρS X σS σX )T N d4 − σX T





(r + λc )(d + λc − λS ) (r ∗ + λc )

−T

with 



X(r ∗ + λc )−T d4 = log K(r + λc )−T



 + ρS  X σS  σX T

√ 1 √ σX T + σX T . 2

The domestic value of this call option is −T    (r + λc )(d + λc − λS ) C4 = S X(d + λc − λS ) N(d4 ) − K S (r ∗ + λc ) √
 × e−(ρS X σS σX )TN d4 − σX T . 

−T

This option can again be hedged in an unusual form by an amount S in stocks, B in foreign cash and B in domestic currency with S = C4 SC X4 , B = −B X. KS  B = X 



(r + λc )(d + λc − λS ) (r ∗ + λc )

−T

√ e−(ρS X σS σX )TN(d4 − σX T )

× (d + λc − λS )−TNd4

√ B = −K(r + λc )−TN(d4 − σS  T ). Table 9 summarizes the main results with respect to the Black and Scholes formula in the presence of information costs. 08:36:08.

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Table 9: The results for the different options using several models in the presence of information costs: Black–Scholes (B–S), Garman–Kohlhagen (G–K), foreign-equity/ foreign-strike (FE/FS), foreign-equity/domestic-strike (FE/DS), fixed-rate-foreignequity (FR/FE), equity-linked-foreign-exchange (FL/FE), and equity-linked-foreignexchange (EL/FE). Type

Asset

Strike

Rate

Distribution-rate

B–S G–K FE/FS FE/DS

S X S X S X

K K KX K

(r + λc ) (r + λc ) (r ∗ + λc ) (r + λc )

FL/FE

S X¯

KX¯

(r + λc )

EL/FE

S X

KS 

(r + λc )(d + λc − λS ) (r ∗ + λc ) ρ = e S  X σS  σX

σ

(d + λc − λS ) (r ∗ + λc ) (d + λc − λS ) (d + λc − λS ) (r + λc )(d + λc − λS ) (r ∗ + λc ) = eρS  X σS  σX

σS σX σS  σS  X

(d + λc − λS )

σX

σS 

In the presence of continuous compounding, the foreign value of the call option is:  −(r+λc )(d+λc −λS )T  e C4∗ = S  e−(d+λc −λS )TN(d4 ) − KS  X e−(r∗ +λc )T √ × e−(ρS X σS σX )T N(d4 − σX T ) with:   −(r∗ +λc )T   √ Xe 1 √ d4 = ln + ρS  X σS  σX T σX T + σX T . −(r+λ )T c Ke 2 The domestic value of the call option is: 

C4 = S Xe

−(d+λc −λS )T



N(d4 ) − K S

e

√ 
× e−(ρS X σS σX )TN d4 − σX T .

−(r+λc )(d+λc −λS )T



e−(r∗ +λc )T

This option can again be hedged in an unusual form by an amount S in stocks, B in foreign cash and B in domestic currency

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with: S = C4 (C4 /S X), B = −B X. KS  e−(r+λc )(d+λc −λS ) −(ρS X σS σX )T e c )T X e−(r∗ +λ √ 
× N d4 − σX T e−(d+λc −λS )TN(d4 ) √ 
B = −Ke−(r+λc )TN d4 − σS  T .

B =

3.5. Pricing Quantos in the Presence of Information Costs: An Alternative Approach The pay-off of a cross-currency contract called quanto is defined with respect to an asset in one country, but it is converted to another currency. Consider for example a call on the CAC 40 index but paid in US dollars. The contract is exposed to the exchange rate risk (because of the two different currencies) and to the price risk of the CAC 40 index. Since the difference between the dollar and euro is small, we denote by S$ the eurodollar exchange rate corresponding to the number of dollars per euro. In 2002, the exchange rate was nearly one for one. Following the analysis in Willmott (1998), we denote by SF the level of the CAC 40 index in France expressed is Euros. Consider the following standard dynamics for the exchange rate: dS$ = µ$ S$ dt + σ$ S$ dX$ and the CAC 40 index dSF = µF SF dt + σF SF dXF . It is possible to construct a hedging portfolio measured in dollars with a long position in the quanto option and a short position in the euro and the CAC 40 index as follows:  = V(S$ , SF , t) − $ S$ − F SF S$ . The term $ S$ is the dollar value corresponding to the number of euros sold. This approach allows the pricing of a contract when the underlying asset value is measured in one currency and it is paid in another currency. Over a small time interval, the change in the portfolio value resulting for the changes in the value of its components and the interest earned on

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the euro can be written as:  ∂2 V ∂2 V 1 ∂2 V 1 ∂V + σF2 SF2 2 + σ$2 S$2 2 + ρσ$ σF S$ SF d = ∂S$ ∂SF 2 ∂t 2 ∂SF ∂S$  − ρσ$ σF F S$ SF − rf $ S$ dt     ∂V ∂V + − $ − F SF dS$ + − F S$ dSF . ∂S$ ∂SF As before, a hedged portfolio a la Black–Scholes can be constructed using the following values for the hedge ratios: ∂V SF ∂V $ = − ∂S$ S$ ∂SF ∂V and F = S1 ∂S . F $ When this portfolio is perfectly hedged, it becomes riskless and must yield the US risk free rate plus the information costs on the assets. Hence, we have 1 1 ∂2 V ∂2 V ∂2 V ∂V + σ$2 S$2 2 + ρσ$ σF S$ SF + σF2 SF2 2 ∂t 2 ∂S$ ∂SN 2 ∂S$ ∂SF ∂V ∂V + S$ (r$ + λS − rf ) + SF (rf − ρσ$ σF ) − (r$ + λV )V = 0. ∂S$ ∂SF At the quanto maturity date t = T , the pay-off is given by

V(S$ , SF , T) = max(SF − K, 0). In this case, it is possible to obtain a solution, which is independent of the exchange rate as V(S$ , SF , t) = W(SF , t). This gives: ∂2 W ∂W 1 ∂W + SF (rF + λS − ρσ$ σF ) − (r$ + λV )V = 0. + σF2 SN2 2 ∂SF ∂t 2 ∂SF A simple comparison with the Black–Scholes equation shows that the pricing of the quanto is equivalent to the valuation of a standard option in the presence of a constant dividend yield equal to r$ + λV − rf − λS + ρσ$ σF . The main difference corresponds to the effects of correlation.

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Table 10:

Simulations of fixed exchange rate foreign equity put, S = 1.1.

S

Price

Delta

Gamma

Vega

Theta

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

22.92389 18.51476 14.61433 11.27509 8.50774 6.28478 4.55084 3.23458 2.25995

−0.92719 −0.83330 −0.72492 −0.61022 −0.49747 −0.39339 −0.30241 −0.22649 −0.16565

0.01728 0.02076 0.02271 0.02296 0.02168 0.01934 0.01641 0.01335 0.01046

0.27261 0.34967 0.41522 0.45869 0.47471 0.46363 0.43025 0.38174 0.32558

−0.00896 −0.01101 −0.01270 −0.01375 −0.01402 −0.01354 −0.01245 −0.01096 −0.00929

S (share price) = 100, K = 100, t = 28/12/2003, T = 27/12/2004, r = 2%, σ = 20%, S ∗ (domestic/foreign) = 1.1, r ∗ = 2%, σs = 15%, ρ = 50%.

Table 11: Simulations of fixed exchange rate foreign equity put. S

Price

Delta

Gamma

Vega

Theta

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

20.83990 16.83160 13.28576 10.25008 7.73431 5.71343 4.13713 2.94053 2.05450

−0.84290 −0.75754 −0.65902 −0.55475 −0.45224 −0.35762 −0.27492 −0.20590 −0.15059

0.01571 0.01887 0.02064 0.02087 0.01971 0.01758 0.01492 0.01213 0.00951

0.24783 0.31788 0.37748 0.41699 0.43156 0.42148 0.39114 0.34704 0.29598

−0.00815 −0.01001 −0.01155 −0.01250 −0.01275 −0.01231 −0.01132 −0.00997 −0.00844

S (share price) = 100, K = 100, t = 28/12/1999, T = 27/12/2000, r = 2%, σ = 20%, S ∗ (domestic/foreign) = 1, r ∗ = 2%, σs = 15%, ρ = 50%.

Table 12: S 80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

Simulations of fixed exchange rate foreign equity put, S = 0.95.

Price

Delta

Gamma

Vega

Theta

19.79790 15.99002 12.62147 9.73758 7.34759 5.42776 3.93027 2.79350 1.95178

−0.80075 −0.71966 −0.62607 −0.52701 −0.42963 −0.33974 −0.26117 −0.19561 −0.14306

0.01493 0.01793 0.01961 0.01983 0.01872 0.01670 0.01417 0.01153 0.00903

0.23544 0.30199 0.35860 0.39614 0.40998 0.40041 0.37158 0.32969 0.28118

−0.00774 −0.00951 −0.01097 −0.01188 −0.01211 −0.01169 −0.01075 −0.00947 −0.00802

S (share price) = 100, K = 100, t = 28/12/2003, T = 27/12/2004, r = 2%, σ = 20%, Bellalah, Mondher. Exotic Derivatives and Risk: Extensions Applications, World Scientific Publishing Co S (domestic/foreign) 0.95,Theory, r 2%, σ and 15%, ρ 50%. 08:36:08.

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Table 13: Simulations of foreign equity call struck in foreign currency. S 96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

Price

Delta

Gamma

Vega

Theta

1.28401 1.57255 1.90129 2.27102 2.68183 3.13312 3.62370 4.15181 4.71521

0.28774 0.32813 0.36932 0.41060 0.45129 0.49078 0.52847 0.56391 0.59674

0.04039 0.04120 0.04128 0.04069 0.03948 0.03769 0.03544 0.03284 0.02999

0.10985 0.11630 0.12098 0.12371 0.12441 0.12313 0.11996 0.11512 0.10886

0.02167 0.02316 0.02434 0.02518 0.02564 0.02574 0.02548 0.02490 0.02405

S = 100, K = 100, t = 05/01/2003, T = 01/03/2004, r = 3%, σ = 20%, S ∗ = exchange rate = domestic/foreign = 0.7.

Table 14: Foreign equity call struck in foreign currency. S 96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

Price

Delta

Gamma

Vega

Theta

1.12351 1.37598 1.66363 1.98714 2.34660 2.74148 3.17074 3.63283 4.12581

0.25177 0.28711 0.32316 0.35928 0.39488 0.42943 0.46241 0.49342 0.52215

0.03534 0.03605 0.03612 0.03560 0.03455 0.03298 0.03101 0.02873 0.02625

0.09612 0.10176 0.10585 0.10824 0.10886 0.10774 0.10497 0.10073 0.09525

0.01896 0.02027 0.02130 0.02203 0.02244 0.02252 0.02230 0.02179 0.02105

S = 100, K = 100, t = 05/01/2003, T = 01/03/2004, r = 3%, σ = 20%, S ∗ = domestic/ foreign = 0.8.

Table 15: Simulations of fixed exchange rate foreign equity call. S 96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

Price

Delta

Gamma

Vega

Theta

3.68977 3.96161 4.24399 4.53671 4.83960 5.15244 5.47500 5.80703 6.14827

0.27164 0.28222 0.29262 0.30284 0.31284 0.32261 0.33212 0.34137 0.35035

0.01058 0.01041 0.01021 0.01000 0.00977 0.00951 0.00925 0.00898 0.00869

0.19666 0.19796 0.19875 0.19901 0.19877 0.19805 0.19685 0.19522 0.19316

0.00749 0.00763 0.00776 0.00787 0.00797 0.00806 0.00813 0.00819 0.00824

S = 100, K = 100, t = 05/01/2003, T = 04/01/2004, r = 4%, σ = 20%, S ∗ = 0.5, r ∗ = 3%,

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Table 16: Simulations of fixed exchange rate foreign equity put. S 96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

Price

Delta

Gamma

Vega

Theta

4.23471 3.99203 3.75987 3.53806 3.32642 3.12473 2.93275 2.75025 2.57696

−0.24290 −0.23232 −0.22191 −0.21169 −0.20169 −0.19192 −0.18241 −0.17316 −0.16418

0.01058 0.01041 0.01021 0.01000 0.00977 0.00951 0.00925 0.00898 0.00869

0.19703 0.19834 0.19912 0.19939 0.19916 0.19844 0.19725 0.19561 0.19356

0.00345 0.00355 0.00364 0.00371 0.00377 0.00381 0.00384 0.00386 0.00386

S = 100, K = 100, t = 05/01/2003, T = 04/01/2004, r = 4%, σ = 20%, S ∗ = 0.5, r ∗ = 3%, σs = 15%, ρ = 0.5.

Table 17: Simulations of fixed exchange rate foreign equity put. S 96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

Price

Delta

Gamma

Vega

Theta

4.23543 3.99272 3.76054 3.53871 3.32704 3.12532 2.93333 2.75080 2.57748

−0.24292 −0.23234 −0.22193 −0.21172 −0.20171 −0.19195 −0.18243 −0.17318 −0.16421

0.01058 0.01041 0.01021 0.01000 0.00977 0.00952 0.00925 0.00898 0.00869

0.19706 0.19837 0.19916 0.19943 0.19919 0.19847 0.19728 0.19565 0.19359

0.00345 0.00355 0.00364 0.00371 0.00377 0.00381 0.00384 0.00386 0.00387

S = 100, K = 100, t = 05/01/2003, T = 04/01/2004, r = 4%, σ = 20%, S ∗ = 0.5, r ∗ = 3%, σs = 15%, ρ = 0.6.

4. Analysis and Valuation of Capped Options 4.1. The Range Forward Contract In a range forward contract, the buyer and the bank agree on two prices, S1 and S2 , at the inception of the contract. At the maturity of the forward contract, the buyer will purchase the foreign currency either at S1 if the spot price is less than S1 , or at S2 if the spot price is greater than S2 , or at the spot price if it is between S1 and S2 . The two prices S1 and S2 are set such that no money changes hands at the inception of the contract.

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Like a range forward contract, a participating forward contract guarantees a minimum exchange rate for a forward sale and a maximum exchange rate for a forward purchase. Besides, the seller (buyer) gets a participation in the foreign currency appreciation (depreciation). Obviously, there is a cost to this participation. Since the contract is structured with no upfront payment, the cost of the seller’s upside participation is that the minimum exchange rate guaranteed through a participating forward sale will necessarily be greater than the outright forward price. Analogously, the maximum exchange rate guaranteed through a participating purchase will necessarily be greater than the outright forward price. A conditional forward purchase contract is similar to an outright forward purchase, except that the long side of the contract has the right to pull out of the forward purchase by paying a fee to the short side of the contract on the maturity date. The contract can also be designed in such a way that there is no cancellation fee simply by guaranteeing a buying price lower than the forward price. Similarly, a conditional forward sale contract is equivalent to an outright forward sale, except that the short side of the forward contract has the right to pull out of the agreement by paying a fee to the long side of the contract. To give an example for the use and valuation of a range forward contract, consider a manager of a US firm intending to buy pounds in three months, when the spot rate is 1.4 dollars per pound and the three-month forward rate is 1.38 dollars per pound. He can buy a range forward contract, taking two rates 1.32 and 1.45, so that the initial contract value is zero. At maturity, if the spot exchange rate is below 1.32, the buyer pays 1.32. If it is above 1.45, he pays 1.45. If the spot exchange rate lays in the interval [1.32, 1.45], he pays the spot rate. At the maturity date, T , the contract’s value is assimilated to a position in a long call, c, and a short put, p. Hence, the contract’s value is V(S, T, K ) = c(S, T, K ) − p(S, T, K ) where S stands for the spot rate. At the contract’s initiation, the strike price is given in a way such that the contract’s value is nil, or V(S, T, K )K=f = c(S, T, f ) − p(S, T, f ) = 0 where f stands for the forward exchange rate. A range forward contract can also be regarded as a combination of two portfolios. The first portfolio corresponds to a long position in a call and a short position in a put. The second portfolio comprises a cap which is

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placed on the range of possible pay-offs, by the sale of a call with a strike price K1 (> f ) and a purchase of a put with a strike price K2 (< f ). Hence the contract’s value is X(S, T, K1 , K2 ) = V(S, T, f ) − c(S, T, K1 ) + p(S, T, K2 ). By construction, the initial premium for the range forward contract is zero. When the buyer takes a strike price, the bank picks the other, so that the contract’s value is zero: X(S, T, K1 , K2 ) = p(S, T, K2 ) − c(S, T, K1 ). Hence, the value of a range forward contract is given by the difference between two currency option prices where ∗

c(S, T, K1 ) = e−r TSN(d1 ) − e−rTK1 N(d2 ) and ∗

p(S, T, K2 ) = e−rTK2 N(−d2 ) − e−r TSN(−d1 ) with


ln(S/Ki ) + r − r ∗ + 21 σ 2 T , = √ σ T 

d1i

√ d2i = d1i − σ T ,

where i = 1, 2.

4.2. The Collar Take, for example, a firm issuing a bond for one year where interest rates are adjusted semi-annually with reference to the LIBOR. If in six months, the LIBOR is above 9% (below 6%), the firm pays (receives) the difference to the bank. Hence, the cost of financing in variable rate ranges between 6% and 9%. The collar can be regarded as a difference between two European option prices. Its value is given by: X(r, T, K1 , K2 ) = p(r, T, K1 ) − c(r, T, K2 ) where: r: six-month LIBOR, K1 : price of a discount bond yielding 9%, K2 : price of a discount bond yielding 6%.

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4.3. Indexed Notes Standard oil issued notes in June 1986 with a nominal value of 1000, and a maturity date of 1990. These notes have an implicit capped option. The additional amount of the indexed note is given by 170 times the amount (if any) by which the price of a barrel of petrol exceeds 25 dollars at maturity date. Hence, if the barrel’s price at that date is 20 dollars, the option is worthless. However, if the barrel’s price lays between 25 and 40 dollars, the option pay-off is up to (40–25) 170. This indexed note is regarded as a sale of a call and a cap on the possible pay-offs at maturity. The embedded option is analyzed as a purchase of a European call with a strike price of 25 and a sale of another call with a strike price of 40. The value of the indexed note is given by X(S, T, K1 , K2 ) = c(S, T, K1 ) − c(S, T, K2 ). Another example of indexed notes, is the index currency note, hereafter, ICON, issued by the long term credit Bank of Japan. Each ICON has a face value of 1000 dollars, payable in 10 years and gives rise to the following payoff [169/S − 1]1000, where S stands for the spot exchange rate yen/dollars. If, at the maturity date, the exchange rate is 169 yen/dollars, the above amount is received. If the exchange rate is 159 yen/dollar, the bearer receives 937.11, (1000–62.89). However, if the exchange rate is below 84.5 dollars, (169/2), the bearer receives nothing. The bearer receives the minimum of zero and a maximum of 1000 and the ICON’s value is given by VICON = B(T) − [c(S, T, K) − c(Y, S, 2K)]

1000 K

with B(T): note’s market value without the option, S: spot exchange rate dollar/yen, K: strike price (dollars/yen) = 1/169, T : maturity date of the “ICON”, c(S, T, K): European currency call (in dollars).

5. Financing with Hybrid Securities: Perls The buyer of a hybrid security (a corporate treasurer of an exporting firm for example) aims at the best trade-off between protection, flexibility and cost.

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5.1. Specificities of the Product The principal characteristics of these notes are as follows: — — — —

Issuer: Sallie Mae Issue: $100.000.000 Interest: (10–7/8%) per annum, payable in US dollars Principal amount: the principal amount of each $1.000 note will be paid at maturity in an amount equal to $2.000–138.950 yen — Investment grade rating: type AAA — Time to maturity: 5 years.

5.2. Cash-Flow Identification Table 18 gives the year 5 principal payment on the security expressed as a function of the yen/dollar and dollar/yen exchange rates. When the exchange rate is above 0.01439 (dollar/yen), the principal payment is zero. For exchange rates below that level, the principal amount is received, increasing to 1000 dollars at 0.0143996, and in the extreme to 2.000, should the yen becomes worthless. Per 1000 dollars face value, the 10–7/8% Sallie Mae reverse perls return 54.357 semi-annually. At maturity, the principal amount is equal to 2000 dollars minus the dollar value of 138.950 yen with a minimum of zero. Table 19 identifies the pay-offs on the combination in year 5. The pay-offs correspond to a straight debt, short a forward contract and long a call option.

Table 18: The principal payment on reverse perls in year 5: the exchange rates. Yen/USD

USD/Yen

Amount in USD

40 60 69.475 92.6338 138.95 277.9 555.8 ∞

0.025000 0.016666 0.014393 0.010796 0.007198 0.003598 0.001799 0.000000

0 0 0 500 1000 1500 1750 2000

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Table 19: The product components. Yen/USD

USD/Yen

Principal

Short contract

Long call

Cumulative

40 60 69.475 92.6338 138.95 277.9 555.8 ∞

0.025000 0.016666 0.014393 0.010796 0.007198 0.003598 0.001799 0.000000

1000 1000 1000 1000 1000 1000 1000 1000

(2473.76) (1315) (1000) (500)

1473.76 315∗ 0 0 0 0 0 0

0 0 0 500 500 1000 1750 2000

500 750 1000

1 − 2000. 315∗ = 1000 (138.950) 60

5.3. Valuation of Perls Following Barnhill and Seale (1990), the pricing of PERLS consists in the valuation components. The coupon amount is $54.375, or 
of the  product (1000) 10 7% /2 . When the yield on the 5-year Sallie-Mae straight bond 8 is 8.2% per annum, the market value is 1.108, or: Bond’s market value =

10

54.375 i=1

1.041i

+

1000 = 1108. 1.04110

The value of the forward contract, VI is VI = 138.950(Fc − Fe ) or VI = 138.950(0.007196 − 0.009074) = −261 dollars where: Fc : forward exchange rate, Fe : equilibrium forward exchange rate at date of issue, or, (0.0071968)(1082)5 (1033)5

S(1+r)t (1+r ∗ )t

= 110.2 yen/dollars

where S: spot exchange rate, r: interest rate for 5 years on the US government securities, r ∗ : interest rate for 5 years on the Japanese government securities.

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The option is priced using a modified version of the Black–Scholes model: ∗

c = e−r TSN(d1 ) − e−rTKN(d2 )  
√ ln(S/K) + r − r ∗ + 21 σ 2 T , d2 = d1 − σ T d1 = √ σ T where: r: U.S. interest rate, (0.082), r ∗ : Japanese interest rate, (0.033), T : time to maturity, (5 years), S: spot price, 138.950 yen or 2000 dollars, σ: the volatility, (12%). Using the above formula, the call price is $1. Knowing the bond value, (1.108), the forward contract value (−$261) and the option value (1), the value of PERLS must be (848).

6. Applications to Bonds, Swaps and the Pricing of Similar Products within Information Uncertainty 6.1. The Pricing of a Swap within Information Uncertainty Since the fixed-interest rate payments are all known in a swap operation, they can be viewed as the sum of zero-coupon bonds. When there are N payments at dates Ti and when the notional principal is one, then the fixed payments can be written as (rs + λB )

N

Z(t; Ti )

i=1

where rs corresponds to the fixed rate and λB is the information costs in this operation. A floating leg of a swap at time Ti corresponds exactly to a deposit of one dollar at time (Ti − τ) and withdrawal of the dollar at time τ. When all the floating legs are added, this gives the value of the floating side in the swap 1 − Z(t; TN ). Hence, the swap value for the receiver of the fixed rate is given by the algebraic sum of the two components (rs +λB ) N i=1 Z(t; Ti )− 1 + Z(t; TN ). When the swap is initiated, the swap rate at which the contract N value is zero. It is given by rs = [1 − Z(t; TN )/ i=1 Z(t; Ti )] − λB . 08:36:08.

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Bootstrapping For an introduction to Bootstrapping in the presence of shadow costs, the reader can refer to the Appendix. When the interest rate rs (Ti ) is given for different maturities, it is possible to use the equation giving the quoted swap rate to get the prices of zerocoupon bonds and the yield curve. Using the following relation for the first maturity date T1 : rs (T1 ) =

1 − Z(t; T1 ) − λB , Z(t; T1 )

this allows the computation of the first discount factor for maturity T1 as Z(t; T1 ) =

1 . 1 + rs (T1 ) + λB

The procedure can be applied to the jth discount factor as j−1 1 − i=1 Z(t; Ti ) . Z(t; Tj ) = 1 + rs (Tj ) + λB The presence of an information cost must be understood with respect to the sunk costs which are necessary to get informed about the specific features of the markets and the expenses necessary to the treatment, analysis and valuation of financial assets.

6.2. The Pricing of Bond Options within Information Uncertainty Consider the pricing of a zero-coupon bond option or an option for which the underlying asset is a zero-coupon bond. The option strike price is K and its maturity date is T . The maturity of the bond is TB ≥ T . The pricing problem needs the valuation of the bond then the pricing of the option. Following the analysis in Willmott (1998), we denote by B(r, t; TB ) the price of the bond. It satisfies the following equation: ∂B 1 2 ∂2 B ∂B + (u − γw) + w − (r + λB )B = 0 2 ∂t 2 ∂r ∂r where λB corresponds to the information cost related to the bond. At maturity, the bond price satisfies the following condition since its value is by

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definition equal to one B(r, TB ; TB ) = 1. The option price V(r, t) must satisfy also the same equation. At maturity, the pay-off of the bond option is given by V(r, T) = max[B(r, t; TB ) − K, 0].

6.3. The Pricing of Caps and Floors within Information Uncertainty The pricing of caps A cap allows the holder to “cap” the increase in the floating interest rate to a given level. At each date ti , the cap allows the payment of a variable interest rate on a given principal amount. The pay-off appears often in the following form: max(rL − rc , 0) where rL corresponds to the floating rate and rc the fixed cap rate. If the frequency of payments corresponds to three months, the rate rL which is applied at time ti is set at time ti−1 . This is the sum of the payments each quarter by each caplet. Each caplet must satisfy the classic equation ∂V ∂V 1 ∂2 V + w2 2 + (u − γw) − (r + λV )V = 0 ∂t 2 ∂r ∂r under the maturity condition V(r, T) = max(r −rc , 0). It is assumed that the actual rate floating rate rL is approximated by the spot rate r. This pay-off reveals that the caplet is a call on the spot rate. The pricing of floors The floor imposes a barrier on the interest rate from below. The floor corresponds to the sum of floorlets, where the cashflow for each floorlet is given by max(rf − rL , 0), where rf corresponds to the floor rate. Again, the floorlet must satisfy the bond equation with V(r, T ) = max(rf − r, 0). The pay-off shows that the floorlet corresponds to a put on the spot rate. The cap/floor parity Using the pay-off of the caplet and the floorlet, when the caplet rate equals the floorlet rate, it is easy to see that the difference between their payoffs must be equal to the difference between the variable rate and the caplet (or floorlet) rate max(rL − rc , 0) − max(rc − rL , 0) = rL − rc .

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This corresponds also to the difference between a variable rate and a fixed rate as for one payment in a swap transaction. Hence, it is possible to state the following parity relationship cap = floor + swap.

6.4. The Pricing of Caplet and a Floorlet Option within Information Uncertainty Since the cash-flow corresponding to a caplet at time ti is given by max(rL − rc , 0) but the rate rL is set at time ti−1 , this cash-flow is equivalent to the 1 cash-flow at ti−1 , 1+r max(rL − rc , 0). This latter cashflow can be written
L 1+rc as max 1 − 1+rL , 0 . 1+rc . Hence, Or, the price at ti−1 of a bond paying (1 + rc ) at time ti is 1+r L the caplet can be viewed as a put expiring at time ti−1 on a bond maturing at time ti . Market practice It is possible to value the cap in a Black (1976) context. In this case, the caplet price is given by c = e−(r

∗ +λ )(t −t) c i

(F(t, ti−1 , ti )N(d1 ) − rc N(d2 ))

with d1 =

log(F/rc ) + 21 σ 2 (ti − ti−1 ) , √ σ ti − ti−1

√ d2 = d1 − σ ti − ti−1 ,

where — — — — —

F(t, ti−1 , ti ): forward rate today between ti−1 and ti , rc : strike price (the cap rate), r ∗ : yield for the maturity ti − t, σ: volatility of the interest rate in the period ti − ti−1 , λc : information cost in the option market.

In the same context, the price of a floorlet is given by e−(r

∗ +λ )(t −t) c i

(−F(t, ti−1 , ti )N(−d1 ) − rc N(−d2 ))

where the different parameters have the same meaning as before.

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6.5. The Pricing of Swaptions, Captions and Floortions within Information Uncertainty The Black (1976) model can be used for the valuation of European swaptions. The underlying asset rf corresponds to the fixed leg of a par swap with maturity date TS . At time T , the option pay-off is given by: max(rf − rE , 0) × present value of future cashflows where rE refers to the strike rate. Following the analysis in Black (1976), the payer swaption formula in this context is given by  −2(TS −T )   1 −(r+λc )(T −t) 1 c= e 1− 1+ F (FN(d1 ) − rE N(d2 )) F 2 with d1 =

log(F/rE ) + 21 σ 2 (T − t) , √ σ T −t

√ d2 = d1 − σ T − t,

where F : forward swap rate, TS : maturity of the swap, σ: volatility of the par rate, rE : strike price. The receiver swaption formula is:  −2(TS −T)   1 −(r+λp )(T −t) 1 p= e 1− 1+ F (rE N(−d2 ) + FN(−d1 )). F 2 This assumes that interest payments in the swap are exchanged every six months.

7. The Pricing of Equity Swaps and Swaptions The analysis in this section is based on the work of Chance and Rich (1998).

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7.1. The Setting The equity factor is defined as a total return index whose value at time j, is I(j). The value at time j of a zero-coupon bond with one dollar face value maturing at k is B(j, k). Each swap corresponds to a transaction in which future payments will appear at times n + 1, n + 2, n + m where n denotes the begining of the current period and n ≤ j < n + 1. When the swap is priced at its initiation date, j = n and this is time 0.

7.2. Pricing Equity Swaps with a Fixed Notional Principal The stream of payments for the party paying the fixed rate and receiving the index return is: At time n + 1 : [I(n + 1)/I(n)] − (1 + R) At time n + 2 : [I(n + 2)/I(n + 1)] − (1 + R) ... At time n + m : [I(n + m)/I(n + m − 1)] − (1 + R). Hence, he receives the percentage change in the index and pays a fixed rate per period R. We denote by V(j; n, n + m) the value of the swap representing the present value of the stream of payments in which the previous payment was at time n with a maturity n + m. It is possible to replicate the payments at time j with n < j < n + 1 as follows. The first cash flow is reproduced by investing I(j)/I(n) dollars in the index and borrowing (1 + R)B(j, n + 1). At n+1, the cash-flow becomes [I(j)/I(n)][I(n + 1)/I(j)]−(1+R) = [I(n + 1)/I(n)] − (1 + R). The second cashflow is reproduced by borrowing (1 + R)B(j, n + 2) and investing B(j, n + 1) in the riskless asset. At time n+1, the riskless asset is worth one dollar which is invested in the index. At n + 2, the investment in the equity index is I(n + 2)/I(n + 1) and the debt due is (1 + R). This procedure allows to reproduce the remaining cash payments.

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At time j, the value of the swap corresponds to the value of these combined transactions: m

I(j) V(j; n, n + m) = − B(j, n + m) − R B(j; n + i). (1) I(n) i=1 This equation is more general than the formula in Jarrow and Turnbull (1995, 1997) since it applies at all times and not only at the start of the swap. The equilibrium swap rate at the start of the swap is found by simply letting j at the start, time n or 0. Setting the last equation to zero and solving for the fixed rate gives the equilibrium swap rate: 1 − B(0, m) R = m . i=1 B(0, i)

(2)

When time n does not correspond to the start date, then before the payment is made, the value of the swap is:

I(n + 1) B(n + 1; n + i). − B(n + 1, n + m) − R I(n) i=1 m

V(n + 1; n, n + m) =

Immediately after the payment is made, the swap value is: V(n + 1; n, n + m) = 1 − B(n + 1, n + m) − R

m

B(n + 1; n + i).

i=2

Substituting Eq. (2) into Eq. (1) gives the value of the swap today: m I(j) i=1 B(j; n + i) V(j; n, n + m) = − B(j, n + m) − [1 − B(0, m)]  m I(n) i=1 B(0, i) which is less affected by the absolute level of the index and the term structure and more by their relative levels. It is important to note that the fixed rate on an equity swap is the same as the fixed rate on a plain vanilla interest rate swap. In fact, if you consider a plain vanilla swap in which the present value of a series of floating payments less the present value of a series of fixed payments at a rate F is: 1 − B(n, n + m) − F

m

B(n; n + i)

i=1

when set to zero and solved for F , this gives the same formula as (2).

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7.3. Pay Floating, Receive Equity Return on Fixed Notional Principal Some equity swaps use a floating rate instead of a fixed rate. A pay floating, receive equity, swap can be decomposed into a pay-fixed, receive-equity, swap plus a plain vanilla pay-floating, receive-fixed interest rate, swap. Equation (1) gives the value of the first swap. The value of the second swap is: R

m

B(j; n + i) + B(j, n + m) − [r(n, n + 1) + 1]B(j, n + 1)

i=1

where r(n, n + 1) corresponds to the floating rate set at n that will be paid at n + 1. Using this gives: V(j; n, n + m) =

I(j) − [r(n, n + 1) + 1]B(j, n + 1). I(n)

(3)

Just after a payment is made, this expression reduces to zero because I(n) =1 I(n)

and [r(n, n + 1) + 1]B(n, n + 1) = 1.

7.4. Pricing the Two-Way Equity Swap In this swap form, both parties pay an equity return. It is assumed that notional principal amounts are 1 and returns are based on two domestic equity indexes I1 and I2 . The first payment is:   I1 (n + 1) I2 (n + 1) I1 (n + 1) I2 (n + 1) −1− −1 = − . I1 (n) I2 (n) I1 (n) I2 (n) The second payment is: I1 (n + 2) I2 (n + 2) − . I1 (n + 1) I2 (n + 1) The last payment would be: I2 (n + m) I1 (n + m) − . I1 (n + m − 1) I2 (n + m − 1)

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The value of the swap at time j corresponds to the value of the initial long and short positions necessary to duplicate the first cash flow only: V(j; n, n + m) =

I1 (j) I2 (j) − . I1 (n) I2 (n)

The remaining cash flows can be reproduced with a self-financing strategy of shorting the second index and using the proceeds to buy the first index. The value of the swap is zero at the start date 0 = j = n. Two-way equity swaps have zero value immediately after payment at each reset date.

7.5. Pricing a Cross-Currency Two-Way Equity Swap Consider the valuation of a swap in which a party pays a domestic equity index return and receives a currency-adjusted return on a foreign index. The domestic index is I2 and the foreign is I1 . Define the spot exchange rates at time n and (n + 1) as units of the domestic currency per unit of foreign currency as S(n) and S(n + 1). The value of the next payment in domestic currency is: I1 (n + 1)S(n + 1) I2 (n + 1) . − I2 (n) I1 (n)S(n) The pricing of the cross-currency two-way equity swap is similar to the pricing of the standard two-way equity swap when the underlying index is defined in terms of its value times the exchange rate, i.e. I1 (j)S(j). In this context, all swap payments beyond the first have zero value and: V(j; n, n + m) =

I1 (j)S(j) I2 (j) − . I1 (n)S(n) I2 (n)

At initiation, (0 = j = n) and the value is zero.

7.6. Pricing the Equity Swap with Variable Notional Principal Many equity swaps are designed with a variable notional principal. Chance and Rich (1998) examine equity swap pricing with the notional principal set at one dollar at the begining by changing the amount according

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to the return on the underlying equity index. They study the case of a payfixed, receive equity, swap. When the notional principal on payment (n + 1) is one dollar, the first payment is: 

 I(n + 1) ($1) −1−R . I(n) The second payment is: 

I(n + 1) I(n)



   I(n + 2) I(n + 1) I(n + 2) −1−R = − (1 + R) I(n + 1) I(n) I(n)

and the last payment is: 

 I(n + m) −1−R I(n + m − 1)   I(n + m) I(n + m − 1) = − (1 + R) . I(n) I(n)

I(n + m − 1) I(n)



In this case, the overall value of the swap at time j corresponds to the sum of the values all the transactions executed at time j:

m

I(j) m − (1 + R) B(n + i − 1; n + i) V(j; n, n + m) = I(n) i=2 − (1 + R)B(j, n + 1).

(4)

At the start of the swap, time 0 = j = n. Setting the swap value equal to 0 and soling gives R.

7.7. Pricing the Capped Equity Swap When structuring a capped equity swap, a limit is imposed on the return paid on the equity leg. The cap is European since it is imposed only at the swap settlement dates.

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Consider the stream of payments on a capped equity swap with fixed notional principal: 

 I(n + 1) min , X − (1 + R) I(n)   I(n + 2) min , X − (1 + R) I(n + 1)   I(n + m) min , X − (1 + R) I(n + m − 1) where X is the per period cap rate (a constant). The first pay-off is (I(n + 1)/I(n)) − 1 − R, if (I(n + 1)/I(n)) ≤ X and X − 1 − R if (I(n + 1)/I(n)) > X. This pay-off is equivalent to: I(n + 1) 1 − (1 + R) − max[0, I(n + 1) − XI(n)]. I(n) I(n) This corresponds to the pay-off of a standard equity swap less the pay-off from 1/I(n) call options. Hence, a capped equity swap can be seen as a standard equity swap overlayed with a series of short call positions. These calls are defined as forward start options. The value of an option at j for the index I(j), the strike price XI(n) and and a time to expiration n + 1 − j is written as c(I(j), XI(n), n + 1 − j). The value of the calls is shown to be: 1 c(I(n + 1), XI(n + 1), 1). I(n + 1) This can be written as: 

 I(n + 1) I(n + 1) c ,X , 1 = c(1, X, 1) I(n + 1) I(n + 1) which is the amount owed on the loan. The value of this swap corresponds to the value of a standard equity swap plus the value of these options at time

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j, or:

I(j) − B(j, n + m) − R B(j; n + i) I(n) i=1 m

V(j; n, n + m) =

1 c(I(j), XI(n), n + 1 − j) I(n) − B(j, n + 1)c(1, X, 1) − · · · − B(j, n + m − 1) × c(1, X, 1) −

which can be restated as:

I(j) V(j; n, n + m) = B(j; n + i) − B(j, n + m) − R I(n) i=1 m

1 c(I(j), XI(n), n − 1 − j) I(n) m−1

B(j, n + i). − c(1, X, 1)

−

(5)

i=1

As before, at the start of the swap, R is determined by setting time 0 = j = n and the value to 0.

7.8. Pricing the Knock-Out Barrier Equity Swap The floating-rate payer in a barrier equity swap pays the equity rate of return each period as long as it never violates the barrier rate over the period. The payments on an equity swap with a barrier H and fixed principal at times (n + 1), (n + 2) and (n + m) are:  if barrier is not hit from time n  I(n + 1) − (1 + R), to n + 1 Payoff at n + 1 I(n)  X − R, otherwise  if barrier is not hit from time  I(n + 2) − (1 + R), n + 1 to n + 2 Payoff at n + 2 I(n + 1)  otherwise X − R,  

I(n + m) − (1 + R), Payoff at n + m I(n + m − 1)  X − R,

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The barrier equity swap can also be seen as a standard equity swap overlayed with a series of written in-barrier options and another series of long recovery options. The barrier equiy swap value corresponds to the value of a standard equity swap, less the value of the barrier option overlay, plus the value of the recovery option overlay:

I(j) − B(j, n + m) − R B(j; n + i) I(n) i=1 m

V(j; n, n + m) =

1 [cIN-BAR (I(j), I(n)R∗ , I(n)H ∗ ; n + 1 − j) I(n) − PIN-BAR (I(j), I(n)R∗ , I(n)H ∗ ; n + 1 − j)] n+m−1

 B(j, n + i) cIN-BAR (1, R∗ , H ∗ ; 1) −

−

i=1

− PIN-BAR (1, R∗ , H ∗ ; 1) + (X − R)D(I(j), I(n)H ∗ , n + 1 − j) + (X − R) n+m−1

B(j, n + i)D(1, H ∗ ; 1). × i=1

The first three terms on the right-hand side correspond to the value of a standard equity swap. The last two terms correspond to the cumulative value of the recovery options. The other terms correspond to the current value of the barrier option.

7.9. Pricing Equity Swaptions A swaption is an option to enter into a swap. A payer swaption is an option to enter into a swap as a fixed-rate payer, floating rate receiver. A receiver swaption is an option to enter into a swap as a fixed-rate receiver, floating rate payer. Exercise of a swaption creates an annuity of fixed payments. Consider for example a payer swaption with a 5% strike maturing in one year. The underlying swap has a term of 5 years. At the maturity of the swaption, if the fixed rate on 5-year swaps is higher than 5%, the holder would exercise and enter into a 5-year swap, paying 5%. Assume that at that date, 5-year swaps in the market have a fixed rate of 6%. It is possible for the holder to enter into an other swap in the market, offsetting the floating

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payments and creating a 5-year annuity of 1%. In this context, the value at expiration of the 5-year annuity corresponds to the swaption value at expiration. The swap holder can also exercise the 5-year swaption without entering into an offsetting swap. The two parties can also agree to settle in cash. When the underlying swap starts at time 0 = n and terminates at m, the swap value at expiration is SW(0, m) = max(0, R − K)

m

B(0, i)

i=1

where K is the strike and the swap rate is given by Eq. (2). Upon exercise, the holder of the swaption pays a fixed rate of K and enters into a swap. The holder can also enter into an offsetting swap, receiving a fixed rate R and paying the equity return. Equity swaps are similar to total return swaps. The difference is that a bond instead of a stock is used.

Summary This chapter stresses the wide applicability of the option pricing theory when it comes to protecting or overcoming financial assets or a foreign currency position. The current trend in financial institutions is to propose tailored conditional profiles that are suited to match the needs and expectations of customers. Several contracts have been described above to fulfill this goal. Garman and Kohlhagen (1983) extended the Black and Scholes (1973) formula for stock options to the valuation of options on currencies where the foreign interest rate replaces the dividend yield. When an investor wants to link a strategy in a foreign stock and a currency, he can use at least four different type of options: a foreign equity option struck in foreign currency, a foreign equity option struck in domestic currency, fixed exchange rate foreign equity options known also as quanto options or an equity-linked foreign exchange option. In a range forward contract, the buyer and the bank agree on two prices at the inception of the contract. At maturity, the buyer buys the foreign currency either at S1 if the spot price is less than S1 , or at S2 if the spot price is greater than S2 , or at the spot price if it is between S1 and S2 . The two prices S1 and S2 are set such that no money changes hands at the inception of the contract. Like a range forward contract, a participating forward contract guarantees a minimum exchange rate for a forward sale and a maximum exchange rate for a forward purchase. Since the contract is structured with no upfront payment, the cost of the seller’s

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upside participation is that the minimum exchange rate guaranteed through a participating forward sale will necessarily be greater than the outright forward price. Since the fixed-interest rate payments are all known in a swap operation, they can be viewed as the sum of zero-coupon bonds. A floating leg of a swap at time Ti corresponds exactly to a deposit of one dollar at time (Ti − τ) and withdrawal of the dollar at time τ. A cap allows the holder to “cap” the increase in the floating interest rate to a given level. At each date ti , the cap allows the payment of a variable interest rate on a given principal amount. The Black (1976) model can be used for the valuation of European swaptions. In an index amortizing swap, the principal amount is amortized with respect to a specified index. When the frequency of payments is for example three months, then at each reset date, each quarter, the principal amount falls and there is an exchange of interest payments on the outstanding principal. This swap is path-dependent. Like other swaps, an equity swap is a transaction between two parties. One party promises to pay the return on a stock or an equity index and the other promises to make either a fixed payment, a floating-rate interest payment, or the return on another stock or equity index. Jarrow and Turnbull (1995, 1997) provide a preference free formula for the fixed rate of an equity swap. Chance and Rich (1998) present several formulas for equity swaps for the cases of fixed and variable notional principals, two sided equity payments, cross-currency equity swaps, capped equity swaps, barrier equity swaps, etc. Unlike an interest rate swap, where the rate is set at the begining of the period, the return on the equity index for an equity swap is determined at the end of each period. Some equity swaps use a floating rate instead of a fixed rate. A pay floating, receive equity, swap can be decomposed into a pay-fixed, receive-equity, swap plus a plain vanilla pay-floating, receive-fixed interest rate, swap. When structuring a capped equity swap, a limit is imposed on the return paid on the equity leg. The cap is European since it is imposed only at the swap settlement dates. The floating-rate payer in a barrier equity swap pays the equity rate of return each period as long as it never violates the barrier rate over the period. The barrier equity swap can also be seen as a standard equity swap overlayed with a series of written in-barrier options and an other series of long recovery options. The barrier equiy swap value corresponds to the value of a standard equity swap, less the value of the barrier option overlay, plus the value of the recovery option overlay. A swaption is an option to enter into a swap. A payer swaption is an option to enter into a swap as a fixed-rate payer, floating rate receiver. A receiver swaption is an option to enter into a swap as a fixed-rate receiver, floating

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rate payer. Exercise of a swaption creates an annuity of fixed payments. This chapter presents and analyzes equity-linked Forex options, quantos, range forward contracts, indexed notes, perls and other hybrid securities. These contracts have some characteristics shared by caps, floors and collars and their valuation sometimes reduces to the valuation of standard options. Some examples are given and simulation results are provided. The analysis is extended to several other contracts.

Appendix: Forward Rates, Bootstrapping and Information Consider the prices of zero-coupon bonds Z(t, T). The implied forward rate which is assumed to apply in the future corresponds to the curve of a timedependent interest rate which is consistent with the market price of interestrate sensitive instruments. Following the analysis in Willmott (1998), the rate r(τ) corresponding to this situation satisfies the following relationship T − t (r(τ)+λB )dτ Z(t; T) = e . Differentiating and rearranging terms gives the forward rate at time t that applies for maturity T : r(T) = −

∂ (log Z(t; T)) − λB . ∂T

Since the forward rate differs each day for each maturity, the forward rate is often denoted by F(t, T) or F(t; T) = −

∂ (log Z(t; T)) − λB . ∂T

It is possible to get an implied forward rate curve using the market prices of different zero-coupon bonds ZiM which are ranked by their maturities. Using the price of the first bond Z1M = e−(y1 +λB )(T1 −t) , gives the rate to be used process between
in the discounting 
now T1 ,  and date y1 = − log Z1M /(T1 − t) − λB or y1 + λB = − log Z1M /(T1 − t) . This rate can be applied in the discounting process during the period from now to T1 . Knowing the rate between now and T1 , it is possible to compute the rate between T1 and T2 using the price Z2M = e−(y1 +λB )(T1 −t) e−(y2 +λB )(T2 −T1 ) .

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This gives the rate to be applied between T1 and T2 or
 log Z2M /Z1M − λB y2 = − T2 − T1 
 or y2 + λB = − log Z2M /Z1M /(T2 − T1 ) . This method of bootsrapping allows the construction of the forward rate curve in the presence of incomplete information.

Bibliography Barnhill, T and W Seale (1990). Financing with hybrid securities having commodity option and forward contract characteristics. Advances in Futures and Options Research, 4, 137–151. Bellalah, M (2001). Market imperfections; information costs and the valuation of derivatives: some general results. International Journal of Finance, 13(3), 1895–1927. Bellalah, M, Ma Bellalah and R Portait (2001). The cost of capital in international finance. International Journal of Finance, 13(3), 1958–1973. Bellalah, M and JL Prigent (2001). Pricing standard and exotic options in the presence of a finite mixture of Gaussian distributions. International Journal of Finance, 13(3), 1975–2000. Bellalah, M, JL Prigent and C Villa (2001). Skew without skewness: asymmetric smiles; information costs and stochastic volatilities. International Journal of Finance, 13(2), 1826–1837. Bellalah, M and F Selmi (2001). On the quadratic criteria for hedging under transaction costs. International Journal of Finance, 13(3), 2001–2020. Black, F (1976). The pricing of commodity contracts. Journal of Financial Economics, 79(3), 167–179. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659. Chance, D and D Rich (1998). The pricing of equity swaps and swaptions. Journal of Derivatives, Summer, 19–31. Garman, M and S Kohlhagen (1983). Foreign currency option values. Journal of International Money and Finance, 2, 231–237. Jarrow, R and S Turnbull (1995). Pricing derivatives on financial securities subject to credit risk. Journal of Finance, L(1), March. Jarrow, R and S Turnbull (1997). When swaps are dropped. Risk, May, 70–77. Rubinstein, M and E Reiner (1991a). Breaking down the barriers. Risk, 4(8), 28–35. Rubinstein, M and E Reiner (1991b). Unscrambling the binary code. Risk, 4(8), 75–83. Willmott, P (1998). Derivatives, John Wiley and Sons.

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Chapter 7

Binaries, Barriers and Their Applications

This chapter is organized as follows: 1. Section 2 presents some examples of simple and complex binaries and barrier options. 2. Section 3 develops the valuation framework for binary barrier options with and without information costs. 3. Section 4 presents the formulas for the valuation of inside and outside barrier options. 4. Section 5 develops a framework for the valuation of barrier options by accounting for the effects of information costs. 5. Section 6 provides an alternative approach for the valuation of outside barrier options by accounting also for the impact of information costs. 6. Section 7 develops the framework for the analysis and valuation of standard and exotic options using simple binary options. Several applications are given for soft binary options, down-and-out call options, switch options, corridor options, knock-out options, etc. 7. Section 8 develops a framework for the analysis and valuation of continuous strike options and soft barrier options by accounting for information uncertainty.

1. Introduction innovations in OTC options markets involve not only certain R ecent relations between the underlying asset price and the strike price but also the number of time units for which a certain condition is satisfied. This corresponds, for example, to financial assets for which a certain condition is satisfied. This is the case for financial assets which are traded within Bellalah, Mondher. Exotic Derivatives and Risk: Theory, Extensions and267 Applications, World Scientific Publishing Co 08:36:21.

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a specified range. Standard calls and puts which have been traded for a long time are not the simplest and easiest financial products to value. For example, the value of a standard option can be obtained as a limit of a sum of binary or digital options. In fact, binary options which are considered as exotic options are much simpler to value than standard options. An option which pays out a fixed amount if the underlying asset price hits a certain predetermined level is a digital option, a bet option or a binary option. Digital options have fixed pay-outs and are either “on” or “off”. In their simplest forms, the pay-off of these options is cash or nothing or asset or nothing. A binary call is a bet on whether the asset price would be above the strike price. A binary put is a bet on whether the asset price would be below the strike price. The value of a simple binary or digital call is given by discounting its pay-off under the usual Black–Scholes arguments of risk neutrality. The pay-off of this option is given by an amount A if the underlying asset price is greater than the strike price, and nothing otherwise. This pay-off corresponds to that of vanilla binaries or digitals. Structured products with embedded digitals are much more interesting than vanilla digitals. Barrier options are path-dependent options which are in life when they knock in and disappear when they knock out. They are referred to as knock-ins or knock-outs. Barrier options can be ranked into four categories. Regular barrier options which come into existence when the barrier is out-of-the-money with respect to the current underlying asset price. Barrier options allow the option holder to go “out” or “in” some specified levels of the underlying asset only on specified days during the option’s life. That day may be the last day of each month, of each quarter, the Friday before the last Monday of each month, and so on. These specificities make up-and-out and down-and-out options more expensive than standard options. Also, they make up-and-in and down-and-in options cheaper than standard options. These options may be European, American or Quasi-American allowing exercise, respectively, at the maturity date, at any time before maturity or at some specified times before maturity. Reverse barrier options which come into existence when the barrier is in-the-money with respect to the current underlying asset price. Double barrier options which come into existence when the barrier option has knock-in or knock-out levels in either side with respect to the underlying asset price. Continuous or soft barrier options are barrier options which do not appear or disappear suddenly. For the analysis of information costs and valuation, we can refer to Bellalah (2001), Bellalah, Prigent and Villa (2001), M. Bellalah, Ma. Bellalah and R. Portait (2001), Bellalah and Prigent (2001), Bellalah and Selmi (2001), etc.

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2. Analysis of Binaries and Barriers 2.1. Standard Binary Options An option which pays out a fixed amount if the underlying asset price hits a certain predetermined level is a digital option. It is sometimes refered to as a bet option or a binary option. Hence, digital options have fixed pay-outs and are either “on” or “off”. These options are somewhat a straight bet on the position of the underlying asset price with respect to the strike price. In their simplest forms, the pay-off of these options is cash or nothing or asset or nothing. In this spirit, a binary call is a bet on whether the asset price would be above the strike price. A binary put is a bet on whether the asset price would be below the strike price. Roughly speaking digital options can be ranked into two categories: all or nothing digitals and one-touch digitals. All-or-nothing options An all-or-nothing digital option is an option which pays out a set amount if the underlying asset is above or below a certain specified level at the option’s maturity date. An all-or-nothing call (put) is an option giving the right to its holder to receive a pre-determined amount, “the all”, if the underlying asset goes above (below) the strike price. How far is the underlying asset price above or below the strike price is not important since the pay-off will be “all” or “nothing” at the maturity date. One touch all-or-nothing option A one-touch digital option is an option which pays out a set amount if the underlying asset price hits a certain specified level at any time during the option’s life. A one touch all-or-nothing call (put) is an option giving the right to its holder to receive a pre-determined amount, “the all”, if the underlying asset goes above (below) the strike price at any time during the option’s life. How far is the underlying asset price above or below the strike price is irrelevant since the pay-off will be “all” or “nothing”.

2.2. Complex Binaries and Range Options Barrier options are options that either cease to exist or come into existence when some pre-specified asset price barrier is hit during the option’s life. A down-and-out call is “knocked out” when the asset price falls to some pre-specified level before the option maturity. Rubinstein and Reiner (1991a,b) provided valuation equations for a whole family of barrier options.

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The up-and-out put In the late 1980s, up-and-out put options emerge while down-and-out call options are traded even before 1970. An up-and-out option known also as a knock-out option, an over-andout option, an over-the-top option, a barrier option, a vanishing option, or an extinguishing option is simply a put option with a special feature: if the underlying asset rises above a certain level during the option’s life, known as the knock-out boundary, the option becomes worthless. Hence, an upand-out put is an option which becomes out when the underlying asset price reaches a certain level, known as the “out” level or knock-out boundary. The up-and-in put The up-and-in put is the inverse of an up-and-out put. If at any time during the option’s life, the underlying asset rises above a certain prespecified level, called the “in” level, the option is worth something. Otherwise its value is zero. If the underlying asset reaches or exceeds the “in” level, the option becomes a standard put otherwise it is worthless. The down-and-out call The down-and-out call is defined in a similar way as a down-and-out put, except for the “out” level. If the underlying asset price stays above that level, the down-and-out call becomes a standard call and if it falls below that level the call is worthless. The up-and-out call An up-and-out call is a call that ceases to exist whenever the underlying asset price reaches the knock-out level, which is above the current underlying asset price and the strike price. The price of an up-and-out-call is very low when compared to standard options because its pay-out is no longer infinite and is limited by a barrier above the strike price. Since, the boundary or the knock-out level is known in advance, it is possible to hedge and value these options. The down-and-in call The down-and-in call is the inverse of a down-and-out call. It is a contingent claim which becomes a standard call when the underlying asset price goes below the “in” level and it is worthless whenever the underlying asset does not go below that level. Using barrier options, it is possible to construct several forms of synthetic forward structures in currency OTC markets. Synthetic forward structures can be presented in currency markets in the

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forms of trigger forward contracts, at maturity trigger forward contracts and forward extra contracts. A trigger forward contract This is a synthetic forward contract which disappears if a certain predetermined rate trades at any time during the contract’s life. This synthetic forward contract can be created by long and short positions in barrier put and barrier call options. At-maturity trigger forward contract This is a synthetic forward contract which disappears if the spot rate trades below (or above) a certain pre-determined rate at the maturity date. This synthetic forward contract can be created by long and short positions in barrier put and barrier calls. Forward extra contract The investor has protection from a long option position with a predetermined strike price for zero cost unless a specified level is hit. If this specified level is traded, then the right to exercise the option becomes an obligation through a synthetic forward contract.

2.3. The Ins and Outs of Barrier Options Derman and Kani (1993) denote a barrier above the current underlying asset as an up barrier; if it is ever crossed, it is from below. They call a barrier below the current underlying asset a down barrier; if it is ever crossed, it is from above. There are two types of barrier options: in options and out options. A knockin option or an in barrier option gives a certain pay-off only if the underlying asset ends in the money and if the barrier is crossed before the maturity date. The in barrier is knocked in when the underlying assset crosses the barrier. In this case, it becomes a standard option of the same type. The option is worthless when the stock does not cross the barrier. A knockout option or an out barrier option gives a certain pay-off only if the underlying asset ends in the money and if the barrier is never crossed before the maturity date. As long as the underlying asset never crosses the barrier, the out barrier option remains a standard option of the same type. The option is worthless (knocked out) when the stock crosses the barrier. Using the definition of a barrier option, it is possible to define up-and-out, up-and-in, down-and-out and down-and-in options. Table 1 shows different

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Table 1: Barrier options and the effect of the barrier on the pay-off. Type

Location Barrier-crossed

Down-and-out call Down-and-in call Up-and-out call Up-and-in call Down-and-out put Down-and-in put Up-and-out put Up-and-in put

S S

Worthless Standard call Worthless Standard call Worthless Standard put Worthless Standard put

Not-crossed Standard call Worthless Standard call Worthless Standard put Worthless Standard put Worthless

types of barrier options as well as the effect of crossing the barrier on the pay-off. The premium paid for a barrier option is in general lower than that of a standard option. This is one of the basic reasons behind the use of barrier options rather than standard options. Two other main reasons are often cited in the literature. The first is that the pay-off of a barrier option may match closely the beliefs of investors regarding the market timing. The second reason is that barrier options may match more closely the hedging needs of investors when compared to standard options.

2.4. Special Features of Barrier Options Managing the risk of an option position with barrier options is more complicated than with standard options. The management of a position using a delta-hedge is possible but is more complicated. The following property is verified between barrier options. The value of a down-and-in call (put) plus the value of a down-and-out call (put) is equivalent to that of a corresponding standard call (put). The value of an up-and-in call (put) plus the value of an up-and-out call (put) is equivalent to that of a corresponding standard call (put). The value of an up-and-in call when the strike is above the barrier is equivalent to that of a standard call since all future stock paths lead to being knocked in. The value of a down-and-in put when the strike is below the in barrier is equivalent to that of a standard put.

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2.5. Some Related Options A binary up-and-in call has a single barrier and no strike price. This option pays the buyer a one-time fixed amount, the first time the underlying asset reaches the barrier from below. If the underlying asset has not crossed the barrier, the option is worthless. A binary up-and-in put has a single barrier and no strike price. This option pays the buyer a one-time fixed amount, the first time the underlying asset reaches the barrier from above. If the underlying asset has not crossed the barrier, the option is worthless at maturity. A capped European-style call presents a strike price and a cap barrier above the strike price. The pay-off of a capped European-style call is similar to that of a standard call with the same characteristics, with a main difference: if the underlying asset ever reaches the barrier from below, the buyer receives immediately a cash amount equal to the difference between the barrier level and the strike price. This capped call is termed non-deferred because the cash is received instantaneously. Otherwise, the cash is received at maturity and the option is said a deferred capped call. A floored European-style put presents a strike price and a floor barrier below the strike price. The pay-off of a floored European-style put is similar to that of a standard put with the same characteristics, with a main difference: if the underlying asset ever reaches the barrier from above, the buyer receives immediately a cash amount equal to the difference between the strike price and the barrier level. This floored put is termed non-deferred because the cash is received instantaneously. Otherwise, the cash is received at maturity and the option is said to be a deferred floored put.

3. Standard Valuation of Binary Barrier Options Following Rubinstein and Reiner (1991a,b), we denote by: R: 1 plus the riskless interest rate r, d: 1 plus the instantaneous pay-out rate, H: barrier level, S(τ): price of the underlying asset after elapsed time τ, St : price of the underlying asset at expiration t, η and φ: binary variables taking the value 1 or −1, and K: the strike price.

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3.1. Path-Independent Binary Options 3.1.1. Standard Cash-or-Nothing Options In their simplest forms, the pay-off of a binary call is nothing when the underlying asset terminal price, St is below the strike price, K and is a predetermined amount, A if the underlying asset terminal price is above the strike price. The pay-off of the binary call is: ccon = 0,

if St ≤ K,

ccon = A,

else.

The pay-off of a binary put is nothing when the underlying asset terminal price, St is above the strike price, K and is a predetermined amount, A if the underlying asset terminal price is below the strike price. The pay-off of the binary put is pcon = 0,

if St ≥ K,

pcon = A,

if else.

These options can be valued with respect to classic formulas of Black– Scholes. The values of standard options are given by √ (1) C = φSd −tN(φx) − φKR−tN(φx − φσ t)
 Sd −t log KR −t 1 √ x= + σ t √ 2 σ t with φ = 1 for a call and −1 for a put. The Black–Scholes formula comprises two parts. The first term, d −t φSN(φx), corresponds to the present value of the underlying asset price conditional upon exercising the option. The second term, φKR−tN(φx − √ φσ t) refers to the present value of the strike price times the probability of exercising the option. The value of a cash-or-nothing call is √ ccon = AR−tN(x − σ t). The value of a cash-or-nothing put is

√ pcon = AR−tN(−x + σ t).

In the presence of a cost of carry b, the value of a cash-or-nothing call is ccon = Ae−rTN(x ). −rT   The value of a cash-or-nothing √ put is pcon = Ae N(−x ) where x = 2 [log(S/K)] + (b − (1/2)σ )T/σ T . 08:36:21.

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3.1.2. Cash-or-Nothing Options with Shadow Costs We denote by λS and λ the information costs associated to S and the option, respectively. The values of standard options with information costs are given by √ C = e(λS −λ+d)t φSN(φx) − φKe−(λ+r)tN(φx − φσ t) x=

[log(S/K) + (r + λS − d)t] 1 √ + σ t √ 2 σ t

with φ = 1 for a call and −1 for a put. The first term in this formula, e(λS −λ+d)t φSN(φx) corresponds to the present value of the underlying asset price conditional upon exercising the √ −(λ+r)t option. The second term, φKe N(φx − φσ t) indicates the present value of the strike price times the probability of exercising the option. The value of a cash-or-nothing call in the presence of shadow costs is given by √ ccon = Ae(λS −λ)tN(x − σ t). The value of a cash-or-nothing put is

√ pcon = Ae(λS −λ)tN(−x + σ t).

3.1.3. Standard Asset-or-Nothing Options These binary options are similar to cash-or-nothing options except that in their pay-off, the predetermined amount is replaced by the terminal asset value. The pay-off of the asset-or-nothing call is caon = 0,

if St ≤ K

caon = St ,

else.

The value of this option is given by the present value of the underlying asset price conditional upon exercising the call or caon = Sd −tN(x). The pay-off of the asset-or-nothing put is nothing when the underlying asset terminal price, St is above the strike price, and is the terminal asset price, St if the underlying asset terminal price is below the strike price paon = 0,

if St ≥ K

paon = St ,

else.

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The value of this option is given by the present value of the underlying asset price conditional upon exercising the put or paon = Sd −tN(−x). In the presence of a cost of carry b, the value of an asset-or-nothing call is caon = Se(b−r)tN(x ). The value of an asset-or-nothing put is paon = Se(b−r)tN(−x ) where 2 )t √ . x = [log(S/K)]+(b+(1/2)σ σ t

3.1.4. Asset-or-Nothing Options with Information Costs The value of the asset-or-nothing call is √ caon = Se(λS −λ+d)tN(x − σ t). The value of the asset-or-nothing put is paon = Se(λS −λ−d)tN(−x). Tables 2–5 provide some simulations of option values and the Greek letters. The reader can make some comments regarding the evolution of these parameters. In the presence of a cost of carry, equal to the difference between the domestic interest rate r and the foreign interest rate, r ∗ , these options can be priced in the same way. Tables 6 and 7 give the values of these options and the associated Greek letters. Table 2: Cash-or-nothing options. Barrier

Price

Delta

Gamma

Vega

Theta

100.00 101.00 102.00 103.00 104.00 105.00 110.00 120.00 130.00 140.00

0.49005 0.47065 0.45150 0.43262 0.41406 0.39585 0.31099 0.17803 0.09350 0.04577

0.01950 0.01947 0.01940 0.01929 0.01913 0.01893 0.01742 0.01290 0.00829 0.00477

−0.00002 0.00002 0.00007 0.00012 0.00016 0.00021 0.00039 0.00058 0.00054 0.00041

−0.00390 −0.00294 −0.00198 −0.00104 −0.00011 0.00078 0.00474 0.00909 0.00917 0.00708

0.00011 0.00008 0.00005 0.00003 0.00000 −0.00002 −0.00013 −0.00025 −0.00025 −0.00019

S = 100, t = 17/12/2003, T = 18/12/2004, r = 2%, σ = 20%, barrier = 140.

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Table 3: Cash-or-nothing options. S

Price

Delta

Gamma

Vega

Theta

96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

0.62605 0.64456 0.66247 0.67978 0.69647 0.71253 0.72794 0.74271 0.75683

0.01851 0.01793 0.01732 0.01669 0.01605 0.01541 0.01476 0.01411 0.01346

−0.00059 −0.00061 −0.00063 −0.00064 −0.00065 −0.00065 −0.00065 −0.00065 −0.00064

−0.01027 −0.01097 −0.01160 −0.01216 −0.01265 −0.01307 −0.01341 −0.01369 −0.01390

−0.00019 −0.00021 −0.00023 −0.00025 −0.00027 −0.00029 −0.00030 −0.00032 −0.00033

S = 100, t = 05/01/2003, T = 05/01/2004, r = 3%, σ = 20%, barrier = 90.

Table 4: Asset-or-nothing options. S

Price

Delta

Gamma

Vega

Theta

96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

32.04676 34.22618 36.46076 38.74601 41.07730 43.44986 45.85890 48.29956 50.76702

2.17828 2.23380 2.28478 2.33107 2.37257 2.40920 2.44095 2.46781 2.48984

0.05551 0.05098 0.04629 0.04150 0.03664 0.03175 0.02686 0.02202 0.01724

1.09187 1.03383 0.96919 0.89846 0.82219 0.74100 0.65550 0.56637 0.47427

0.04444 0.04330 0.04195 0.04039 0.03864 0.03672 0.03465 0.03243 0.03009

S = 100, t = 05/01/2003, T = 05/01/2004, r = 3%, σ = 20%, barrier = 110.

3.1.5. Standard Gap Options Gap options are structured to give the following pay-off for a call cgap = 0,

if St ≤ K

cgap = St − A,

else.

The “gap” refers to the difference (A−K). Note that the pay-off of a gap call corresponds to the difference between the pay-offs of an asset-or-nothing call and a cash-or-nothing . Therefore, its value is given by √ cgap = Sd −tN(x) − AR−tN(x − σ t).

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Table 5: Asset-or-nothing options. Barrier

Price

Delta

Gamma

Vega

Theta

100.00 101.00 102.00 103.00 104.00 105.00 110.00 120.00 130.00 140.00 150.00

57.94737 55.99851 54.05440 52.11964 50.19862 48.29551 39.17721 23.92946 13.39969 6.98079 3.42846

2.52913 2.52674 2.51958 2.50773 2.49134 2.47053 2.30761 1.78728 1.21145 0.73818 0.41329

0.02286 0.02768 0.03248 0.03723 0.04189 0.04646 0.06657 0.08763 0.08329 0.06478 0.04380

0.00194 0.09883 0.19597 0.29265 0.38822 0.48204 0.90634 0.40286 0.41011 0.12724 0.77475

−0.00001 −0.00267 −0.00534 −0.00799 −0.01062 −0.01319 −0.02483 −0.03839 −0.03846 −0.03062 −0.02094

S = 100, t = 17/12/2003, T = 18/12/2004, r = 2%, σ = 20%, barrier = 140.

Table 6:

Digital FX-cash-or-nothing options.

S

Price

Delta

Gamma

Vega

Theta

0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04

0.00843 0.00969 0.01110 0.01266 0.01438 0.01629 0.01839 0.02069 0.02320

0.12559 0.14020 0.15593 0.17278 0.19078 0.20992 0.23020 0.25162 0.27415

0.34624 0.31487 0.28279 0.24997 0.21641 0.18211 0.14707 0.11130 0.07482

0.00253 0.00279 0.00306 0.00334 0.00363 0.00393 0.00425 0.00457 0.00489

0.00843 0.00969 0.01110 0.01266 0.01438 0.01629 0.01839 0.02069 0.02320

S = 1, t = 07/02/2003, T = 07/02/2004, r = 3%, r ∗ = 4%, σ = 20%, barrier = 1.5.

This formula is like that of a standard call except for the cash amount replacing sometimes the strike price. The pay-off of a gap option put is pgap = 0,

if St ≥ K

pgap = St − A,

else.

Note that the pay-off of a gap put corresponds to the difference between the pay-offs of an asset-or-nothing put and a cash-or-nothing. Hence, its value is given by √ pgap = −Sd −tN(−x) + AR−tN(−x + σ t).

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Table 7:

279

Digital FX-asset-or-nothing options.

S

Price

Delta

Gamma

Vega

Theta

0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04

0.01355 0.01559 0.01787 0.02040 0.02321 0.02632 0.02974 0.03350 0.03762

0.20342 0.22741 0.25327 0.28106 0.31079 0.34249 0.37616 0.41181 0.44941

1.38891 1.34333 1.29632 1.24785 1.19789 1.14644 1.09349 1.03906 0.98315

0.00415 0.00458 0.00503 0.00551 0.00600 0.00652 0.00706 0.00761 0.00817

0.01355 0.01559 0.01787 0.02040 0.02321 0.02632 0.02974 0.03350 0.03762

S = 1, t = 07/02/2003, T = 07/02/2004, r = 3%, r∗ = 4%, σ = 20%, barrier = 1.5.

This formula is like that of a standard put except for the cash amount replacing sometimes the strike price. Gap options can be defined with respect to two different strike prices K1 and K2 . The call’s pay-off is zero if S ≤ K1 and is S − K2 if S > K1 . The call’s pay-off is zero if S ≥ K1 and is K2 − S if S < K1 . Using the analysis in Rubinstein and Reiner (1991), the call formula is: cgap = Se(b−r)tN(d1 ) − K2 e−rtN(d2 ) d1 =

log(S/K1 ) + (b + (1/2)σ 2 )t , √ σ t

√ d2 = d1 − σ t.

The put formula is: pgap = K2 e−rtN(−d2 ) − Se(b−r)tN(−d1 ) d1 =

log(S/K1 ) + (b + (1/2)σ 2 )t , √ σ t

√ d2 = d1 − σ t.

3.1.6. Gap Options with Shadow Costs The value of a gap call is

√ cgap = e(λS −λ−d)tSN(x) − Ae−(λ+r)tN(x − σ t).

The value of the gap put is

√ pgap = −e(λS −λ−d)tSN(−x) + Ae−(λ+r)tN(−x + σ t).

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3.1.7. Supershares The pay-off from a supershare option is 0 if KL > S > KH and KSL otherwise. In this setting, the formula for a supershare option is: csupershare =

Se(b−r)t [N(d1 ) − N(d2 )] KL

d1 =

log(S/KL ) + (b + (1/2)σ 2 )t , √ σ t

d2 =

log(S/KH ) + (b + (1/2)σ 2 )t . √ σ t

Table 8: Supershares options. S

Price

Delta

Gamma

Vega

Theta

96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

0.82553 0.83492 0.84275 0.84904 0.85378 0.85701 0.85875 0.85902 0.85787

0.00942 0.00786 0.00630 0.00475 0.00323 0.00173 0.00026 −0.00117 −0.00255

−0.00156 −0.00156 −0.00155 −0.00153 −0.00150 −0.00147 −0.00143 −0.00138 −0.00134

−0.02912 −0.02920 −0.02912 −0.02890 −0.02852 −0.02800 −0.02734 −0.02654 −0.02562

−0.00079 −0.00080 −0.00081 −0.00082 −0.00082 −0.00082 −0.00081 −0.00080 −0.00079

S = 100, t = 05/01/2003, T = 05/01/2004, r = 3%, σ = 20%, lower strike = 80, higher strike = 120.

Table 9: Supershares options. S

Price

Delta

Gamma

Vega

Theta

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

0.47854 0.61242 0.72712 0.81090 0.85761 0.86660 0.84166 0.78941 0.71782

0.02767 0.02533 0.02014 0.01315 0.00550 −0.00179 −0.00798 −0.01266 −0.01571

−0.00020 −0.00084 −0.00129 −0.00151 −0.00151 −0.00134 −0.00107 −0.00074 −0.00042

−0.00870 −0.01745 −0.02465 −0.02880 −0.02919 −0.02592 −0.01970 −0.01158 −0.00272

0.00024 0.00048 0.00068 0.00079 0.00080 0.00071 0.00054 0.00031 0.00007

S = 100, t = 21/12/2003, T = 21/12/2004, r = 2%, σ = 20%, lower strike = 80, upper strike = 120. 08:36:21.

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3.2. Path-Dependent Binary Barrier Options Following Rubinstein and Reiner (1991), let f(u) be the density of the risk-neutral underlying asset return u. Let g(u) stand for the density of the natural logarithm of the underlying asset return when it hits the barrier but ends up below the knock-out level at expiration (given that the underlying asset starts at S above the barrier H). Let h(τ) be the density for the first time τ when the underlying asset crosses the barrier. The density function f is given by:   1 (s − µt) 1 1 2 f(s) = √ exp − v , v = √ , µ = r − d − σ2. 2 2 σ t σ 2t The density function g is given by:   2 1 2 exp(2µα/σ ) exp − v , α = log(H/S ), g(u) = √ 2 σ 2πt v =

(u − 2ηα − ηµt) . √ σ t

The density function h(τ) is given by:   1 2 ηα exp − v , h(τ) = − √ 2 στ 2πt

v=−

ηα − ηµτ . √ σ τ

The last density is known as the first passage time density. We denote by 1 x = √ [ln(S/K) + νσ 2 t], σ t

1 x1 = √ [ln(S/H) + νσ 2 t]. σ t

1 y = √ [ln(H 2 /SK) + νσ 2 t], σ t

1 y1 = √ [ln(H/S) + νσ 2 t]. σ t

1 z = √ [ln(H/S) + a1 σ 2 t], σ t ν =1+

µ , σ2

a=

µ , σ2

a1 =

µ2 + r + σ 2 . σ2

As in Rubinstein and Reiner (1991), 28 types of path-dependent binary options are valued with 44 different formulas. We define the following 08:36:21.

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notations for path-dependent binary barrier options: √ [1A] = SN(φx), [1C] = Ke−rtN(φx − φσ t), √ [2A] = SN(φx1 ), [2C] = Ke−rtN(φx1 − φσ t),  2(ν−1)  2ν √ H −rt H N(ηy), [3C] = Ke N(ηy − ησ t), [3A] = S S S  2(ν−1)  2ν √ H H N(ηy1 ), [4C] = Ke−rt N(ηy1 − ησ t). [4A] = S S S Using the above notations, Tables 10–13 give the ‘in’ binary barrier call and put value formulae for in-cash or nothing (Tables 10 and 11) and in-asset or nothing (Tables 12 and 13). Tables 14–17 give the ‘out’ binary barrier call and put value formulae for out-cash or nothing (Tables 14 and 15) and out-asset or nothing (Tables 16 and 17). Tables 18–22 provide simulations results of option values. Table 10: Simulation results for down (or up) and in cash or nothing call. Option

Condition

Down Down Down Up Up Up

S>H K>H KH KH KH KH KH KH KH K H and K < H. For a down-and-in call when K > H, the underlying asset price can end up above the strike price while having first reached the barrier. This

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gives the holder a positive pay-off. Also, it can end up above the barrier without ever having hit the barrier before maturity. This gives the holder a rebate. This may be written as: If (conditional on) S(τ) ≤ H for some τ ≤ t, St > K, pay-off = St − K. If (conditional on) S(τ) ≤ H for some τ ≤ t, St ≤ K, pay-off = 0. For an up-and-in call, when (K < H), the underlying asset price can end up above the barrier having first hit the barrier. This gives the holder a positive pay-off. Also, it can end up below the barrier but above the strike price. This gives the holder nothing. The holder receives the rebate when the underlying asset price ends up above the barrier without touching it before expiration. The difference between the down-and-in call and the up-and-in call is that in the latter case the underlying asset price start out below the barrier. Using notations [1]–[6], Tables 26 and 27 give the down (or up)-and-in call and put formulae. Table 26: Down (or up)-and-in call. Option

Condition

Call-price

Down

S>H

Down Down Up

K>H KH

Down Down Up

K>H KH

Down Down Up

K>H KH K H if ∀τ ≤ t, S(τ) ≤ H if ∃τ ≤ t, S(τ) < H if ∀τ ≤ t, S(τ) ≥ H

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when the barrier is hit. Since the time the barrier is hit is unknown, the valuation of down-and-out options also needs the knowledge of the density of the first passage time τ for the barrier to be hit by the underlying asset. Since we need to value the rebate associated with the down-and-out option, its present value is given by its expected value discounted by the interest rate raised to the power of the first passage of time as in the formula [6 ]:  . [6 ] = Rh r −τ h(τ)dτ .

 = Rh

H S

a+a1

 N(φz) +

H S

a−a1

√ N(ηz − 2ηa1 σ t)

where η = 1 if the barrier is hit from above and −1 if it is approached from below.

5. Valuation of Barrier Options with Shadow Costs of Incomplete Information We use the same notation as before. Hence, the density functions f , g and h(τ) in the presence of a cost λ are given by:   1 (u − µt) 1 1 2 f(u) = √ exp − v , v = √ , µ = r + λS − d − σ 2 2 2 σ t σ 2πt   exp(2µα/σ 2 ) 1  g(u) = exp − v 2 , α = log(H/S) √ 2 σ 2πt (u − 2ηα − ηµt) √ σ t   ηα − ηµτ ηα 1 2 h(τ) = − √ exp − v , v = − . √ 2 σ τ στ 2πt v =

We denote by: x= y= z=

1 √ [ln(S/K) + νσ 2 t], x1 σ t 1 √ [ln(H 2 /SK) + νσ 2 t], σ t 1 √ [ln(H/S) + a1 σ 2 t] σ t

ν =1+

µ , σ2

a=

µ , σ2

a1 =

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=

1 √ [ln(S/H) + νσ 2 t] σ t 1 y1 = σ √ [ln(H/S) + νσ 2 t] t

µ2 +r+λS +σ 2 . σ2

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The following notations are used for path-dependent binary barrier options: [1A] [1C] [2A] [2C] [3A] [3C] [4A] [4C]

= e(λS −λ−d)tSN(φx), √ = Ke−(r+λ)tN(φx − φσ t) = e(λS −λ−d)tSN(φx1 ), √ = Ke−(r+λ)tN(φx1 − φσ t)  2ν = e(λS −λ−d)tS HS N(ηy),  2(ν−1) √ = Ke−(r+λ)t HS N(ηy − ησ t)  2ν = e(λS −λ−d)tS HS N(ηy1 ),  2(ν−1) √ = Ke−(r+λ)t HS N(ηy1 − ησ t).

The following notations are still valid for the valuation of in-barrier and out-barrier options: √ t) [1] = φSd −tN(φx) − φKR−tN(φx − φσ √ [2] = φSd −tN(φx1 ) − φKR−tN(φx1 − φσ t)  2ν  2(ν−1) √ [3] = φS HS N(ηy) − φKR−t HS N(ηy − ησ t)  2ν  2(ν−1) √ [4] = φS HS N(ηy1 ) − φKR−t HS N(ηy1 − ησ t)  H 2(ν−1) √ √ [5] = KR−t N(ηy1 − ησ t)] h [N(ηx1 − ησ t) − S  a+b  a−a1 √ N(φz) + HS N(ηz − 2ηa1 σ t)] [6] = Rh [ HS where the binary variables η and φ take the value 1 and −1. It is convenient to note that there are parity relations between options as in the standard case without information costs. The same tables as before provide the appropriate formulas. Tables 30–38 provide some simulation results and the Greek letters for some options. Table 30:

Down-and-in call.

S

Price

Delta

Gamma

Vega

Theta

96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

0.50904 0.46021 0.41602 0.37601 0.33978 0.30699 0.27730 0.25042 0.22610

−0.04892 −0.04426 −0.04005 −0.03624 −0.03279 −0.02967 −0.02685 −0.02429 −0.02197

0.00465 0.00421 0.00381 0.00345 0.00312 0.00283 0.00256 0.00232 0.00210

0.08303 0.07737 0.07207 0.06710 0.06245 0.05810 0.05403 0.05022 0.04666

0.00213 0.00197 0.00183 0.00169 0.00156 0.00145 0.00134 0.00124 0.00115

S = 100, t = 05/01/2003, T = 05/01/2004, r = 3%, σ = 20%, barrier = 80, rebate = 10.

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Table 31:

Simulation results for down-and-in put options.

S

Price

Delta

Gamma

Vega

Theta

96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

6.81652 6.30271 5.82014 5.36775 4.94446 4.54909 4.18045 3.83730 3.51841

−0.51444 −0.48304 −0.45344 −0.42344 −0.39537 −0.36851 −0.34290 −0.31854 −0.29544

0.03140 0.03036 0.02924 0.02807 0.02686 0.02561 0.02436 0.02310 0.02185

0.57295 0.56929 0.56326 0.55507 0.54494 0.53308 0.51973 0.50509 0.48937

0.01153 0.01163 0.01167 0.01164 0.01155 0.01141 0.01122 0.01100 0.01073

S = 100, t = 05/01/2003, T = 05/01/2004, r = 3%, σ = 20%, barrier = 80, rebate = 10.

Table 32:

Simulation results for down-and-out call options.

S

Price

Delta

Gamma

Vega

Theta

96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

10.42997 10.72166 11.04693 11.40527 11.79601 12.21842 12.67164 13.15476 13.66680

0.29101 0.32478 0.35801 0.39059 0.42241 0.45337 0.48341 0.51247 0.54050

0.03377 0.03323 0.03258 0.03182 0.03097 0.03004 0.02906 0.02803 0.02697

0.60069 0.60860 0.61358 0.61583 0.61552 0.61287 0.60807 0.60132 0.59281

0.01851 0.01893 0.01927 0.01954 0.01974 0.01987 0.01995 0.01997 0.01993

S = 100, t = 05/01/2003, T = 05/01/2004, r = 3%, σ = 20%, barrier = 80, rebate = 10.

Table 33: Simulation results for down-and-out put options. S

Price

Delta

Gamma

Vega

Theta

96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

2.08868 2.07981 2.06333 2.03982 2.00990 1.97419 1.93333 1.88792 1.83857

−0.00871 −0.01637 −0.02344 −0.02989 −0.03570 −0.04089 −0.04545 −0.04940 −0.05276

−0.00766 −0.00707 −0.00645 −0.00581 −0.00519 −0.00456 −0.00395 −0.00336 −0.00279

−0.13568 −0.12952 −0.12228 −0.11414 −0.10526 −0.09581 −0.08594 −0.07580 −0.06551

−0.00434 −0.00421 −0.00404 −0.00384 −0.00362 −0.00337 −0.00312 −0.00285 −0.00258

S = 100, t = 05/01/2003, T = 05/01/2004, r = 3% , σ = 20%, barrier = 80, rebate = 10.

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Table 34: Simulation results for up-and-in call options. S

Price

Delta

Gamma

Vega

Theta

96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

8.73359 9.48673 10.27542 11.09904 11.95681 12.84781 13.77101 14.72527 15.70931

0.75242 0.78817 0.82327 0.85760 0.89100 0.92336 0.95455 0.98448 1.01304

0.03574 0.03511 0.03432 0.03340 0.03236 0.03119 0.02993 0.02856 0.02712

0.69351 0.70087 0.70530 0.70675 0.70522 0.70074 0.69335 0.68313 0.67020

0.02331 0.02373 0.02406 0.02430 0.02445 0.02451 0.02447 0.02435 0.02414

S = 100, K = 100, t = 05/01/2003, T = 05/01/2004, r = 3%, σ = 20%, barrier = 120, rebate = 10.

Table 35: Simulation results for up-and-in put options. S

Price

Delta

Gamma

Vega

Theta

96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

0.41315 0.45617 0.50241 0.55200 0.60509 0.66179 0.72224 0.78655 0.85484

0.04295 0.04619 0.04956 0.05307 0.05670 0.06046 0.06435 0.06836 0.07248

0.00324 0.00337 0.00350 0.00363 0.00376 0.00389 0.00401 0.00412 0.00424

0.06655 0.07151 0.07671 0.08213 0.08778 0.09366 0.09977 0.10611 0.11268

0.00185 0.00198 0.00211 0.00224 0.00238 0.00252 0.00267 0.00281 0.00296

S = 100, K = 100, t = 05/01/2003, T = 05/01/2004, r = 3%, σ = 20%, barrier = 120, rebate = 10.

Table 36: Simulation results for up-and-out call options. S

Price

Delta

Gamma

Vega

Theta

96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

3.93234 4.16560 4.40308 4.64439 4.88914 5.13698 5.38750 5.64034 5.89515

0.23317 0.23742 0.24127 0.24474 0.24783 0.25054 0.25286 0.25483 0.25645

0.00425 0.00385 0.00347 0.00309 0.00270 0.00233 0.00197 0.00162 0.00129

0.09258 0.08793 0.08310 0.07812 0.07306 0.06796 0.06285 0.05779 0.05281

0.00384 0.00372 0.00359 0.00345 0.00330 0.00315 0.00299 0.00283 0.00266

S = 100, K = 100, t = 05/01/2003, T = 05/01/2004, r = 3%, σ = 20%, barrier = 120, rebate = 10.

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Table 37:

297

Simulation results for up-and-out put options.

S

Price

Delta

Gamma

Vega

Theta

96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

8.36234 7.89395 7.44469 7.01403 6.60140 6.20615 5.82763 5.46511 5.11786

−0.46878 −0.44954 −0.43084 −0.41273 −0.39524 −0.37844 −0.36236 −0.34703 −0.33246

0.01924 0.01870 0.01811 0.01748 0.01680 0.01608 0.01533 0.01457 0.01378

0.35683 0.35442 0.35055 0.34525 0.33852 0.33042 0.32097 0.31025 0.29829

0.00546 0.00558 0.00565 0.00568 0.00568 0.00563 0.00555 0.00544 0.00528

S = 100, K = 100, t = 05/01/2003, T = 05/01/2004, r = 3%, σ = 20%, barrier = 120, rebate = 10.

Table 38: Simulation results for up-and-in call options. Barrier

Price

Delta

Gamma

Vega

Theta

100.00 101.00 102.00 103.00 104.00 105.00 110.00 120.00 130.00 140.00

0.00000 8.94280 8.94258 8.94160 8.93905 8.93388 8.82450 7.80831 5.82548 3.74254

0.00000 0.57957 0.57969 0.57998 0.58052 0.58130 0.58924 0.59429 0.52437 0.39296

0.00000 −0.57957 0.02527 0.02528 0.02530 0.02535 0.02605 0.03115 0.03587 0.03406

0.00000 0.39207 0.39210 0.39225 0.39262 0.39336 0.40814 0.51318 0.61734 0.60027

0.00000 −0.01068 −0.01068 −0.01068 −0.01069 −0.01071 −0.01112 −0.01401 −0.01683 −0.01631

S = 100, K = 100, t = 17/12/2003, T = 18/12/2004, r = 2%, σ = 20%.

6. Outside Barrier Options: An Alternative Approach In classic barrier options, an “inside” barrier is defined with respect to the underlying asset price. Some other barrier options are defined with respect to a second variable, an “outside” barrier which determines whether the option is knocked in or out.

6.1. Valuation of Outside Barrier Options Using the nature of the barrier, ‘in’ or ‘out’, and its location with respect to the underlying asset ‘up’ or ‘down’, as with inside barrier options, it is 08:36:21.

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possible to structure four different outside barrier call options and four different outside barrier put options. Recall that an ‘in’ option plus an ‘out’ option with the same strike price, barrier and maturity date have the same pay-off as an otherwise identical standard option. So there is no use in valuing the eight options separately. In fact, it is sufficient to value the ‘in’ or the ‘out’ options to get the prices of the others by the stated relationship. Hence, the value of a down-and-in call is given by the difference between the price of a standard call and that of the corresponding down-andout call. The valuation of options with an outside barrier in a Black–Scholes economy is analyzed in Heynen and Kat (1994). They consider these options as options on two underlying assets S and V where S defines the actual option pay-off, called the ‘pay-off variable’, and V stands for the ‘barrier variable’, defining whether the option is knocked ‘in’ or ‘out’. Following Heynen and Kat (1994), we assume the following dynamics for the two underlying assets: d ln( SSt ) = µ1 dt+σ1 dW1 , d ln( VV0t ) = µ2 dt+ σ2 dW2 . We denote by    √ ln(S0 /K) + µ1 + 21 σ12 T , d2 = d1 − σ1 T d1 = √ σ1 T 2ρ 2ρ d1 = d1 + √ ln(H/V0 ), d2 = d2 + √ ln(H/V0 ) σ1 T σ1 T √ 1 e1 = √ [ln(H/V0 ) − (µ2 + ρσ1 σ2 )T ], e2 = e1 + ρσ1 T σ2 T 2 2 e1 = e1 − √ ln(H/V0 ), e2 = e2 + √ ln(H/V0 ). σ2 T σ2 T We denote by τHV the first passage of time for V through the barrier H and by p ↑ (u, τHV > T), the joint density of ln(ST /S0 ) and V not hitting the barrier H from above (below p ↓ (u, τHV > T)) during the option’s life. It can be shown that p ↓ (u < x, τHV > T)   2µ2 ln(H/V0 ) x − µ1 T ln(H/V0 ) − µ2 T 2 , φρ − e σ2 =N √ ,φ √ σ T σ2 T  1 ln(H/V0 ) + µ2 T x − µ1 T − 2ρ ln(H/V0 ) , −φ , φρ . ×N √ √ σ1 T σ2 T

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The option pricing formulas can be obtained by taking the discounted expected pay-off under the appropriate probability measure. Hence, the value of a down-and-out outside call is given by  ∞ −rT cDOO = e (S0 eu − K)p ↓ (u, τHV > T)du. ln(K/S0)

The value of a down-and-out outside put is  ln(K/S0) −rT (K − S0 eu )p ↓ (u, τHV > T)du. pDOO = e −∞

The value of an up-and-out outside call is given by the following integral  ∞ −rT (S0 eu − K)p ↑ (u, τHV > T)du. cUOO = e ln(K/S0)

The value of an up-and-out outside put is given by  ln(K/S0) −rT (K − S0 eu )p ↑ (u, τHV > T)du. pUOO = e −∞

Using some transformations and tedious calculations, it can be shown that the values of down-and-out and up-and-out calls and puts are given by:

µ +ρσ 2 ηS0 N(ηd1 , φe1 , −ηφρ) − e − ηKe

−rT

2

2

σ22

2

ln(H/V0 )

 N(ηd2 , φe2 , −ηφρ) − e

N(ηd1 , φe1 , −ηφρ)

2

µ2 σ22

ln(H/V0 )



N(ηd2 , φe2 , −ηφρ)

where η takes the value 1 for the call and −1 for the put and φ takes the value 1 for an up-and-out option and −1 for a down-and-out option. The correlation coefficient between the two processes is very important in this formula. When ρ = 0, the formula reduces to a standard Black– Scholes call which multiplies a probability factor. For in-options, the probability corresponds to that of knocking the option in. For out-options, the probability corresponds to that of not knocking the option out. When ρ = 1, the formula reduces to that of an inside barrier option. When the pay-off and the barrier variables are negatively correlated, there is a higher probability that a down-and-out call is knocked-out as the pay-off variable moves in-the-money area. A higher correlation coefficient yields a higher option price. For a down-and-in call the higher the correlation coefficient, the lower the option price.

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6.2. Outside Barrier Options: An Alternative Approach in the Presence of Shadow Costs We assume the following dynamics: d ln( SSt ) = µ1 dt + σ1 dW1 , d ln( VV0t ) = µ2 dt + σ2 dW2 . We denote by √ [ln(S0 /K) + (µ1 + 21 σ12 )T ] , d2 = d1 − σ1 T √ σ1 T 2ρ 2ρ d1 = d1 + √ ln(H/V0 ), d2 = d2 + √ ln(H/V0 ) σ1 T σ1 T √ 1 e1 = √ [ln(H/V0 ) − (µ2 + ρσ1 σ2 )T ], e2 = e1 + ρσ1 T σ2 T 2 2 e1 = e1 − √ ln(H/V0 ), e2 = e2 + √ ln(H/V0 ). σ2 T σ2 T

d1 =

We also denote by τHV the first passage of time for V through the barrier H and by p|(u, τHV > T), the joint density of ln(ST /S0 ) and V not hitting the barrier H from above (below p|(u, τHV > T)) during the option’s life. It can be shown that   x − µ1 T ln(H/V0 ) − µ2 T V p|(ux, τH T) = N , φρ √ ,φ √ σ2 T σ1 T   2µ2 ln(H/V0 ) ln(H/V0 ) + µ2 T x − µ1 T − 2ρ ln(H/V0 ) σ22 , −φ , φρ . −e N √ √ σ1 T σ2 T The value of a down-and-out outside call is  ∞ −(r+λ)T cDOO = e (S0 eu − K)p ↓ (u, τHV > T )du. ln(K/S0)

The value of a down-and-out outside put is  ln(K/S0) −(r+λ)T (K − S0 eu )p ↓ (u, τHV > T )du. pDOO = e −∞

The value of an up-and-out outside call is given by the following integral:  ∞ −(r+λ)T (S0 eu − K)p ↑ (u, τHV > T )du. cUOO = e ln(K/S0)

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The value of an up-and-out outside put is given by  ln(K/S0) −(r+λ)T (K − S0 eu )p ↑ (u, τHV > T )du. pUOO = e −∞

The values of down-and-out and up-and-out calls and puts are: µ +ρσ 2 ηS0 N(ηd1 , φe1 , −ηφρ) − e − ηKe

−rT

2

2

σ22

2

ln(H/V0 )

 N(ηd2 , φe2 , −ηφρ) − e

N(ηd1 , φe1 , −ηφρ)

2

µ2 σ22

ln(H/V0 )



N(ηd2 , φe2 , −ηφρ)

where η takes the value 1 for the call and −1 for the put and φ takes the value 1 for an up-and-out option and −1 for a down-and-out option.

7. Analysis and Valuation of Standard and Exotic Options Using Binary Options The pay-off of a binary option is given by an amount A if the binary condition 1(ST >K) is satisfied. Using the standard Black–Scholes approach, the value of a binary call is given by   ln(S0 /K) µ √ −T −T + CBI = r AE[1(ST >K) ] = r AN T √ σ σ T with µ = ln(R) − 21 σ 2 . We show how to represent and value standard options, down-and-in options, switch options, corridors and knock-out range options in terms of elementary digital options.

7.1. Valuation of Soft Binary Options 7.1.1. Example 1: Standard Options Consider a sequence of strike prices Ki increasing by an amount x, i.e. K0 = K,

K1 = K + x,

K2 = K + 2x, . . . , Ki = K + ix.

If we take this sequence, the amount x and add the corresponding binary conditions, the following pay-off is obtained for an infinite number

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 of strike prices: ∞ i=0 x1(ST >K+ix) . When x tends to zero, i.e. in the limit, the above condition can be rewritten as  1(ST >x) dx. x>K

Note that the value of this integral is (ST −K)1(ST >K) where the indicator variable takes the value 1 if ST > K and 0 if ST < K. This condition is simply the pay-off of a standard call. If we take several multiplicative and additive combinations of binary options, then it is possible to represent the pay-offs of exotic options in terms of digital options.

7.1.2. Example 2: Down-and-Out Call Options The down-and-out call with a strike price K and a knock-out barrier H can be represented by the condition: [ST − K]1(ST >K) 1(Inf{τ,0≤τ≤T,Sτ ≥H}) .  The condition 1(Inf{τ,0≤τ≤T,Sτ ≥H}) is equivalent to t∈[0,T ]∩Q 1(ST ≥H) where Q is the set of all rational numbers. This representation allows one to obtain the prices of down-and-out call options.

7.1.3. Example 3: Switch Options By contrast to the preceding examples where time is fixed and summation is done over K, now the strike price is fixed and summation is done over time. When summation is done over K, we refer to the derivative assets as “quantitative options”. When summation is done over time, we refer to the derivative assets as “qualitative options”. This terminology is introduced in Pechil (1995). If we fix the strike price and use an increasing number of time units, τi as summation index, then the choice of a pay-out amount Aτ, and an increasing sequence τ0 = 0, τ1 = τ, . . . , τi = iτ with nnτ ≤ T < (n + 1)τ gives the following discrete time condition A i=0 1(Siτ>K ) τ. When τ tends to zero, i.e. in the  T limit, the above condition can be rewritten in a continuous time as A τ=0 1(Siτ >K) dτ. In order to value the switch option in a Black–Scholes economy, suppose that a certain time T0 elapsed, and the option has paid at least an amount Aτ0 with τ0 < T0 . The

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value of the discrete switch option is given by

 −(T −T0 ) Cdso = r E[1(Siτ >K) ]τ . Aτ0 + A iτ∈[T0 ,T ]

 This gives the option value Cdso = r −(T −T0 ) [Aτ0 + A iτ∈[T0 ,T ] ]

ln SKT 0 µ N + [iτ − T0 ]τ . √ σ σ iτ − T0 The value of the continuous switch option is given by    T −(T −T0 ) Aτ0 + AE 1Sτ >K dτ . Ccso = r τ=T0

Using Fubini’s theorem for the calculation of the term under expectation, we have 

 T T ln SKT 0 µ E 1(Sτ >K) dτ = N + τ − T0 dτ . √ σ σ τ − T0 T0 τ=T0 The value of the switch option is given in Pechil (1995). When Sτ0 ≤ K, the option value is Ccso = r −(T −T0 ) Aτ0 + r −(T −T0 ) A(T − T0 )N(d1 ) σ σ2 − r −(T −T0 ) AK∗ N(d1 ) − r −(T −T0 ) A 2 N(d1 ) µ 2µ σ 2 2K∗ µ e σ N(d2 ) 2µ2 √   1 2 −(T −T0 ) σ T − T0 +r A e−d1 /2 A. √ µ 2π When Sτ0 > K, the option value is + r −(T −T0 ) A

Ccso = r −(T −T0 ) Aτ0 + e−(r+λ)(T −T0 ) A(T − T0 )N(d1 ) σ2 σ − r −(T −T0 ) AK∗ N(d1 ) + r −(T −T0 ) A 2 N(−d1 ) µ 2µ σ 2 2K∗ µ e σ N(−d2 ) 2µ2 √   1 2 −(T −T0 ) σ T − T0 +r A e−d1 /2 A √ µ 2π − r −(T −T0 ) A

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where

µ K∗ T − T0 − √ , σ T − T0 K∗ µ d2 = − T − T0 − √ , σ T − T0 d1 =

  1 K . K = ln σ ST0 ∗

Note that this formula is useful for the valuation of corridor options.

7.1.4. Example 4: Corridor Options Recall that a corridor option gives the right to its holder to receive a set amount for each day the underlying asset trades within a given range. The pay-off of a corridor is determined by the product of an amount A by the time units τ such that Siτ lies in the range [K1 , K2 ]. It can be shown that the pay-off of a corridor is given by the difference between two switch options. The following condition presents the pay-off for the corridor: A

n 

1(K1 K, the option value is Ccso = e−(r+λ)(T −T0 ) Aτ0 + e−(r+λ)(T −T0 ) A(T − T0 )N(d1 ) σ − e−(r+λ)(T −T0 ) AK∗ N(d1 ) µ σ2 σ 2 2K∗ µ −(r+λ)(T −T0 ) e σ N(−d2 ) N(−d ) − e A 1 2µ2 2µ2 √   1 2 −(r+λ)(T −T0 ) σ T − T0 +e A e−d1 /2A √ µ 2π

+ e−(r+λ)(T −T0 ) A

where d1 =

µ K∗ T − T0 − √ , σ T − T0   K 1 ∗ K = ln . σ ST0

d2 = −

µ K∗ T − T0 − √ , σ T − T0

8. Continuous Strike and Continuous Barrier Options Continuous barrier options or soft barrier options are barrier options which do not appear or disappear suddenly, since they stay alive to reflect the minimum or maximum level of the underlying asset through time. Options with continuous strike prices and barriers help to overcome some hedging problems that arise in the management of portfolios with in and out options. This is made possible since these options offer a gradual rather than an instantaneous knock-out.

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Example A call with a strike range of 40 to 50 has at the maturity date a similar pay-off as a portfolio of calls for which the strike prices range from 40 to 50. A continuous barrier option is a barrier option for which the barrier extends from 80 to 90 and which has similar characteristics as a portfolio of barrier options for which the barriers range evenly between 80 and 90. A continuous strike option is an option in which the contract stipulations allow a continuous strike price.

8.1. Valuation of the Continuous Strike Option The continuous strike option, CSO can be hedged with a portfolio of European options in a static hedging strategy. It can also be hedged in a dynamic strategy with a position of  units of the underlying asset and borrowing a cash amount. Since the option can be hedged, it can be easily valued in a B–S context using the expected discounted value. We use the following notation: St : σ: r: r∗ :

value of the underlying asset at maturity, volatility of the natural log of the return of the underlying asset, risk-free rate in numeraire currency for the same maturity, risk-free rate in foreign currency or continuous dividend yield for the same maturity, H: barrier for a discrete barrier option, U: upper limit of the barrier range of a soft barrier option, L: lower limit of the barrier range of a soft barrier option. The value of a continuous strike call option is given by: √ 1 ∗ 2 ∗ CSO = [S 2 e(r−2r +σ t) N(x + σ t) − 2KSe−r t N(x) 2 √ + K2 e−rtN(x − σ t)] [ln(

S0

)+(r−r ∗ + 1 σ 2 )t]

K 2 √ with x = . σ t Since the option pay-off involves S 2 , the above formula can be used to value and replicate power options. The CSO does not have an upper limit on the strike price.

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8.2. Continuous Strike Option with Shadow Costs The value of a continuous strike call option is given by: √ 1 2 ∗ ∗ CSO = [S 2 e(r−2r +σ t) N(x + σ t) − 2KSe−r t N(x) 2 √ + K2 e−rt N(x − σ t)] with x =

[ln( K0 )+(r+λS −r ∗ + 21 σ 2 )t] √ . σ t S

8.3. Continuous Strike Range Options The continuous strike range option, CSRO is an option for which there is a lower and an upper limit on the strike price. The pay-off of a continuous strike range option is similar to that of a portfolio with a long position in a CSO with a strike price KL and a short position in a CSO with strike price KU . The pay-off of a continuous strike range option is CSRO =

1 1 max[St − KL , 0]2 − max[St − KU , 0]2 . 2 2

The pay-off to this option can be easily replicated by one of the following strategies: — buy a CSO with a strike price KL and sell a CSO with a strike price KU . — an equally weighted portfolio of European calls with strike prices varying from KL to KU .

8.4. Soft Barrier Options A barrier option can be transformed to a continuous barrier option by keeping a single strike price and allowing for proportional knock-in and knockout of the barrier. This allows the option to get gradually in or out. Example When K > H and the initial underlying asset price is 100, we can define a barrier range from 75 to 90 to a down-and-out call. If the lowest underlying asset price attained has been 90 or above, the option is still alive. If the lowest underlying asset price attained has been 75 or less, the option is worthless.

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If the lowest underlying asset price attained has been 87, then 20%, (90 − 87)/(90 − 75) of the option disappears. The value of a barrier range option can be easily obtained by integrating the formula given in Rubinstein and Reiner (1991) between the upper and lower limit of the range option. Recall that the value of a down-and-in call with K > H, in the absence of rebate, is given by √ ∗ CDI (K > H) = Se−r t (H/S)2νN(y) − Ke−rt (H/S)(2ν−2)N(y − σ t) with:

     2 1 1 2 (r − r ∗ ) 1 H y = √ ln + . +ν+ σ t , ν= SK 2 σ2 2 σ t The value of the soft barrier range down-and-in call option for (K > H) is given by  U 1 CDI (K > H)dH CSBRDI = (U − L) L or   (ν+0.5) 1 −r ∗ t −2ν (SK) Se S A1 CSBRDI = (U − L) 2(ν + 0.5) 

A1 =

λ+ 21

2

(U ) SK



N(y1 (U)) − µs N(y2 (U))



λ+ 21

(L2 ) − SK and



N(y1 (L)) − µs N(y2 (L)) 



ν− 21



(SK)  CSBRDI = Ke−rt S −2(ν−1)   2 λ + 21 with

 A2 = 

 2

λ− 21



(U ) SK

N(y3 (U)) − µK N(y4 (U))



λ− 21

(L2 ) − SK

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 N(y3 (L)) − µK N(y4 (L))

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with

√ 1 y1 (H ) = √ ln(H 2 /SK) + λσ t σ t   √ 1 y2 (H ) = y1 (H ) − λ + σ t 2 √ 1 y3 (H ) = √ ln(H 2 /SK) + (λ − 1)σ t σ t   √ 1 y4 (H ) = y3 (H ) − ν − σ t 2

and µS = e

    − 21 σ 2 t λ− 21 λ+ 21

,

µK = e

    − 21 σ 2 t λ− 21 λ+1+ 21

.

Using the parity relationship, the difference between this formula and the European Black–Scholes call gives the European down-and-out barrier range call formula.

8.5. Soft Barrier Options with Shadow Costs The value of a down-and-in call with K > H, in the absence of rebate is  2ν (λS −λ−r ∗ )t H CDI (K > H ) = Se N(y) S  (2ν−2) √ −(r+λ)t H N(y − σ t) − Ke S with     2  1 2 (r + λS − r ∗ ) 1 H 1 + ν+ σ t , ν= + . y = √ ln SK 2 σ2 2 σ t The value of the soft barrier range down-and-in call option for (K > H) is 

 (SK) ν + 21 1 (λS −λ−r ∗ )t −2ν  A1 S Se CSBRDI = (U − L) 2 ν + 21 

ν+ 21

(U 2 ) A1 = SK −

2



N(y1 (U)) − µS N(y2 (U))

ν+ 21

(L ) SK

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N(y1 (L)) − µS N(y2 (L))

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and the value of:

 −rt −2(ν−1)

CSBRDI = Ke S with

 A2 = 

 2

ν− 21

N(y3 (U)) − µK N(y4 (U))



−

 (SK) ν − 21  A2 2 ν + 21



(U ) SK

2

311

ν− 21





(L ) SK

N(y3 (L)) − µK N(y4 (L))

with

   2 √ √ 1 1 H + νσ t, y2 (H) = y1 (H) − ν + σ t y1 (H) = √ ln SK 2 σ t    2 √ √ 1 1 H + (ν − 1)σ t, y4 (H) = y3 (H) − ν − σ t y3 (H) = √ ln SK 2 σ t and µS = e

    − 21 σ 2 t ν− 21 ν+ 21

,

µK = e

    − 21 σ 2 t ν− 21 ν+1+ 21

.

Using the parity relationship, the difference between this formula and the European Black–Scholes call gives the European down-and-out barrier range call formula. Table 39: Single barrier FX-up-and-out call. S

Price

Delta

Gamma

Vega

Theta

0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04

0.04500 0.04799 0.05102 0.05408 0.05715 0.06022 0.06328 0.06631 0.06929

0.29878 0.30298 0.30586 0.30734 0.30734 0.30584 0.30277 0.29813 0.29189

−0.29878 −0.30298 −0.30586 −0.30734 −0.30734 −0.30584 −0.30277 −0.29813 −0.29189

0.00097 0.00076 0.00052 0.00026 −0.00002 −0.00032 −0.00065 −0.00099 −0.00135

0.04500 0.04799 0.05102 0.05408 0.05715 0.06022 0.06328 0.06631 0.06929

S = 1, K = 1, t = 07/02/2003, T = 07/02/2004, r = 3%, r ∗ = 4%, σ = 20%, barrier = 1.5. 08:36:21.

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Table 40:

Single barrier FX-up-and-out put.

S

Price

Delta

Gamma

Vega

Theta

0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04

0.10195 0.09665 0.09156 0.08666 0.08195 0.07743 0.07311 0.06896 0.06500

−0.52961 −0.50989 −0.49029 −0.47086 −0.45167 −0.43274 −0.41412 −0.39582 −0.37789

0.52961 0.50989 0.49029 0.47086 0.45167 0.43274 0.41412 0.39582 0.37789

0.00364 0.00370 0.00376 0.00380 0.00383 0.00386 0.00387 0.00387 0.00387

0.10195 0.09665 0.09156 0.08666 0.08195 0.07743 0.07311 0.06896 0.06500

S = 1, K = 1, t = 07/02/2003, T = 07/02/2004, r = 3%, r∗ = 4%, σ = 20%, barrier = 1.5.

Table 41:

Single barrier FX-up-and-in call.

S

Price

Delta

Gamma

Vega

Theta

0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04

0.00884 0.01016 0.01165 0.01329 0.01512 0.01714 0.01936 0.02179 0.02446

0.13231 0.14783 0.16455 0.18251 0.20170 0.22214 0.24384 0.26679 0.29097

−0.13231 −0.14783 −0.16455 −0.18251 −0.20170 −0.22214 −0.24384 −0.26679 −0.29097

0.00267 0.00295 0.00324 0.00354 0.00385 0.00418 0.00452 0.00487 0.00522

0.00884 0.01016 0.01165 0.01329 0.01512 0.01714 0.01936 0.02179 0.02446

S = 1, K = 1, t = 07/02/2003, T = 07/02/2004, r = 3%, r∗ = 4%, σ = 20%, barrier = 1.5.

Table 42:

Single barrier FX-up-and-in put.

S

Price

Delta

Gamma

Vega

Theta

0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.00001 0.00002 0.00002 0.00002 0.00003 0.00004 0.00004 0.00005 0.00006

−0.00001 −0.00002 −0.00002 −0.00002 −0.00003 −0.00004 −0.00004 −0.00005 −0.00006

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

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Table 43: Single barrier FX-down-and-out call. S

Price

Delta

Gamma

Vega

Theta

0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04

0.05384 0.05815 0.06266 0.06737 0.07227 0.07736 0.08264 0.08810 0.09375

0.43109 0.45081 0.47042 0.48984 0.50904 0.52798 0.54661 0.56492 0.58286

0.52931 0.50961 0.49003 0.47063 0.45145 0.43253 0.41391 0.39562 0.37769

0.00364 0.00370 0.00376 0.00380 0.00383 0.00386 0.00387 0.00388 0.00387

0.05384 0.05815 0.06266 0.06737 0.07227 0.07736 0.08264 0.08810 0.09375

S = 1, K = 1, t = 07/02/2003, T = 07/02/2004, r = 3%, r∗ = 4%, σ = 20%, barrier = 0.5.

Table 44:

Single barrier FX-down-and-in put.

S

Price

Delta

Gamma

Vega

Theta

0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04

0.00088 0.00074 0.00062 0.00052 0.00044 0.00036 0.00030 0.00025 0.00021

−0.01429 −0.01202 −0.01010 −0.00848 −0.00711 −0.00596 −0.00499 −0.00418 −0.00350

0.01429 0.01202 0.01010 0.00848 0.00711 0.00596 0.00499 0.00418 0.00350

0.00051 0.00044 0.00038 0.00033 0.00029 0.00025 0.00021 0.00018 0.00016

0.00088 0.00074 0.00062 0.00052 0.00044 0.00036 0.00030 0.00025 0.00021

S = 1, K = 1, t = 07/02/2003, T = 07/02/2004, r = 3%, r∗ = 4%, σ = 20%, barrier = 0.5.

Table 45: Single barrier FX-down-and-out put. S

Price

Delta

Gamma

Vega

Theta

0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04

0.10106 0.09591 0.09094 0.08614 0.08152 0.07707 0.07280 0.06871 0.06479

−0.51531 −0.49786 −0.48017 −0.46236 −0.44453 −0.42675 −0.40908 −0.39159 −0.37433

0.51503 0.49760 0.47993 0.46215 0.44434 0.42657 0.40892 0.39144 0.37420

0.00313 0.00326 0.00337 0.00347 0.00355 0.00361 0.00366 0.00369 0.00372

0.10106 0.09591 0.09094 0.08614 0.08152 0.07707 0.07280 0.06871 0.06479

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Summary An all-or-nothing call (put) is an option giving the right to its holder to receive a pre-determined amount, “the all”, if the underlying asset goes above (below) the strike price. How far is the underlying asset price above or below the strike price is not important since the pay-off will be “all” or “nothing” at the maturity date. A one touch all-or-nothing call (put) is an option giving the right to its holder to receive a pre-determined amount, “the all”, if the underlying asset goes above (below) the strike price at any time during the option’s life. Barrier options are options that either cease to exist or come into existence when some pre-specified asset price barrier is hit during the option’s life. A down-and-out call is “knocked out” when the asset price falls to some pre-specified level before the option maturity. Rubinstein and Reiner (1991) provided valuation equations for a whole family of barrier options. Barrier options have pay-offs that depend on two market levels: the strike and the barrier. They are used by several investors who want to gain exposure to future market scenarios. Barrier options are a modified version of standard call and put options. They can be characterized by a barrier level and a strike level. The pay-off of these options depend upon whether or not the underlying asset ever crosses a barrier level during the option’s life. These options account sometimes for a cash rebate associated with crossing the specified barrier. The rebate is paid in general to the holder as a sort of consolation prize when the option is “killed”. Two types of barrier options are studied: in options and out options. A knockin option or an in barrier option gives a certain pay-off only if the underlying asset ends in the money and if the barrier is crossed before the maturity date. The in barrier is knocked in when the underlying assset crosses the barrier. In this case, it becomes a standard option of the same type. The option is worthless when the stock does not cross the barrier. A knockout option or an out barrier option gives a certain pay-off only if the underlying asset ends in the money and if the barrier is never crossed before the maturity date. As long as the underlying asset never crosses the barrier, the out barrier option remains a standard option of the same type. The option is worthless (knocked out) when the stock crosses the barrier. A binary up-and-in call has a single barrier and no strike price. This options pays the buyer a one-time fixed amount, the first time the underlying

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asset reaches the barrier from below. If the underlying asset has not crossed the barrier, the option is worthless. A binary up-and-in put has a single barrier and no strike price. This options pays the buyer a one-time fixed amount, the first time the underlying asset reaches the barrier from above. If the underlying asset has not crossed the barrier, the option is worthless at maturity. A capped European-style call presents a strike price and a cap barrier above the strike price. The pay-off of a capped European-style call is similar to that of a standard call with the same characteristics, with a main difference: if the underlying asset ever reaches the barrier from below, the buyer receives immediately a cash amount equal to the difference between the barrier level and the strike price. This capped call is termed non-deferred because the cash is received instantaneously. Otherwise, the cash is received at maturity and the option is said a deferred capped call. A floored European-style put presents a strike price and a floor barrier below the strike price. The pay-off of a floored European-style put is similar to that of a standard put with the same characteristics, with a main difference: if the underlying asset ever reaches the barrier from above, the buyer receives immediately a cash amount equal to the difference between the strike price and the barrier level. This floored put is termed non-deferred because the cash is received instantaneously. Otherwise, the cash is received at maturity and the option is said to be a deferred floored put. Options with continuous strike prices and barriers help to overcome some hedging problems that arise in the management of portfolios with in and out options. This is made possible since these options offer a gradual rather than an instantaneous knock-out. This chapter analyzes simple and complex digital options, digital range options and barrier options. It turns out that binary options are useful in the representation of the pay-offs of simple and complex options. Using the representation in Pechil, the chapter analyzes and provides simple valuation formulas for down-and-out options, switch options, corridors and knock-out range options. Second, analytic formulas and barriers are presented. Also, a framework for the analysis and valuation of continuous strike options, continuous strike range options and soft barrier options is presented.

Questions 1. What are the definitions of standard binary options? 2. What are the definitions of complex binaries and range options?

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3. What are the specific features of path independent binary options? 4. Define standard cash-or-nothing options, standard asset-or-nothing options, standard gap options and supershares. 5. What are the definitions of path-dependent binary barrier options? 6. What are the specific features of soft binary options? 7. What are the specific features of continuous strike option, continuous strike range options and soft barrier options?

Bibliography Bellalah, M (2001). Market imperfections; information costs and the valuation of derivatives: some general results. International Journal of Finance, 13(3), 1895–1927. Bellalah, M, Ma Bellalah and R Portait (2001). The cost of capital in international finance. International Journal of Finance, 13(3), 1958–1973. Bellalah, M and JL Prigent (2001). Pricing standard and exotic options in the presence of a finite mixture of Gaussian distributions. International Journal of Finance, 13(3), 1975–2000. Bellalah, M, JL Prigent and C Villa (2001). Skew without skewness: asymmetric smiles; information costs and stochastic volatilities. International Journal of Finance, 13(2), 1826–1837. Bellalah, M and F Selmi (2001). On the quadratic criteria for hedging under transaction costs. International Journal of Finance, 13(3), 2001–2020. Derman, E and I Kani (1993). The ins and outs of barrier options, Goldman Sachs. Quantitative Strategies Research Notes. Heynen, R and H Kat (1994). Crossing barriers. Risk, 7(6), 45–58. Pechil, A (1995). Classified information. Risk, 8(6), June, 59–61. Rubinstein, M and E Reiner (1991a). Breaking down the barriers. Risk, 4(8), 28–35. Rubinstein, M and E Reiner (1991b). Unscrambling the binary code. Risk, 4(8), 75–83.

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Chapter 8

Lookback Options, Double Lookback Options and Their Applications

This chapter is organized as follows: 1. Section 2 provides some definitions and examples of lookback options. 2. Section 3 develops valuation formulas for European lookback options in a Black–Scholes context. In particular, standard lookbacks, extreme options, limited risk options and partial lookback options are valued. Also, simulation results are provided. 3. Section 4 provides valuation formulas for lookback options with shadow costs of incomplete information. 4. Section 5 presents valuation formulas for an exotic timing option with and without the presence of information costs. 5. Section 6 develops a general context for the valuation of double lookback options, lookbacks on two assets and semilookbacks on two assets. The analysis is extended to account for the effects of information costs.

1. Introduction ome financial contracts allow the holder to receive, at the maturity date, a pay-off depending on the maximum or the minimum of the realized values of the underlying asset during the life of the contract. The underlying asset may be a spot asset, a forward or a futures contract, commodities, indices and so on. These options are negotiated either in organized markets or in the OTC markets and may be embedded in some contracts issued by financial institutions and firms. Hence, a lookback call is defined as an option whose pay-off strike price corresponds to the minimum price recorded by the underlying asset during the option’s life. A lookback put is defined as

S

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an option whose strike price corresponds to the maximum price recorded by the underlying asset during the option’s life. These options are interesting but more expensive than standard options. Therefore, partial lookback options are designed to reduce the costs attached to “full” lookbacks while preseving their main characteristics. Goldman, Sosin and Gatto (1979) and Goldman, Sosin and Shepp (1979) analyzed and valued standard lookback options which entitle the holder to buy the underlying asset at its realized minimum value during a certain period. Harrison and Kreps (1979) and Harrison and Pliska (1981) showed that in complete markets, it is possible to implement a self-financing strategy to duplicate the pay-off of an option using the underlying asset and riskless discount bonds. The current option value is given by its expected pay-off discounted to the present under the appropriate probability measure. Hence, these options can be priced in a Black–Scholes economy using martingale techniques. They can also be valued using lattice approaches and numerical techniques. Whatever approach is used, the specificities of these options are reflected in their terminal and intermediate pay-offs. The exotic timing option gives its holder the difference between the maximum value recorded during the option’s life and an initial value based on underlying asset price at the time of initiation. Although the payoff on the exotic option corresponds to a lookback option, a major advantage is that, in contrast to the lookback option, the holder does not run the risk of zero value when the maximum value is achieved at maturity. The exotic timing option has some appeal to investors, especially those who beleive they have special skills at market timing, because even a small rise in the asset price allows the opportunity for the investor to lock-in a profit. We provide an analytic valuation formula for the pricing of an exotic timing option, which has the characteristics of a lookback and a timing option. The option’s sensitivities are studied and prices are simulated. This option is introduced in the literature by Bellalah and Prigent (1997). The payoffs of double lookback options depend on the maximum and/or the minimum prices of one or two related assets. Double lookback options include calls and puts for which the underlying asset corresponds to the difference between the maximum and the minimum prices of two correlated assets during a specified period. Double lookback options are mainly options for which the payoff is linked to two traded assets. The payoffs depend on the extremal prices of one and/or two assets during a specified period. For example, call and put options are traded on the spread between the maximum and minimum price of Xerox stock over a given specified period. Also, the option to receive the maximum of General

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Motors’ stock price at the maximum of Ford’s stock price is an example of a double lookback option. Another example is given by the option to receive the minimum of IBM’s stock price at the minimum of Digital’s stock price over a given period. These examples of double lookbacks can be defined using the underlying asset price or its return. Double lookback options allow investors to bet on the difference between the extreme values of two underlying assets. Double lookback options may be cheaper than standard lookback options.

2. Analysis of Lookback Options In this section, we describe some lookback options with respect to their terminal and intermediate pay-offs. These options are often structured in order to offer, for a higher initial premium, some or all the potential value of the option during its life.

2.1. A Standard Lookback Option A lookback option gives its holder the right to buy (sell) a fixed amount of an asset at the best price which occurs over the option’s life. Hence, a lookback call (put) involves the right to buy (sell) at the lowest (highest) price.

2.2. The Floating Strike Lookback For a floating strike lookback, the strike price is unknown before the maturity date. For a lookback call, the strike price corresponds to the minimum price realized by the underlying asset during the option’s life. For a lookback put, the strike price corresponds to the maximum realized price by the underlying asset during the same period. Hence, a floating strike lookback call gives the right to the holder to buy at the minimum. A floating strike lookback put gives the right to the holder to sell at the maximum. When the underlying asset price drops and then rises, the call will pay-off from the lowest price realized. Since the strike price drops with the underlying asset value, these options offer a suitable solution to the market entry problem and are more expensive than standard options. When the initial underlying asset price corresponds to the minimum value recorded during the option’s life, then the floating strike lookback call pay-off is equal to that of a standard call.

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2.3. Fixed-Strike Lookbacks For fixed-strike lookbacks, the strike price is known in advance. The call option pay-off is given by the difference between the highest value of the underlying asset during the option’s life and the fixed strike price. The put option pay-off is given by the difference between the fixed strike price and the lowest value of the underlying asset recorded during the option’s life. When the final underlying asset price is the maximum value recorded during the option’s life, then the fixed strike lookback call’s pay-off is equal to that of a standard call.

2.4. Lookback Strategies The value of path-dependent options is a function of the underlying asset, time, the strike price, and a function f(.) specifying the option in question. Most lookback options are defined as calls or puts for which the pay-off can be written in the following form for the call: C = max[0, ST − K, f(.) − K] and the following pay-off for the put: P = max[0, K − ST , K − f(.)] where f(.) is the value of the underlying asset, determined in a way defined in the issue of the financial asset. Hence, a lookback option, is an option for which the function f(.) corresponds to the maximum or the minimum of all values attained by the underlying asset during the option’s life. The maximum or the minimum can be calculated in a continuous time framework as it can be sampled at different times. An average strike lookback call (put) is defined by the function f(.) giving the minimum (maximum) of the underlying asset value over the option’s life. A lookback call strategy is constructed as follows: — Long a synthetic forward contract at spot, i.e. buying a European call and selling a European put, — Long a lookback put. The pay-off of this strategy is the maximum value of S which can be achieved by exercising the lookback put and by paying the strike price K, which corresponds to the synthetic forward contract. A lookback put strategy is constructed as follows:

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— Short a synthetic forward contract at spot, i.e. selling a European call and buying a European put, — Long a lookback call.

3. Analytic Formulas for Lookback Options When the dynamics of the underlying asset are given by the following familiar equation: dSt = rSt dt + σSt dWt . 2

Define: St = S0 e(r−(σ /2))t+σWt . Using the following notations for the maximum and the minimum over the interval [t1 , t2 ], Mtt12 = max{Ss /s ∈ [t1 , t2 ]}, mtt21 = min{Ss /s ∈ [t1 , t2 ]} then 
σ2 t + σWt , Xt = ln(St /S0 ) = r − 2 Yt = ln(M0t /S0 ) = max{Xs /s ∈ [0, t]}, yt = ln(mt0 /S0 ) = min{Xs /s ∈ [0, t]}. Option prices are computed at time 0 and option contracts are assumed to have been initiated at time T0 ≤ 0.

3.1. Standard Lookback Options Since the pay-off of a standard European lookback call at the maturity date is given by ST − mTT0 , its current value is √ C = S0 N(d  ) − e−rT m0T0N(d  − σ T )    − 2r2

2
  σ √ σ 2r   S0 S0  0 N −d  + T − erTN(−d  ) + e−rT 2r σ mT0 with 1 d = √ σ T 

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S0 ln m0T0



 σ2 + rT + T . 2

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Since the pay-off of a standard lookback put is (MT00 − ST ), its current value is √ P = −S0 N(−d  ) + e−rTMT00 N(−d  + σ T )    − 2r2 
2  σ √ σ S0    N d − (2r/σ) T + erTN(d  ) S0 − + e−rT 0 2r MT0 with: 1 d = √ σ T



S0 ln MT00



 σ2 + rT + T . 2

Note that these options correspond to ordinary options as in Black–Scholes formulas plus another term corresponding to the specificities of their pay-offs.

3.2. Options on Extrema On the maximum The pay-off of a call on the maximum at maturity T is (MT00 − K)+ . When K ≥ MT00 , the current call price is √ CM = S0 N(d) − e−rTKN(d − σ T )   

 
2
− 2r2 √ σ 2r σ S 0 + e−rT N d− T + erTN(d) S0 − 2r K σ with


  S0 σ2 1 ln + rT + T . d= √ K 2 σ T

When K < MT00 , the call’s value is

√ CM = e−rT(MT00 − K) + S0 N(d  ) − e−rTMT00N(d  − σ T )    − 2r2
 
2
 σ σ 2r √ S0    S0 − N d − T + erTN(d  ) . + e−rT 0 2r σ MT0

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On the minimum The pay-off of a put on the minimum at the maturity date T is (K − m0T0 )+ . When K < m0T0 , the put’s price is √ p = −S0 N(−d) + e−rTKN(−d + σ T )   

 
− 2r2
2 √ σ 2r S σ 0 + e−rT S0  N −d + T − erTN(−d) . σ 2r K When K ≥ m0T0 , the put’s value is

√ p = e−rT(K − m0T0 ) − S0 N(−d  ) + e−rT m0T0 N(−d  + σ T )     − 2r2

2 σ √ 2r S0 σ   N −d  − T − erTN(−d  ) . S0 − + e−rT 0 2r σ mT0

3.3. Limited Risk Options When m and K are constant, then the pay-off of a limited risk call at T is (ST − K)+ 1(MTT

0

≤m) .

The call’s current value is nil when MT00 > m. When MT00 ≤ m, the call’s current value is √ √ Clr = S0 [N(d) − N(dm )] − e−rTK[N(d − σ T ) − N(dm − σ T )]
(−2r/σ 2 ) 
 √ S0 r√ +m N 2dm − d − 2 T −σ T m σ

 √ 2r − N dm − +σ T σ 

 
 S0 − 2r2 r√ r√ −rT −e K T − N dm − 2 T e σ N 2dm − d − 2 m σ σ with


  S0 σ2 1 ln dm = √ + rT + T . m 2 σ T

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The pay-off of a limited risk put is (K − ST )+ 1(mTT When When

m0T0 m0T0

0

≥m) .

< m, the put is worthless. ≥ m, the put’s current value is:

√ √ p = −S0 [N(−d) − N(−dm )] + e−rTK[N(−d + σ T )−N(−dm + σ T )]
(−2r/σ 2 ) 

  √ S0 2r √ −m T +σ T N −2dm + d + σ m

  √ r√ − N −dm + 2 T +σ T σ
− 2r2 +1 
 S0 σ 2r √ −rT N −2dm + d + T −e K m σ

 2r √ − N −dm + T . σ These options are issued in foreign exchange markets and also in stock index markets.

3.4. Partial Lookback Options The pay-off of this option is (ST − ηmTT0 )+ with η > 1. The current value of a partial lookback call is: 

 √ ln(η) ln(η) − ηe−rT m0T0N d  − √ − σ T Cpl = S0 N d  − √ σ T σ T   2r  − 2

2
 σ 2r √ σ ln(η)  S0 −rT  +e ηS0 ×  0 N −d − √ + T 2r σ mT0 σ T 

− erT η

2r σ2



  ln(η)  N −d  − √  . σ T


The pay-off of a partial lookback put is (ηMTT0 − ST )+ with 0 < η < 1. When η = 1, these options become standard lookback options.

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The current value of a partial lookback put is  

√ ln(η) ln(η) − ηe−rTMT00 N d  + √ + σ T p = −S0 N −d  + √ σ T σ T   2r  


2 σ2 2r √ ln(η) σ  S0  − e−rT − N d + T ηS0  √ 2r σ MT00 σ T   
2r ln(η)  − erT η σ 2 N d  + √ . σ T When η = 1, these options become standard lookback options. Tables 1–11 provide simulations values of lookback option values and the Greek letters for different parameters. The reader can compare the evolution of the different values.

4. Valuation of Single Lookback Options with Shadow Costs of Incomplete Information Using the previous notations for the maximum and the minimum over the interval [t1 , t2 ], Mtt12 , mtt21 , the formulas for Xt , Yt and yt and the values of λS and λ as the information costs associated to S and the option, it is possible to derive the appropriate formulas for these options in the presence of shadow costs of incomplete information.

4.1. Standard Lookback Options Since the pay-off of a standard European lookback call at the maturity date is given by ST − mTT0 , its current value is √ c = S0 e(λS −λ)TN1 (d  ) − e−(r+λ)T m0T0 N1 (d  − σ T )  − 2(r+λ2 S ) 
σ 2 σ S 0 + e−(r+λ)T S0  2(r + λS ) m0T0  

 √ ) 2(r + λ S × N1 −d  + T − e(r+λS )TN1 (−d  ) σ

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Table 1:

Simulations values of standard lookback calls.

S

Price

Delta

Gamma

Vega

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

12.69868 14.07381 16.47870 19.69548 23.51395 27.75571 32.28247 36.99416 41.82216

0.15888 0.38520 0.56967 0.71022 0.81140 0.88085 0.92664 0.95584 0.97392

0.04822 0.04043 0.03155 0.02317 0.01616 0.01081 0.00697 0.00436 0.00266

0.55516 0.55539 0.50224 0.42028 0.33046 0.24686 0.17669 0.12199 0.08168

Theta −0.01517 −0.01516 −0.01370 −0.01144 −0.00897 −0.00668 −0.00477 −0.00328 −0.00219

S = 100, t = 27/12/2003, T = 27/12/2004, r = 2%, σ = 20%, historical minimum for call values = 80.

Table 2:

Simulations values of standard lookback calls.

S

Price

Delta

Gamma

Vega

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

12.69868 13.49235 14.28601 15.59962 17.84962 20.86480 24.47425 28.52559 32.89347

0.15873 0.15873 0.15888 0.36200 0.53253 0.66806 0.77083 0.84569 0.89836

−0.00000 −0.00000 0.04293 0.03687 0.02989 0.02305 0.01702 0.01213 0.00837

0.55516 0.58986 0.62455 0.62826 0.58207 0.50555 0.41656 0.32846 0.24952

Theta −0.01517 −0.01611 −0.01706 −0.01716 −0.01588 −0.01377 −0.01132 −0.00891 −0.00675

S = 100, t = 27/12/2003, T = 27/12/2004, r = 2%, σ = 20%, historical minimum = 90.

Table 3:

Simulations values of standard lookback call values.

S

Price

Delta

Gamma

Vega

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

12.69868 13.49235 14.28601 15.07968 16.36714 18.55002 21.47525 24.98807 28.94772

0.15873 0.15873 0.15873 0.15888 0.35208 0.51623 0.64894 0.75170 0.82838

−0.00000 −0.00000 0.00000 0.04070 0.03530 0.02906 0.02285 0.01727 0.01262

0.55516 0.58986 0.62455 0.65925 0.66445 0.62140 0.54770 0.45979 0.37051

Theta −0.01517 −0.01611 −0.01706 −0.01801 −0.01814 −0.01695 −0.01492 −0.01250 −0.01005

S = 100, t = 27/12/2003, T = 27/12/2004, r = 2%, σ = 20%, historical minimum = 95.

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Table 4:

327

Simulations values of standard lookback puts.

S

Price

Delta

Gamma

Vega

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

28.92734 25.01663 21.75307 19.28817 17.72846 17.12260 17.46068 18.25435 19.04802

−0.83534 −0.72290 −0.57713 −0.40492 −0.21698 −0.02532 −0.15873 −0.15873 −0.15873

0.01967 0.02664 0.03263 0.03671 0.03841 0.03777 0.00000 −0.00000 0.00000

0.25039 0.38810 0.54161 0.69230 0.82263 0.92079 0.98284 1.02752 1.07219

Theta −0.00677 −0.01053 −0.01473 −0.01886 −0.02243 −0.02512 −0.02682 −0.02804 −0.02926

S = 100, t = 27/12/2003, T = 27/12/2004, r = 2%, σ = 20%, historical maximum = 110.

Table 5:

Simulations values of standard lookback puts.

S

Price

Delta

Gamma

Vega

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

38.01884 33.49836 29.34733 25.71130 22.73054 20.52049 19.15725 18.67119 19.04802

−0.93185 −0.87182 −0.78352 −0.66593 −0.52219 −0.35894 −0.18498 −0.00954 −0.15873

0.00992 0.01531 0.02122 0.02682 0.03135 0.03424 0.03525 0.03448 0.00000

0.12473 0.21957 0.34497 0.49218 0.64701 0.79350 0.91792 1.01164 1.07219

Theta −0.00335 −0.00593 −0.00934 −0.01337 −0.01761 −0.02162 −0.02503 −0.02760 −0.02926

S = 100, t = 27/12/2003, T = 27/12/2004, r = 2%, σ = 20%, historical maximum = 120.

Table 6:

Simulations values of standard lookback puts.

S

Price

Delta

Gamma

Vega

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

47.55623 42.75170 38.13436 33.80880 29.89601 26.52029 23.79438 21.80628 20.61068

−0.97383 −0.94526 −0.89807 −0.82794 −0.73279 −0.61345 −0.47366 −0.31940 −0.15786

0.00448 0.00779 0.01209 0.01701 0.02201 0.02649 0.02991 0.03196 0.03254

0.05582 0.11062 0.19417 0.30734 0.44522 0.59768 0.75150 0.89330 1.01234

Theta −0.00149 −0.00297 −0.00523 −0.00831 −0.01207 −0.01625 −0.02046 −0.02435 −0.02761

S = 100, t = 27/12/2003, T = 27/12/2004, r = 2%, σ = 20%, historical maximum = 130.

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Table 7:

Simulations values of calls on the maximum.

S

Price

Delta

Gamma

Vega

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

8.06811 7.74191 8.00442 9.22543 11.69841 15.58275 20.88919 27.50227 35.22391

−0.09904 −0.01904 0.13695 0.36185 0.63323 0.92138 1.19775 1.44123 1.63996

0.01022 0.02508 0.04001 0.05146 0.05726 0.05704 0.05187 0.04353 0.03400

−0.15882 −0.06462 0.08615 0.25819 0.41077 0.51410 0.55793 0.55089 0.51374

Theta 0.00585 0.00372 −0.00008 −0.00457 −0.00855 −0.01109 −0.01176 −0.01068 −0.00835

S = 100, K = 100, t = 27/12/1999, T = 27/12/2000, r = 2%, σ = 20%, max = 110.

Table 8:

Simulations values of calls on the maximum.

S

Price

Delta

Gamma

Vega

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

18.52257 17.95397 17.47973 17.37340 17.95200 19.51133 22.26935 26.33387 31.69861

−0.10929 −0.11207 −0.06825 0.03671 0.20512 0.42644 1.68091 1.94522 1.19775

−0.00330 0.00443 0.01583 0.02877 0.04056 0.04898 0.05282 0.05206 0.04761

−0.19324 −0.19326 −0.13123 −0.00763 0.15624 0.32685 0.47121 0.56729 0.60865

Theta 0.00621 0.00670 0.00549 0.00250 −0.00170 −0.00617 −0.00992 −0.01224 −0.01283

S = 100, K = 100, t = 27/12/2003, T = 27/12/2004, r = 2%, σ = 20%, max = 120.

Table 9:

Simulations values of calls on the maximum.

S

Price

Delta

Gamma

Vega

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

28.85299 28.41059 27.85611 27.31163 26.98107 27.12358 28.00943 29.87310 32.87710

−0.07345 −0.10232 −0.11542 −0.09544 −0.02786 0.09429 0.26837 0.48306 0.72151

−0.00625 −0.00443 0.00081 0.00935 0.02001 0.03091 0.04017 0.04638 0.04891

−0.14927 −0.19867 −0.21655 −0.18207 −0.08771 0.05668 0.22714 0.39382 0.53027

S = 100, K = 100, t = 27/12/2003, T = 27/12/2004, r = 2%, σ = 20%, max = 130.

08:36:30.

Theta 0.00455 0.00628 0.00724 0.00680 0.00467 0.00107 −0.00335 −0.00773 −0.01125

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Table 10:

329

Simulations values of puts on the minimum.

S

Price

Delta

Gamma

Vega

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

30.71318 27.08831 24.49320 22.70998 21.52845 20.77020 20.29697 20.00866 19.83665

−0.84114 −0.61480 −0.43033 −0.28978 −0.18860 −0.11915 −0.07336 −0.04416 −0.02608

0.04824 0.04043 0.03155 0.02317 0.01616 0.01081 0.00697 0.00436 0.00266

0.55516 0.55539 0.50224 0.42028 0.33046 0.24686 0.17669 0.12199 0.08168

Theta −0.01711 −0.01634 −0.01439 −0.01184 −0.00920 −0.00681 −0.00484 −0.00332 −0.00221

S = 100, K = 100, t = 27/12/2003, T = 27/12/2004, r = 2%, σ = 20%, min = 80.

Table 11:

Simulations values of puts on the minimum.

S

Price

Delta

Gamma

Vega

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

31.30641 26.50684 22.84951 20.17319 18.28533 16.99665 16.14220 15.58996 15.24097

−1.08140 −0.84114 −0.62701 −0.44972 −0.31154 −0.20925 −0.13677 −0.08729 −0.05456

0.04938 0.04544 0.03858 0.03073 0.02316 0.01666 0.01152 0.00770 0.00501

0.51767 0.58986 0.59192 0.54236 0.46307 0.37340 0.28715 0.21219 0.15157

Theta −0.01736 −0.01817 −0.01745 −0.01558 −0.01308 −0.01043 −0.00795 −0.00583 −0.00414

S = 100, K = 100, t = 27/12/2003, T = 27/12/2004, r = 2%, σ = 20%, min = 85.



with: d =

1 √ σ T


 ln mS00 + (r + λS )T + T0

 σ2 T 2

. Since the pay-off of a

standard lookback put at the maturity date is given by (MT00 −ST ), its current value is √ p = −S0 e(λS −λ)TN1 (−d  ) + e−(r+λ)T MT00 N1 (−d  + σ T )   −(2(r+λS )/σ 2 ) 
2 S σ 0 −(r+λ)T +e S0 − 2(r + λS ) MT00 
 
√ 2(r + λ ) S × N1 d  − T + e(r+λS )TN1 (d  ) σ 08:36:30.

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with: 1 d = √ σ T



S0 ln MT00



 1 2 + (r + λS )T + σ T . 2

4.2. Options on Extrema Options on the maximum The pay-off of a call on the maximum at the maturity date T is (MT00 − K)+ . When K ≥ MT00 , the current call price is √ CM = S0 e(λS −λ)TN(d) − e−(r+λ)TKN1 (d − σ T )  
− 2(r+λ2 S ) 2 σ S0 σ S0 − + e−(r+λ)T 2(r + λS ) K   

2(r + λS ) √ T + e(r+λS )TN1 (d) × N1 d − σ with:


  S0 1 2 ln + (r + λS )T + 1/2σ T . d= √ K σ T

When K < MT00 , the call’s value is:

CM = e−(r+λ)T (MT00 − K) + S0 e(λS −λ)TN1 (d  ) √ − e−(r+λ)TMT00N1 (d  − σ T ) −(2(r+λS )/σ 2 )   
2 σ S 0 + e−(r+λ)T S0 − 2(r + λS ) MT00 
 
2(r + λS ) √  (r+λS )T  × N1 d − T +e N1 (d ) . σ

Options on the minimum The pay-off of a put on the minimum at the maturity date T is (K − m0T0 )+ . When K < m0T0 , the put’s price is √ Pm = −S0 e(λS −λ)TN1 (−d) + e−(r+λ)TKN1 (−d + σ T )  
− 2(r+λ2 S ) 2 σ S0 σ S0 + e−(r+λ)T 2(r + λS ) K   

2(r + λS ) √ × N1 −d + T − e(r+λS )TN1 (−d) . σ 08:36:30.

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When K ≥ m0T0 , the put’s value is p = e−(r+λ)T (K − m0T0 ) − S0 e(λS −λ)TN1 (−d  ) √ + e−(r+λ)T m0T0N1 (−d  + σ T ) 
−(2(r+λS )/σ 2 ) σ2 S0 −(r+λ)T +e S0 2(r + λS ) m0T0  
2(r + λS ) √ × N1 −d  + T − e(r+λS )TN1 (−d  ) . σ

4.3. Limited Risk Options When m and K are constant, then the pay-off of a limited risk call at time T is (ST − K)+ 1(MTT ≤m) . 0

The call’s current value is nil when MT00 > m. When MT00 ≤ m, the call’s current value is: Clr = S0 e(λS −λ)T [N(d) − N1 (dm )] √ √ − e−(r+λ)T K[N1 (d − σ T ) − N(dm − σ T )] 

− 2(r+λ2 S ) √ σ (r + λS ) √ S0 e(λS −λ)T N1 2dm − d − 2 T −σ T +m m σ 

 √ 2(r + λS ) − N1 dm − +σ T σ
  
S0 − 2(r+λ2 S ) (r + λS ) √ −(r+λ)T σ −e e K T N1 2dm − d − 2 m σ 
(r + λS ) √ − N1 dm − 2 T σ with


  S0 1 σ2 ln + (r + λS )T + T . dm = √ m 2 σ T

The pay-off at the maturity date of a limited risk put is (K − ST )+ 1(mTT

0

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When m0T0 < m, the put is worthless. When m0T0 ≥ m, the put’s current value is: Plr = −S0 e(λS −λ)T [N1 (−d) − N1 (−dm )] √ √ + e−(r+λ)TK[N1 (−d + σ T ) − N1 (−dm + σ T )] 
 

− 2(r+λ2 S ) σ (r + λS ) √ S0 (λS −λ)T e T N1 −2dm + d + 2 −m m σ+σ 

 √ 2(r + λS ) − N1 −dm + +σ T σ
− 2(r+λ2 S ) 
 σ S0 2(r + λS ) √ −(r+λ)T K T N1 −2dm + d + −e m σ 
2(r + λS ) √ − N1 −dm + T . σ These options are issued in foreign exchange markets and also in stock index markets.

4.4. Partial Lookback Options These options differ from lookback options since at maturity date their pay-off is (ST − ηmTT0 )+ with η > 1. The current value of a partial lookback call is: 
ln(η) (λS −λ)T  Cpl = S0 e N1 d − √ σ T 
√ ln(η) −(r+λ)T 0  − ηe mT0N1 d − √ − σ T σ T 
2 σ ηS0 + e−(r+λ)T 2(r + λS ) − 2(r+λ2 S )

 σ S0 ln(η) 2(r + λS ) √  × N1 −d − √ + T σ m0T0 σ T −e

(r+λS )T

η

2(r+λS ) σ2

08:36:30.

N1




ln(η) −d − √ σ T 

 .

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At the maturity date, the pay-off of a partial lookback put is (ηMTT0 − ST )+ with 0 < η < 1. The current value of a partial lookback put is: 
ln(η) (λS −λ)T  Ppl = −S0 e N1 −d + √ σ T 
√ ln(η) −(r+λ)T 0  + ηe MT0N1 −d + √ + σ T σ T 
σ2 − e−(r+λ)T ηS0 2(r + λS ) 


− 2(r+λ2 S )




ln(η) N1 d + √ − × σ T 
2(r+λS ) ln(η)  (r+λS )T 2 . η σ N1 d + √ −e σ T S0 MT00

σ






  2(r + λS ) √ T σ

When η = 1, these options become standard lookback options.

5. The Exotic Timing Option This section derives the value of an exotic option which gives its holder the difference between the maximum value of the underlying asset during the option’s life and the initial asset value. Since the highest price of the asset during the option’s life cannot be less than the initial price, this difference cannot be less than zero. The key difference between the exotic timing and lookback put options is that the value of the maximum difference is based on initial asset price for the former and a terminal price for the latter.

5.1. Analysis and Valuation The main difference between lookbacks and the exotic timing option can be simply illustrated using a 3-month option on the CAC 40 index. Assume the CAC 40 is at 3800 and over a subsequent 90 day period it reaches 4200. This latter value corresponds to the maximum. The net change is +400 points multiplied by the size of the contract, 10 euro, which gives a total gain of 4000 euro. An investor who buys an exotic option contract will

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have 4000 euro. Now, assume that the investor who believes he has superior market timing skills buys the lookback option on day 30 and also buys an exotic timing option on the same day when the index is at 3400. At maturity, the pay-off from the lookback is zero. The payoff from the exotic timing option is given by the net change (+800 points) multiplied by the size of the contract, 200 euro, which gives a total gain of 8000 euro. Assume 0 denotes the initial time; T denotes the contract’s maturity date; and, τ denotes the present time. The remaining time to maturity is t = T − τ. S represents the underlying asset price; r, the riskless interest rate; and σ, the instantaneous standard deviation. The maximum value of the underlying asset in a given time interval is M(τ) = max S(δ)0≤δ≤τ . When pricing this exotic timing option, the value of an asset paying the realized maximum over a given period must be determined. Then, the current value of the underlying asset is substracted from it. In a risk neutral world, the option value is given by Cmax (τ) = e−rt EQ [(M(T ) − S(T )) + (S(T ) − S(0))|F τ ] where the conditional expectation EQ is taken with respect to the risk-neutral probability Q and the available information Fτ . The pay-off corresponds to that of a lookback put initiated at time 0 and a forward contract. The exotic timing option value is: Cmax (τ) = P(τ) + [S(τ) − e−rt S(0)] where P(τ) is the lookback put given by Eq. (14) in Conze and Viswanathan (1991). At τ = 0, this last equation is rewritten as √ Cmax (0) = S(0)(1 − e−rT ) − S(0)N(−d1 ) + e−rTS(0)N(−d1 + σ T )

  2  2r √ rT −rT σ + S(0)e e N(d1 ) − N d1 − T 2r σ where d1 =

(r+ 21 σ 2 T) √ σ T

and N(.) is the cumulative normal density function.

5.2. Simulations and Option Characteristics It is useful to compare the exotic timing option sensitivities with those of lookbacks and standard puts. For the sake of clarity, the exotic timing option is rewritten as Cmax (0) = P(0) + S(0)(1 − e−rT ). This expression demonstrates that the exotic option is more expensive than lookback puts and standard puts with a strike price S(0). When the time to maturity tends to ∞, Cmax (0) − P(0) = S(0). This result demonstrates that a long position in a perpetual exotic timing option and a short position in a perpetual lookback

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put is equivalent to a position in the initial underlying asset. The option’s delta is greater than that of a lookback put and a standard put Cmax (0) = P(0) + (1 − e−rT ). The gammas of the option and the lookback put are nil in this context. The option’s derivative with respect to the volatility parameter is positive and equals that of a lookback put. It is given by √ Cmax (0) = (S(0)e−rT σ/r)[erTN(d1 ) − (N(d1 − (2r/σ) T ))]. The option’s derivative with respect to the interest rate parameter is greater than that of a lookback put ρCmax (0) = ρP(0) + S(0)T e−rT . The option’s derivative with respect to the time to maturity is positive and greater than that of a lookback put Cmax (0) = P(0) + rS(0)e−rT . For illustrative purposes, the above formula is used to simulate the exotic timing option values. When the underlying asset price S is 100, the results for different levels of r, σ and T are reported in Table 12. Simulations show that exotic option values are increasing functions of the underlying asset price, the interest rate, the volatility parameter and time to maturity.

5.3. Valuation in the Presence of Shadow Costs of Incomplete Information In a risk neutral world, the option value is given by Cmax (τ) = e−(r+λ)tEQ [(M(T ) − S(T )) + (S(T ) − S(0))|Fτ ] Table 12: Comparison of exotic timing option values, Cmax (0) with lookback put option values, P(0).

r

T = 0.25 Cmax (0)

σ = 0.2 P(0)

0.10 0.15 0.20 0.25 0.30

8.49 8.82 9.29 9.89 10.59

6.02 5.13 4.41 3.83 3.36

r = 0.1 T T Cmax (0) 0.25 0.40 0.50 0.75 1.00

08:36:30.

8.49 10.98 12.48 15.88 18.98

σ = 0.2 P(0) 6.02 7.058 7.607 8.654 9.463

r = 0.1 T = 0.25 σ Cmax (0) 0.20 0.25 0.30 0.35 0.40

8.49 10.54 12.68 14.88 17.06

σ P(0) 6.02 8.071 10.211 12.411 14.591

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where the conditional expectation EQ is taken with respect to the risk-neutral probability Q and the available information Fτ . The payoff corresponds to that of a lookback put initiated at time 0 and a forward contract. The exotic timing option value is Cmax (τ) = P(τ) + [S(τ) − e−rt S(0)] where P(τ) is the lookback put given by Eq. (14) in Conze and Viswanathan (1991). At τ = 0, the formula becomes Cmax (0) = S(0)e(λS −λ)T (1 − e−(r+λS )T ) − S(0)e(λS −λ)TN(−d1 ) 
√ σ2 + e−(r+λ)TS(0)N(−d1 + σ T ) + S(0)e−(r+λ)T 2(r + λS ) √ × [e(r+λS )T N(d1 ) − N(d1 − (2(r + λS )/σ) T )] where (r + λS + d1 = √ σ T

σ2 )T 2

.

6. The Valuation of Double Lookback Options with Information Costs He, Keirstead and Rebholz (1998) offer some mathematical results which are important for the valuation of double lookbacks. They present various joint density/distribution functions of the extreme values of two correlated Brownian motions. They also provide different formulas for double lookbacks and double barriers. We follow their analysis and introduce information costs in the valuation of these options.

6.1. The Main Mathematical Results for Double Lookbacks The valuation of double lookback options is based on the knowledge of the joint probability density/distribution functions of extreme values of two correlated Brownian motions. Consider two Brownian motions with drifts: Xi (t) = αi t + σi wi (t),

t≥0

where αi and σi are constants and wi is a Brownian motion. The correlation between the two Brownians is given by cov(w1 (t), w2 (t)) = ρt. 08:36:30.

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We denote the running minimum and maximum of Xi by: Xi (t) = min(0≤s≤t) Xi (s), Xi (t) = max(0≤s≤t) Xi (s). For ease of exposition, let us define: P(X1 (t) ∈ dx, X1 (t) ≤ x1 ) = g(x, x1 , t, α1 )dx, P(X1 (t) ≤ x1 ) = G(x1 , t, α1 ),

x ≤ x1 ,

x1 ≥ 0

x1 ≥ 0

P(X1 (t) ∈ dx, Xi (t) ≥ x1 , X1 (t) ≤ x2 ) = g(x, x1 , x2 , t, α1 )dx x1 ≤ 0 ≤ x2 ,

x ∈ [x1 , x2 ]

P(X1 (t) ≤ x1 , X1 (t) ≤ x2 ) = G(x, x1 , x2 , t, α1 ), x1 ≤ 0 ≤ x2 , x ∈ [x1 , x2 ]. If you consider the extremes of −X1 in lieu of X1 and adjust the drift from α1 to α2 , you obtain: P(X1 (t) ∈ dx, X1 (t) ≥ x1 ) = g(−x, −x1 , t, −α1 ), x ≥ x1 , x1 ≤ 0, P(X1 (t) ≥ x1 ) = G(−x1 , t, −α1 ), x1 ≤ 0. In this context, He et al. (1998) show that the probability density/distribution functions for the maximum (or the minimum) of a Brownian motion with constant drift is given by:
 −4x12 −4x1 x 1 x − α1 t  2 g(x, x1 , t, α1 ) = √ φ , x ≤ x1 , x1 ≥ 0 1 − e 2σ1 t √ σ1 t σ1 t

  2x1 α1 x1 − α1 t −x1 − α1 t σ12 ˆ G(x1 , t, α1 ) = N N −e , x1 ≥ 0 √ √ σ1 t σ1 t where φ(.) is the standard normal density and N(.) is the corresponding normal distribution function. In the same context, the joint probability density function of the maximum, minimum and end point of a Brownian motion with a constant drift for x ∈ [x1 , x2 ] with x1 ≤ 0, x2 ≥ 0 is: 2 α1 x α1 t − 2 σ12 2σ1

g+ − (x, x1 , x2 ) = e 

 ∞  1 x − 2n(x2 − x1 ) x − 2n(x2 − x1 ) − 2x1 − . √ √ √ σ t σ1 t σ1 t n=−∞ 1 This density can also be written in an equivalent form as:


2 g+ − (x, x1 , x2 ) = e x2 − x1

2 α1 x α1 t − 2 σ12 2σ1



∞  n=−∞

e

 x − x1 dx. × sin(nπ) x2 x1 08:36:30.




−

n2 π2 σ12 t 2(x2 −x1 )2




−x1 sin nπ x2 − x1



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He et al. (1998) show also that the joint probability distribution function of the maximum and minimum of a Brownian motion with constant drift is:
 ∞ 2nα1 (x2 −x1 )   x2 − α1 t − 2n(x2 − x1 ) σ12 N e G+ − (x1 , x2 , t, α1 ) = √ σ1 t N=−∞ 
x1 − α1 t − 2n(x2 − x1 ) − N √ σ1 t
 2x1 α1  x2 − α1 t − 2n(x2 − x1 ) − 2x1 σ12 N −e √ σ1 t
 x1 − α1 t − 2n(x2 − x1 ) − 2x1 − N . √ σ1 t They develop a theorem on the probability density/distribution functions of the extreme values of two correlated Brownian motions. This theorem is fundamental for the pricing of double lookback options. Define: P(X1 (t) ∈ dx1 , X2 (t) ∈ dx2 , X1 (t) ≥ m1 , X2 (t) ≥ m2 ) = p(x1 , x2 , t, m1 , m2 , α1 , α2 , σ1 , σ2 , ρ)dx1 dx2 then: for x1 ≥ m1 ,

x2 ≥ m2 ,

m1 ≤ 0,

m2 ≤ 0

p(x1 , x2 , t, m1 , m2 , α1 , α2 , σ1 , σ2 , ρ) =

ea1 x1 +a2 x2 +bt h(x1 , x2 , t, m1 , m2 , α1 , α2 , σ1 , σ2 , ρ)  σ1 σ2 1 − ρ2

where: h(x1 , x2 , t, m1 , m2 , α1 , α2 , σ1 , σ2 , ρ) 
 ∞
nπθ0 nπθ rr0 2  −(r 2 + r0 )2 nπ sin sin Iβ e = βt n=1 2t β β t and: a1 =

α1 σ2 − ρα2 σ1 , (1 − ρ2 )σ12 σ2

a2 =

α2 σ1 − ρα1 σ2 (1 − ρ2 )σ22 σ1

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1 z1 =  1 − ρ2




x1 − m1 σ1




−ρ

x2 − m2 σ2

 ,


 m1 ρm2 1 m2 − , z20 = − z10 =  + σ1 σ2 σ2 1 − ρ2   r = z21 + z22 , r0 = z210 + z220

z2 =

339

x 2 − m2 σ2

z2 , θ ∈ [0, β] z1  z20 r0 = z210 + z220 , tan θ0 = , θ0 ∈ [0, β]. z10 The second part of their theorem shows that for x1 ≥ m1 , x2 ≤ M2 , m1 ≤ 0, M2 ≥ 0, tan θ =

P(X1 (t) ∈ dx1 , X2 ∈ dx2 , X1 (t) ≥ m1 , X2 (t) ≤ M2 ) p(x1 , −x2 , t, m1 , −M2 , α1 , −α2 , σ1 , σ2 , −ρ)dx1 dx2 . The third part of their theorem shows that for x1 ≤ M1 , x2 ≤ M2 , M1 ≥ 0, M2 ≥ 0, P(X1 (t) ∈ dx1 , X2 ∈ dx2 , X1 (t) ≤ M1 , X2 ≤ M2 ) p(−x1 , −x2 , t, −M1 , −M2 , −α1 , −α2 , σ1 , σ2 , ρ)dx1 dx2 . He et al. (1998) obtain also a result which is useful for the pricing of semilookback options. The above results can be used in the valuation of several types of double lookbacks.

6.2. The Valuation of Double Lookback Options with Information Costs The above theorems and results are very useful for the valuation of double lookback options, as well as semilookback options. Consider the following prices of a riskless bond and two stocks in a Black and Scholes economy: B(t) = ert


(µ1 −q1 −

S1 (t) = S1 (0)e


(µ2 −q2 −

S2 (t) = S2 (0)e

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 t+σ1 w1 (t))  t+σ2 w2 (t))

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where r is the riskless rate, µi the expected return of the stock i, qi the dividend yield of stock i. Under the risk neutral probability with information costs, we have:
(r+λS1 −q1 −

S1 (t) = S1 (0)e


(r+λS2 −q2 −

S2 (t) = S2 (0)e

σ12 2 σ12 2



t+σ1 w∗1 (t))



t+σ2 w∗2 (t))

where λSi is the information costs associated to the stock i. Now, we can apply the previous results on the various distributions and densities by letting αi = r + λSi − qi . We consider the pricing of lookbacks at time 0 for a maturity date T . The lookback period runs from t ∗ to T . We define the running minimum and maximum of stock prices Si with i = 1, 2 as: Si (t) = ∗min Si (s), t ≤s≤t

S i (t) = max Si (s). ∗ t ≤s≤t

The payoff of a general lookback option at T is given by: H(S1 (T ), S 1 (T ), S 1 (T ), S2 (T ), S 2 (T ), S 2 (T )) where H is a continuous function.

6.2.1. Lookback on the Spread of One Asset Consider a European call/put on the spread between the maximum and minimum of a single stock price referred to as a lookback spread. The payoff functions of these spreads are given by: max[0, (S 1 (T ) − S 1 (T )) − K],

max[0, K − (S 1 (T ) − S 1 (T ))].

These options are contingent upon the maximum dispersion of the stock prices realized over a given period of time. Using the joint density/distribution functions for the maximum and minimum of a single Brownian motion: G+ − (x1 , x2 ) = P(X1 (t) ≥ x1 , X1 ≤ x2 ). Define: VS P (x1 , x2 ) = max[0, (S1 (0)emax(M1 ,x2 ) − S1 (0)emin(m1 ,x1 ) )].

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The price at time 0 of a lookback spread call is:  ∞  0 −∂G+ − (x1 , x2 ) . dx1 dx2 VS P (x1 , x2 ) CS P = e−(r+λC )t ∂x1 , ∂x2 0 −∞

6.2.2. The Valuation of Lookbacks on Two Assets It is possible to define at least three types of double lookbacks whose payoffs depend on the extreme values of two assets following correlated geometric Brownian motions. The first type corresponds to double maxima. The second type corresponds to double minima. The third type corresponds to double lookback spreads. Double maxima lookbacks are calls or puts on the difference between the maximum of two assets S1 and S2 . Their payoffs are given by: max[0, (aS 1 (T ) − bS 2 (T )) − K] max[0, K − (aS 1 (T ) − bS 2 (T ))] where a and b are positive constants chosen by investors. When the strike price is zero, the double maxima corresponds to a call on the maximum of S1 at the maximum of S2 . Double minima lookbacks are calls or puts on the difference between the minimum of two assets S1 and S2 . Their payoffs are given by: max[0, (aS 1 (T ) − bS 2 (T )) − K] max[0, K − (aS 1 (T ) − bS 2 (T ))]. When the strike price is zero, the double minima call is equivalent to an option to sell the minimum of S1 for the minimum of S2 . Double lookback spreads are calls or puts on the spread between the maximum S1 and the minimum of S2 . Their payoffs are given by: max[0, (aS 1 (T ) − bS 2 (T ))] max[0, K − (aS 1 (T ) − bS 2 (T ))]. In order to value these double lookback options, define the following payoffs: VD max (x1 , x2 ) = max[0, aS1 (0)emax(M1 ,x1 ) − bS2 (0)emax(M2 ,x2 ) − K] VD min (x1 , x2 ) = max[0, aS1 (0)emin(m1 ,x1 ) − bS2 (0)emin(m2 ,x2 ) − K] VD LS (x1 , x2 ) = max[0, aS1 (0)emax(M1 ,x1 ) − bS2 (0)emin(m1 ,x2 ) − K].

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In this context, the prices for double maxima, double minima and double lookback spread options are given by: −(r+λC )t



CD max = e



∞

∞

dx1 0

dx2 0

−∂2PX1(t) ≤ x1 , X2(t) ≤ x2 ∂x1 ∂2  0 dx1 dx2

× VD max (x1 , x2 ) −(r+λC )t

CD min = e



0

−∞

−∞

−∂P 2 X1(t) ≥ x1 , X2(t) ≥ x2 ∂x1 ∂x2  0 dx1 dx2

× VD min (x1 , x2 ) −(r+λC )t

CD LS = e



0

−∞

× VD LS (x1 , x2 )

−∞

−∂P 2 X1(t) ≥ x1 , X2(t) ≤ x2 . ∂x1 ∂x2

6.2.3. The Valuation of Semilookbacks on Two Assets Lookback options whose payoffs depend on the extreme value of one asset and the final value of an other asset are referred to as semilookbacks. This is the case for example for an option to buy the maximum of S1 at the final value of S2 or an option to sell the minimum of S2 at the final value of S1 . The probability density functions necessary for the valuation of semilookbacks are given by: P(X1 (t) ≤ M1 , X1 (t) ∈ dx1 , X2 (t) ∈ dx2 ) = u+ 0 (x1 , x2 , t, .)dx1 dx2 P(X1 (t) ∈ dx1 , X2 (t) ∈ dx2 , X2 (t) ≥ m2 ) = u0 − (x1 , x2 , t, .)dx1 dx2 P(X1 (t) ∈ dx1 , X2 (t) ∈ dx2 ) = f+ 0 (x1 , x2 , t, .)dx1 dx2 P(X1 (t) ∈ dx1 , X2 (t) ∈ dx2 ) = f− 0 (x1 , x2 , t, .)dx1 dx2 . Consider the following payoffs for call options: VS max (x1 , x2 ) = max[0, S1 (0)emax(M1 ,x1 ) − S2 (0)ex2 − K] VS min (x1 , x2 ) = max[0, S1 (0)ex1 − S2 (0)emin(m2 ,x2 ) − K].

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The values of semilookback calls are given by:  ∞  ∞ −(r+λC )t CS max = e dx2 VS max (x1 , x2 )f+ 0 (x1 , x2 , T ) dx1 −∞

o

CS min = e−(r+λC )t





∞

−∞

dx1

0

−∞

dx2 VS min (x1 , x2 )f0 − (x1 , x2 , T ).

Tables 13–20 provide simulations of double barrier options values and the corresponding Greek letters. The reader can compare the dynamics of the different parameters. Table 13:

Simulations values of double knock-in calls.

S

Price

Delta

Gamma

Vega

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

1.60329 2.78348 4.30595 6.27121 8.76446 11.81552 15.38947 19.40142 23.74210

0.20696 0.26729 0.34541 0.44387 0.55466 0.66476 0.76210 0.83913 0.89355

0.01119 0.01393 0.01819 0.02157 0.02251 0.02081 0.01717 0.01268 0.00836

0.14663 0.19562 0.28365 0.37842 0.44490 0.46177 0.42660 0.35230 0.25939

Theta −0.00394 −0.00527 −0.00771 −0.01036 −0.01222 −0.01270 −0.01172 −0.00966 −0.00707

S = 100, K = 100, t = 28/12/2003, T = 27/12/2004, r = 2%, σ = 20%, lower barrier = 95, upper barrier = 120.

Table 14: Simulations values of double knock-in puts. S

Price

Delta

Gamma

Vega

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

19.07174 15.05862 11.82941 9.29107 7.25958 5.56340 4.09701 2.82577 1.76196

−0.87764 −0.72377 −0.57117 −0.45042 −0.36791 −0.31397 −0.27356 −0.23431 −0.19013

0.02863 0.03196 0.02722 0.01942 0.01261 0.00869 0.00765 0.00836 0.00958

0.34325 0.45909 0.45796 0.37842 0.28031 0.21126 0.19144 0.21480 0.25939

Theta −0.00947 −0.01268 −0.01262 −0.01036 −0.00760 −0.00566 −0.00512 −0.00580 −0.00707

S = 100, K = 100, t = 28/12/2003, T = 27/12/2004, r = 2%, σ = 20%, lower barrier = 95, upper barrier = 120.

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Table 15:

Simulations values of double knock-out calls.

S

Price

Delta

Gamma

Vega

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

0.00000 0.00000 0.00000 0.00000 0.15157 0.23221 0.21938 0.12904 0.00000

0.00000 0.00000 0.00000 0.03372 0.02466 0.00679 −0.01137 −0.02345 −0.00000

0.00000 0.00000 0.00000 −0.00068 −0.00312 −0.00386 −0.00304 −0.00137 0.00000

0.00000 0.00000 0.00000 −0.00000 −0.05385 −0.08199 −0.07697 −0.04502 −0.00000

Theta 0.00000 0.00000 0.00000 0.00000 0.00152 0.00231 0.00216 0.00127 0.00000

S = 100, K = 100, t = 28/12/2003, T = 27/12/2004, r = 2%, σ = 20%, lower barrier = 95, upper barrier = 120.

Table 16: Simulations values of double knock-out puts. S

Price

Delta

Gamma

Vega

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

0.37548 0.50717 0.33809 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.05761 −0.00608 −0.05683 0.00000 0.00000 0.00000 0.00000 0.00000 0.05703

−0.01187 −0.01228 −0.00614 0.00000 0.00000 0.00000 0.00000 0.00000 −0.00086

−0.13233 −0.17730 −0.11730 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

Theta 0.00372 0.00499 0.00330 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

S = 100, K = 100, t = 28/12/2003, T = 27/12/2004, r = 2%, σ = 20%, lower barrier = 95, upper barrier = 120.

Summary A lookback option gives its holder the right to buy (sell) a fixed amount of an asset at the best price which occurs over the option’s life. Hence, a lookback call (put) involves the right to buy (sell) at the lowest (highest) price. For a floating strike lookback, the strike price is unknown before the maturity date. For a lookback call, the strike price corresponds to the minimum price realized by the underlying asset during the option’s life. For a

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Table 17:

345

Simulations values of double knock-out calls.

S

Price

Delta

Gamma

Vega

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

0.00000 0.00000 0.00000 3.63929 7.25452 10.95156 14.75816 18.63577 22.49399

0.00000 0.00000 0.74145 0.72054 0.72904 0.75067 0.77068 0.77731 0.76161

0.00000 0.00000 −0.00734 −0.00010 0.00381 0.00455 0.00272 −0.00112 −0.00641

0.00000 0.00000 0.00000 0.06611 0.11503 0.12470 0.08289 −0.01610 −0.17318

Theta 0.00000 0.00000 −0.00000 −0.00184 −0.00320 −0.00350 −0.00238 0.00029 0.00456

S = 100, K = 100, t = 27/12/2003, T = 27/12/2004, r = 2%, σ = 20%, lower barrier = 90, upper barrier = 200.

Table 18: Simulations values of double knock-out puts. S

Price

Delta

Gamma

Vega

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

12.97597 6.41239 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

−1.31245 −1.30487 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

−0.00248 0.00708 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

−0.13063 −0.05505 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

Theta 0.00355 0.00150 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

S = 100, K = 100, t = 27/12/2003, T = 27/12/2004, r = 2%, σ = 20%, lower barrier = 90, upper barrier = 200.

lookback put, the strike price corresponds to the maximum realized price by the underlying asset during the same period. For fixed-strike lookbacks, the strike price is known in advance. The call option pay-off is given by the difference between the highest value of the underlying asset during the option’s life and the fixed strike price. The put option pay-off is given by the difference between the fixed strike price and the lowest value of the underlying asset recorded during the option’s life. The value of path-dependent options depend on the underlying asset, time, the strike price, and on a function specifying the option in question. The exotic timing option has some appeal to investors, especially those who

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Table 19:

Simulations values of double knock-out calls.

S

Price

Delta

Gamma

Vega

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

0.00000 0.00000 0.00000 2.77383 5.27547 7.38733 8.98368 9.95060 10.21283

0.00000 0.00000 0.57454 0.53122 0.46536 0.37489 0.25951 0.12438 −0.02041

0.00000 0.00000 −0.00687 −0.01130 −0.01603 −0.02117 −0.02571 −0.02857 −0.02900

0.00000 0.00000 −0.00000 −0.13438 −0.27899 −0.44395 −0.61569 −0.76632 −0.86538

Theta 0.00000 0.00000 0.00000 0.00362 0.00754 0.01205 0.01679 0.02097 0.02374

S = 100, K = 100, t = 27/12/2003, T = 27/12/2004, r = 2%, σ = 20%, lower barrier = 90, upper barrier = 150.

Table 20:

Simulations values of double knock-out calls.

S

Price

Delta

Gamma

Vega

96.00 97.00 98.00 99.00 100.00 101.00 102.00 103.00 104.00

6.12736 6.41323 6.69416 6.96875 7.23558 7.49328 7.74048 7.97588 8.19821

0.28596 0.28102 0.27466 0.26688 0.25770 0.24715 0.23527 0.22212 0.20776

−0.00494 −0.00636 −0.00778 −0.00918 −0.01055 −0.01188 −0.01315 −0.01436 −0.01550

−0.08859 −0.12207 −0.15689 −0.19283 −0.22967 −0.26715 −0.30504 −0.34308 −0.38101

Theta 0.00060 −0.00021 −0.00108 −0.00198 −0.00291 −0.00388 −0.00487 −0.00588 −0.00689

S = 100, K = 100, t = 05/01/2003, T = 01/03/2004, r = 4%, σ = 20%, lower barrier = 5, upper barrier = 150.

believe that they have special skills at market timing, since the option is at-the-money when it is purchased and will finish in-the-money if the asset price moves higher any time during the life of the option. The exotic timing option can be created as an independent instrument or can be embedded in bond issues or equity issues where the underlying assets may be as diverse as an index price, a share of stock, a currency, an interest rate, a commodity or any other tradable asset. Using standard arbitrage arguments, we provide a simple analytic formula for the valuation of an exotic timing option. This option has some specificities when compared to lookback options. The option price sensitivities

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are given and options prices are simulated. The well known lookback strategies can be easily implemented using the exotic timing option. He, Keirstead and Rebholz (1998) provide some important results for the valuation of double lookbacks. They present various joint density/distribution functions of the extreme values of two correlated Brownian motions. They also provide different formulas for double lookbacks and double barriers. This chapter defines and analyzes by simple examples several forms of path-dependent options, i.e. full and partial lookback options with floating and fixed strike prices. First, standard lookback options, extreme options, limited risk options and partial lookback options are analyzed and valued. Simulation results are presented.

Questions 1. 2. 3. 4. 5. 6. 7. 8. 9.

What is a standard lookback option? What is a floating strike lookback? What is a fixed-strike lookback? What is an option on the maximum? What is a partial lookback option? What is an exotic timing option? What is a double lookback? What is a lookback on a spread? What is a semilookback on two assets?

Bibliography Bellalah, M and JL Prigent (1997). A note on the valuation of an exotic timing option. Journal of Futures Markets, 17(4), 483–487. Conze, A and R Viswanathan (1991). Path dependent options: the case of lookback options. Journal of Finance, 46, 1893–1907. Goldman, M, H Sosin and M Gatto (1979). Path dependent options, buy at the low, sell at the high. Journal of Finance, 34, 1111–1127. Harrison, JM and D Kreps (1979). Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory, 20, 381–408. Harrison, JM and S Pliska (1981). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and Their Applications, 11, 215–260. He, H, W Keirstead and J Rebholz (1998). Double lookback options. Mathematical Finance, 8, 201–228.

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Chapter 9

Asian and Flexible Asian Options and Their Applications

This chapter is organized as follows: 1. Section 2 reviews the important results regarding the valuation and hedging of Asian options. It presents simulation results of option values, since the approaches used for the valuation of Asian options are also used in the valuation of basket options. 2. Section 3 presents the framework for the valuation of these options under incomplete information. The main formulas are provided in this context. 3. In Section 4, flexible Asian options are studied in different contexts. The general context for the analysis and valuation of flexible arithmetic and geometric Asian options is presented. An analytic approximation is provided for flexible arithmetic Asian options. The solution is based on an extension of the method used to approximate standard arithmetic Asian options with their corresponding geometric Asian options. Most studies approximating arithmetic Asian options with geometric Asian options use either an arbitrarily fixed number of moments (Levy and Turnbull, 1992; Turnbull and Wakeman, 1991) or a reduced effective strike price (Vorst, 1992). The study of Zhang (1995) is based on a Taylor series approximation method to the corresponding geometric mean.

1. Introduction

T

he deregulation of financial services, the competition pressure, the absence of talents for financial products and the ease with which banks can mimic their competitors’ ranges of products irreversibly push financial

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institutions into an endless and vital quest for the ultimate option package, hybrid security or swap for their customers. An increased emphasis has been placed on the ability to design new products which bring more effective solutions to increasingly complex financial problems. Lookback, compound, contingent, knock-out and balloon options now belong to the ever-increasing lexicon of exotic assets. Quite recently, a new class of derivative products has been added to this already impressive list, namely the Asian options, also called average rate options. These options are issued with a special feature: their pay-offs depend on the average price of the underlying asset over a fixed period, leading up to the maturity date. These options appear in a straightforward form or may be implicit in a bond contract. Average price options or Asian options, like many other exotic options, are gaining in popularity in the foreign exchange market, interest rate and commodity market. Asian options are path-dependent options and their final pay-off is a function of the average values of the underlying asset in the past. These financial innovations are traded in over-the-counter markets and enable investors to accomplish several hedging strategies. Asian options allow the hedging of a series of cash-flows. The averaging process gives a means of overcoming the problem of manipulating the underlying asset price on a particular day. Average value options contribute to the reduction of price manipulation of the underlying asset price at the maturity date. Examples of these options include commoditylinked bond contracts and average currency options. Commodity-linked bond contracts give the right to the holder to receive the average value of the underlying commodity over a certain period or the nominal value of the bond, whichever is higher. Hence, the holder of a commoditylinked bond has a position in a straightforward bond and an option on the average value of the commodity. The strike price is given by the bond’s nominal value. An average currency call on sterling, for example, gives the right to its holder to receive at the maturity date the maximum of two quantities: zero or the difference between the arithmetic daily exchange rate averaged over some specified period and the strike price. When the average is based on the geometric mean, the valuation of Asian options is simple and closed-form solutions of the Black–Scholes type are easily obtained. This is due to the fact that the product of log-normal prices is itself log-normal. However, when the average is based on the arithmetic mean, the valuation of Asian

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options is rather complex since the sum of log-normal components has no explicit representation. Therefore, it is rather difficult to get a closed-form solution as in the Black–Scholes case. Risk management with exotic options has given rise to basket options, which are called simply options on a basket of assets or currencies. Asian options, based on average values of the underlying asset, are actively traded in the over-the-counter market. The pay-off of these options depends on the way the average is calculated and on a prespecified observation frequency. When the average is arithmetic, the option is an arithmetic Asian option. When the average is geometric, the option is a geometric Asian option. Flexible Asian options are extensions of standard Asian options which are more flexible with regard to the weighting scheme. For example, for a travel company, an exporter or any economic agent facing a seasonal exchange rate risk, leading him to attribute heavier weights for weeks or months with greater cash flows, he is better off (in hedging) with flexible Asian options than with standard Asians. In fact, in the latter case, the weighting is equal while in the former a heavier weight can be assigned for periods with higher cash flows. There are two kinds of flexible Asian options: arithmetic and geometric. For the analysis of information costs and valuation, we can refer to Bellalah (2001), Bellalah, Prigent and Villa (2001), M. Bellalah, Ma. Bellalah and LR. Portait (2001), Bellalah and Prigent (2001), Bellalah and Selmi (2001), etc.

2. The Average Price Options: Analysis and Valuation Asian options are path-dependent options whose pay-off is based on an average. In some cases, the underlying asset of the option is an average and in other cases, the strike price is a floating one. Asian options are the appropriate hedging instruments for traders who want to transact continuously over finite time horizons. Asian options provide an effective way of capping costs or placing floors on aggregate profits. In this section, five different approaches to the valuation of an average value option are presented: the approach of Kemma and Vorst, K–V (1990), that of Turnbull and Wakeman, T–W (1991), that of Conze and Viswanathan, C–V (1991), that of Curran (1992), and that of Bouaziz, Briys and Crouhy (1994).

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2.1. Analysis of Average Price Options We use the following notation: T : option’s maturity date, t: current time, d: continuous yield on the underlying asset, [t0 , tN ]: the time interval over which the average is calculated. The average is often calculated at different points in the interval, ti ti = t0 + iH,

for i = 0, 1, 2, . . . , N

where H = (tN − t0 )/N. We denote by A(t), the running average defined as follows: A(t) = 1
m i=0 S(ti ) for tm ≤ t ≤ tm+1 with 0 ≤ m ≤ N and A(0) = 0 for t < t0 . m+1

2.1.1. The Standard Put–Call Parity Condition In this context, A(tN ) is the arithmetic average of N + 1 prices. These prices are considered at equal intervals of time H between [t0 , tN ]. The time interval may be a day, a week or a month. This is the basic form of average prices. The pay-off of the average call option is c(A(t), K, t) = max[A(tN )−K, 0]. The pay-off of the average put option is p(A(t), K, t) = max[K−A(tN ), 0]. In general, it is sufficient to value one of these options to get the value of the other, since as with standard European options, there is a put–call parity relationship between average options.

2.1.2. The Standard Put–Call Parity Condition The put–call parity condition can be written for tm ≤ t ≤ tm+1 as p(A(t), K, t) = c(A(t), K, t)   eg(H−ξ) [1 − eg(N−m)H ] −rτ A(t)(m + 1) +e + S(t) −K (N + 1) (N + 1)(1 − egH ) where g = r − d, ξ = (t − tm ), τ = (tN − t). When t < t0 , this relationship becomes   eg(t0 −t) − eg(t0 −t+H(N+1)) −rτ −K . S(t) p(A(t), K, t) = c(A(t), K, t) + e (N + 1)(1 − egH )

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2.2. The Valuation Approaches As in most option pricing problems, it is assumed that the dynamics of the underlying asset are represented by dS(t) = µS(t)dt + σS(t)dW where µ and σ are the drift and the volatility parameters. Under this assumption,√S(tj ) can be expressed in terms of S(tj−1 ) as 1 2 S(tj ) = S(tj−1 )e(µ− 2 σ )H+σ HYj . Where Yj follows a normal distribution with a zero mean and a unit standard deviation. Using the risk-neutral approach, the average option value is given by its expected pay-off discounted to the present c(S(t), K, t) = e−rτE∗ [max[M(tN ) − K, 0]] where E is the expectation operator conditioned on [A(t) − S(t)] at time t under the risk-adjusted density function. This means that µ is replaced by (r − d) in the dynamics of the underlying asset price, i.e. dS(t) = (r − d)S(t)dt + σS(t)dW. Hence, to value the average option, we have to determine the distribution of M(tN ). If we denote by f ∗ (.) the density function, then the average call value is c(S(t), K, t) = e−rτE[max[M(tN ) − K, 0]]  ∞ [M(t) − K]f ∗ (w)dw. = K ∗

Since the function f (.) is non-standard, numerical methods and MonteCarlo techniques must be used to value this integral. The main drawback of these techniques is that they are time-consuming. An alternative approach is proposed by Levy and Turnbull (1992). They advanced the following analytic approximation: 1 2

E∗ [max[M(tN ) − K, 0]] = eα+ 2 ν N(d1 ) − KN(d2 ) 2

) and d2 = d1 − ν. where d1 = (α−ln(K)+ν ν If M(t) is distributed as its corresponding geometric average, i.e.

G(t) = [S(ti )S(ti+1 )S(ti+2 ), . . . , S(tN )](1/(N+1))

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then α and ν correspond to the mean and standard deviation of ln(G(t)) which are not easily calculated. The approaches presented here are attempts to approximate the risk-neutral probability distribution of the arithmetic mean price. We present now the different approches to the valuation of an average value option: the approaches of Kemma and Vorst, KV (1990), Turnbull and Wakeman, TW (1991), Conze and Viswanathan, CV (1991) and Curran (1992).

2.2.1. The Kemma and Vorst Approach Using arbitrage arguments, Kemma and Vorst (1990) presented a dynamic hedging strategy, from which the value of the average option can be obtained. They showed that the average option’s price is always less or equal to the price of a standard European option. The average option price Using Monte-Carlo simulation methods, they calculated the price of an arithmetic average option and presented the following formula for the option value when the geometric mean is used:    1 d∗ E[max(G(T ) − K, 0)] = e S(T0 )N(d) − KN d − σ (T − T0 ) 3 where ∗

d =

ln

 S(T0 ) K

+ 21 (r + 16 σ 2 )(T − T0 )

σ 13 (T − T0 )

S(Ti ) is the stock price at different instants Ti , for i  = 0, . . . , n, G(T) is the geometric average for S with G(T) = [ ni=0 S(Ti )](1/(n+1)) . A geometric average rate option The geometric average option can be priced in the Kemma and Vorst (1990) model using the following formula: c = Se(bA −r)TN(d1 ) − Ke−rTN(d2 ) with



 S 1 2 1 + bA + σ T d1 = ln √ K 2 σA T √ σ d2 = d1 − σA T , σA = √ . T 08:36:40.

and

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The adjusted cost of carry is given by bA = 21 (b − formula is:

σ2 ). 6

For the put, the

c = −Se(bA −r)TN(−d1 ) + Ke−rTN(−d2 ).

2.2.2. The Turnbull and Wakeman Approach Turnbull and Wakeman (1991) presented an algorithm for the valuation of European arithmetic options. When testing it against Monte-Carlo methods, they found it to be accurate and not time-consuming. They also derived closed-form solutions for the pricing of European geometric options. Contrary to the results of Kemma and Vorst (1990), Turnbull and Wakeman (1991) proved the following result: when the option’s maturity is less than the averaging period, the price of an average value option can be greater than that of a standard European option. Turnbull and Wakeman (1991) provided an approximation for the mean M1 and the variance M2 to be consistent with the moments of the arithmetic average. c = Se(bA −r)TN(d1 ) − Ke−rTN(d2 ) with:



 S 1 2 1 + bA + σ T2 d1 = ln √ K 2 σA T2

and 



ln(M2 ) ln(M1 ) − 2bA , bA = . T T We denote by T the original time to maturity and by τ the time to the beginning of the average period. The exact first and second moments for the arithmetic average are given by: ebT − ebτ M1 = b(T − τ) d2 = d1 − σA T2 ,

σA =

2

M2 =

2e2b+σ T (b + σ 2 )(2b + σ 2 )(T − τ)2

2 2e2b+σ τ eb(T −τ) 1 . + − b(T − τ)2 2b + σ 2 (b + σ 2 )

When the option is into the average period, the strike price is modified to: K = 08:36:40.

T T1 K − SA . T2 T2

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In this expression, the term SA corresponds to the average asset price during the realized time period T1 (T1 = T − T2 ).

2.2.3. The Conze and Viswanathan Approach Following Conze and Viswanathan, let MT1 ,T be the asset’s average value in the interval [T1 , T ], σ its volatility and r the riskless interest rate of interest. When T1 < T , the pay-off at maturity of a European average call is max[MT1 ,T − K, 0]. The pay-off of a European average put in the same context is max [K − MT1 ,T , 0]. If we denote by Z and  the following quantities: T

1 − (T −T

2

T (T −T1 )

r( 2(TT−T ) −T)+σ 2

3T1 −T T2 12 (T −T1 )2

Z = MT1 ,0 S0 e

T and  = √σ3 T −T then, using standard arbitrage arguments, the average 1 call value when T1 ≤ 0, is 1)

1

C = ZN(d1 ) − Ke−rTN(d2 ) √ d2 =  T .

Z ln( K ) + rT + 21 2 T , d1 = √  T The average put value is

P = −ZN(−d1 ) + Ke−rT N(−d2 ) Z √ ln K + rT + 21 2 T , d2 =  T . d1 = √  T  In the same context, let Z and  to be the following expressions: 

Z = S0 e

−r



T −T1 2



−σ 2



T −T1 12



,

and

σ 1   = √ √ T + 2T1 . 3 T When T1 ≥ 0, the average call value is C = Z N(d  1 ) − Ke−rTN(d  2 )

d1 =

ln

  Z K



+ rT + 21  2 T , √  T

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The average put value is P  = −Z N(−d1 ) + Ke−rTN(−d2 )

d1

=

ln

  Z K



+ rT + 21  2 T , √  T

√ d2 = d1 −  T .

2.2.4. The Curran Approach Curran (1992) proposed a technique known as “conditioning” in the valuation of average options. When implementing this technique, the integration is performed across all possible geometric mean prices for which the probability density is simple. The price in the absence of a cost of carry Using the expected pay-off, conditional on that geometric mean price, the following formula is presented for an average Asian option:     n (µ − ln K∗ ) σxi −rT 1 (µi + 21 σi2 ) P =e e N + n i=1 σx σx    (µ − ln(K∗ )) − KN σx 2

with µi = ln(S0 ) + (r − d − σ2 )ti µ = ln(S0 ) + (r − d − (n − 1)(t/2)σi2 = σ 2 )[t1 + t(i − 1)],    i(i − 1) 2 2 σxi = σ t1 + t (i − 1) − 2n    (n − 1)(2n − 1) σx2 = σ 2 t1 + t 6n K∗ = 2K −

n 

e

σ2 )(t1 2

+



 σx2 σx2 µi + 2i (ln(K)−µ) + σi2 + 2i σx

2σx

i=1

where n is the number of averaging points, t1 is the time of first averaging point and t is the time between averaging points. It seems that the geometric conditioning method, gives better results than the previous methods.

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The price in the presence of a cost of carry In the presence of a cost of carry b, the price of an Asian option based on the geometric conditioning approach can be written as:    (µ − ln K∗ ) σxi −rT 1 n (µi + 21 σi2 ) N +  e P =e σx σx n i=1  ∗  (µ − ln(K )) − KN σx with



σ2 µi = ln(S0 ) + b − 2

σ2 µ = ln(S0 ) + b − 2



ti

t t1 + (n − 1) 2

σi2 = σ 2 [t1 + t(i − 1)]  σx2i

=σ

2



i(i − 1) t1 + t (i − 1) − 2n



   (n − 1)(2n − 1) σx2 = σ 2 t1 + t 6n K∗ = 2K −

n 

e

 

σx2 σx2 µi + 2i (ln(K)−µ) + σi2 + 2i σx

2σx

.

i=1

The two modifications with respect to the previous formula appear in the terms µi and µ.

2.2.5. The Bouaziz, Briys and Crouhy Approach Bouaziz, Briys and Crouhy (1994) present a “closed-form solution” for a European Asian option whose strike price is an average. The formula applies to “plain vanilla” average options (those for which the time interval taken into account for the strike average calculation is the life of the option) and forward-starting average options. The formula relies upon a slight linear approximation. Although some previous contributions in the literature

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already use approximation techniques, this approach derives a formal upper bound to the approximation error. It seems that the closed-form solution performs quite well and is obviously computationally efficient. In this model, the Asian option is written on a generic underlying asset with a maturity date T . The option is forward-starting since its strike is computed as an arithmetic average of the underlying asset prices over the period [T − A, T ]  1 T K= S(u)du A T −A where A is a time instant after date 0 of issuance of the option. The case of a plain vanilla Asian option corresponds to T − A = 0. The price of the underlying asset is generated by the familiar equation: dS = µ dt + σ dz. S The price of the forward-starting Asian option at time t, Ce (S, K, t) is given by Ce (S, K, t) = e−r(T −t) E[(ST − K)+ |Ft ] where K is defined as before and E(.|Ft ) is the conditional expectation operator with respect to the transformed risk-neural probability measure Q which replaces P. Two time windows are considered: [0, T − A] and [T − A, A]. In the time window prior to T − A, applying the iterated conditional expectations (see Duffie, 1992) to this last equation gives Ce (S, K, t) = e−(r)(T −t) E[E[(ST − K)+ |FT −A ]|Ft ]. At this level Taylor expansions and a linearization are used. After some computations, the closed form solution to the pricing of the Asian option over [0, T − A] is found to be √ √  √   3ˆr A σ 2 A −3ˆrA/8σ 2 r −rA Aˆ e C (S, K, t) = St e N + e 2 2σ 6π 2

2

r , and ν = Aσ3 . with rˆ = r − σ2 , m = Aˆ 2 In the time window posterior to T − A, the strike price is gradually revealed and investors know the sequence [ST −A , ST ] and Mt which is equal

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to A1

t T −A

359

S(u)du. The strike price can be split into components as of time t:    1 t 1 T 1 t S(u)du = S(u)du + S(u)du. A T −A A T −A A t

The pricing problem can be reformulated as follows: 

+     1 T −r(T −t) E ST − Mt − S(u)du Ft . C(S, K, t) = e A t Using Taylor expansions and the same linearization procedure gives the desired formula:  

 ν − m2 ν m −r(T −t) 2 C(S, K, t) = St e e mN √ + 2π ν −t) , and ν = Aσ3 . for t > T − A with m = 1 − TA−t + r(T − t) − rˆ (T2A The corresponding put formula is given by   1 − e−r(T −t) −r(T −t) P(St , K, t) = C(S, K, t) + Mt e − 1 St . + A 2

2

2.3. Asian Options and Path-Dependent Quantities in an Integral Representation Consider the standard diffusion process for an underlying asset S described by a log-normal random walk: dS = µS dt + σS dX. Consider a derivative security for which the pay-off at maturity date T depends on the path followed by the underlying asset between time zero and maturity. It is possible to represent the path-dependent quantity by an integral of a function f(S, t). Assume for example the following representation T I(T ) = 0 f(S, τ)dτ. Assume that the derivative asset price is a function of the underlying asset S, time t and a function of an independent variable I called the state variable. It is also possible to know the stochastic differt ential equation satisfied by I by incrementing t by dt in I(t) = 0 f(S, τ)dτ to obtain dI = f(S, t)dt. For an Asian option with a pay-off depending on the  taverage value of the asset, the arithmetic average can be written as I = 0 S dτ. The pay-off of this arithmetic avarage option can be written

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as max[ TI − S, 0]. For an Asian option, the geometric average value of the t asset can be written as I = 0 log(S)dτ. Continuous sampling: The pricing equation Consider the pricing of a derivative security for which the pay-off depends on S, I and time. Following the analysis in Willmott (1998), it is possible to construct a hedge portfolio using a long position in the derivative security and a short position in the underlying asset = V(S, I, t) − S. Over a small interval of time, the change in value of this hedging portfolio can be written as

∂V 1 2 2 ∂2 V ∂V ∂V d = + σ S dI + −  dS. dt + ∂t 2 ∂S 2 ∂I ∂S When  = ∂V and dI = f(S, t)dt, then the change in portfolio value for ∂S the hedging purpose can be written as

∂V 1 2 2 ∂2 V ∂V dt. + f(S, t) + σ S d = ∂S 2 ∂I ∂t 2 Since the change in value is riskless, it must earn the riskless rate plus the information costs on the assets. This gives the following equation: ∂V 1 ∂2 V ∂V ∂V + (r + λS )S − (r + λV )V = 0 + σ 2 S 2 2 + f(S, t) 2 ∂S ∂I ∂S ∂t where λi refers to information costs. This equation must be solved under the following condition: V(S, I, T ) = P(S, I ). Example t For an arithmetic Asian option, we have: I = 0 S dτ. The following equation must be solved to obtain the value of this option under incomplete information: ∂V 1 ∂2 V ∂V ∂V + σ2S2 2 + S + (r + λS )S − (r + λV )V = 0. ∂t 2 ∂S ∂I ∂S Tables 1–4 provide some simulations of Asian option values and the corresponding Greek letters. The reader can comment on the different tables.

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Table 1:

361

Simulations of the values of discrete geometric average calls.

S

Price

Delta

Gamma

Vega

Theta

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

0.67982 1.45815 2.73786 4.61182 7.11101 10.20411 13.81397 17.84061 22.18204

0.11415 0.20191 0.31334 0.43750 0.56126 0.67352 0.76722 0.84015 0.89352

0.01531 0.02068 0.02423 0.02520 0.02370 0.02036 0.01625 0.01215 0.00859

0.12546 0.19298 0.25541 0.29743 0.31028 0.29414 0.25620 0.20665 0.15496

−0.00139 −0.00204 −0.00252 −0.00264 −0.00232 −0.00159 −0.00060 −0.00050 −0.00157

S = 100, K = 100, t = 27/12/2003, T = 27/12/2004, r = 2%, σ = 20%, start of averaging period = 27/06/2004, realized average price = 80.

Table 2:

Simulations of the values of discrete geometric average calls.

S

Price

Delta

Gamma

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

5.68881 8.84946 12.62537 16.85791 21.39842 26.13054 30.97400 35.87854 40.81497

0.56126 0.69884 0.80627 0.88186 0.93056 0.95970 0.97608 0.98481 0.98926

0.02953 0.02408 0.01757 0.01167 0.00716 0.00411 0.00223 0.00115 0.00057

Vega 0.24823 0.22905 0.18579 0.13413 0.08625 0.04824 0.02117 0.00336 −0.00772

Theta −0.00185 −0.00109 −0.00005 0.00105 0.00202 0.00280 0.00340 0.00384 0.00417

S = 100, K = 80, t = 27/12/2003, T = 27/12/2004, r = 2%, σ = 20%, start of averaging period = 27/06/2004, realized average price = 80.

3. The Average Price Options: Analysis and Valuation Under Shadow Costs of Incomplete Information 3.1. Analysis of Average Price Options The pay-off of the average call option is c(A(t), K, t) = max[A(tN ) − K, 0]. The pay-off of the average put option is p(A(t), K, t) = max[K − A(tN ), 0]. 08:36:40.

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Table 3:

Simulations of the values of discrete geometric average puts.

S

Price

Delta

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

19.22733 15.03897 11.35199 8.25926 5.79177 3.91819 2.56136 1.62131 0.99606

−0.87919 −0.79143 −0.68000 −0.55583 −0.43208 −0.31982 −0.22611 −0.15319 −0.09981

Gamma

Vega

Theta

0.01531 0.02068 0.02423 0.02520 0.02370 0.02036 0.01625 0.01215 0.00859

0.13881 0.20716 0.27042 0.31328 0.32697 0.31165 0.27456 0.22584 0.17498

−0.00429 −0.00512 −0.00578 −0.00609 −0.00594 −0.00540 −0.00459 −0.00368 −0.00278

S = 100, K = 100, t = 27/12/2003, T = 27/12/2004, r = 2%, σ = 20%, start of averaging period = 27/06/2004, realized average price = 80.

Table 4:

Simulations of the values of discrete geometric average puts.

S

Price

Delta

80.00 85.00 90.00 95.00 100.00 105.00 110.00 115.00 120.00

4.63342 2.82738 1.63661 0.90246 0.47628 0.24171 0.11849 0.05634 0.02608

−0.43208 −0.29450 −0.18706 −0.11148 −0.06278 −0.03364 −0.01726 −0.00853 −0.00408

Gamma

Vega

Theta

0.02953 0.02408 0.01757 0.01167 0.00716 0.00411 0.00223 0.00115 0.00057

0.26157 0.24324 0.20081 0.14998 0.10293 0.06576 0.03952 0.02255 0.01230

−0.00476 −0.00418 −0.00331 −0.00240 −0.00161 −0.00101 −0.00059 −0.00033 −0.00018

S = 100, K = 80, t = 27/12/2003, T = 27/12/2004, r = 2%, σ = 20%, start of averaging period = 27/06/2004, realized average price = 80.

The put–call parity condition with shadow costs In the presence of shadow costs of incomplete information, the put–call parity condition can be written for tm ≤ t ≤ tm+1 as p(A(t), K, t) = c(A(t), K, t)   eg(H−ξ) [1 − eg(N−m)H ] −(r+λ)τ A(t)(m + 1) +S(t) −K +e (N + 1) (N + 1)(1 − egH ) where g = r + λS − d, 08:36:40.

ξ = (t − tm ),

τ = (tN − t).

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When t < t0 , this relationship becomes p(A(t), K, t) = c(A(t), K, t)   eg(t0 −t) − eg(t0 −t+H(N+1)) −(r+λ)τ −K . S(t) +e (N + 1)(1 − egH )

3.2. The Valuation Approaches Using the risk-neutral approach, the average option value is given by its expected pay-off discounted to the present: c(S(t), K, t) = e−(r+λ)τE∗ [max[M(tN ) − K, 0]] where E is the expectation operator conditioned on [A(t) − S(t)] at time t under the risk-adjusted density function. This means that µ is replaced by (r + λS − d) in the dynamics of the underlying asset price, i.e. dS(t) = (r + λS − d)S(t)dt + σS(t)dW. Hence, to value the average option, we have to determine the distribution of M(tN ). When f ∗ (.) is the density function, the average call value is c(S(t), K, t) = e−(r+λ)τE[max[M(tN ) − K, 0]]  ∞ [M(t) − K]f ∗ (w)dw. = K

An alternative approach is proposed by Levy and Turnbull (1992). They advanced the following analytic approximation: 1 2

E∗ [max[M(tN ) − K, 0]] = eα+ 2 ν N(d1 ) − KN(d2 ) where d1 =

(α − ln(K) + ν2 ) ν

and

d2 = d1 − ν.

3.2.1. The Extension of the Kemma and Vorst Approach to Account for Information Costs The option value in the presence of shadow costs when the geometric mean is used:    1 ∗ E[max(G(T) − K, 0)] = ed tS(T0 )N(d) − KN d − σ (T − T0 ) 3

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where d ∗

=

ln

 S(T0 ) K

+

1 2



 r + λS + 16 σ 2 (T − T0 )/ σ 13 (T − T0 ) ,



S(Ti ) is the stock price at different instants  Ti , for i = 0, . . . , n, G(T) is the geometric average for S with G(T) = [ ni=0 S(Ti )](1/(n+1)) .

3.2.2. The Extension of the Conze and Viswanathan Approach to Account for Information Costs Following Conze and Viswanathan, let MT1 ,T be the asset’s average value in the interval [T1 , T ], σ its volatility and r the riskless interest rate of interest. When T1 < T , the pay-off at maturity of a European average call is max[MT1 ,T − K, 0]. The pay-off of a European average put in the same context is max[K − MT1 ,T , 0]. If we denote by Z and  the following quantities: T

1 − (T −T

T (T −T )

(r+λS )(

T2

−T)+σ 2

T2

3T1 −T

2(T −T1 ) 12 (T −T1 )2 Z = MT1 ,0 1 S0 1 e

T where λi refers to information cost. and  = √σ3 T −T 1 Using standard arbitrage arguments, the average call value when T1 ≤ 0, is )

C = Ze(λS −λ)TN(d1 ) − Ke−(r+λ)TN(d2 ) where

Z

+ (r + λS )T + 21 2 T √  T The average put value is d1 =

ln

K

√ and d2 =  T .

P = −Ze(λS −λ)TN(−d1 ) + Ke−(r+λ)TN(−d2 ) d1 =

Z ) + (r + λS )T + 21 2 T ln( K √  T √ d2 =  T .

In the same context, let Z and  to be the following expressions: Z = √ T −T1 2 T −T1 S0 e−(r+λS )( 2 )−σ ( 12 ) and  = √σ3 √1T T + 2T1 . When T1 ≥ 0, the average call value is C = Z e(λS −λ)TN(d  1 ) − Ke−(r+λ)TN(d  2 )

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where d1

=

ln

  Z K



+ (r + λS )T + 21  2 T , √  T

and

365

√ d2 = d1 −  T .

The average put value is P  = −Z e(λS −λ)TN(−d1 ) + Ke−(r+λ)TN(−d2 ) 

d1



ln( ZK ) + (r + λS )T + 21  2 T , = √  T

√ d2 = d1 −  T .

3.2.3. The Extension of the Curran Approach to Account for Information Costs Using the expected pay-off, conditional on that geometric mean price, the following formula is presented for an average Asian option:  n   (µ − ln K∗ ) σxi 1  (µi + 1 σ 2 ) −(r+λ)T 2 i N e + P =e n i=1 σx σx   (µ − ln(K∗ )) −KN σx with

σ2 ti µi = ln(S0 ) + r + λS − d − 2



σ2 t µ = ln(S0 ) + r + λS − d − t1 + (n − 1) 2 2 2 2 σi = σ [t1 + t(i − 1)]    i(i − 1) σx2i = σ 2 t1 + t (i − 1) − 2n    (n − 1)(2n − 1) 2 2 σx = σ t1 + t 6n ∗

K = 2K −

n 

e

 

σx2 σx µi + 2i (ln(K)−µ) + σi2 + 2i σx

2σx

i=1

where n is the number of averaging points, t1 is the time of first averaging point and t is the time between averaging points. It seems that the geometric conditioning method, gives better results than the previous methods. 08:36:40.

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3.2.4. The Extension of the Bouaziz, Briys and Crouhy Approach to Account for Information Costs In this model, the Asian option is written on a generic underlying asset with a maturity date T . The price of the forward-starting Asian option at time t, Ce (S, K, t) is given by Ce (S, K, t) = e−(r+λ)(T −t) E[(ST − K)+ |Ft ] where K is defined as in Eq. (21) and E(.|Ft ) is the conditional expectation operator with respect to the transformed risk-neutral probability measure Q which replaces P. Two time windows are considered: [0, T − A] and [T − A, A]. In the time window prior to T − A, applying the iterated conditional expectations (see Duffie, 1992) to this last equation gives Ce (S, K, t) = e−(r+λ)(T −t) E[E[(ST − K)+ |FT −A ]|Ft ]. At this level Taylor expansions and a linearization are used. After some computations, the closed form solution to the pricing of the Asian option over [0, T − A] is found to be: √ √  √   2A 3ˆ r A σ Aˆ r 2 Ce (S, K, t) = St e−(r+λ)A N + e−3ˆrA/8σ 2 2σ 6π with σ2 Aˆr Aσ 2 , m= , and ν = . 2 2 3 The pricing problem can be reformulated as follows: 

+    T 1  C(S, K, t) = e−(r+λ)(T −t) E ST − Mt − S(u)du  Ft .  A t rˆ = r + λS −

Using Taylor expansions and the same linearization procedure gives the desired formula:  

 m ν − m2 ν −(r+λ)(T −t) C(S, K, t) = St e e 2 mN √ + 2π ν for t > T − A with m = 1 − ((T − t)/A) + r(T − t) − rˆ ((T − t)2 /2A), 2 and ν = Aσ3 . The corresponding put formula is given by   1 − e−(r+λS )(T −t) −(r+λ)(T −t) P(St , K, t) = C(S, K, t) + Mt e + − 1 St . A 08:36:40.

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4. Analysis andValuation of Flexible Asian Options Before valuing flexible Asian options, we introduce the flexible weighted average and flexible average geometric and arithmetic averages. Following Zhang (1995), the general weighted average (GWA) is given by GWA(n, α) =

n 

W(n, α, i)P(i)

i=1


for i = 1, 2, . . . , n with W(n, α, i) = iα ni=1 /iα where α: a weighting positive parameter, P(i): the ith observation. The main advantage of the GWA is its flexibility. The flexible geometric average, which  is a simple extension of the standard geometric average, is given by ni=0 Siw(i) with w(i) = W(n, α, i) where n is the number of observations and Si is the ith observation. Note that the log of the FGA corresponds to a flexible arithmetic average of the log of the underlying prices. The flexible arithmetic average, which is a simple
extension of the standard arithmetic average, is given by FAA(n) = ni=1 w(i)Sn .

4.1. Flexible Asian Options We use the following notations: T : option’s maturity date, t: current time, τ: time remaining to maturity, ξ: a binary indicator which is 1 (−1) for a call (a put), f, g, a: refer, respectively, to flexible, geometric and arithmetic, n: number of observations specified in the option contract, h: observation frequency or the time interval between two consecutive observations, j: number of observations already done, (n − 1)h: length of the averaging period, (τ − (n − i))h: the starting time of the averaging period, ρij : correlation coefficient between assets i and j. When j = 0, the averaging period does not start and τ > (n − 1)h. When j = n, τ = 0, and the option is at its maturity date. When 1 ≤ j < n, the option lies in the averaging period.

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At the option’s maturity date, the pay-off of the option based on the flexible geometric average is given by P gf = max[ξ fga (n) − ξK, 0]. Under the assumption of log-normality, the value of the flexible geometric Asian option is given by its expected pay-off discounted to the present under the appropriate probability. The general formula given in Zhang (1995) is:



fg f f f Cj = ξSAf (j)N ξdn−j + ξσ Tn−j − ξKe−rτN(ξdn−j ) f

f 1 2 f (Tµ,n−j −Tn−j )]

where Af (j) = Bf (j)e[−r(τ−Tµ,n−j )− 2 σ j i=1 ( S[τ−(n−i)h] )w(i) , for 1 ≤ j ≤ n S

, Bf (0) = 1, Bf (j) =

f

f

dn−j =

[ln(S/K) + (r − 21 σ 2 )Tµ,n−j + ln(Bf (j))]

f (σ Tn−j )

f Tµ,n−j

=

n 

w(i)[τ − (n − i)h]

i=j+1

f Tn−j

=

n 

w (i)[τ − (n − i)h] + 2 2

i=j+1

i−1 n  

w(i)w(k)[τ − (n − k)h].

i=j+2 k=j+1

In this formula, the weighted average of the returns of the already passed observations is Bf (j). f The term Tµ,n−j is assimilated to the mean function. f The term Tn−j is regarded as the effective variance function. This formula reduces to that in Kemma and Vorst (1990) and Turnbull and Wakeman (1991) for α = 0 and w(i) = 1/n for all i = 1 to n.

4.2. Approximating Flexible Arithmetic Asian Options At the maturity date, the pay-off of the option based on the flexible arithmetic average is given by P af = max[ξ faa (n) − ξK, 0].

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Using the approximations given above and discounting the option pay-off, the flexible arithmetic Asian option price is given by



fa f f fa Cj = ξSκnf Af (j)N ξdn−j + ξσ Tn−j − ξKe−rτ N(ξdn−j ) with f

f

[ln(Sκn /K) + (r − 21 σ 2 )Tµ,n−j + ln(Bf (j))] d (n − j) =

f (σ Tn−j ) f

f

where κn is approximated by 1 1 κnf = 1 + E(fn ) + [E(fn )2 + var(fn )] 2 4 with E(fn ) = ν2 var[i|w(i)]   n n−1 n   2 iw(i)[1 − w(i)] − 2 iw(i) w(j) +σ h i=1

i=1

j=i+1

f

mean function with flexible where E(n ) is the

weights w(i) for i = 1 to n, with var[i|w(i)] = ni=1 (i−M)2 w(i), M = ni=1 iw(i), ν = h(r− 21 σ 2 ) and f f f var[n ] = 2ν2 var[i|w(i)][E(n ) − 21 ν2 var[i|w(i)]] + 4σ 2 hQ − [E(n )]2
n

n−1 n 2 2 2 with Q = i=1 i(i − M) [w(i)] + 2 i=1 i(i − M) w(i) j=i+1 (i − M)w(j). Note that this formula can be obtained directly from the preceeding formula by replacing the current asset price by the approximation factor κf (j).

Summary Asian options are path-dependent options whose pay-off is based on an average. In some cases, the underlying asset of the option is an average and in other cases, the strike price is a floating one. The ‘Asian contagion’ has reached not only the currency and commodity options markets; it has spilled over into the equity market. Although the original use of averaging was in the commodity sector to prevent any market squeeze just

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before expiry, corporate managers quickly figured out the kind of poison pills that could result from an astute use of averaging. Next to staggered board elections, supermajority provisions and dual class recapitalization, one can now find the Asian warrant. Asian options are the appropriate hedging instruments for traders who want to transact continuously over finite time horizons. Indeed, Asian options provide an effective way of capping costs or placing floors on aggregate profits. We present different approches for the valuation of an average value option: the approaches of Kemma and Vorst (1990), Turnbull and Wakeman (1991), Conze and Viswanathan (1991) and Curran (1992). Using arbitrage arguments, Kemma and Vorst (1990) presented a dynamic hedging strategy, from which the value of the average option can be obtained. They showed that the average option’s price is always less or equal to the price of a standard European option. Turnbull and Wakeman (1991) presented an algorithm for the valuation of European arithmetic options. When testing it against Monte-Carlo methods, they found it to be accurate and not time-consuming. They also derived closed-form solutions for the pricing of European geometric options. Contrary to the results of Kemma and Vorst (1990), Turnbull and Wakeman (1991) proved the following result: when the option’s maturity is less than the averaging period, the price of an average value option can be greater than that of a standard European option. Curran (1992) proposed a technique known as “conditioning” in the valuation of average options. When implementing this technique, the integration is performed across all possible geometric mean prices for which the probability density is simple. In this chapter, Asian options are analyzed and valued. First, we propose the different approaches for the valuation of arithmetic and geometric Asian options. Second, we present the general context for the analysis and valuation of basket options. Finally, we introduce the concept of flexible arithmetic and geometric mean to the valuation of flexible arithmetic and geometric Asian options.

Questions 1. 2. 3. 4.

What are the specific features of average price options? What are the main valuation approaches for these options? What are the definitions of flexible Asian options? How are flexible arithmetic Asian options approximated?

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Bibliography Bellalah, M (2001). Market imperfections; information costs and the valuation of derivatives: some general results. International Journal of Finance, 13(3), 1895–1927. Bellalah, M, Ma Bellalah and R Portait (2001). The cost of capital in international finance. International Journal of Finance, 13(3), 1958–1973. Bellalah, M and JL Prigent (2001). Pricing standard and exotic options in the presence of a finite mixture of Gaussian distributions. International Journal of Finance, 13(3), 1975–2000. Bellalah, M, JL Prigent and C Villa (2001). Skew without skewness: asymmetric smiles; information costs and stochastic volatilities. International Journal of Finance, 13(2), 1826–1837. Bellalah, M and F Selmi (2001). On the quadratic criteria for hedging under transaction costs. International Journal of Finance, 13(3), 2001–2020. Bouaziz, L, E Briys and M Crouhy (1994). The pricing of forward-starting Asian options. Journal of Banking and Finance, 18(5), 823–839. Conze, A and R Viswanathan (1991). Path dependent options: the case of lookback options. Journal of Finance, 46, 1893–1907. Curran, M (1992). Beyond average intelligence. Risk, 5(10), 60–62. Duffie, D (1992). Dynamic Asset Pricing Theory, Princeton University Press. Kemma, AGZ and ACF Vorst (1990). A pricing method for options based on average asset values. Journal of Banking and Finance, 14,113–129. Levy, E and S Turnbull (1992). Average intelligence. Risk, 5(2), 53–59. Turnbull, S and L Wakeman (1991). A quick algorithm for pricing European average options. Journal of Financial and Quantitative Analysis, 28, 1–20. Vorst, T (1992). Analytic Boundaries of Average Exchange Rate Options, Econometric Institute, Erasmus. Willmott, P (1998). Derivatives, John Wiley and Sons. Zhang, PG (1995). Flexible arithmetic Asian options. Journal of Derivatives, 3 (Spring), 53–63.

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Chapter 10

Steps, Parisian and Static Hedging of Exotic Options

This chapter is organized as follows: 1. Section 2 reviews the main difficulties in using classic barrier options and proposes a general context for step options. The analysis is focused on simple arithmetic and linear step options, delayed barrier options and other occupation time derivatives. The analysis accounts for the effects of information costs. 2. Section 3 presents the framework for the analysis and valuation of Parisian options and cumulative Parisian options. The analysis is extended to account for the effects of information costs. 3. Section 4 develops static hedges for many exotic options using standard options. The method is based on a relationship between European options with different strike prices as in Carr, Ellis and Gupta (1998).

1. Introduction or over 15 years, the over-the-counter (OTC) markets have expanded F more rapidly than their exchange forebears. Outside of financial institutions, the natural users of OTC derivatives are corporate treasurers and investment managers. A great variety of instruments is now available. The growth in volume and variety of derivative instruments has been vertiginous. These instruments are fashioned from forward contracts and standard options. The OTC contracts are negotiated between two parties away from an organized exchange, even though a broker may act as an intermediary. These

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contracts are often tailored to meet the precise requirements of both the parties. Despite the tendency for banks to invent exotic options which may serve little economic purpose, it is surely indeniable that the characteristics of risk and return of options have proved to be an attractive building block for OTC transactions. A large variety of barrier options are traded in equity, foreign exchange and fixed income markets. These options have become popular over the last several years in over-the-counter options markets. The valuation of standard knock-in and knock-out options in closed-form formulas appear in Merton (1973), Rubinstein and Reiner (1991), Derman and Kani (1993), etc. Barrier options are attractive since they are often cheaper than vanilla options. By including a barrier provision, the investor can eliminate paying for some scenarios he feels are unlikely. Besides, the premium reduction can be substantial when volatility is high. However, it is important to note that several problems appear in the management of these options because of the discontinuity at the barrier level. A short-term price movement through the barrier can extinguish the option. This can lead to the loss of the entire investment. The standard barrier options present at least four main disadvantages. The first is that option buyers stand to loose their money due to a short-term price spike through the barrier. The second is the discontinuity of the delta around the barrier, which creates hedging problems for option dealers. The third is the emergence of a conflict of interest between option dealers and option buyers, leading to some manipulations. The fourth reason is the increased volatility around barrier level. These options can lead sometimes to market manipulation. If large positions of knock-out options are accumulated for a same level of a barrier, options sellers can attempt to drive the price of the underlying instrument through the barrier. This leads to massive losses for barrier options buyers. This situation appears in several markets. A new type of option, called the Parisian barrier option or Parisian option is introduced by Chesney, Jeanblanc and Yor (1995). The main difference between a Parisian option and a barrier option is that the holder of a Parisian option loses his option only when it stays long enough under or above a barrier. Carr, Ellis and Gupta (1998) propose a generalization result of the put–call symmetry (PCS) relationship. They develop a method for the valuation and static hedging of some exotic options. Following the analysis in Bowie and Carr (1994), static portfolios are created of standard options whose values match the payoffs of the option at maturity. The proposed PCS can be seen as an extension of the put–call parity.

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2. Step Options 2.1. Risk Management Problems with Classic Barrier Options The following examples illustrate a conflict of interest between investors and dealers. Example 1 According to the Wall Street Journal (May 5, 1995), knock-out options can roll even the mammoth foreign-exchange markets. In the last past year, many Japanese exporters moved to hedge against a falling dollar with currency options. The exporters chose these options because they were confident that the dollar would fall no further than a knock-out level of 95 yen. Once the dollar plunged through 95 yen, there is a substantial loss. The dollar then tumbled as the Japanese companies “which had lost their hedges, scrambled to cover” their large exposures by dumping dollars. Dealers think that pitched battles often erupt around knock-out barriers, with traders hollering across the trading floor of looming billion-dollar transactions. In a few minutes, all is over. Example 2 According to Derivatives Week (November 6, 1995), “Some U.S. players are keen to include a statement (in the standardized trade confirmation for barrier options) alerting counterparties to that fact they may be involved in other trades that could move the market and extinguish the barrier”. Such situations prompted market participants to appeal for regulation. Since the delta of a barrier option is discontinuous and the gamma explodes to infinity near the barrier, this creates severe hedging problems at that level. In general, dealers establish static positions in a series of standard options to approximate the hedge for a range of underlying asset prices. However, when the underlying asset price nears the barrier, this creates the rebalancing of static hedges. This leads to an increase in market volatility and in the cost of hedging barrier options. Several other examples of problems created by the discrete nature of barrier options are provided for example in the Wall Street Journal (February 15, 1995) regarding the manipulation in the Venezuelan bond market by U.S. institutions. A modification of the barrier provision to retain as much of the premium savings as possible and to achieve continuity at the barrier is a desirable property. To introduce finite knock-out rates in barrier options, we consider

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the step options. The model proposed by Linetsky (1999) for barrier options is an extension of a Brownian particle that reaches a prespecified barrier level. The extension considers a Brownian motion with killing at finite rate ρ below the barrier. This means
that the probability that a Brownian particle T survives until time T is exp(−ρ 0 H(B−St )dt) where H(x) is the Heaviside step function. The time integral corresponds to the occupation time below the barrier until time T , which is denoted by τB− times the killing rate ρ. This exponential can be viewed as a knock-out discount factor with a knock-out rate ρ. The finite knock-out rate can make the option payoff and the delta continuous at the barrier. Using this definition, Linetsky develops a pricing methodology for the family of occupation time derivatives parameterized by knock-out rates. These options are called step options or gradual knock-in and knockout options. Example 3 The payoff of a down-and-out proportional step call is: exp(−ρτB− ) max[ST − K, 0]. Example 4 The payoff of a simple step option with simple principal amortization below the barrier is: max[1 − ρτB− , 0], max[ST − K, 0]. The payoff of a delayed barrier options that knock out after the underlying spends a predetermined amount of time below the barrier is: 1(τB− B max[ST − K, 0]

(1)

where LT is the lowest price of the underlying asset between the contract inception t = 0 and expiration t = T with t in the interval (0, T ). The 12:15:56.

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indicator function 1LT is equal to 1 if LT > B (the barrier is never hit) and zero otherwise. The payoff of this down-and-out call in the presence of a knock-out rate ρ is: exp(−ρτB− ) max[ST − K, 0]

(2)

where τB− is the amount of time during the option life that the underlying asset price was lower than a prespecified barrier level B, or occupation time below the barrier until time T . The option lifetime can be written as: τB− + τB+ = T where the occupation time below (above) the barrier are given by:  T − H(B − St )dt τB =

(3)

(4a)

0

τB+ =



T

H(St − B)dt

(4b)

0

where H(x) is the Heaviside step function taking the value 1 if x ≥ 0 and zero otherwise. The payoff in Eq. (2) corresponds to proportional step options (or geometric or exponential step options). These options amortize − their principal based on the occupation time. The discount factor e−ρτB is the proportional knockout factor. When ρ → ∞, the payoff (2) corresponds to that of an otherwise identical standard barrier option (1). When ρ → 0, the payoff (2) corresponds to that of standard call. In the same context, the payoff of an up-and-out step call at expiration is: exp(−ρτB+ ) max[ST − K, 0].

(5a)

The payoff of a down-and-out step put at expiration is: exp(−ρτB− ) max[K − ST , 0].

(5b)

The payoff of an up-and-out step put at expiration is: exp(−ρτB+ ) max[K − ST , 0].

(5c)

The buyer of a down-and-out (up-and-out) proportional step option is penalized at rate ρ for the time spent by the underlying asset price below (above) the barrier. Let us define a 90% knock-out time TB− (TB+ ) as the occupation time below (above) the barrier needed to reduce the option principal by 12:15:56.

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90%. In this context, the buyer of the down-and-out step option receives 10% of an otherwise identical option if the underlying asset spent time TB− below the barrier during the contract’s life (e−ρTB = 0.1 and TB = ln( 10 )). ρ −ρ A daily knock-out factor β defined as exp( 250 ) can be used as a measure of the knock-out speed. The option payoff is discounted by this factor for each trading day the underlying asset spends below the barrier. The payoff (2) can be written now as −

βnB max[ST − K, 0]

(6)

n− B

where is defined as the total number of trading days the underlying asset price spent below the barrier during the option’s life. Note that the term β is 1 for standard options and zero for barrier options. The terms τB− and n− B represent occupation times measured in years and trading days, n−

B respectively, τB− = 250 . The advantages of step options lie in the ability to structure contracts with any desired knock-out rate. By choosing a finite rate, the holder assures himself that the option will never lose its entire value because of a short-term price spike through the barrier. The dealer has the possibility to hedge step options by trading the underlying asset because the option’s delta is continuous at the barrier. Step options with finite knock-out rates provide buyers and sellers with risk management advantages when compared to standard barrier options.

2.2.2. Simple Arithmetic and Linear Step Options −

The specific choice of the knock-out factor e−ρτB in Eq. (2) corresponds to a proportional principal amortization in the presence of a knock-out rate below the barrier. It is possible to choose a simple principal amortization: max[1 − ρτB− , 0].

(7)

In this case, the payoff of a down-and-out simple step call (or arithmetic or linear step call) is given by: max[1 − ρτB− , 0] max[ST − K, 0].

(8)

The knock-out time TB− of a simple step option can be defined as the minimum occupation time below the barrier required to reduce the option principal to zero or: TB− = 1/ρ. The ratio of knock-out time to the option life is: α=

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(9)

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Given the same fraction of the option principal lost in the first trading day below the barrier, it is expected that a simple step option will knock out faster than a proportional option. If ρ = 0.2 per trading day, this means that a simple step option will lose its entire principal in five days. A proportional step option will lose its principal more and more slowly because of the effect of the proportional amortization. Example Consider a six-month proportional option with an effective knock-out time of five days. Hence, after the first five days spent below the barrier, the option loses 90% of its principal even if there are several months left to expiration. A simple knock-out option will lose all of its principal over the first five trading days below the barrier. The other simple step knock-out options The payoff of an up-and-out simple step call is: max[1 − ρτB+ , 0] max[ST − K, 0].

(10a)

The payoff of a down-and-out simple step put is: max[1 − ρτB− , 0] max[K − ST , 0].

(10b)

The payoff of an up-and-out simple step put is: max[1 − ρτB+ , 0] max[K − ST , 0].

(10c)

The simple step knock-in options Simple step knock-in options are defined in such a way that the sum of an in option and the corresponding out option is equal to a standard option. The payoff of a down-and-in simple step call is: min[ρτB− , 1] max[ST − K, 0].

(11a)

The payoff of an up-and-in simple step call is: min[ρτB+ , 1] max[ST − K, 0].

(11b)

The payoff of a down-and-in simple step put is: min[ρτB− , 1] max[K − ST , 0].

(11c)

The payoff of an up-and-in simple step put is: min[ρτB+ , 1] max[K − ST , 0].

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The delayed barrier options An interesting example of an occupation time derivative is given by a downand-out call that knocks out when the occupation time below the barrier B exceeds a specified fraction α between 0 and 1 of the life of the option T . The payoff of a delayed barrier option or cumulative Parisian option is given by: 1(τB− L is: Cod (x, T ) ≤ xe−(δ+λ

C −λS )T

N(d1 ) − Ke−(r+λ

Cod (x, T ) ≥ xe−(δ+λ

N(d1 ) − Ke−(r+λ

C )T

C −λS )T

N(d2 )

C )T

N(d2 )

 −2 x N(y) − xe L  2−2 √ −δT −(r+λC )T x N(y − σ T ). + Ke e L −(λC −λS )T

The value of the put if x > L is: Pod (x, T ) ≤ Ke−(r+λ

N(−d2 ) − xe−(δ+λ

C −λS )T

N(−d1 )

Pod (x, T ) ≥ Ke−(r+λ

N(−d2 ) − xe−(δ+λ

N(−d1 )

C )T

C −λS )T

C )T

 2−2 √ x N(−y + σ T ) L  −2 −(λC −λS )T x N(−y) + xe L − Ke−δT e−(r+λ

C )T

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with 1 d1 = √ σ T √ d2 = d1 − σ T ,



 ln

=

x K



 σ2T 2  2  √ L 1 y = √ ln + σ T . xK σ T

+ (r + λS − δ)T +

1 r + λS , + 2 σ2

When D = 0, the Parisian option is equivalent to a knock out option. When D = T , the Parisian option is equivalent to a standard option.

3.3. The Valuation of Parisian Options The dynamics of the underlying asset under the risk neutral probability Q are given by: dSt = St [(r + λS − δ)dt + σ dWt ],

S0 = x

(37)

or    σ2 S t + σWt . St = x exp r + λ − δ − 2 If we denote by:   1 σ2 S m= r+λ −δ− , σ 2

  1 L b = ln σ x

then St = x exp(σ(mt + Wt )).

3.3.1. Parisian Out-Option Down-and-out option − Let us denote byH˜ L,D (S) the first time at which the age of an excursion below the level L for St is greater than or equal to D: − S H˜ L,D (S) = inf{t ≥ 0 | ISt T φ(xe 2

(41)

0,− − = Hb,D = inf{t ≥ 0 | IZt T (x exp(σZT ) − K) exp mZT − 2 Using the (r, m)-discounted value, the value of this option can be written as: + − ∗Cod (x, T ; K, L, D; r, δ) = EP [IHb,D >T (x exp(σZT ) − K) exp(mZT )].

b. Parisian down-and-out put In the same context, the value of the Parisian down-and-out put is: + − ∗Pod (x, T ; K, L, D; r, δ) = EP [IHb,D >T (K − x exp(σZT )) exp(mZT )].

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Up-and-out option For an up-and-out option, the (r, m)-discounted value is: σZT + EP [IHb,D ) exp(mZT )] >T φ(xe

where + = inf{t ≥ 0 | IZt >b (t − gb,t ) ≥ D} Hb,D gb,t = sup{u < t | Zu = b}.

3.3.2. Parisian In-Option There is a parity relationship between in and out options. If we denote the (r, m)-discounted value of a down-and-in call by + − ∗Cid (x, T ; K, L, D; r, δ) = EP [I{Hb,D ≤T } (x exp(σZT ) − K) exp(mZT )]

then, we have ∗Cod (x, T ; K, L, D; r, δ) = ∗BS(x, T ) − ∗Cid (x, T ; K, L, D; r, δ) (43) where the Black and Scholes price is     m2 C T EP [(x exp(σZT )−K)+ exp(mZT )]. BS(x, T ) = exp − r + λ + 2 The values of up-and-out and up-and-in calls: + + ∗Ciu (x, T ; K, L, D; r, δ) = EP [I(Hb,D ≤T ) (x exp(σZT ) − K) exp(mZT )]

verify the following relationship: ∗Cou (x, T ; K, L, D; r, δ) = ∗BS(x, T ) − ∗Ciu (x, T ; K, L, D; r, δ). The case for which x = L For the case x = L (or b = 0), the following notation is used for excursions below L (with x = L or b = 0): Co− (x, T ; K, D; r, δ) = e−(r+λ

EQ [(ST − K)+ IH,L,D,(S)>T ].

C )T

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Or using the notation (42): − ∗Co− (x, T ; K, D; r, δ) = EP [(x exp(σZT ) − K)+ exp(mZT )I(H0,D >T ) ]. (44)

This case is referred to as a zero down-and-out call. For excursions above L, we have: Co+ (x, T ; K, D; r, δ) = e−(r+λ

EQ [(ST − K)+ I(H˜ +

C )T

L,D (S)>T )

].

(45)

Case b > 0 When b > 0 (or L > x), this reduces to the case b = 0. The first hitting time of b is Tb = inf{t ≥ 0 | Bt = b}. Its law is: µb (du) = 

b2

be− 2u √ 2πu3



du and its Laplace transform is:

γ2 E exp − Tb 2

 = exp −bγ,

for γ ≥ 0.

Following Chesney et al. (1995), the (r, m)-discounted value of a down-andout call is expressed in terms of zero down-and-out calls in the following proposition:  D d ∗Co (x, T ; K, L, D; r, δ) = µb (du) ∗ Co− (x, T − u; K, D; r, δ) 0

where: + − ∗Co− (x, T ; K, D; r, δ) = EP [IH0,D >T (x exp(σZT ) − K) exp(mZT )].

When b > 0 and Tb ≤ T , then the (r, m)-discounted value of an up-andin call is expressed in terms of zero down-and-in calls in the following proposition:  T u ∗Ci (x, T ; K, L, D; r, δ) = µb (du) ∗ Ci+ (x, T − u; K, D; r, δ) 0

where: + + ∗Ci+ (x, T ) = EP [I{H0,D ≤T } (x exp(σZT ) − K) exp(mZT )]

is a zero up-and-in call.

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Case b < 0 In this case, the same approach gives the following formulae:  T d ∗Ci (x, T ; K, L, D; r, δ) = µb (du) ∗ Ci− (x, T − u; K, D; r, δ) 0

where:

 + − ∗Ci− (x, T ) = EP I{H0,D (x exp(σZ ) − K) exp(mZ ) . T T ≤T }

4. Static Hedging of Exotic Options 4.1. The Put–Call Symmetry We denote by C(K), P(K) the prices of European call and put options with a strike price K at time 0. We denote by B the price of a pure discount bond paying 1 at the maturity date T . For a forward delivery, at time T , the put–call parity can be written as: C(K) = [F − K]B + P(K).

(46)

Certain restrictions are imposed on the process of the underlying asset price whose drift is assumed to be zero. The volatility of the forward price is a known function of the forward price and time t: σ(Ft , t) = σ(F 2 /Ft , t),

for all Ft  0

and

t ∈ [0, T ]

where F is the current forward price. The symmetry condition is satisfied in Black (1976) model. We denote by Xt = ln(Ft /F) and let v(x, t) = σ(Ft , t). The equivalent condition is given by v(x, t) = v(−x, t) for all x ∈
and t ∈ [0, T ]. Carr et al. (1998) show that in frictionless markets, with no arbitrage, zero drift, the following relationship (PCS) holds: C(K)K−1/2 = P(H)H −1/2

(47)

where the geometric mean of the call strike K and the put strike H is the forward price F : (KH)−1/2 = F . Carr et al. (1998) illustrate the PCS result using the following example. When the forward price is 12, a call struck at 16 has the same value as 4/3 puts struck at 9. The PCS relation shows that a call struck at twice the current forward price has twice the value of a put struck at half the current forward.

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4.2. Single Barrier Options Path-dependent options with a single barrier can be represented in terms of standard options. This result can be obtained using the put–call symmetry. Consider the valuation and hedging of knockout calls. These calls are knocked out the first time the underlying hits a specified barrier. Knock ins become standard calls when the barrier is hit, otherwise expire worthless. The following parity relationship applies between in and out options: OC(K, H) = C(K) − IC(K, H)

(48)

where IC(K, H) is an in call and OC(K, H) is an out call with a strike K and a barrier H.

4.2.1. Down-and-Out Calls A down-and-out call (DOC) with strike K and barrier H < K is worthless when H is hit at any time during the option life. If the barrier is not hit before maturity, the payoff is equal to that of a standard call struck at K. It is possible to hedge the down-and-out call by matching its terminal payoff and the payoff along the barrier.The terminal payoff is matched by buying a standard call, C(K). Now, the option values along the barrier must be studied. When F = H, the DOC is worthless while the current hedge C(K) has positive value. In this case, we must sell off an instrument with the same value as the European call when the forward price is at the barrier. Using PCS when F = H, this gives: C(K) = KH −1 P(H 2 K−1 ). This needs to write KH −1 European puts struck at H 2 K−1 . Hence, the complete replicating portfolio for a DOC is a buy-and-hold strategy in standard options which is bought at the initiation of the option: DOC(K, H) = C(K) − KH −1 P(H 2 K−1 ),

H ≺ K.

If the barrier is hit before the maturity date, the replicating portfolio must be liquidated with PCS guaranteeing that the proceeds from selling a call are offset by the cost of buying back the puts. If the barrier is not hit before maturity, the written puts have no value since H 2 K−1 < H when H < K. The long call provides the desired payoff.

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4.2.2. Up-and-Out Calls The knockout barrier of an up-and-out call (UOC) is set above the current forward price. When (H ≤ K), the UOC is worthless since it is knocked out before it can have a positive payoff. The UOC has an intrinsic value before it knocks out when H > K. The replicating portfolio comprises a European call struck at K. To obtain zero value at the barrier level, Eqs. (46) and (48) are used with up and in securities: UOC(K, H) = C(K) − UIP(K, H) − (H − K)UIB(H),

H  K, F. (49)

By definition, the up-and-in bond UIB(B) pays 1 dollar at maturity if the barrier H has been hit before then. The main advantage in representing an up-and-out call in terms of up-and-in puts and bonds is that Eq. (49) applies for all continuous time processes for the underlying asset. The PCS can be applied to show that the UIP(K, H) can be replicated by KH −1 calls struck at H 2 K−1 . Carr et al. (1998) show that an UIB(H) can be replicated by buying two binary calls (BC) struck at H and H −1 European calls struck at H: UIB(H) = 2BC(H) + H −1 C(H),

H  F.

Recall that a binary call pays one dollar at maturity if the underlying asset finishes above H then. Equation (49) can be written in terms of standard and binary calls: UOC(K, H) = C(K) − KH −1 C(H 2 K−1) − (H − K)[2BC(H) + H −1 C(H)],

H  K, F.

It is also well known that binary calls can be synthesized by an infinite number of vertical spreads of standard calls: BC(H) = lim n[C(H) − C(H + n−1 )]. n↑∞

Hence, the up-and-out call can be replicated using only standard calls.

4.3. Multiple Barrier Options 4.3.1. Double Knockout Calls Consider a call in the presence of two barriers. The call knocks out if either barrier is hit. Assume that the strike price and the forward price are 12:15:56.

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both between the two barriers. The parity relationship between a double knock-out call (O2 C) and a double knock in call (I2 C) knocks in if either barrier is hit): O2 C(K, L, H) = C(K) − I2 C(K, L, H) where L and H correspond to the lower and higher barrier. The payoff of a double knock out call can be replicated using portfolios of standard options. A double knock out O2 C(K, L, H) seems to be a combination of a DOC(K, L) and an UOC(K, H). The payoff of a double knock out call is zero if either barrier is hit and the standard call at maturity if neither barrier is hit. These payoffs can be obtained using a portfolio of a call knocking out at the lower barrier and a call knocking out at the higher barrier. This is true when the knock out of one option knocked out the other. The replicating portfolio for the double knock out O2 C(K, L, H) can be constructed using first a long standard call. If we expect that the forward price attains the lower barrier L before it reaches the higher barrier H then the value of the call C(K) can be nullified along the barrier L by selling KL−1 puts struck at L2 K−1 . Then the replicating portfolio is O2 C(K, L, H) ≈ C(K) − KL−1 P(L2 K−1 ). If we expect that the forward price attains the higher barrier H first, then the replicating portfolio is O2 C(K, L, H) ≈ C(K) − KH −1 C(H 2 K−1 ) − (H − K)[2BC(H) + H −1 C(H)]. Since it is impossible to know in advance which barrier will be hit first, the combination of the two portfolios gives O2 C(K, L, H) ≈ C(K) − DIC(K, L) − UIC(K, H) or O2 C(K, L, H) = C(K) − KL−1 P(L2 K−1 ) − KH −1 C(H 2 K−1 ) − (H − K)[2BC(H) + H −1 C(H)].

(50)

A gap call (put) is an asset-or-nothing option that pays the underlying asset price S if it is above (below) a strike price K, and zero otherwise. The two following parity relationships apply for gap calls and puts: GC(K) = K × BC(K) + C(K) and GP(K) = K × BP(K) − P(K). A binary call (put) is a cash-or-nothing option paying a dollar if the underlying asset price is above (below) the strike price and zero otherwise. 12:15:56.

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Carr et al. (1998) provide the following binary put–call symmetry relationship in the presence of no arbitrage, zero drift and a deterministic volatility: K1/2 BC(K) = GP(H)H −1/2 ,

H 1/2 BP(H) = GC(K)K−1/2

where the geometric mean of the binary call strike K and the binary put strike H is the forward price: (KH)1/2 = F . Using this results, Carr et al. (1998) provide the following approximation for the double knock out option: O2 C(K, L, H) = C(K) − L−1 (KP(L2 K−1 ) − HP(L2 KH −2 )) − H −1 (KC(H 2 K−1 ) − LC(H 2 KL−2 )) − (H − K)[2BC(H) + H −1 C(H) − 2L−1 GP(L2 H) − L−1 P(L2 H −1 )].

(51)

Even if Eq. (51) is a better approximation than Eq. (50), we can use a replicating portfolio for the double knock-out call as an infinite sum: O C(K, L, H) = C(K) − 2

∞

[L−1 (HL−1 )n (KP(L2 K−1 (LH −1 )2n )

n=0

− HP(K(LH −1 )2(n+1) )) + H −1 (LH −1 )n (KC(H 2 K−1 (HL−1 )2n ) − LC(K(HL−1 )2(n+1) )) + 2(H − K)(HL−1 )n [BC(H(HL−1 )2n ) − L−1 GP(L(LH −1 )2n+1 )] + (H − K)[H −1 (LH −1 )n C(H(HL−1 )2n ) − L−1 (HL−1 )n P(L(LH −1 )2n+1 )]]. A good approximation of the option value can be obtained for a small number n.

4.3.2. Roll-Down Calls A roll-down call (RDC) involves two barriers which are below the initial underlying asset price and the strike price. If the nearer barrier H1 is not hit before the maturity date, then the roll-down call shows the same terminal

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pay-off as a standard call struck at K0 . If the nearer barrier H1 is hit before the maturity date, then the strike price is rolled down to the barrier and a new out barrier becomes active at H2 < H1 . Carr et al. (1998) consider a decreasing sequence H1 , , H2 , . . . , Hn of positive barrier levels set below F > H1 and strike prices K0 , K1 , . . . , Kn with Ki > Hi . At initiation, the extended roll-down call can be represented as a function of down-and-out calls as n

ERDC(Ki , Hi ) = DOC(K0 , H1 ) + [DOC(Ki , Hi+1 ) i=1

− DOC(Ki , Hi )].

(52)

For example, a standard roll-down corresponds to the case n = 1 and K1 = H1 . The hedge can be implemented as follows. If the underlying asset does not hit the barrier H1 , the DOC(K0 , H1 ) gives the desired payoff, (FT − K0 )+ . In this case, the knock-out calls in the sum cancel each other. If the underlying asset hits the barrier H1 , the DOC(K0 , H1 ) disappears. This is the case also for the DOC(K0 , H1 ). When F = H1 , the position is n

[DOC(Ki , Hi+1 ) ERDC(Ki , Hi ) = DOC(K1 , H2 ) + i=2

− DOC(Ki , Hi )].

(53)

The DOC(Ki , Hi+1 ) gives the desired payoff between two barriers Hi and Hi+1 if the next barrier is never hit. In the same context, the DOCs in the sum roll down the strike to Ki+1 if the barrier is hit. When PCS is verified at each barrier, the extended roll-down call value for F > H1 is ERDC(Ki , Hi ) = C(K0 ) − K0 H1−1 P(H12 K0−1 ) n

−1 2 + [Ki Hi−1 P(Hi2 Ki−1 ) − Ki Hi+1 P(Hi+1 Ki−1 )]. i=1

(54) The replicating strategy shows that the investor holds a call struck at or above the highest untouched barrier and puts struck at or below the barrier. If the barrier is never hit, the call gives the desired payoff at expiry. The puts have no value. Hence, when a barrier Hi is hit for the first time, the strategy consists in selling the call struck at Ki−1 and buying back Ki−1 Hi−1 puts

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−1 struck at Hi2 Ki−1 , selling Ki Hi−1 puts struck at Hi2 Ki−1 and buying the call struck at Ki . PCS guarantees that these transitions are self-financing. When n = 1 and K1 = H1 in Eq. (54), this gives the standard roll-down call.

4.3.3. Ratchet Calls A ratchet call (RC) can be seen as an extended roll-down call, with strikes set at the barriers, which never knocks out completely. This can be done by having the only purpose of the lowest barrier be to ratchet down the strike. A ratchet call with an initial strike K0 can be synthesized by setting the strikes Ki in the ERDC(Ki , Hi ) equal to the barriers Hi for i between 1 and (n − 1). When the last spread of down-and-out calls DOC(Hn , Hn+1 ) − DOC(Hn , Hn ) is replaced in Eq. (52) with a down-and-in call DIC(Hn , Hn ), the ratchet call can be valued using barrier calls: n−1

RC(K0 , Hi ) = DOC(K0 , H1 ) + [DOC(Hi , Hi+1 ) i=1

− DOC(Hi , Hi )] + DIC(Hn , Hn ),

F  H1 .

When DOC(K, H) = C(K) − DIC(K, H) and DIC(H, H) = P(H) are replaced, this equation becomes RC(K0 , Hi ) = DOC(K0 , H1 ) +

n−1

[P(Hi )

i=1

− DIC(Hi , Hi+1 )] + P(Hn ),

F  H1 .

When the PCS is verified at each barrier, the ratchet call is given as a function of standard options: RC(K0 , Hi ) = C(K0 ) − K0 H1−1 P(H12 K0−1 ) +

n−1

−1 2 [P(Hi ) − Hi Hi+1 P(Hi+1 Hi ) + P(Hn )],

F  H1 .

i=1

Hedding with this replicating portfolio is equivalent to the extended roll-down call hedge: the position is changed at each barrier.

4.3.4. Lookback Calls A floating strike lookback call (LC) can be seen as a ratchet call, except that there is a continuum of rolldown barriers extending from the initial forward 12:15:56.

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price to the origin. This is elaborated in a way such that the strike price corresponds to the minimum price over the life of the option. The lower bound for a floating strike lookback call is given by: LC  RC(F, Hi ) = DOC(F, H1 ) +

n−1

[P(Hi )

i=1

− DIC(Hi , Hi+1 )] + P(Hn ). When PCS is verified at each barrier, the lower bound can be given in the form of standard options: LC  C(F) −

FH1−1 P(H12 F −1 )

n−1

+ [P(Hi ) i=1

−

−1 2 P(Hi+1 Hi−1 )] Hi Hi+1

+ P(Hn ).

The upper bound for a floating strike lookback call (LC) is given in terms of down-and-in bonds by LC  C(H1 ) − P(H1 ) +

n−1

[(Hi − Hi+1 )DIB(Hi )] + Hn DIB(Hn ).

i=1

When each barrier Hi is reached for the first time, the down-and-in bonds ratchet down the delivery price of the synthetic forward C(H − 1) − P(H1 ) by Hi − Hi+1 dollars. When PCS is verified at each barrier, it can be applied to represent the down-and-in bonds in terms of standard options.

Summary The creation of OTC market products is initially controlled by a few players, but then becomes available to a wide range of participants. The “commoditization” of instruments and the increased competition reduce profit margins. The inevitable result is the search for ever more products from which to make money. At some stage, products become more and more complex requiring more and more expenses in information acquisition. The problems of liquidity, transparency, commissions and charges are specific features of the OTC markets. Since the Parisian option is defined with respect to a specified barrier, it is possible to define Parisian out options and Parisian in options. For 12:15:56.

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Parisian out options, the holder of a down-and-out option loses his option if the underlying asset price reaches a level L and stays constantly below that level for a time period longer than a fixed number D referred to as the option window. Otherwise, the option holder receives a payoff φ(ST ) = (x − K)+ for a call ((K − x)+ for a put). The holder of an up-and-out option loses his option if there is an excursion above the level L which is older than D. For Parisian in options, the holder of a down-and-in option receives the payoff only if there is an excursion below the level L which is older than D. The holder of an up and in option loses receives the payoff only if there is an excursion above the level L which remains longer than D. This chapter has also studied a family of path-dependent options contingent on occupation times of the underlying asset process. Step options are parameterized by knock-out factors. Their payoffs are defined as payoffs of otherwise identical standard options discounted by knock out factors e−ρτB or max(1 − ρτB , 0) where τB refers to the occupation time below (above) a specified barrier level B for down (up) step contracts. When ρ approaches zero, step options become standard options. When ρ is very high, step options become standard barrier options. This option can be hedged continuously since its delta is a continuous function of the underlying asset price at the barrier. Step options provide an attractive alternative to standard barrier options. In fact, they offer desired premium savings and their deltas are continuous at the barrier. Hedging exotic options with a static portfolio of standard instruments simplifies the risk management of complex derivatives. In fact, the static portfolio is easy to construct and maintain over time. A static hedge can replicate the payoffs of the path-dependent option when carrying costs are zero. This technique applies to several path-dependent options. The fundamental result in the analysis of Carr et al. (1998) concerns the creation of a replicating portfolio using the put–call symmetry. This symmetry allows the design of simple portfolios to mimic the values of standard options along barriers. This allows an extension of the put–call parity to different strike prices.

Questions 1. What are the main risk management problems with classic barrier options? 2. What are the definitions of simple arithmetic and linear step options? 3. What are the definitions of general occupation time derivatives?

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4. What are the specific features of simple step options, delayed barrier options and other occupation time derivatives? 5. What are the definitions Parisian in and out options? 6. What are the specific features of cumulative Parisian options?

Bibliography Black, F (1976). Studies of stock price volatility changes. Proceedings of the American Statistical Association, 177–181. Bowie, J and P Carr (1994). Static simplicity. Risk, 7(9), 45–49. Carr, P, K Ellis and V Gupta (1998). Static hedging of exotic options. Journal of Finance, 53, 1165–1190. Chesney, M, M Jeanblanc and M Yor (1995). Brownian excursions and Parisian barrier options. Advances in Applied Probability, 29, 165–184. Derman, E and I Kani (1993). The ins and outs of barrier options. Goldman Sachs, Quantitative Strategies Research Notes. Linetsky, V (1999). Step options. Mathematical Finance, 9, 55–96. Merton, R (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183. Rubinstein, M and E Reiner (1991a). Breaking down the barriers. Risk, 4(8), 28–35. Rubinstein, M and E Reiner (1991b). Unscrambling the binary code. Risk, 4(8), 75–83.

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Chapter 11

Value at Risk: Basic Concepts and Applications in Risk Management

This chapter is organized as follows: 1. Section 2 presents the definition of the value at risk concept, the risk measurement framework and RiskMetrics. 2. Section 3 studies the statistical and probability foundation of the value at risk concept. 3. Section 4 presents a more advanced approach to value at risk using RiskMetrics. 4. Section 5 explains the VaR of nonlinear positions and presents the delta approximation and the delta–gamma approximation. 5. Section 6 studies the main limits and the validity of VaR estimates. 6. Section 7 develops some simple extensions of the VaR concept. 7. Section 8 presents the foundations of simulation methods and their specific features. In particular, we study historical simulation and structured Monte-Carlo methods. 8. Section 9 studies VaR methods and special events as market crashes. 9. Section 10 is devoted to the applications of the risk measures and the study of the reporting management system. 10. The Appendix explains the VaR concept for nonlinear positions.

1. Introduction the risks for a financial market participant or a financial instiM easuring tution has become the main focus of modern finance theory. The interest in measuring market risk and monitoring the positions is a consequence of securitization and the need to measure performance. Value-at-Risk is a Bellalah, Mondher. Exotic Derivatives and Risk: Theory, Extensions and403 Applications, World Scientific Publishing Co 12:16:03.

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measure of the maximum potential change in the value of a portfolio of a financial institution or financial instruments with a given probability over a prespecified horizon. The VaR concept gives an answer to the following question: How much the investor will lose with x% probability over a specified period of time? Riskmetrics is a set of tools allowing the users to estimate their exposure to market risk under the “Value-at-risk framework”. The market risk corresponds to the potential changes in value of a position as a consequence of changes in market prices. J.P. Morgan develops the RiskMetrics of VaR methodologies and publish them in a technical document via Internet. The integration of VaR in modern financial management requires that all positions to be marked to market and needs the estimation of future variability of the market value. VaR is used by dealers, non-financial corporations, institutional investors, bank and securities firm regulators and securities and exchange commissions. In 1995, the Basle Committee on Banking Supervision proposed allowing banks to determine their market risk capital requirements by implementing the bank’s VaR model. In this case, the committee specifies the parameters to load into the VaR model. Full valuation models are based on revaluing the portfolio on different scenarios. These scenarios can be generated using historical simulation, distributions if returns generated from a set of varaince–covariance matrixes, etc. This process is known as stress testing. These methods account for the whole distribution of returns instead of a single VaR number. However, they are time consuming. The second difference between VaR approaches is how market movements are estimated. RiskMetrics assumes at the beginning conditional normality to estimate market movement. However, this approach was refined and accounts now for higher order moments of the distribution of returns (kurtosis and leptokurtosis). Since most VaR models deal either with the non-normality of security returns or with their heteroscedasticity, Barone-Adesi, Bourgoin and Giannopoulos (1998) propose a modified historical simulation approach that accounts for both effects. The historical simulation is often based on a normal distribution of past returns. These returns are used for current asset prices to simulate their future returns. In this context, the VaR of a portfolio is determined after different paths have been explored. The possible clustering of large returns and the resulting fluctuations in daily volatility can lead to some errors in the determination of the confidence levels of VaR. This is

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true for all models that ignore clustering. In particular, this is true for the VaR calculation based on a variance–covariance matrix and Monte-Carlo methods. Many practioners think that the VaR number can be used to aid managers in the understanding of their risk position. In practice, the management of an individual trader’s book position requires more careful considerations of the risk parameter sensitivities than the single VaR number. This is important for the management of the option price sensitivities or Greek letters. The European Union’s Capital Adequacy Directive, recognizes VaR models as a valid model for the determination of capital requirements for foreign exchange risks and other market risk capital requirements. The U.S. Federal Reserve allows banks to use their internal VaR models to determine the capital they should hold for market risk over a given period. The SEC allows publicly traded firms in the U.S. to select VaR as one of three methods for calculating VaR and disclosing potential loss in earnings or cash flows. There is much debate about the appropriate uses of VaR and the way it is calculated. The value of a firm is given by its discounted expected cashflows. However, some firms can be valued directly with respect to their current values of assets and some other firms must be valued with respect to their growth opportunities. In this last case, the value of the firm may be closely related to its growth opportunities (or options).

2. Value at Risk and RiskMetrics 2.1. The Risk Measurement Framework and Risk Metrics Measuring the risks for a financial market participant or a financial institution has become the main focus of modern finance theory. The interest in measuring market risk and monitoring the positions is a consequence of securitization and the need to measure performance. The main advantages of VaR management are that: — it incorporates the mark-to-market approach uniformly, — it relies on a much shorter horizon forcast of market variables.

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The integration of VaR in modern financial management requires that all positions to be marked to market and needs the estimation of future variability of the market value.

2.2. The Definition of Risk Risk is often defined as the degree of uncertainty regarding future returns. This global definition of risk can be extended to define different kinds of risks according to the source of the underlying uncertainty. The operational risk indicates the possible errors in settling transactions or in instructing payments. The credit risk refers to the potential loss resulting from the inability of a firm to fund its illiquid assets. Market risk refers to the deviations of future earnings due to changes in market conditions.

2.3. Value-at-Risk: Definition Value-at-Risk is a measure of the maximum potential change in the value of a portfolio of a financial institution or financial instruments with a given probability over a prespecified horizon. The VaR concept gives an answer to the following question: How much the investor will lose with x% probability over a specified period of time? The following two examples are adapted from RiskMetrics. If an investor estimates that there is a 95% chance that the Euro/USD exchange rate will not fall by more than 2% of its current value over the next day, he can determine the maximum potential loss on for example, USD 100 million Euro/USD position. Example 1 Consider a USD-based firm which holds an Euro 105 million FX position. The manager wants to calculate the VaR over a one day horizon. He thinks that there is a 5% chance that the loss will be higher than what VaR projected. The exposure to market risk must be determined in a first step. The exposure corresponds to the market value of the position in the investor base currency (the U.S. firm). If the foreign exchange rate is 1.05 Euro/USD, the market value of the position is USD 100 million or (105/1.05). The VaR of the position in USD is determined in a second step. The VaR is given by 1.65 times the standard deviation. It is approximately equal to the market value of the position times the estimated volatility. If the estimated

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volatility is 0.55 %, then: FX risk = 100 million (1.65)(0.55%) = 907.500 dollars. Hence, in 95% of the time, the firm will not lose more than 907,500 over the next day. Example 2 Consider now a USD-based firm which holds an Euro 105 million position in a 10-year government bond. The manager wants to calculate the VaR over a one day horizon. He thinks that there is a 5% chance of understating the realized loss. In a first step, the exposure to market risk must be determined. The exposure corresponds to the interest rate risk on the bond and the FX risk resulting from the Euro. The market value of the position is still USD 100 million which is at risk to the two market risk factors (interest rates, exchange rates, etc.) The estimation of the interest rate risk in a second step needs the calculation of the standard deviation on a 10-year European bond in Euro. Suppose this standard deviation is equal to 0.58%. In this case, we have: Interest rate risk = 100 million dollars (1.65)(0.58%) = 957,000 dollars. The estimation of the FX Risk is given by: FX risk = 100 million dollars (1.65)(0.55) = 907,500 dollars. It is important to note that the total risk of the bond must account for the return on the Euro/USD exchange rate and the return on the 10-year European bond. Assume the correlation is equal to −0.3. The total risk of the position is:
2 2 VaR = σIrate + σFX + 2ρIrate,FX σIrate σFX or VaR =

 0.9572 + 0.90752 + 2(−0.3)(0.957)(0.9075).

3. Statistical and Probability Foundation Is the distribution of return constant over time? The time series reveal volatility clustering since periods of large returns are clustered and distinct from periods of small returns, which are also

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clustered. This shows clearly a change in variances referred to as heteroscedasticity. Researchers investigate alternative modeling methods other than the normal distribution. These models use either unconditional (timeindependent) or conditional distributions of returns (time-dependent). The first class corresponds for example to the normal distribution, finitevariance symmetric and asymmetric stable Paretian distributions, etc. The second class of models corresponds to GARCH and stochastic volatility models for example.

3.1. Properties of the Normal Distribution The normal distribution is a parametric distribution because it depends on several parameters. In RiskMetrics, the VaR of a single asset (at time t) is given by VaRt = [1 − e(−1.65σt|t−1 ) ]Vt−1 . The common approximation used is VaRt = 1.65σt|t−1 Vt−1 with: Vt−1 : marked-to-market value of the instrument, σt|t−1 : standard deviation of continuously compounded returns for time t made at time t − 1. The normal probability density function for a random variable rt is: (rt −µ) 2 1 f(rt ) =  e−0.5[ σ ] 2 (2πσ )

with: µ = E[Rt ] : the mean of the random variable, σ 2 = E[(Rt − µ)2 ] : the variance of the random variable. The skewness characterizes the asymmetry around the mean: s3 = E[(Rt − µ)3 ]. It is equal to zero for a normal distribution. The skewness coefficient is given by s3 /σ 3 . The kurtosis characterizes the flatness of a given distribution: s4 = E[(Rt − µ)4 ].

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The kurtosis coefficient is given by (s4 /σ 4 ). It is equal to 3 for a normal distribution.

3.2. Using Percentiles or Quantiles to Measure Market Risk The percentile or quantile corresponds to a magnitude (the dollar amount at risk) and is given by the following formula for a continuous probability α distribution: p = −∞ f(r)dr; where f(r) corresponds to the probability density function. The fifth percentile is the value such that 95% of the observations lie above it. If we define r˜t as r˜t = (rt − µt )/σt , then r˜t is normal with mean 0 and a unit variance. Example Suppose an investor wants to find the 5% percentile of rt under the normal distribution. Since probability (r˜t < −1.65) = 5%, or probability (r˜t = (rt −µt ) < −1.65) = 5%, then probability (rt < −1.65σt + µt ) = 5%. This σt equation says that there is a 5% probability that an observed return at time t is less than −1.65 times its standard deviation plus its mean. When µt = 0, we obtain the classic result for short term horizon VaR calculation: probability (rt < −1.65σt ) = 5%.

3.3. Aggregation in the Normal Model An important property of the normal distribution is that the sum of normal random variables is also normally distributed. Consider the case of a portfolio with three assets. The return on the portfolio is: rp,t = w1 r1,t + w2 r2,t + w3 r3,t . Each return can be described as a random walk: r1,t = µ1 σ1,t ξ1,t , r2,t = µ2 σ2,t ξ2,t , r3,t = µ3 σ3,t ξ3,t . If the ξt are multivariate normally (MVN) distributed then ξt follows MVN(µt , Rt ) where parameter matrix Rt is the correlation matrix of (ξ1,t , ξ2,t , ξ3,t ). In this case, the formula for the mean and the variance of the portfolio are given by: µp,t = w1 µ1 + w2 µ2 + w3 µ3 2 2 2 2 2 σp,t = w21 σp,t + w22 σp,t + w23 σp,t + 2w1 w2 σ1,2,t 2 2 + 2w2 w3 σ2,3,t . + 2w1 w3 σ1,3,t

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3.4. The Risk Metrics Model of Financial Returns The return variances are heteroscedastic and autocorrelated. Return covariances are autocorrelated. Consider a set of N securities, i : 1, . . . , N. Returns are generated according to the following model: ri,t = σi,t ξi,t , where ξi,t follows a N(0, 1) and the ξi,t follow an MVN(0, Rt ), where Rt is 2 an (N)(N) time dependent correlation matrix. The variance of each term σi,t and the correlation of returns ρij,t are a function of time. The property that the return distribution is normal given a time dependent mean and correlation matrix assumes that returns follow a conditional normal distribution (conditional on time).

3.5. The Choice of the Horizon Several models consider a horizon of one day and 95–99% confidence interval in the measurement of the amount of risk for the institution. This horizon assumes implicitly that markets and assets are very liquid and allow the different participants to unwind their positions in one day. This approximation is mainly valid for market trading activities. The main argument behind the use of a one day horizon is that this period can account for the risk control effects when it is multiplied by the square root of the horizon period. However, this is true only for products with linear payoffs such as forwards and swaps. In fact, the extrapolation may not be efficient for options. It is possible to use longer horizons and to implement scenario analysis and Monte-Carlo methods in what is known to be stress testing. However, since pricing models are based on some assumptions which might be invalid during market stress, this method can to lead to “inaccurate” results. The correlation between financial prices and aggregation Several VaR models use the historic correlations among the different risk factors with respect to the main results in modern portfolio theory. However, the historic correlations are unstable and other measures can be used. The following Tables 1 and 2 provide the VaR of a single asset at the 1% level.

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Table 1: VaR of a single asset at the 1% level. Alpha (%) 1 1 1 1 1 1 1

1-Alpha (%)

Z_alpha

Sigma t/(t − 1) (%)

V(t − 1)

VaR

99 99 99 99 99 99 99

2.3263 2.3263 2.3263 2.3263 2.3263 2.3263 2.3263

10 11 12 13 15 20 25

100 100 100 100 100 100 100

23.263 25.59 27.916 30.242 34.895 46.527 58.159

Table 2: VaR of a single asset at the 5% level. Alpha (%) 5 5 5 5 5 5 5

1-Alpha (%)

Z_alpha

Sigma t/(t − 1) (%)

V(t − 1)

VaR

95 95 95 95 95 95 95

1.6449 1.6449 1.6449 1.6449 1.6449 1.6449 1.6449

10 11 12 13 15 20 25

100 100 100 100 100 100 100

16.449 18.093 19.738 21.383 24.673 32.897 41.121

4. A More Advanced Approach to Value at Risk Using Risk Metrics The VaR corresponds to a number indicating the potential change in the future value of a given portfolio. In the process of calculating the VaR, the manager must specify the horizon for the calculation as well as the “degree of confidence” chosen. VaR calculations can also be done without resorting to the standard deviation.

4.1. The VaR of a Portfolio in Risk Metrics Consider a manager who wants to compute the VaR of a portfolio over a one-day with a 5% chance that the actual loss in the portfolio’s value is higher than the VaR estimate. In this case, the VaR calculation is performed in four steps.

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The first step determines the current value of the portfolio V0 on a markto-market basis. The second step determines the future value of the portfolio, V1 = V0 er˜ where r˜ refers to the portfolio’s return over the horizon. The third step estimates the one-day return on the portfolio, r˜ in a way such that there is a 5% chance that the actual return will be less than r˜ , i.e., probability (r < r˜ ) = 5%. The fourth step determines the portfolio’s future “worse case” value, V˜ 1 as: V˜ 1 = V0 er˜ . In this context, the VaR estimate is: V0 − V˜ 1 . This VaR estimate can also be written as: V0 (1 − er˜ ). When r˜ is small, er˜ is nearly equal to (1 + r˜ ). In this case, VaR is nearly V0 r˜ . RiskMetrics gives an estimation of r˜ . Example Consider a potfolio with a current marked-to-market value V0 = USD 700 million. The determination of the VaR requires first the one day forecast of the mean. J.P. Morgan assumes that this mean is zero over one day. Then, we must calculate the standard deviation of the returns of the portfolio. If the return on the portfolio is distributed conditionally normal, then it is assumed that the change in value of the portfolio is approximated by its delta. The other greeks can also be used to appreciate the change in value. The second approach involves creating a large number of possible rate scenarios and revalues the portfolio under each scenario. VaR is defined in this context as the fifth percentile of the distribution of value changes.

4.2. Alternative Approaches to Risk Estimation More than one VaR model is used in practice since practitioners have selected an approach based on their specific needs. The models implemented differ on the way changes in the values of a portfolio are estimated as a reaction to market movements. They differ also on the way the potential market movements are estimated. Basically, there are two approaches to estimate the change in the portfolio’s value as a result of market movements: the analytic approach and the simulation approach. The analytical sensitivity approach is based on the following equation: Estimated value change = f(position sensitivity factor, estimated rate/price change). 12:16:03.

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The position sensitivity factor describes the relationship between the value (the price) of an instrument and its underlying rate or price. The above equation can be written as: Estimated value change = position sensitivity factor (estimated rate/price change). This linear relationship estimates, for example, the change in the bond’s value from its duration. It estimates also the change in the option value from its delta. This initial approach in RiskMetrics has been extended to account for nonlinear relationships between the market value of an instrument and rate changes. This accounts, for example, for the gamma risk when analyzing the change in the option price. In this new version, the change in value can be written using simple approximations as: Estimated value change = (position sensitivity 1 × estimated rate/price change) + 0.5(position sensitivity 2 × estimated rate/price change)2 + · · · The analytical approach requires that positions be summarized in order to apply the estimated rate change. The aggregation process is referred to as mapping. There are at least two methods to approximate nonlinear VaR.

4.3. The Delta and Gamma Approximation In this method, the changes in the option value are estimated via a linear model. The approximation assumes that the exchange rate does not change significantly. This approximation results from the fact that the delta is a linear approximation of a nonlinear approximation between the option and its underlying asset.

4.3.1. The Delta Approximation The VaR of a portfolio of options can be determined using the “greeks”. The basic method uses an option pricing model to obtain the delta. This delta is used to determine the amount of a market factor that must be held to compensate for a change in the underlying asset price. The present value of the delta hedge position in the underlying is included in the determination of

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the portfolio variance. This method is efficient only for very small changes in the underlying asset price. In fact, the delta is a linear measure only for very small changes in the underlying asset price over very small intervals of time. Since the VaR is concerned with the effects of large changes in the underlying asset price, the linearity may lead to an inappropriate assesement of market risk measures. In this method, the changes in the option value are estimated via a linear model. The approximation assumes that the exchange rate does not change significantly. This approximation results from the fact that the delta is a linear approximation of a nonlinear approximation between the option and its underlying asset. The return on the above portfolio can be written as follows: rp = r1y + rEuro/USD + δrEuro/USD with: r1y : the price return on the one year European interest rates, rEuro/USD : the return on the Euro/USD exchange rate, and δ: the option’s delta. Assuming that the portfolio return is normally distributed, the VaR at the 95% level is:
2 + (1 + δ)2 + σ 2 VaR = 1.65 σ1y Euro/USD + 2(1 + δ)ρ1y,Euro/USD σ1y σEuro/USD .

4.3.2. The Delta–Gamma Approximation The approximation based on the gamma term accounts for nonlinear effects of changes in the spot rate. The introduction of the gamma term accounts for the nonsymmetric returns in the distribution. It reflects the skewness. Since the distribution is no more symmetric, we must use another number than 1.65 for the 95 percentile VaR. The estimation method is improved when the second derivatives of the delta (the gamma) is used in the risk measure. Since the option price function is nonlinear for different prices of the underlying asset, a risk measure including gamma may also lead to an inaccurate measure of market risk for significant changes in the underlying asset price. However, the simultaneous use of delta and gamma can improve risk estimation. The analytic framework for determining the VaR of a portfolio of options can adequately measure price level risk. It is possible with some difficulties to include the vega and the option time decay (theta) in the VaR calculation using an analytical framework. The approximation based on the gamma term 12:16:03.

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accounts for nonlinear effects of changes in the spot rate. The return on the above portfolio can be written as follows: rp = r1y + rEuro/USD + δrEuro/USD + 0.5PEuro/USD (rEuro/USD )2 where: : the gamma of the option, PEuro/USD : the value of the exchange rate dem/usd when the VaR forecast is made. The introduction of the gamma term accounts for the nonsymmetric returns in the distribution. It reflects the skewness. Since the distribution is no more symmetric, we must use another number than 1.65 for the 95% percentile VaR. In this case, it is possible to calculate the fifth percentile of the return distribution for rp by determining the first four moments of rp : the mean, the variance, the skewness and the kurtosis. Then, we find a distribution whose first four moments match those of rp ’s. Tables 3 to 5 illustrate the computation of VaR for a portfolio of 21 assets. Table 3: Weights, drifts, and volatility for a portfolio with 21 assets. Stocks

Weights (%)

Drift (%)

Volatility (%)

A B C D E F G H I J K L M N O P Q R S T

6.945 6.254 7.537 1.507 3.868 0.987 3.305 0.180 5.770 1.165 0.000 2.518 4.018 5.248 3.103 1.951 3.844 1.902 0.000 4.702

0.79 1.00 2.80 0.27 2.00 2.35 1.00 1.00 2.00 2.34 1.00 1.00 1.33 1.00 0.50 0.59 1.00 0.50 2.47 2.11

20.00 21.00 22.00 23.00 24.00 25.00 26.00 27.00 28.00 29.00 20.00 20.00 21.00 22.00 23.00 24.00 25.00 26.00 20.00 20.00

Risk free asset

35.198

0.00

0.00

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Table 4: VaR of the portfolio for a horizon of 292 days (1 year). Alpha

1-Alpha

Z_alpha

VAR

Horizon time (days)

5%

95%

1.6449

8%

292.00

5. The VaR and Nonlinear Positions Nonlinear positions correspond, for example, to a portfolio of options. RiskMetrics provides two approaches to compute VaR of nonlinear positions. The first is an analytical approximation. The second is a structured MonteCarlo simulation. The first approach is based on Taylor series expansion. This approximation gives a relationship between the return on the position and the return on the underlying rates. This approach assumes that the change in value of the portfolio is approximated by its delta. The other greeks can also be used to appreciate the change in value. The second approach involves creating a large number of possible rate scenarios and revalues the portfolio under each scenario. VaR is defined in this context as the fifth percentile of the distribution of value changes. There are at least two methods to approximate nonlinear VaR. For more details, the interested reader can refer to the Appendix.

6. The Limits and Validity of VaR Estimates Hoppe (1998) explains how variance-based statistical methods are variably unreliable and that the unreliability is related to the sample size in a counterintuitive manner and to the holding period. Several assumptions underpin the use of linear, variance-based statistics to describe the volatility of distributions of market returns. Research has proved that the assumption that distributions of raw market returns are normally and independently distributed (NID) is incorrect. The main attraction of standard deviation is that the properties of the normal distribution are easily understood. When the mean and the variance of a normal distribution are known, it is possible to give precise probability statements regarding the location of values in that distribution. In this context, it is possible to assert that the probability that a randomly selected value will be more or less than 1.645 standard deviations away from the mean is 0.10 with half of the probability 0.05 in each tail. The VaR estimates depend on this property of the normal distribution. Since the VaR estimates are concerned with extreme probabilities (1% or

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Correlation matrix for the portfolio. T5

T6

T7

T8

T9

T10

T11

1.000 0.086 0.116 0.114 0.108 0.167 0.064 0.119 0.037 0.108 0.191 0.114 0.094 0.055 0.066 0.072 0.051 0.060 0.189 0.099 0.000

0.086 1.000 0.096 0.094 0.089 0.137 0.051 0.098 0.028 0.089 0.161 0.093 0.076 0.041 0.051 0.056 0.038 0.045 0.160 0.080 0.000

0.116 0.096 1.000 0.137 0.130 0.200 0.074 0.142 0.040 0.130 0.234 0.136 0.111 0.060 0.075 0.082 0.055 0.066 0.232 0.117 0.000

0.114 0.094 0.137 1.000 0.144 0.222 0.082 0.158 0.044 0.144 0.336 0.151 0.123 0.065 0.082 0.090 0.061 0.072 0.259 0.129 0.000

108 0.089 0.130 0.144 1.000 0.234 0.087 0.166 0.047 0.152 0.295 0.159 0.130 0.069 0.087 0.095 0.064 0.076 0.273 0.136 0.000

0.167 0.137 0.200 0.222 0.234 1.000 0.146 0.280 0.079 0.255 0.493 0.268 0.219 0.118 0.148 0.162 0.110 0.130 0.458 0.230 0.000

0.064 0.051 0.074 0.082 0.087 0.146 1.000 0.126 0.041 0.115 0.214 0.121 0.100 0.057 0.070 0.076 0.054 0.063 0.201 0.103 0.000

0.119 0.098 0.142 0.158 0.166 0.280 0.126 1.000 0.072 0.226 0.428 0.236 0.194 0.107 0.132 0.145 0.099 0.117 0.403 0.201 0.000

0.037 0.028 0.040 0.044 0.047 0.079 0.041 0.072 1.000 0.072 0.126 0.075 0.064 0.041 0.047 0.051 0.039 0.043 0.119 0.064 0.000

0.108 0.089 0.130 0.144 0.152 0.255 0.115 0.226 0.072 1.000 0.429 0.238 0.196 0.109 0.134 0.147 0.101 0.119 0.406 0.200 0.000

0.191 0.161 0.234 0.336 0.295 0.493 0.214 0.428 0.126 0.429 1.000 0.479 0.391 0.207 0.261 0.288 0.192 0.229 0.831 0.400 0.000

(Continued )

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T16

T17

T18

T19

T20

T21

0.114 0.093 0.136 0.151 0.159 0.268 0.121 0.236 0.075 0.238 0.479 1.000 0.270 0.150 0.185 0.203 0.139 0.164 0.559 0.268 0.000

0.094 0.076 0.111 0.123 0.130 0.219 0.100 0.194 0.064 0.196 0.391 0.270 1.000 0.137 0.167 0.182 0.128 0.149 0.485 0.234 0.000

0.055 0.041 0.060 0.065 0.069 0.118 0.057 0.107 0.041 0.109 0.207 0.150 0.137 1.000 0.111 0.120 0.091 0.104 0.270 0.135 0.000

0.066 0.051 0.075 0.082 0.087 0.148 0.070 0.132 0.047 0.134 0.261 0.185 0.167 0.111 1.000 0.150 0.111 0.129 0.347 0.171 0.000

0.072 0.056 0.082 0.090 0.095 0.162 0.076 0.145 0.051 0.147 0.288 0.203 0.182 0.120 0.150 1.000 0.126 0.150 0.392 0.193 0.000

0.051 0.038 0.055 0.061 0.064 0.110 0.054 0.099 0.039 0.101 0.192 0.139 0.128 0.091 0.111 0.126 1.000 0.119 0.276 0.141 0.000

0.060 0.045 0.066 0.072 0.076 0.130 0.063 0.117 0.043 0.119 0.229 0.164 0.149 0.104 0.129 0.150 0.119 1.000 0.336 0.170 0.000

0.189 0.160 0.232 0.259 0.273 0.458 0.201 0.403 0.119 0.406 0.831 0.559 0.485 0.270 0.347 0.392 0.276 0.336 1.000 0.537 0.000

0.099 0.080 0.117 0.129 0.136 0.230 0.103 0.201 0.064 0.200 0.400 0.268 0.234 0.135 0.171 0.193 0.141 0.170 0.537 1.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

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T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 T14 T15 T16 T17 T18 T19 T20 T21

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5% for example), how robust is the standard deviation at the extremes in the face of violation of the main assumptions? Reliability of real standard deviations and VaR estimates The main question in risk management is: what is the best estimate of the risk today? Hoppe (1998) tested VaR estimates for a two-component portfolio composed of equal dollar-long positions in the S&P 500 and the 30-year US Treasury bond using the daily near-month futures price series from 1991 to 1996. The main results in his study present some implications regarding the VaR concept. The findings imply that asserting a VaR probability estimate with twodecimal-place the 0.10, 0.05 or 0.01 level misrepresents the precision regardless of sample size, holding period or asset class. He asserts that “the putative precision of a VaR probability estimate with two significant digits to the right of the decimal point is deceptive”. The alternative proposed by Hoppe (1998) is to device risk estimation techniques that avoid several problems in linear variance-based statistics.

6.1. The Parametric Case This corresponds to the characterisation of the model assuming the normal distributions as the basic statistical model when calculating the VaR. Following Stahl (1997), we denote by: V(F ): VaR as a function of the underlying distribution F , V(G): VaR as a function of the underlying distribution G, which is an element of the family of non-parametric distributions. Stahl (1997) shows that the following ratio is bounded by a constant c:V(G)/F(G) ≤ c or V(G) ≤ cV(F ). This estimation is valid for all distributions with finite variance. It gives some insight into the amount of model risk which may be caused by the assumption of normality in VaR models. The VaR formula at the 99% level can be written as: √ VaRφ = VM σZ0.99 δ T (1) where: VM : market value of the position, φ: standard normal distribution,

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σ: standard deviation, δ: sensitivity factor, T : horizon, and Z0.99 : 99th percentile of the standard normal distribution. The Tchebyshev’s inequality can be used to define an upper bound for the 99th percentile for an arbitrary distribution. This allows to appreciate the potential impact of a mispecified normal distribution model. The inequality is:  2 1 P(|X − µ| > kσ) ≤ k where µ is the mean of the variable X. This inequality shows that the probability of X’s future realisations lying outside the interval [−∞, µ+kσ] is at most 1/k2 . Taking the complementary event, that X takes values inside the same interval, gives an upper bound (1 − (1/k)2 ). In this context, (µ + kσ) is considered as a quartile of X of a level not less than (1 − (1/k)2 ). When µ = 0, this gives:   2  1 ≤ kσ. (2) F −1 1 − k The solution to (1 − (1/k)2 ) = 0.99 is k = 10. Hence, 10σ is an upper bound for the 99th percentile of F . The VaR for a model using an arbitrary distribution F is: √ VaRF = F −1 (0.99)δVM T . The application of inequality (2) to F −1 (0.99) gives the upper bound for the ratio: VaRF 10σ F −1 (0.99) ≤ = 4.29. = −1 VaRφ σφ (0.99) 2.33σ This scaling factor of 4.29 applies to any distribution with a finite variance. It represents an upper bound for the 1% confidence level. For the 5% level, the upper bound is 2.72. The above argument applies also a portfolio of assets. We denote by: VaRFport : VaR of a portfolio under an arbitrary distribution F , φ VaRport : VaR of a portfolio under a normal distribution φ.

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φ

Using the above analysis, it is clear that: VaRFport /VaRport ≤ 4.29 for all levels of confidence between 1% and 5%. This result is a consequence of the following inequality:   VaRFport = V F C(V F ) ≤ 4.29V φ C(4.29V φ )  φ = 4.29 V φ C(V φ ) = 4.29 VaRport with: V F : n-dimensional VaR vector on n assets, C: correlation matrix, and  : transposition operator.

6.2. The Non-Parametric Case Non-parametric methods such as historical simulation and Monte-Carlo simulation reduce the model risk stemming from a misspecified distribution. The consequences of violating the underlying stationary assumption are studied in Stahl (1997). The main assumption is a switch from a distribution F1 to a distribution F2 at time s for a time series of T realizations. Let us denote, respectively, by σ12 and σ12 the variances of F1 and F2 . The empirical distribution function estimated from all T observations can be close to the mixture FM of the distributions F1 and F2 mixed in weights (1−p) and p. In this case, it is possible to compare the VaR calculated under FM to the VaR calculated under the actual distribution F2 . Applying the same reasoning as above, Stahl (1997) shows that the range of all scaling factors must be between 2.5 and 4. This confirms the results in the parametric case.

7. Simple Extensions of the VaR Concept 7.1. VaR for Central Banks Blejer and Schumacher (1998) stress the view that currency crises arise from central bank financial vulnerabilty and propose a methodology based on the concept of VaR to assess the solvency of a country’s central bank. The understanding of crises requires a shift in emphasis from analyzing the sustainability of a regime towards assessing its vulnerability. Dornsbusch (1998) suggests also that IMF members must put in place some sort of “VaR” analysis by focusing on the maximum potential losses,

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the VaR methodology highlights the consequences of bad scenarios. The VaR approach to current crises summarizes in a single indicator the effects of inconsistent policies reflected in central bank positions and the consequences of factors such as the public’s expectatons of the sustainability of a nominal regime. The central bank portfolio The formula presented by Blejer and Schumacher (1998) can be directly used to measure the potential losses for the central bank arising from its activities. The actual calculation of the VaR for a central bank involves some initial decisions regarding the time horizon, the confidence interval, etc. The merit of the VaR approach in this context is to focus attention, utilizing the concepts of solvency, vulnerability and risk as they pertain to the monetary authority. The value of a central bank portfolio is represented by the equation:   L 2 S S ¯ ¯ V = Rδ ∗ S + Dδ + γG + Fd δ − H − C δ ∗ ∗S − Fχ δ ∗ −P δ, , σA A A where: ¯ the gross stock of international reserves (capital plus interest earnings); R: ¯ D: the net domestic debt denominated in the domestic currency (loans to the financial sector, including re-discounts, Y¯ , minus the central bank’s ¯ i.e. D ¯ = Y¯ − B¯ and Y¯ and D ¯ include future domestic liabilities B). cashflows; G: the stock of loans and advances to the government; Fd : the long leg of the forward denominated in the domestic currency; H S = G + R + D: the monetary base; CS : the outstanding stock of foreign debt (capital plus interest earnings); F S : the short leg of the forward denominated in a foreign currency; P(d, L/A, sA2 )A: the implicit or explicit financial sector guarantee, expressed as the value of the put option times the size of bank assets, A. The value of the put is a function of the domestic interest rate, the leverage ratio, L/A, and the volatility of bank assets; S: the spot exchange rate; g: the price of the government liabilities to the central bank; ∗ d ∗ = e−(i ) : the price of the international zero-coupon bond with maturity 1; ∗∗ d = e−(i ) : the price of a domestic zero coupon bond denominated in the domestic currency with maturity 1; and ∗∗ d ∗∗ = e−(i ) : the price of the country’s foreign currency denominated zerocoupon bond with maturity 1. 12:16:03.

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This equation assumes that all positions have a maturity equal to one unit of time. The equation shows the relationships between the economic value of the assets of the central bank and the liabilities. The equation allows a direct evaluation of the evolution of central bank solvency. In fact, the equation represents equity in terms of the exposure of the central bank positions to risk factors. Blejer and Schumacher (1998) derive the central bank VaR using the variance–covariance approach. The derivation assumes that: i∗ = i + Eds + a(uncovered interest rate parity with country risk) where Eds corresponds to the expected depreciation within one period and a is country credit risk: i∗∗ = i∗ + a and g = 0, i.e. the government is not expected to repay its debt to the central bank. The VAR portfolio can be expressed as follows:

VARCB

   VARS         VARi∗           VARα  = [VARS , VAR∗i VARα VAREds VARG VARC ][cor.matrix]   VAREds         VARG        VARC

where: ¯ ∗ − CS δ∗∗ − Fχ δ∗ ] VARS = σS K[Rδ is the VAR due to changes in the exchange rate.  ∗ ¯ − Fχ S) + e−(i∗ +Eds+α) VARi∗ = σi∗ k e−i (RS    ∂P ∗ +α) S −(i ¯ + Fd − C S A −e × D ∂δ is the VAR due to changes in the international interest rate.    ∂P −(i∗ +Eds+α) ¯ + Fd − A D VAREds = σEds k e ∂δ is the VAR due to changes in expected depreciation. VARα = σα ∗ ∗ k[e−(i +Eds+α) (D +Fd − ∂P A) +e−(i +α) CS S] is the VAR due to changes in ∂δ the country risk. VARG = sG K is the VAR due to changes in public expenditure funded with central bank loans. VARc = sc kd ∗∗ S is the VAR due to

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changes in the foreign debt. In this expression, k indicates the chosen confidence level. This VAR can be directly used to measure the potential losses of the central bank arising from its activities and exogenous prices. For further analysis of the formula, the reader can refer to Blejer and Schumacher (1998).

7.2. VaR: One Step Beyond Two main approaches have been used for VaR calculation. The first approach is based on an analytic formula which describes the confidence interval. This approach has the advantages of limited programming effort. The second approach or the simulation approach is based on multiple runs of possible price outcomes. This approach can give more precise results at a significant computational cost. The initial standard approach used to evaluate the VaR makes a Taylor series expansion of the market-to-market (MTM) to the first order. It assumes that all market prices have correlated normal distributions. In the first order approximation, the vegas can be added without changing the model logic. In this case, the VaR implementation needs the volatilities’ variances and correlations between themselves and with every financial variable used (interest rate, foreign exchange rate, etc.). According to Cardenas et al. (1997), from a practical point of view, “the VaR takes the market-to-market MTM as a “Black box”: if this box takes volatilities as inputs, then the VaR must reflect the fluctuations of these variables in the same way as it does for interest rates”. Cardenas et al. (1997) show that the MTM may be expressed as the sum of independent random variables, each being normal or the square of a normal variable. Their result can be used to imply the cumulative distribution. The knowledge of this distribution allows to deduce the confidence interval by determining the MTM for which the cumulative distribution is equal to the chosen confidence level. The extension of the standard VaR seems to be powerful. For more details, see Cardenas et al. (1997).

7.3. VaR by Increments The estimation of the overall VaR of a portfolio must account for the impact of changes in individual positions on overall VaR. This point has been studied in several papers by Garman (1996a, b, c). Dowd (1998) present changes of overall VaR and incremental VaR or IVAR to appreciate the change in a portfolio on the overall VaR.

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An approximate solution for incremental VaR When a manager is interested in the incremental VaR associated with a small position in an asset A, he can compute it as follows: IVAR for a position in asset A = VaR (portfolio with marginal position in A) − VaR (portfolio without marginal position in A)

(3)

where the VaR of the first and second portfolios can be calculated using the same method. The computation of IVAR in this way is slow and cumbersome. Therefore Dowd (1998) presents an approximate solution for IVAR. The starting point of his approach is to estimate the variance of the rate of return to the new portfolio using the rate of return to the old portfolio and the rate of return to the new asset as: σp2new = (1 + a)−2 [σp2 + a2 σA2 + 2aσA,p ] or:
σpnew = (1 + a)−1 σp2 + a2 σA2 + 2aσA,p

(4)

where: σpnew : standard deviation of the rate of return to the new portfolio, σp : standard deviation of the rate of return to the old portfolio, and σA,p : covariance between the rates of return to the new asset and the old portfolio, and a: size of the new asset position relative to the existing portfolio. When a is small, it is possible to neglect the term a2 σA2 to obtain:
σpnew = (1 + a)−1 σp2 + 2aσA,p . (5) When returns are iid, the VaR of the old portfolio is given by: VaRold = −ασp W

(6)

where α is the parameter reflecting the confidence interval (1.65 for a 95% confidence level) and W is the portfolio size. Using Eq. (6), the VaR of the new portfolio can be obtained by multiplying (5) by −α(1 + a)W , substituting Eq. (6) into the result and rearranging the terms gives:
VaRnew = (VaRold )2 + 2aα2 σA,p W 2 . (7)

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Replacing Eq. (7) in (3) gives the IVAR. To obtain an analytic solution for the IVAR, we square Eq. (7) and then substract (VaRold )2 from both sides to obtain: (VaRnew )2 − (VaRold )2 = 2aα2 σA,p W 2

(8)

which can also be written as:  2  2    VaRnew − VaRold = VaRnew + VaRold VaRnew − VaRold   = VaRnew + VaRold IVAR   = IVAR + 2 VaRold IVAR = IVAR2 + 2 VaRold IVAR

(9)

where IVAR = VaRnew − VaRold . As before, for a small a, IVAR2 is small relative to ((2 VaRold ) IVAR) and can be neglected. The equality between this equation and the right hand side of (8) gives: IVAR = aα2 σA,p W 2 /VaRold .

(10)

Substituting the old portfolio VaR [Eq. (4)] allows to write (10) as: IVAR = −aWασA,p /σp = aβA,p VARold

(11)

where βA,p = σA,p /σp2 . This equation of the IVAR shows that IVAR corresponds to the relative size of the new position a times the asset’s portfolio beta βA,p times the old portfolio VaR. Since the manager knows the old VaR and a, he has only to compute the covariance between the new asset and the portfolio to estimate the IVAR. This formula can be used in real time.

7.4. The Concept of CVAR Garman (1997) introduces the concept of CVAR, based upon the VaRdelta (or DelVAR). The relationship beween the VaRdelta and the VaR is similar to that between an option and its delta. The concept measures the sensitivity of VaR to a unit cash-flow. He gives a definition of the concept of “component VaR”, CVAR. This concept presents at least three properties: if the components partition of the portfolio, the CVAR must add up to the portfolio vaR; if the component were to be deleted from the portfolio, the CVAR shows how the portfolio VaR will change, and CVAR is negative for components that can hedge the rest of the portfolio.

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7.5. Coherent Risk Measures The use of the VaR method needs to fix in advance a probability level (for example 5%) and the measure of risk of the position is an amount q0.05 such that the net worth of the position X at some future date T is smaller than q0.05 with the 5% probability. The number q0.05 or the 5% quantile of X is known as “5% VaR of X”. It is sometimes denoted as VaR0.05 (X). When q0.05 > 0, no additional capital is required. When q0.05 < 0, then |q0.05 | would be the additional initial capital required to hold the position, under the assumption that the capital is invested safely. Artzner et al. (1997) address several difficulties with the VaR approach. They propose a set of characteristics that must be satisfied by any “coherent” risk measure. The approach gives an answer to the following question: how bad is bad? This leads to measure the negative of the average future net worth X given that X is below a quantile qε , i.e.: ρ(X) = E[−X| ≤ VaRε (X)]. This measure can be useful when adding two positions. In this context, an acceptable position is defined as a position for which the number ρ(X) is negative. Artzner et al. (1997) conclude that the generalized scenarios method is a universal coherent risk measurement method. For an application of the method, see Artzner et al. (1997).

8. Simulation Methods and Their Specific Features 8.1. Historical Simulation The historical simulation approach applied in the full valuation model does not require explicit assumptions about the distribution of assets returns. It quantifies risk by replicating one specific path of market evolution. This approach values the portfolios under a number of different historical time windows. Historical simulation must account for market conditions Since most VaR models deal either with the non-normality of security returns or with their heteroscedasticity, Barone-Adesi et al. (1998) propose a modified historical simulation approach that accounts for both effects. The historical simulation is often based on a normal distribution of past returns. These returns are used for current asset prices to simulate their

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future returns. In this context, the VaR of a portfolio is determined after different paths have been explored. The possible clustering of large returns and the resulting fluctuations in daily volatility can lead to some errors in the determination of the confidence levels of VaR. This is true for all models that ignore clustering. In particular, this is true for the VaR calculation based on a variance–covariance matrix and Monte-Carlo methods. To overcome this problem, Barone-Adesi et al. (1998) propose a historical simulation consistent with the clustering of large returns. They model the volatility of the portfolio returns as an asymmetric Garch process (generalized autoregressive conditional heteroscedasticity) and allow the postive and the negative returns to have different impacts on volatility. This method allows to obtain standardized residuals by dividing the historic daily returns by the Garch volatility. To adjust the standardized residuals to current market conditions, they multiply a randomly selected standardized residual by the Garch forecast of the next day’s volatility. This operation allows to simulate the return of the portfolio for the next day. They use the simulated return to update the Garch forecast for the following day. In the same way, this is multiplied by a newly selected standardized residual to simulate the return for the second day. They implement this recursive procedure over an horizon of 10 days in order to generate a portfolio of returns and volatilities. This procedure is repeated in order to generate a batch of sample paths for the portfolio returns. They choose a confidence band for the corresponding portfolio values by considering the empirical distribution of values at each time. The worst case is identified over the next 10 days as the lower 1% area. The implementation of the procedure does not require the correlation matrix of security returns.

8.2. Structured Monte-Carlo Simulation Several paths are generated to quantify risk. The returns are defined by specifying a stochastic process. Since analytical VaR for portfolios whose profit and loss distributions are not normally distributed, it is possible to use Monte-Carlo methods based on scenarios driven by the volatility and correlation estimates. The methodology can be implemented in three steps. The first step: the scenario generation In this step, we generate a large number of future price scenarios using the volatility and correlation estimates for the underlying portfolio. 12:16:03.

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The second step: portfolio valuation In this step, a portfolio value is calculated for each scenario. The third step: summary In this step, the results of the simulation are reported in the form of a portfolio distribution or a particular risk measure. Example Using the volatility and correlation estimates , the method can be applied to generate a large number of scenarios (1,000 for example) of Euro one-year and Euro/USD exchange rates for one week. You can represent the actual distribution of the assets in the portfolio and the correlation coefficients between the returns on the different assets. The set of interest and foreign exchange rates calculated under simulation can then be used to revalue the portfolio. It is now possible to represent the payouts of the instruments in the portfolio (the bond as a function of the yield and the option as a function of the exchange rate). At this level, the manager can analyze the distribution of values and select the VaR using the appropriate percentile.

8.3. Monte-Carlo and Computational Problems VaR models compute one or more percentiles of the distribution for profit for a given portfolio of market sensitive instruments. The traditional simulation techniques revalue the trading portfolio hundreds of time in historical simulation to generate the VaR. With a Monte-Carlo simulation, the trading portfolio must be revalued thousands of times. The determination of VaR during a trading day leads to an operational problem and a calculational problem. The operational problem results from the fact that the transaction database reflects the portfolio as it exists at the time of the report, while some transactions do not still enter the transactions database. The calculational problem is that VaR needs a great deal of computer time. The use of Monte-Carlo simulation can be improved and applied to the determination of intra-day VaR reporting. This approach appears in Jamshidian and Zhu (1997) and Frye (1998). Frye (1998) proposes an organization of the calculation into three parts. The first step is overnight when the most computer resources are available. In this step, every transaction is valued over a wide grid of hypothetical market outcomes. The second step values each new transaction as it occurs over the same grid of outcomes. 12:16:03.

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The proposed method uses in the first and the second step a small number of statistically determined factors rather than changes in all market variables. The third step implements the Monte-Carlo simulation at report time. For each outcome, a linear interpolation uses the results of steps 1 and 2 to imply an appropriate profit or loss. This leads to a complete picture of the distribution of profits including all the desired percentiles. This step approximates portfolio value rather than doing an exact valuation. The intraday Monte-Carlo simulation in Frye (1998) uses three ingredients: — the values of the closing portfolio under several hypothetical scenarios, — the probability structure of the relevant market variables, — some linear interpolations. This procedure is more efficient than the conventional Monte-Carlo simulation since it is “relatively” not time-consuming.

8.4. Monte-Carlo: The JP Morgan Methodology The Monte-Carlo method is based on three steps: the scenario generation, the portfolio valuation and a summary of the results as a risk measure (or portfolio distribution).

8.4.1. Scenario Generation

√

The price of an instrument for an horizon of t days is given by: Pt = P0 eσ tY where Y is a standard normal random variable and σ is the one day volatility of the instrument. This equation is used to generate future prices. In practice, to generate n normal variates with unit variance and correlations given by a n(n) matrix , the main idea is to generate n independent variates and to combine them to achieve the desired correlations. The procedure is as follows. The Cholesky factorization is used to decompose  into a lower triangular matrix A in a way to get  = AA . Then an n(1) vector Z of independent standard normal variates is generated. Denoting Z = AY , the elements of Z will each have unit variance and will be correlated according to . This procedure allows to generate random variates with arbitrary scenarios. Example: The case of two independent variates Consider the case of two variates with correlation matrix:   1 ρ = . ρ 1 12:16:03.

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The Cholesky factorization of this matrix is:   1  0 A= . (1 − ρ2 ) ρ When Y is a 2(1) vector with independent standard normal variables Y1 and Y2 , then the elements of Z = AY are:  Z1 = Y1 and Z2 = ρY1 + (1 − ρ2 )Y2 . Since the variance of Z1 is 1, then the variance of Z2 is:  Var(Z2 ) = ρ2 var(Y1 ) + ( (1 − ρ2 )2 ) var(Y2 ) = 1. In this case, the expected value of Z1 Z2 is ρ. This procedure can generate random variates with arbitrary correlations. Consider the application of the procedure in the following example to generate asset prices in t days. Example We denote by P01 and P02 the current prices of two assets with daily volatilities σ1 and σ2 and a correlation coefficient ρ. The first step is to generate correlated standard √ normal variates Z1 and √ Z2 and compute the future prices using: Pt1 = P01 eσ1 tZ1 and Pt2 = P02 eσ2 tZ2 . This procedure must be repeated several times in order to generate a collection of scenarios. The value of the portfolio must be calculated in a second step for each of the scenarios.

8.4.2. Portfolio Valuation The methodology in JP Morgan is based on three parametric alternatives for portfolio valuation using the securities values and their underlying assets: full valuation, linear approximation and higher order approximation. Full valuation In each scenario, an option pricing model (like the Black and Scholes (1973) model) is applied to generate option prices. Linear approximation The delta-approximation is used to estimate the change in the option value via a linear model. Given an initial option value V0 and an initial underlying asset variable R0 , the future option value V1 is approximated at a future value R1 , by: V1 = V0 + δ(V1 − V0 ) 12:16:03.

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where δ is the standard first derivative of the option price with respect to the underlying asset price. Higher order approximation The delta-approximation can be improved by including the gamma and the theta using the following formulas: 1 V1 = V0 + δ(V1 − V0 ) + (V1 − V0 )2 2 and 1 V1 = V0 + δ(V1 − V0 ) + (V1 − V0 )2 − θt. 2

8.4.3. Filling the Missing Data The missing data affect the estimates of volatility and correlation. RiskMetrics applies the expectation maximization algorithm (EM algorithm) to fill in the missing prices and economic variables. The EM algorithm has been used by Dempster, Laird and Rubin (1977), Bellalah and Lavielle (2003), etc. The following steps are used. First, it is assumed that at each point in time, the cross-section of returns are multivariate normally distributed with mean µ and covariance matrix . Second, it estimates the mean and covariance matrix using the available data. Third, it replaces the missing data by their respective conditional expectations. This means that the missing data’s expected values are used given the current estimates of the mean, the covariance and the observed data.

8.5. The Properties of Covariance Matrices and VaR: The Normal Mixture RiskMetrics applies a correlation matrix to compute the VaR of an arbitrary portfolio. The correlation matrix is used to compute the portfolio’s standard deviation. The VaR estimate is calculated as a multiple of that standard deviation. The portfolio’s variance must be non-negative. RiskMetrics applies a normal mixture model of returns to measure the tails of selected return distributions. The normal mixture model assumes that returns are generated as follows: rt = σ1,t ε1,t + σ1,t δt ε2,t with rt : time t return, ε1,t : a normally distributed random variable with mean 0 and variance 1, ε2,t : a normally distributed

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2 , δt a variable taking the random variable with mean µ2,t and variance σ2,t value 1 with probability p and 0 with probability (1 − p). σ1,t : volatility given by the RiskMetrics model. The normal mixture model assumes that daily returns standardized by the RiskMetrics volatility forecasts are generated by the following model: r˜t = ε1,t + δt ε2,t . When δt = 0, the return is generated from a standard normal distribution with mean 0 and variance 1. When δt = 1, the return is generated from a standard normal distribution 2 with mean σ2,t and variance (1 + σ2,t ). Although the assumed mixture is composed of normal distributions, the mixture distribution itself is not normal. When constructing the VaR, this model uses the standard RiskMetrics volatility.

9. VaR Methods and Special Events 9.1. Simple VaR Methods Do Not Account for Market Crashes According to Paul-Choudhury (1996), market crashes and options pose essentially the same problem when calculating VaR. Recall that in the original form of JP Morgan’s RiskMetrics, the returns on the underlying portfolio are normally distributed. In this context, 90% of the returns fall within more or less 1.65 standard deviations (the fifth and 95th percentiles) of the mean return. The VaR gives in this setting the maximum probable loss on a portfolio under market conditions. One of the methods to overcome the normality assumption and to account for real world distributions is to resort to stress testing and scenario analysis. Besides for a portfolio of options or derivative assets with nonlinear payoffs, the problem is more complicated. In this case, the use of the option Greek letters is important. For a portfolio of options, the nonlinearity is important near the money and for longer maturity options. When the effects of gamma risk (the derivative of the delta) are included in the calculation, the portfolio returns become asymmetric or skewed. In this case, it is possible to include the effects of the gamma for example by calculating the VaR on the basis of a skewed distribution of returns using one of the three following solutions. The first considers an approximation of the skewed distribution by a deformed normal distribution. The second needs the calculation of the fifth and 95th percentiles from the skewed distribution.

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The third method fits the skewed distribution to a more general family of distributions with known statistical characteristic measures. Many banks prefer historical simulations or the Monte-Carlo method to the analytic approximations to VaR since these approximations might ignore the effects of gamma and crashes. However, the problem of simulation is that it might be extremely time-consuming and costly in terms of computer power. For small users the refinements to be made by implemeting simulation methods may be outweighted by the difficulty of running the method. When a confidence interval of 99% is used in the VaR calculation, the losses on a portfolio due to market risk should not exceed the VaR number on more than one day out of every 100. The corresponding frequency for a 95% confidence interval is one day in 20 nearly (or in a trading month). In practice, the 1% of market moves which are not accounted for in the VaR calculation contain events like stock market crashes. For this reason, a manager must also identify his risks in stress situations and simulate extreme market moves over a given range of different scenarios. The use of Monte-Carlo methods in this context is independent of the normality assumption of returns. For example, a confidence interval of 99% can account for the events up to 2.33 standard deviations away from the mean asset return.

9.2. The VaR and the Event Risk The wide applicability of VaR is well documented and its limitations are well identified. The VaR concept measures market risk for a stable market context where future price changes reflect the historical movements of asset prices. However, when a major event appears, the standard VaR may be useless. This limitation is overcome by modifying standard VaR to account for specific types of market risk.

9.2.1. The Definition of an Event Standard VaR models are based on historical simulation or on the estimation of a variance–covariance matrix. The standard definition of event risk refers to the expected catastrophic loss from, for example, a market crash. This event is an infrequent event. This definition requires certain conclusions about the returns that are in the extreme limits of the return distribution. An event can be defined by large extreme price moves. Anticipated events

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are regular or not. Regular anticipated events include, for example, the outcomes of government reports on financial statistics. Irregular anticipated events are rather sporadic like stock market crashes.

9.2.2. Measuring Event Risk The classic method to measure event risk is scenario analysis and stress testing. A scenario is formulated where a given asset will fall by 15% in one day, then a portfolio is valued under the scenario to determine the effects on its value. This approach does not assign a probability to the 15% fall. Zangary (1997) proposes a model allowing a modification of database VaR estimates using the decider personal view. His model is based on the idea that observed returns are a mixture of event and normal returns or a mixture model. In his model, the normal returns are standardized as in the baseline model of RiskMetrics. The event return distribution is modelled in two steps. In the first step, a prior distribution is assumed for event returns. At this level, an assumption is used to assert, for example, that event returns are normally distributed with a zero mean and a 10 standard deviation. In the second step, the historical data are used to update this assumption by estimating the mixture model from historical returns. The use of a mixture model is important. In fact, the model has two main specific features. The first is that it uses data more efficiently by identifying non-event return data and using the event returns for inference. The second feature is that the inclusion of a prior distribution on event returns enables to incorporate intuition into VaR estimates. An example is given in Zangary (1997) in which the manager defines an event as a 10 standard deviation move. This view is incorporated in the mixture model by setting the prior standard deviation of the event return distribution to 10. The mixture model is defined as follows: return = event indicator (return from event distribution) + return from benchmark distribution. The event indicator is one if the event appears on a given day and is zero otherwise. The event days can be specified or determined directly from the data. In the last case, the data tell us the probability that a given day is an event. This model provides a framework for incorporating event risk into a standard VaR model based on a variance–covariance matrix. The user must classify the days as event and non-event. A specific VaR model is given for different combinations of observed returns. The mixture model with an emphasis on data can be used on days when regular events occur to get VaR estimates.

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10. Applying the Risk Measures, the Reporting Management System and the VaR 10.1. Applying the Risk Measures The different measures of market risk can be applied in different contexts to measure market risks, to check the valuation models, to evaluate the performance, to estimate capital levels required to support risk taking, etc. Setting limits using the VaR concept shows how much money could be lost. This allows also managers to allocate risk to the areas which they feel offer the most potential. Financial institutions have to meet capital requirements to cover the market risks that they incur as a consequence of their normal operations. The international standards for market risk based capital requirements are the European Community (which issued a binding Capital adequacy Directive (EC-CAD)) and the Basel Committee on Banking Supervision at the Bank for International Settlements (which provides proposals on the use of banks internal models). To make a comparison between the standard data sets, and RiskMetrics Regulatory Dataset, the following formula can be used: √ VBasel = 2.33/1.65VRiskMetricsRD 10 with VRiskMetricsRD : volatilities provided in RiskMetrics Regulatory Dataset, VBasel : volatilities suggested by Basel Committeee for use in internal models.

10.2. The Reporting Management System and the VaR According to Fong and Casella (1998), a risk management reporting system must identify the risk factors that make up the uncertainty of returns. For fixed income portfolios, the different sources of risk include the level of the interest rate, the spreads over government rates, the volatility of interest rates, the measure of exchange rate changes, etc. If we denote by P the price of the transaction i = 1 and by Di , . . . , Dn the exposures or sensitivities for each risk factor F , the change in the transaction value due to a change in the risk factors can be written to a first approximation as:  P =− Di Fi P i=1 n

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where: P: the price of the transaction i, F : a risk factor, and D: exposure to a risk factor. This rough approximation assumes that the price change is a linear function of the factor changes and that the changes in the factors are instantaneous. If we denote by Fi = Fi the changes in value of each risk factor and by Y the specific risk of each transaction, then a more appropriate basis for the price sensitivity of a transaction is: P = A −

n  i=1

1 Di Xi + Ci Xi2 + Y 2 i=1 n

(13)

where Di corresponds to a linear exposure to the factors (the dollar duration of interest rate exposure) and Ci is the quadratic exposures of the transaction value to the factors (dollar convexity measures of interest rate exposure). Let Pi and Pi stand for the prices of the transaction calculated under the assumption that the factor Fi is modified by Fi and −Fi . Using a finite factor change (rather than an instantaneous factor change), Eq. (13) can be written as: Pi − Pi 2Fi

(14)

Pi + Pi − 2P Fi2

(15)

Di = − and: Ci = − with:

√ Fi = σi 3

(16)

where σi stands for the volatility of Fi over the interval t. Using Eqs. (14), (15) and (16), it is possible to construct a global response curve using a range of t. This allows a better characterisation of the price sensitivity of a transaction to a range of perturbations. The quantity A in Eq. (13) is equal to: 1 Ci σi2 2 i=1 n

µ−

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where the expected return is given by: µ = E(P).

(18)

Hence, the price sensitivities can be analyzed with respect to interest rate changes. Portfolio level analysis When the factor changes X follow a jointly normal distribution, the first three moments of P can be specified. The portfolio mean is given by Eq. (18). The portfolio variance is: σ 2 = VAR(P) =

n n  

1  Ci Cj σij2 + s2 . 2 i=1 j=1 n

Di Dj σij +

i=1 j=1

n

(19)

The portfolio skewness is: µ3 = E(P − µ)3 = 3

n  n n  

+

Di Dj Ck σik σjk

i=1 j=1 k=1 n  n n  

Ci Cj Ck σij σjk σki .

(20)

i=1 j=1 k=1

These three measures can be used in VAR calculation. Recall that VAR corresponds to the maximum loss expected with a given probability for a specified time horizon. The loss in the portfolio is measured by Prob[P ≤ −VAR] = α where VAR is the VAR at confidence level (1 − α). When the gamma distribution is used as a proxy for the probability distribution, this gives the following VAR formula: VAR = k(γ)σ. The skewness of the distribution is γ = µ3 /σ 3 .

Summary The VaR corresponds to a number indicating the potential change in the future value of a given portfolio. In the process of calculating the VaR, the manager must specify the horizon for the calculation as well as the “degree of confidence” chosen. VaR calculations can also be done without resorting to the standard deviation. Nonlinear positions correspond, for example, to a portfolio of options. The VaR of a portfolio of options can be determined using the “greeks”. The basic method uses an option pricing model to obtain

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the delta. This delta is used to determine the amount of a market factor that must be held to compensate for a change in the underlying asset price. The present value of the delta hedge position in the underlying is included in the determination of the portfolio variance. This method is efficient only for very small changes in the underlying asset price. In fact, the delta is a linear measure only for very small changes in the underlying asset price over very small intervals of time. Since the VaR is concerned with the effects of large changes in the underlying asset price, the linearity may lead to an inappropriate assessment of market risk measures. The estimation method is improved when the second derivatives of the delta (the gamma) is used in the risk measure. Since the option price function is nonlinear for different prices of the underlying asset, a risk measure including gamma may also lead to an inaccurate measure of market risk for significant changes in the underlying asset price. However, the simultaneous use of delta and gamma can improve risk estimation. The analytic framework for determining the VaR of a portfolio of options can adequately measure price level risk. It is possible with some difficulties to include the vega and the option time decay (theta) in the VaR calculation using an analytical framework. Rouvinez (1997) develops two analytic methods in the determination of accurate VaR composed of nonlinear instruments (like options): the delta–gamma method and the delta–gamma maximum loss method. The first method called the percentile approach determines the level which should not be exceeded for a given percentage of the time. The second method called the optimization approach computes the maximum loss resulting from a change in market conditions within a given range around the current value. VaR models can be used by banks and were the key to the Basel Committee on Banking Supervision’s acceptance of in-house models for the determination of capital requirements. The Basel Committee standards require that banks use their own models for determining the capital caushin for market risk by multiplying their VaR figures by three. Stahl (1997) proposes some justifications of this multiplier using a probability approach. Several in-house models are based on VaR and models that combine the main features in portfolio theory models, option pricing models and statistical models. First experiences with VaR models reveal that model errors tend to be cumulative, leading to a considerable underestimation of the true risk. The simplifying assumptions concern the return distributions of financial assets (often the normal distribution is used) while in reality distributions present fat tails, are skewed and not stationary. Therefore, regulators and bank risk managers must be well aware of the dangers that

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come with the simplifying assumptions of the models or what is known as the model risk. VaR models compute one or more percentiles of the distribution for profit for a given portfolio of market sensitive instruments. The traditional simulation techniques revalue the trading portfolio hundreds of time in historical simulation to generate the VaR. With a Monte-Carlo simulation, the trading portfolio must be revalued thousands of times. The determination of VaR during a trading day leads to an operational problem and a calculational problem. The operational problem results from the fact that the transaction database reflects the portfolio as it exists at the time of the report, while some transactions do not still enter the transactions database. The calculational problem is that VaR needs a great deal of computer time. The use of Monte-Carlo simulation can be improved and applied to the determination of intra-day VaR reporting. The wide applicability of VaR is well documented and its limitations are well identified. The VaR concept measures market risk for a stable market context where future price changes reflect the historical movements of asset prices. However, when a major event appears, the standard VaR may be useless. This limitation is overcome by modifying standard VaR to account for specific types of market risk. Fong and Casella (1998) discuss a simple context that can provide the foundations for a risk management reporting system. Risk analysis represents a scientific method for the quantification of risk for a single asset or a portfolio of securities. Risk analysis must account for the risks of individual assets as well as the aggregated risk involved in a portfolio. Hence, the desired reports for risk management must include a detailed breakdown of the individual components of a portfolio as well as the portfolio as a whole. This chapter develops the Value-at-Risk concept and its applications. The analysis of the limits of the VaR measure allows to extend the analysis to the simulation methods in the VaR computation. Some applications of the risk measures are also given.

Questions 1. 2. 3. 4. 5. 6.

What is the Value-at-Risk concept? Describe the risk measurement framework and RiskMetrics. Describe the RiskMetrics model for financial returns. How is the VaR of a portfolio computed in RiskMetrics? What is the problem with the VaR of nonlinear positions? What are the limits of VaR estimates?

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What are the foundations of simulation methods? What is meant by a historical simulation? Describe the structured Monte-Carlo simulation. How are the VaR methods applied in special events?

Appendix: VaR and Nonlinear Positions Rouvinez (1997) develops two analytic methods in the determination of accurate VaR composed of nonlinear instruments (like options): the deltagamma method and the delta-gamma maximum loss method. The first method called the percentile approach determines the level which should not be exceeded for a given percentage of the time. The second method called the optimization approach computes the maximum loss resulting from a change in market conditions within a given range around the current value. Consider a portfolio P depending on m market risk factors (rates) Si with i varying from 1 to m. The innovations Si are jointly normally distributed with a symmetric covariance matrix . The return on the portfolio P or the return function depends on the innovations Si . The evaluation of P(S) is in general approximated by the second order Taylor expansion: 1 P(S) = δ S + S  γS + O(S 3 ) + · · · 2 where δ(m-dimensional vector) and γ(m × m) dimensional symmetric matrix correspond to the first and second derivatives.

A.1 The Percentile Approach For a given confidence level c, the VaR is Prob{P ≤ −VAR} = c which corresponds to the quantile or percentile of order (1 − c) of the distribution. −1 It can be written as VAR = −FP (1 − c) where FP (X) stands for the cumulative density function (CFD) of P. Following Rouvinez (1997), when the return function is linear or P(S) = δ S we have: √ √ P ∼ N(0, σδ2 ), with σδ = δ δ t. In this context, the quantile of this distribution is the product of α (the quantile of standard normal distribution) and the standard deviation VAR = √ √  α δ δ t. For a confidence level of 99%, α = 2.32.

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This delta-normal VAR is adequate for portfolios with linear payoffs. When the innovations are jointly normally distributed, it is possible to make a linear transformation L as: S ∗ = (O A−1 )S = LS where A is a matrix such that t = AA where A is the Cholesky decomposition of t. The matrix O is the orthogonal matrix composed of the eigenvectors of the matrix γA = A γA. This transformation leads to m new uncorrelated innovations S ∗ with a covariance matrix: ∗ = L(t)L = OA−1 (t)A−1 O = I. The probability density function of jointly normal distributions is proportional to: exp[S ∗ (∗ )−1 S ∗ ] = exp[S ∗ IS ∗ ]   m m   ∗ 2 (Si ) = exp[(Si∗ )2 ]. = exp i=1

i=1

If we define new sensitivities as: δ∗ = L−1 δ = (OA )δ and γ ∗ = L−1 γL−1 = O (A γA)O then: δ∗ S ∗ = δ L−1 LS = δ S and S ∗ γ ∗ S ∗ = S  L (L−1 γL−1 )LS = S  γS. The new return function is: 1 P(S ∗ ) = δ∗ S ∗ + S ∗ γ ∗ S ∗ . 2 The linear transformation renders the new innovations independent so the returns can be written as a sum of independent returns: P(S ∗ ) =

m  i=1

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The above analysis considers only a linear payout function. To account for the quadratic term in the payout function, the return functions can be written as:   δ∗i 2 (δ∗i )2 1 ∗ ∗ Pi = γi,i Si + ∗ − ∗ γi,i 2γi,i 2 ∗ for γi,i different from zero. Using some calculations and the moment generating function, it is possible to obtain the exact variance, skewness and kurtosis of the distribution of returns: µ3 µ4 σ 2 = µ2 , v1 = 3 and v2 = 4 . σ σ

For normal distributions, v1 = 0 and v2 = 3. For a portfolio with low optionality, a normal distribution of returns can be assumed and the following mean–variance approximation can be used to give a reasonable estimate of VaR: VAR = |µ1 − ασ|. This approximation does not work for portfolios with strongly nonlinear payoffs. In this case, a bound for the VAR can be calculated using for example the Tchebychev inequality: Prob{|X − µ1 | > sσ} ≤

1 . s2

This gives for s−2 = 1 − c:

     1   σ . VAR ≤ µ1 −  1−c 

Another useful one tail inequality given in Rohatgi (1976) is: Prob{X − µ1 < s} ≤

µ2 , µ2 + s2

for s < 0.

This gives the following improved bound for the VAR:      c σ  . VAR ≤ µ1 − 1−c Using a more general inequality in Rohatgi (1976): Prob{|X − µ1 | > sσ} ≤

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For s > 1, this gives the new bound for the VAR:         c  VAR ≤ µ1 − 1 + (v2 − 1) σ  . 1−c   This improves on Tchebychev’s inequality when c > 1 − v−1 2 . To compute the exact solution to the problem, Rouvinez (1997) shows ! ∞ −1 that VAR = −FP (1 − c) where Fx (x) = 21 − −∞ Im exp[−isx] φx (s) ds. 2πs

A.2 The Optimization Approach This method looks for the worst event within a given set of scenarios. This method can be considered as a generalization of stress-testing. The definition of the set of scenarios Si allows to compute the returns for the portfolio P. The VaR is considered as the absolute value of the maximum loss. A confidence region is defined using a set of constraints on the The " innovations. ∗ 2 2 sum of the squares yields a random variable: (S ∗ )2 = m (S ) ∼ χm i i=1 distributed as a chi-squared with m degrees of freedom. Using β as the quantile of order c for the chi-squared distribution, β satisfies Prob{(S ∗ )2 < β} = c. Hence β = 9.21 when c = 99% and m = 2. This expression gives the confidence region: D = {S ∗ |(S ∗ )2 < β}. Using the definition of S ∗ gives: (S ∗ )2 = S ∗S ∗ = S  (t)−1 S. This method can be illustrated for linear instruments by looking for the solutions to the optimization problem, min[δ S] subject to: S  −1 S ≤ βt where β is the quantile for a χ2 distribution with m degrees of freedom. Using the objective function: G(S, λ) = δ S + (S  −1 S − βt) gives: ∂G = δ + 2−1 S = 0 ∂(S) 1 δ. so S(λ) = − 2

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Replacing in the constraint gives S  −1 S ≤ βt so  = √  1/2 δ δ/βt. Replacing in S gives the scenario:  βt δ S = δ δ √ √ √ and the return P = − β δ δ t. It is convenient to note that the VaR obtained with delta optimisation only differs from the delta normal VaR by a factor. For the extensions of this method, the reader can refer to Rouvinez (1997).

Bibliography Artzner, P, F Delbaen, JM Eber and D Heath (1997). Thinking coherently. Risk, 10(11), November, 68–71. Barone-Adesi, G, F Bourgoin and K Giannopoulos (1998). Don’t look back. Risk, 42, August, 100–103. Bellalah and Lavielle (2003). A decomposition of empirical distributions with applications to the valuation of derivative assets. Multinational Finance Journal. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659. Blejer, M and L Schumacher (1998). VaR for central banks. Risk, 10(11), October, 65–69. Cardenas, J, E Fruchard, E Koehler, C Miche and I Thamazeau (1997). VaR: one step beyond. Risk, 10(10), October. Dempster, A, N Laird and D Rubin (1977). Maximum-likelihood from incomplete data via the EM algorithm. Journal R. S. Soc. B, 39, 1–38. Dornsbusch, R (1998). Capital Controls: An Idea Whose Time is Gone. Princeton University Press, Princeton. Dowd, K (1998). VaR by increments. Risk, November, 31–32. Fong, G and M Casella (1998). Total report. Risk, April, 60. Frye, J (1998). Monte Carlo by day. Risk, November, 66–71. Garman, M (1996a). Making VaR proactive. Financial Engineering Associates. Garman, M (1996b). Making VaR more flexible. Derivatives Strategy, April, 52–53. Garman, M (1996c). Improving on VaR. Risk, May, 61–63. Garman, M (1997). Taking VaR to pieces. Risk, 10(10), October, 70–71. Hoppe, R (1998). VaR and the unreal world. Risk, July, 45–50. Jamshidian and Zhu (1997). Scenario simulation model. Finance and Stochastics, January. Paul-Choudhury, S (1996). Optional extras. Risk, June, 23–25. Rohatgi, VK (1976). An introduction to probability theory and mathematical statistics. Wiley Interscience, 100–103. Rouvinez, C (1997). Going Greek with VAR. Risk, 10(2), February, 57–65. Stahl, G (1997). Three cheers. Risk, 10(5), May, 67–69. Zangary, P (1997). Catering for an event. Risk, 10(7), July, 34–35.

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Chapter 12

Credit Risk and Credit Valuation: The Basic Concepts

This chapter is organized as follows: 1. Section 2 is an introduction to credit valuation with reference to the CreditMetrics approach. 2. Section 3 studies default and credit quality migration in the CreditMetrics approach. 3. Section 4 presents credit valuation in the KMV approach. 4. Section 5 explains portfolio management of default risk in the KMV approach. 5. Section 6 studies the main approaches in the credit risk literature. 6. Section 7 develops the main uses and abuses of bond default rates. It studies the management of default risk in portfolios of derivatives. 7. Section 8 shows how to manage default risk in portfolios of derivatives. 8. Section 9 presents bond pricing models. It studies default risk and the term structure of credit risk. 9. Section 10 presents a comment on market versus accounting based measures of default risk.

1. Introduction redit risk analysis and credit valuation need some preliminary definitions of the basic concepts. Credit exposure refers to the amount which is subject to changes in value upon a change in credit quality or a loss in the event of default. The average shortfall corresponds to the expected loss given that a loss occurs or exceeds a given level.

C

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The counterparty refers to the partner in a transaction in which each side takes broadly comparable credit risk to the other. Credit scoring refers in general to the estimation of the relative likelihood of default for a firm. The current exposure corresponds to the amount it would cost to replace a transaction now if the counterparty defaults. The default probability refers to the likelihood that an obligor will encounter credit distress within a given period. Credit quality migration refers to the possibility that an obligor with a certain credit rating migrates to any other credit rating by the risk horizon. Migration analysis refers to a technique for the estimation of the likelihood of credit quality migrations. In the analysis of credit derivatives, some terms are often used. A credit event is often defined as an insolvency or payment default, a bankruptcy event, a certain price decline or a rating downgrade for the reference asset. The reference asset may be an actively traded corporate or sovereign bond (or a portfolio of bonds) or widely syndicated loan (or portfolio of loans). The reference rate is an agreed fixed or floating interest rate. The default payment can be the par-post default price of the reference asset, a given percentage of the notional amount of the transaction or a payment of par by the seller in exchange of the default reference asset. Portfolio risk resulting from changes in debt value generated by changes in obligor credit quality is the main task attributed to CreditMetrics. Credit risk appears because the asset’s value (the bond for example) can change depending on the credit quality of the issuer. The value of this asset will decline with a default or a downgrade and will appreciate when the credit quality of the issuer is improved. Credit rating agencies assign an alphabetic or numeric label to rating categories. Altman (1987) defines a default event with reference to missed interest and principal payments. Credit valuation provides a mean to reduce the likelihood of losses through portfolio diversification. The credit risk is in general measured in terms of probabilities using a credit valuation model. The model is based on a theory that describes the link between the attributes of the borrowing institution and its possible bankruptcy. Corporate bonds and liabilities are subject to default risk. The default risk is in general less than 0.5% for a typical high-grade borrower. This risk cannot be hedged away. However, it can be shifted and someone must bear it in the end. This chapter deals with several aspects regarding credit risk and credit valuation.

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2. Introduction to Credit Valuation: The CreditMetrics Approach 2.1. The Portfolio Context of Credit A quantitative portfolio approach to credit risk management allows the study of concentration risk. This risk refers to additional risk resulting from higher exposure to one or several correlated obligors. A portfolio credit risk methodology as the one in CreditMetrics allows to capture simultaneously the benefits of diversification and concentration risks and provides an efficient risk-based capital allocation process. If we consider a bond rated BBB, which matures in n periods, then at the end of the year, the bond stays at BBB, the issuer defaults or it migrates up or down to one of the other categories. Hence, the probabilities that this bond will end up in one of the other categories in a period allow the computation of the bond price under each of the possible rating scenarios. The new present value of the bond can then be calculated from the remaining cash flows under its new ratings. The discount rate is obtained from the forward zero curve which is different for each rating category. The knowledge of the probabilities or likelihoods for the bond to be in a given rating category and the values of the bond in these categories allow the determination of the distribution of value of the bond in one period.

2.2. Obtaining the Distribution of Values for a Portfolio of Two Bonds In the presence of two bonds, the same procedure is applied for each bond. Since each bond can have any of eight values in one year (because of rating migration), the portfolio can take on 64 different values (8 × 8). The portfolio value at the end of the year in each of the 64 states is simply the sum of the values for each bond. The determination of the portfolio value distribution needs the computation of the probability of observing these values. The estimation of the 64 joint likelihoods can be done from historical rating data. If the rating outcomes are independent for the two bonds, then the joint likelihood is the product of the individual likelihoods for each bond (when correlation is zero). In practice, the correlations between rating migrations are not zero. Therefore, a model must be used for the estimation of correlations. The portfolio value distribution is obtained using the

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different 64 values of the portfolio at the year end and the likelihoods of achieving each of these 64 values.

2.3. Different Credit Risk Measures The two measures used in CreditMetrics in the appreciation of credit risk are the standard deviation and the percentile level. These measures reflect potential losses from the same portfolio distribution. The first measure (determination of the standard deviation) needs the computation of the mean value for the portfolio by multiplying the values with the corresponding probabilities and then adding the resulting values. This allows the computation of the standard deviation. The second measure is a specified percentile level which indicates the lowest value that the portfolio will achieve 1% of the time (the first percentile). The likelihood that the actual portfolio value is less than this number is only 1%. The above analysis and the proposed concepts apply to other types of exposures like loans, letters of credit, swaps and forward contracts.

2.4. Stand-Alone or Single Exposure Risk Calculation The procedure used in CreditMetrics for the computation of the credit risk for a single or stand-alone exposure is based on three steps. The first step: Credit rating migration This step estimates changes in value due to up (down) grades and default. The likelihood of credit rating migration is conditioned on the senior unsecured credit rating of the obligor. A transition matrix is conceived using public rating migration data. This allows the determination of the likelihoods of migration to any credit quality state in a given period. The second step: Valuation The values at the risk horizon are determined for the credit quality states. When the credit quality migration corresponds to a default case, the likely residual value is a function on the seniority class of the debt. When the credit quality migration is in another category, the forward zero curves for each rating category are used to revalue the bond’s remaining cashflows. The third step: Credit risk estimation Using the likelihoods and values, it is possible to calculate the risk estimate: the standard deviation or the percentile level.

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2.5. Portfolio Risk Calculation in CreditMetrics The procedure for a stand-alone exposure can be used for a portfolio of two exposures. In general, there are eight possible outcomes for an obligor’s credit quality in one period. In the presence of two obligors, all possible combinations of states can contribute to the joint risk. When the two obligor’s credit rating changes are independent, the joint likelihood is the product of the individual likelihoods. The introduction of correlations needs a model that links firm asset value to firm credit rating. The Merton’s (1974) model or the option theoretic valuation of debt is useful in this context. When the assets value falls in a way such that the value is less than the amount of liabilities outstanding, (the default threshold), then the firm will default. Merton’s (1974) model can include rating changes as well as default threshold. The value of the firm’s assets with respect to these thresholds determines the future rating. The extension of this model allows a link between the firm’s value and the firm’s credit rating. It allows to build the joint probabilities for different obligors.

2.5.1. Portfolio Credit Risk The volatility of value due to credit quality changes can be calculated from the joint states between obligors, i.e., the joint likelihoods and revaluation estimates. The value of the portfolio is simply the sum of the individual values. The standard deviation and the percentile level are calculated as in the stand-alone exposure case. The marginal risk The concept of marginal risk indicates where the risks are concentrated in the portfolio. The marginal risk shows the marginal increase to the portfolio risk as a consequence of adding a new bond to the portfolio. The marginal risk can refer also to the marginal impact on portfolio risk of increasing the exposure by a small amount.

2.5.2. Differing Exposure Type CreditMetrics determines the credit risk for market-driven instruments like swaps and forward contracts. The value of a swap is given by the difference between two components. The first corresponds to the forward risk-free value of the swap cashflows. This component is the same for all forward credit rating states. The second component corresponds to the loss expected

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on a swap, resulting from a default net of recoveries by the counterparty on all the cash flows after the risk horizon. The difference between this second component and the first one allows the revaluation of the swap. The revaluation of the swap is based on the following formula: value of the swap in a period (with rating R) = risk free value in one period − expected loss in period 1 through maturity (with rating R), where R refers to any credit rating category. The expected loss for each forward non-default credit rating is given by: expected loss (with rating R) = average exposure (from period 1 to maturity), Probability of default in period 1 through maturity (with rating R) (1 − recovery fraction). The average exposure calculation is time consuming. The probability of default for each rating category between year 1 and the maturity corresponds to the expected loss calculation. It is calculated using a transition matrix. This method allows the computation of the swap value in each of the non-default credit rating categories. In the case of default during the risk interval, the expected loss in the defaulted state is given by: Expected loss (case of default) = expected exposure (in the first period) (1 − recovery fraction). This expression assumes that the risk interval is very short (one year for example).

3. Default and Credit Quality Migration in the CreditMetrics Approach 3.1. Default Credit rating agencies assign an alphabetic or numeric label to rating categories. Altman (1987) defines a default event with reference to missed interest and principal payments. The likelihood of credit distress is defined in CreditMetrics with respect to default rates. They use credit ratings as an indication of the chance of default and credit rating migration likelihood. They consider that the firm encountered credit distress even in a context when only the subordinated debt realized a default. The methodology in J.P. Morgan assumes that the senior credit rating is the most indicative of ecountering credit distress. Filling probabilities of default with a transition matrix CreditMetrics uses historical default studies to obtain transition matrices which comprise one-year default rates.

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3.2. Credit Quality Migration The value of a firm and its assets changes suggest changes in credit quality. As a firm moves toward bankruptcy, the value of equity falls. Since the credit rating of the firm is given, it is possible to work backward to the “threshold” in asset value that delimits default. In the approach used by J.P. Morgan, the firm default model uses the default likelihood to place a threshold below which default appears. The methodology uses the rating migration probabilities to define thresholds above which the firm would upgrade or downgrade from its current credit rating. The rating migration probabilities are represented by a transition matrix. This matrix is a square table of probabilities which reflect the likelihood of migrating to an other rating category in a period given the obligator’s present credit rating.

3.3. Historical Tabulation CreditMetrics uses a technique to model different volatilities of credit quality migration conditioned on the actual credit rating. Hence, each row in the transition matrix allows the description of a volatility of credit rating changes which is unique to that row’s initial credit rating. Some rating agencies publish tables of cumulative default likelihood over longer holding periods. It is possible to use a cumulative default rate table to get an implied transition matrix which better replicates the default history. In this spirit, CreditMetrics uses a transition matrix to model credit rating migrations. This matrix can be constructed by using a least squares fit to the cumulative default rates. The Markov process is used to model the proceess of default. Using the Markov process, it is possible to generate a cumulative default rates matrix from an imputed transition matrix.

3.4. Recovery Rates Estimating recovery rates In the event of default, the estimation of recovery rates is not an easy task. The bond market prices in an efficient way the future realized liquidation values. Recovery rates of bonds Carty and Lieberman (1996), in Moody’s study, estimate recovery rates for corporate bonds using different seniority classes: senior secured, senior

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unsecured, senior subordinated, subordinated, junior subordinated, etc. The examination of the recovery statistics by seniority class shows that the subordinated classes are appreciably different from one another in their recovery realizations. However, there are no statistically significant difference between secured and unsecured senior debt. Recovery rates of bank facilities Bank facilities include loans, letter of credit and commitments among other things. Carty and Lieberman (1996) and Asarnow and Edwards (1995) consider bank facilities as a seniority class of their own. These studies report estimates of the mean and median recovery rates which are not different by more than 5%. These studies are based on the U.S. bankruptcy experience. Table 1 provides the one-year transition matrix. Table 2 gives the recovery rates for different types of debt. Table 3 gives the credit spreads for different ratings and years.

Table 1:

AAA AA A BBB BB B CCC

Input data 1-year transition matrix.

AAA (%)

AA (%)

A (%)

BBB (%)

BB (%)

B (%)

CCC (%)

Default (%)

Sum (%)

90.00 0.70 0.08 0.03 0.02 0.00 0.17

8.05 91.14 2.32 0.29 0.09 0.09 0.00

0.60 6.30 90.00 6.00 0.34 0.11 0.24

0.05 0.40 4.00 85.00 5.66 0.30 1.00

0.20 0.10 0.70 5.00 84.00 6.34 1.99

0.00 0.20 0.30 1.10 6.15 86.00 8.50

0.00 0.01 1.00 0.09 0.80 3.50 70.00

1.10 1.15 1.60 2.49 2.94 3.66 18.10

100.00 100.00 100.00 100.00 100.00 100.00 100.00

Table 2:

Senior secured Senior unsecured Senior subordinated Subordinated Junior subordinated

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Recovery rates.

Mean (%)

Standard deviation (%)

65.00 45.00 35.00 25.00 15.00

20.00 18.00 16.00 19.00 10.00

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Table 3:

AAA AA A BBB BB B CCC

Credit spreads.

1 (%)

2 (%)

3 (%)

5 (%)

7 (%)

10 (%)

20 (%)

30 (%)

0.12 0.42 0.52 0.62 0.72 0.82 0.92

0.17 0.47 0.77 1.07 1.37 1.67 1.97

0.22 0.52 0.82 1.12 1.42 1.72 2.02

0.27 0.57 0.87 1.17 1.47 1.77 2.07

0.32 0.62 0.92 1.22 1.52 1.82 2.12

0.37 0.67 0.97 1.27 1.57 1.87 2.17

0.42 0.72 1.02 1.32 1.62 1.92 2.22

0.47 0.77 1.07 1.37 1.67 1.97 2.27

3.5. Credit Quality Correlations The estimation of default correlations is not an easy task. Gollinger and Morgan (1993) use time series of default likelihoods to correlate across 42 constructed indices of industry default likelihoods. Stevenson and Fadhil (1995) correlate the default experience across 33 industry groups.

3.5.1. Finding Evidence of Default Correlations The study of the histories of rating changes and defaults reveal the existence of correlations in credit rating changes. CreditMetrics use the following formula to compute the average default correlation from the data:
 σ2 N (µ−µ − 1 2) σ2 ρ= ∼ (N − 1) (µ − µ2 ) where: N: number of names covered in the data, µ: average default rate over the years in the study, and σ: standard deviation of the default rates observed from year to year. The estimation of default correlations is a first step toward the estimation of the joint likelihood of any possible combination of credit quality outcomes. When the credit rating system uses n states (AAA, . . . , default), then between two obligators, there exists n(n) possible joint states for which the likelihoods can be estimated. The estimation of joint credit quality migration likelihoods can be done using credit ratings time series across several firms. This estimation method assumes that all firms with a given credit rating are identical.

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3.5.2. Estimating Credit Quality Correlations through Bond Spreads It is possible to study price histories of corporate bonds in order to estimate credit correlations from historical data. The correlations in the bond price dynamics allow for estimations of correlations of credit quality moves. This approach estimates credit correlation from bond prices and credit spreads. It estimates the correlation in the movements of the credit spreads. Then, it uses a model to link spread movements to credit events. A pricing model as the one proposed by Duffee (1995) allows to infer the probability of the issuer defaulting from the observed bond spread. This procedure is similar to that of estimating the implied volatility from option prices.

3.5.3. Asset Value Model The approach in CreditMetrics is based on a process which drives credit rating changes and the estimation of its parameters. Their approach is based on Merton’s (1974) model in which the firm’s asset value drives the credit rating changes and defaults. The model assumes that there is a certain level such that if the firm’s assets fall below it in the next period, the firm will be unable to satisfy its obligations and will default. The method relates the asset value in one period to the credit rating or default of the firm in one period. The asset values corresponding to changes in rating are referred to as asset value thresholds. The CreditMetrics model assumes that the asset has a mean µ and a volatility σ. This parametrization of the asset value process allows a connection between the asset thresholds and the transition probabilities for a given firm. It also allows the computation of the probability that each event occurs. Hence, for example: Pr(default) = Pr(R < Zdef ) = (Zdef /σ) where  is the cumulative distribution for the standard normal distribution. Hence, there exists asset returns thresholds Zdef such that if R is less than Zdef , the obligor will default. If we know (Zdef /σ), it is possible to obtain Zdef using −1 (p) which gives the level below which a standard normal variable falls with probability p. This procedure is used to determine threshold values for asset return as well as the corresponding transition probabilities for each obligor according to its asset value processes. Tables 4–7 provide different illlustrations of credit VaR computations. Table 4 shows the information about the obligor and the currency used. The

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Table 4: Credit VaR stand alone. User Inputs Obligor information Senior unsecured long term rating Industry group (for credit curves) Base currency (for yield curve) Bond #1 Principal amount of bond Fixed rate coupon (% of par) Maturity (integer years) Seniority standing Seniority (enter 1-to-5) Bond #2 Principal amount of bond Fixed rate coupon (% of par) Maturity (integer years) Seniority standing Seniority (enter 1-to-5) Bond #3 Principal amount of bond Fixed rate coupon (% of par) Maturity (integer years) Seniority standing Seniority (enter 1-to-5)

4 1 4

BBB 0 Euro

1.000 5.50% 5Years 1

Senior secured

1.000 6.00% 4Years 2

Senior unsecured

1.000 8.00% 1Year 2

Senior unsecured

Outputs Expected default losses Due to exposure #1 Due to exposure #2 Due to exposure #3 Total

15.44 10.69 10.69 36.81

Expected change in value Due to exposure #1 Due to exposure #2 Due to exposure #3 Total

(34.08) (18.10) 43.10 (9.08)

Volatility of value Due to exposure #1 Due to exposure #2 Due to exposure #3 Total

63.08 92.47 101.77 256.12

ariation. 12:16:11.

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Table 5: Credit VaR stand alone. User Inputs Obligor information Senior unsecured long term rating Industry group (for credit curves) Base currency (for yield curve) Bond #1 Principal amount of bond Fixed rate coupon (% of par) Maturity (integer years) Seniority standing Seniority (enter 1-to-5) Bond #2 Principal amount of bond Fixed rate coupon (% of par) Maturity (integer years) Seniority standing Seniority (enter 1-to-5) Bond #3 Principal amount of bond Fixed rate coupon (% of par) Maturity (integer years) Seniority standing Seniority (enter 1-to-5)

4 1 4

BBB 0 Euro

1.000 5.50% 4 Years 1

Senior secured

1.000 6.00% 5 Years 2

Senior unsecured

1.000 8.00% 1 Year 5

Junior subordinated

Outputs Expected default losses Due to exposure #1 Due to exposure #2 Due to exposure #3 Total

15.44 10.69 3.56 29.69

Expected change in value Due to exposure #1 Due to exposure #2 Due to exposure #3 Total

(20.11) (18.10) 57.66 19.45

Volatility of value Due to exposure #1 Due to exposure #2 Due to exposure #3 Total

65.00 92.47 146.99 301.01 onds.

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Table 6: Credit VaR stand alone. User Inputs Obligor information Senior unsecured long term rating Industry group (for credit curves) Base currency (for yield curve) Bond #1 Principal amount of bond Fixed rate coupon (% of par) Maturity (integer years) Seniority standing Seniority (enter 1-to-5) Bond #2 Principal amount of bond Fixed rate coupon (% of par) Maturity (integer years) Seniority standing Seniority (enter 1-to-5) Bond #3 Principal amount of bond Fixed rate coupon (% of par) Maturity (integer years) Seniority standing Seniority (enter 1-to-5)

4 1 4

BBB 0 Euro

1.000 5.50% 1 Year 1

Senior secured

1.000 6.00% 5 Years 1

Senior unsecured

1.000 8.00% 1 Year 1

Senior secured

Outputs Expected default losses Due to exposure #1 Due to exposure #2 Due to exposure #3 Total

15.44 15.44 15.44 46.31

Expected change in value Due to exposure #1 Due to exposure #2 Due to exposure #3 Total

(3.37) (13.35) 19.88 3.15

Volatility of value Due to exposure #1 Due to exposure #2 Due to exposure #3 Total

67.30 66.00 70.64 203.83 onds.

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Table 7: Credit VaR stand alone. User Inputs Obligor information Senior unsecured long term rating Industry group (for credit curves) Base currency (for yield curve) Bond #1 Principal amount of bond Fixed rate coupon (% of par) Maturity (integer years) Seniority standing Seniority (enter 1-to-5) Bond #2 Principal amount of bond Fixed rate coupon (% of par) Maturity (integer years) Seniority standing Seniority (enter 1-to-5) Bond #3 Principal amount of bond Fixed rate coupon (% of par) Maturity (integer years) Seniority standing Seniority (enter 1-to-5)

4 1 4

BBB 0 Euro

1.000 5.50% 1 Year 5

Junior subordinated

1.000 6.00% 5 Years 4

Subordinated

1.000 8.00% 1 Year 3

Senior subordinated

Outputs Expected default losses Due to exposure #1 Due to exposure #2 Due to exposure #3 Total

3.56 5.94 8.31 17.81

Expected change in value Due to exposure #1 Due to exposure #2 Due to exposure #3 Total

(15.25) (22.85) 48.53 (10.43)

Volatility of value Due to exposure #1 Due to exposure #2 Due to exposure #3 Total

135.34 121.53 116.74 373.01 bonds.

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necessary information about the different bonds 1, 2 and 3 is used as an input. The output corresponds to the expected default losses, the expected change in value and the volatility of value. A similar analysis is conducted in other tables. The reader can make some comments about the observed results.

3.5.4. Application of Model Outputs The literature on credit pricing includes Das, Sanjiv and Tufano (1996), Jarrow and Turnbull (1995), Merton (1974), etc. The measures of credit risk can have several applications. Prioritizing risk reduction actions When a manager sets priorities for actions, the use of risk statistics allows the reduction of portfolio risk. Credit risk limits The risk statistics can be used for limit setting based on percentage risk. This needs the choice of a specific risk measure to be used. The analysis is concerned with individual exposures and relative measures. The standard deviation is an adequate statistic since it could capture the relative risks of different instruments. Economic capital assessment Credit risk measures can be used to asess the capital which a firm puts at risk by holding a credit portfolio. The analysis is concerned with a portfolio measure. A percentile level can be an appropriate indicator of economic capital. For example, using the first percentile level, the economic capital can be regarded as the level of losses on a portfolio for which we are 99% certain. If the investor is 99% certain of meeting the obligations in the next period, then the first percentile level can be considered as the economic capital allocated to an asset portfolio.

4. Credit Valuation and the KMV Approach Credit valuation provides a mean to reduce the likelihood of losses through portfolio diversification. The credit risk is in general measured in terms of probabilities using a credit valuation model. The model is based on a theory that describes the link between the attributes of the borrowing institution and

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its possible bankruptcy. The approach of KMV for credit valuation is based on option pricing theory. If the model captures the relationship between the state of the institution and the probability of default, it will account for the borrower’s credit standing through time. The model can then be applied to detect potential deterioration of credit using actual market data. Since the firm’s ability to pay its debt depends upon its future market value, the firm can easily raise cash by selling off a fraction of its assets or by issuing debt. The value of the firm depends on the present assessment of its future returns from its business. This value can be obtained from the value of equity and that of bond issues. The value of equity corresponds to the product of share prices by the number of shares outstanding. The value of bond issues is given by the current price (per unit of face value) times the face amount of the issue. Current liabilities are in general valued at their nominal values.

4.1. Loan Default How to value a credit? Consider an all-equity financed firm who issues a commercial paper. The lender buys a claim on the firm’s assets. The value of the levered firm becomes equal to the value of its equity and debt. The value of the company will change randomly in anticipation of its future cash-flows. If at the debt’s maturity date, the asset value is higher than the face value of debt, the stockholders will pay the debt. If the firm’s value is below the amount due on the loan, the firm can face bankruptcy and the stockholders get nothing. The lender’s risk emerges in this situation. This risk can be studied using the probability that the asset value at maturity of the loan falls below the loan balance. The probability of default can be calculated in this context by modelling the dynamics of the asset value. This probability is a function of the expected return on assets, the volatility of the firm’s cash flows, the face value and the maturity of the debt. The probability of default characterizes the occurrence of loss, rather than the expected loss (i.e. the dollar amount). The expected loss can be defined as the probability weighted mean dollar amount of the difference between the nominal value of debt and the actual inflows for the lender. In this context, a formula can be given for the expected loss as the difference between two terms. The first corresponds to the debt nominal value times the probability of default. This represents the expected loss in the event of default. The second term corresponds to the expected amount to be recovered.

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4.2. Debt Structure For a levered firm, the structure of corporate liabilities can include debt of various maturities and current liabilities such as accounts payable. There is a hierarchy of the claims on the firm’s assets. This hierarchy is important since additional debt can lead to a transfer of wealth between the different claimholders. In fact, additional debt can reduce the expected loss for holders of higher priority claims. At the same time, it can increase the expected loss for the holders of claims subordinated to the additional amount of debt. The priority structure and the maturities of the different types of debt are important since the claims maturing earlier, can trigger bankruptcy. In the presence of different types of debts, it is also possible to obtain a formula for the expected loss function.

4.3. Capital Flows, Loan Pricing and Portfolio Diversification The distribution of dividends and cash-flows does not affect the firm’s value in the Modigliani and Miller analysis. However, it can affect the distribution of value between the different classes of claims. The payments of dividends and coupons can decrease the value of the firm’s assets and by the way, they can increase the probability of default as well as the expected loss. Credit valuation allows the pricing of loans, i.e. the interest rate that must be applied for a given loan. Since the value of a loan is a function of the firm’s assets, it is possible to use the option pricing theory for the computation of the loan price. Portfolio diversification can allow a decrease in the probability in large losses. However, it does not reduce the amount of the expected loss on a portfolio. This amount is given by the average of the expected losses on the individual assets, weighted by their relative weights in the portfolio. Using a portfolio with different loans and the main concepts in portfolio theory, it is clear that the dispersion of a diversified portfolio loss around its mean is much smaller. In the limit, a well diversified portfolio would show no deviation of the actual loss from the expected amount. In practice, it is not possible to obtain a well diversified portfolio of loans because individual loans are not completely independent. This result is well known because of common risk or systematic risk that cannot be diversified away. However, the total risk can be reduced to the minimum by eliminating most of the specific risk. This is possible if the loans in the portfolio are not concentrated in any one segment of the market. The degree of diversification can be appreciated by the variance of portfolio losses in a mean–variance context. 12:16:11.

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4.4. On Credit Valuation in the KMV Approach A credit valuation model must be based on a theory that describes the relationship between the characteristics of the firm and its potential bankruptcy. The firm’s value depends on its activity and cashflows. It is given by the sum of its equity and liabilities in the balance sheet. When the value of the firm at the debt maturity date is higher than the face value of debt, the firm pays the bondholders. Otherwise, the firm is in default. In this setting, stockholders get nothing and bondholders take over the firm’s assets. This approach is initiated by Black and Scholes (1973) and Black and Cox (1976). The probability that the asset value at the debt’s maturity date be less than the loan balance can represent a measure of credit risk. This probability characterizes the occurrence of loss. By assimilating the different claims on the firm’s value as options, the firm’s value is shared by all stakeholders. The distribution of the firm’s value between different stakeholders can be effected between the different classes of claims. The main purpose of credit valuation concerns the pricing of debt by accounting for its risks. Since corporate debt represents a claim on the firm’s value, it is possible to use the theory of derivative asset pricing to value this claim. Following the work of Black and Scholes (1973) and Black and Cox (1976), Vasicek (1984) assumes that market value of the firm’s assets A is equal to the market value of equity plus the market values of current liabilities C, the short term debt D, the bonds. Hence, we have: A = C + D + B + S.

(1)

The value of the firm’s assets are described by the following process: dA = µA dt + σA dz

(2)

where: µ: instantneous mean rate of return on the assets, σ: instantaneous standard deviation of the rate of return on the assets, dz: an increment of the Wiener process. Let d be the total amount of distributions (dividends and bond interest) over the term T corresponding to the maturity date of current liabilities. If d is paid at time t = 0, then dA(0) = −F.

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The firm is in default if the value of its assets at date T , A(T) are less than the amount payable at that date A(T) < DT + CT . Using Eqs. (2) and (3), the logarithm of the total asset value at time t = 0 follows a normal distribution with a mean: E(log A(t) | A(0) = A) = log(A − F) + µt − 1/2σ 2 t

(4)

and a variance Var(log A(t) | A(0) = A) = σ 2 t.

(5)

4.4.1. Loan Default Now, we can study the loan default in this context. The short-term loan is considered in default when the value A(T) of the assets is less than the amount payable at maturity of the loan: A(t) < DT + CT . We denote the probability of default by: ρ = P[A(t) < DT + CT | A(0) = A]. This probability can be written as   log(DT + CT ) − log(A − F) − µT + 1/2σ 2 T ρ=N √ σ T

(6)

(7)

where N(.) indicates the cumulative normal distribution function. The loan loss L on the short term debt is given by: L = 0,

if A(T)  DT + CT

= DT + CT − A(T),

if CT  DT + CT

= DT ,

if A(T) < CT .

In this context, the expected loss is  DT +CT  EL = (DT + CT − a)f(a)da +

(8)

Cr

DT f(a)da 0

CT

where f indicates the probability density of A(T) given A(0) = A. The expected loss is given by:   log(DT + CT ) − log(A − F) − µT + 1/2σ 2 T EL = (DT + CT )N √ σ T   log(DT + CT ) − log(A − F) − µT + 1/2σ 2 T µT − (A − F)e N √ σ T 12:16:11.

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 log(DT + CT ) − log(A − F) − µT + 1/2σ 2 T − CT N √ σ T   log CT − log(A − F) − µT + 1/2σ 2 T µT . + (A − F)e N √ σ T

465



(9)

The expected loss on the combined claim represented by the current liabilities and the short-term debt corresponds to the sum of the first two terms. This result can be seen as the face value of this claim times the probability of default less the recovered amount. The negative of the sum of the third and fourth terms indicates the expected loss on the current liabilities alone. The expected loss on the short-term loan is the difference between the expected loss on the combined claim minus the expected loss on the current liabilities.

4.4.2. Loan Pricing Recall from the standard option pricing theory that the dynamics of the value A of an asset are given by dA = µA dt + σA dz. In this context, the value of a derivative asset D obeys the following partial differential equation: Dt + rADA + 1/2σ 2 A2 DAA − rD = 0.

(10)

In this context, the value of a loan D can be computed using this equation and the boundary condition at time t = T : D(T) = DT − L. Using Eq. (8), we have: D(T) = DT ,

if A(T)  DT + CT

= A(T) − CT ,

if CT  A(T) < DT + CT

= 0,

if A(T) < CT .

The solution of Eq. (10) under this boundary condition is: D = (DT − Q)e−rT

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where

 log(DT + CT ) − log(A − F) − rT + 1/2σ 2 T √ σ T   log(DT + CT ) − log(A − F) − rT + 1/2σ 2 T rT − (A − F)e N √ σ T   log(CT ) − log(A − F) − rT + 1/2σ 2 T − CTN √ σ T   log CT − log(A − F) − rT + 1/2σ 2 T rT . + (A − F)e N √ σ T Note that the price of the loan loss, Q, is given by a formula identical to that for the expected loss in Eq. (9). The main difference is that the expected rate of return on assets is replaced by the risk-free interest rate r. When µ = r, then Q = EL. When µ > r, then Q > EL and the loan is priced in such a way that the return on the loan compensates the lender for the expected loss. We denote by β the beta of the asset and by µL the expected rate of return on the loan. If the expected rate of return on assets in excess of the riskless rate is proportional to the asset beta, the expected excess rate of return on the loan will also be proportional to the beta of the loan as βL = (A/D)DA β. Since Eq. (10) is equivalent to the condition 

Q = (DT + CT )N

A DA (µ − r) D it follows that µL − r = βL ((µ − r)/β). This means that the loan return carries a compensation for the systematic portion of the variance of the possible loss. Now, we denote by i the interest rate charged on the loan: µL − r =

D = DT e−iT . The rate premium, corresponding to the difference between the loan interest rate and the riskless rate is given by:   Q 1 . i − r = − log 1 − T DT This result can also be given in terms of simple rates of interest rather than continuous compounding. If we denote by R and I the simple riskless rate

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and simple loan interest rate, then: 1 + RT = erT ; 1 + IT = eiT . In this case, the simple interest rate premium can be written as: I −R=

1Q . TD

If Q were equal to the expected loss, the loan interest rate would exceed the risk-free rate simply by the amount of the expected loss taken as a percentage of the amount advanced and annualized. If investors are risk-averse, the rate premium would in addition to the expected loss incorporate a premium for the systematic risk of the loan.

5. Portfolio Management of Default Risk in the KMV Approach Corporate bonds and liabilities are subject to default risk. The default risk is in general less than 0.5% for a typical high-grade borrower. This risk cannot be hedged away. However, it can be shifted and someone must bear it in the end. Portfolio theory and quantitative methods have been used to compute the amount of risk reduction attainable through diversification for a portfolio of equity. This theory is widely used by practitioners who developed techniques for computing the asset attributes which are fundamental for an actual portfolio management tool. These developments have not been implemented for debt portfolios. KMV developed these methods in its practice with commercial banks in order to measure diversification and to maximize return in debt portfolios.

5.1. The Model of Default Risk When a lender acquires a corporate note, it is as if he is engaged in two transactions. The first is buying a debt obligation. The second is the sale of a put option to the borrower. In fact, when the firm’s assets are less than the face value of the debt, the borrower “puts” the assets to the lender and uses the cash proceeds to pay the note. Hence, the situation of the lender can be represented by a portfolio with two assets: long a risk free bond and short a default option. The probability of default can be determined using option pricing theory and mainly the volatility of the firm’s assets. If you consider, for example, a firm with a market value of 100 million dollars and a debt of 50 million dollars (maturing in one year) in the presence of a given volatility,

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then it is possible to represent the range of possible asset values and their frequencies in a diagram. The diagram representing the future firm asset value and the frequency distribution will show a default point at 50 million (the left hand side of the frequency distribution). Hence, the default risk of a company can be derived from the behavior of its asset values and its liabilities.

5.2. Asset Market Value and Volatility The firm’s equity behavior can also be derived from the firm’s asset values. The position of stockholders can be viewed as a call on the firm’s asset value where the strike price is given by the face value of debt obligations. If at the debt’s maturity date, the firm’s value is higher than the amount of liabilities, stockholders exercise their calls by paying off their obligations. Otherwise, they default. When the market value changes, this induces changes in the value of liabilities depending on the degree of seniority. When the asset value falls to a critical level, the probability of default increases and the market value of the liabilities decreases. In general, when the asset value falls, the volatility of equity increases. Option pricing theory can be used to infer the volatility and asset value.

5.3. Measurement of Portfolio Diversification The diversification of a portfolio of loans can be measured by specifying the range and likelihood of all possible losses. Defaults produce losses and the possibility to recover a fraction of the face value of the loan. In this context, KMV specifies the “loss given default” as one less the expected recovery. In the same context, the expected loss for a single loan corresponds to the probability of default times the loss given default. The unexpected loss is given by the volatility of loss. It is equal to the loss given default times the square root of the product of the probability of default by one minus the same probability. If we denote by: EDF: the probability of default, LGD: the loss given default (as a fraction of the face value of debt), then √ expected loss: EL = (EDF)(LGD). Unexpected loss: UL = LGD × EDF(1 − EDF). The average expected loss corresponds to the average of the expected losses of the portfolio components by accounting for the appropriate weights. The weights are equal to each exposure amount as a percent of the total portfolio exposure. The computation of portfolio diversification is equivalent to the calculation of the portfolio’s unexpected loss using default correlations.

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5.4. Model of Default Correlation The strength of the default relationship between two borrowers can be appreciated by default correlation. When this correlation is zero, the probability that they default simultaneously corresponds to the product of their individual default probabilities. When the correlation varies between zero and one, the higher the correlation, the higher the probability that the two borrowers default at the same time. KMV basic default model assumes that the firm defaults when its market asset value falls below a critical level. Hence , the joint probability of default corresponds to the likelihood that two firms asset values are below the critical values. The option pricing theory is helpful in the appreciation of default correlation between firms using their asset correlation and their individual probability of default. The formula for the default correlation in KMV analysis is: (actual probability of both defaulting−probability of both defaulting if they were independent) Default correlation = . √ EDF1(1 − EDF1)EDF2(1 − EDF2)

5.5. The Likelihood of Large Losses In general, individual debt assets show skewed loss probabilities and the probability of a small loss is much higher than the probability of large losses. The skewness reveals that in general, the actual losses are less than the average losses. The distribution of losses shows in general a small probability of large losses and a large probability of small losses. The knowledge of the actual asymmetric distribution of losses is important because it presents some implications for capital adequacy. In fact, when the frequency distribution of losses is given, it is possible to compute the likelihood of losses which is higher than the amount of capital held against the portfolio. In this context, we can set the probability to the desired level by varying the amount of capital.

5.6. Loan Valuation The value of a loan can be observed in a market place or it can be determined in relation with market prices of traded instruments. Consider, for example, a fixed rate corporate loan. The value of the loan is given by the discounting of its future cashflows at the appropriate rate. In the presence

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of default, the discount rate must account for the expected loss premium and the risk premium. The expected loss premium reflects the actuarial expectation of loss. The risk premium corresponds to the compensation for the non-diversifiable loss risk in the loan. It can be estimated using the borrower’s probability of default. The market price for risk can be computed from data in financial markets. A loan cannot be repaid (event 1) with a probability (EDF) or the firm can repay (event 2) with a probability (1 − EDF). Let us denote by RF the risk free interest rate. If we denote by Y the yield on the loan in the absence of default and by (RF − LGD) the return in the case of default, then the expected return on the loan is given by: E(r) = EDF(RF − LGD) + (1 − EDF)Y , E(r) = RF with Y = RF + LGD × EDF/(1 − EDF). The term LGD(EDF)/(1 − EDF) is referred to as the expected loss premium. In the KMV approach, the yield on the loan must be Y = RF + expected loss premium + risk premium.

5.7. Economic Capital and Fund Management Consider a bank with a portfolio of assets, called the “fund”. The economic capital of a bank is related to the market value of equity. In the absence of default risk, the economic capital can be considered as the excess of the market value of assets over the market value of liabilities. The objectives of a fund manager are to maximize the value of equity, to attain maximal diversification and to maintain capital adequacy. The fund manager can achieve some of its objectives by buying assets at or below market and selling them at or above market. Capital adequacy means the desired leverage must be computed each time a new asset is added to the existing portfolio. Capital adequacy refers to the use of equity funding in order to lower to an acceptable level the risk of default. Maximal diversification refers to the lowest level of portfolio unexpected loss given the level of expected return. Risk contribution and optimal diversification The portfolio management process needs the maximization of diversification for any given level of return. The portfolio management process needs the measurement of the risk contribution of each asset and its return relative to that risk.

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6. The Credit Risk Literature 6.1. The Accounting Analytic Approach Accounting analytic methods estimate firm specific credit quality with respect to financial ratios. The S& P and Moody’s publish historical default likelihood for their letter rating categories. Statistical prediction of the likelihood of default Several models try to build credit quality estimation with reference to statistical techniques. The main approaches used within this context are qualitative dependent variable models, discriminant analysis and neural networks. The linear discriminant analysis tries to categorize between the firms which have defaulted and the others. A classification is done with respect to a statistical estimation approach. This category includes, for example, the Z-score models à la Altman. Several other methods can be used. Examples include logistic regressions, probit/logit analysis, etc. Neural network techniques have also been applied to credit scoring.

6.2. The Option Pricing Approach This approach is initiated mainly by Merton (1974). The firms value evolves randomly and default occurs when the value of the firms assets becomes less than its obligations. This method is applied by KMV. The model gives a continuous numeric value which may be mapped to default likelihoods.

6.3. The Migration Analysis In order to assess risk, it is necessary to predict the potential range of outcomes using historical data. If an obligor is BB today, then chances he will be upgraded or downgraded for a given future horizon must be calculated. RiskMetrics use for this task migration analysis based on transition matrices. These matrices are published regularily. This approach is initiated by Altman (1993). Since then several authors have used migration analysis to better estimate allowance for loan and lease losses or expected default losses. CreditMetrics calculates the volatility of value due to credit quality changes, i.e. the potential of unexpected losses. Their approach is an extension of the models in Jarrow, Lando and Turnbull (1994), Das, Sanjiv and Tufano (1996), Ginzburg, Maloney and Willner (1993) among others.

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6.4. Expected Losses The volatility of losses or unexpected losses is in general more difficult to estimate with comparison to expected losses. Some institutions assumed that their specific portfolio will present the same correlation effects as some index portfolio which can be the total credit market or a sector index. In this context, two approaches have been proposed. The first refers to the historical default probability. The second refers to the volatility of holding period returns.

6.5. The Historical Default Volatility This approach used several assumptions to explain the volatility of default rates. The volatility of holding period returns The volatilities of total holding period returns are studied because the volatility of default events does not represent all the credit risk. The main studies in this context are conducted by Wagner (1996), Asarnow (1996), etc. The general method referred to as the RAROC approach is based on the tracking of a benchmark corporate bond (or index). The estimated volatility of value is used to proxy for the volatility of some exposure under analysis. The main problem with this approach is its inability to estimate infrequent credit events such as upgrades, downgrades and default. The CreditMetrics approach uses long term estimate of migration likelihood. This is a main advantage with respect to the standard approach because it does not depend on observation within some recent sample. CreditMetrics shows the construction of volatility across credit quality categories. The relative frequencies are represented as a function of the bond values. Bonds which belong to a credit rating category have volatility of value due to day-to-day credit spread fluctuations. These fluctuations are estimated by the RAROC approach. The probabilistic CreditMetrics approach is based on the assumption that all migrations might have been realized where each is weighted by the likelihoods of migration. The likelihoods of migration are estimated from long term data.

6.6. A Portfolio View The modern portfolio theory is applied by several institutions to estimate the portfolio effects of credit risk in a mean-variance framework. Stevenson

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and Fadhil (1995) constructed industry indices of default experience. They use the industry level estimate default correlation to appreciate the correlations between the indices. Gollinger and Morgan (1993) used time series of default likelihoods to estimate default correlations across industry indices. KMV used Merton’s model (1974) to estimate the value of the firm’s debt in the context of the option pricing theory.

7. Uses and Abuses of Bond Default Rates The manager of a bond portfolio subject to default risk is interested in the average expected loss. The credit risk for the bond portfolio can be defined as the possible range of losses around the average expected loss. The main studies in this field try to estimate the default rate for a given rating class, the annual variance of the default rate, and the probability to move from an initial rating to any possible rating. The estimate of this latter probability can be done in a square table referred to as the transition matrix.

7.1. Examples Example 1. Computation of the expected loss of a loan Consider a portfolio of loans with different rating classes from 1 to 6. The manager knows that bonds rated 3 are equivalent to bonds rated BBB. The historical default rate of BBB bonds is 0.15% and the probable loss in the event of default is 30%. In this case, it is possible to compute the expected loss of each loan per dollar of exposure as: Expected loss of each loan per dollar of exposure = (default probability)(loss given default) = 0.15(0.30) = 0.045%. This calculation must be repeated for each class of loans and summing gives the expected dollar loss for the entire portfolio. Example 2. A simple approach for default risk The default risk for a single loan over a horizon of one year can be calculated by assuming two outcomes: default and non-default. The loss in the event of default is 30%. The volatility of the loss (the unexpected loss) is given by the following formula:  Unexpected loss = (loss given default) DF(1 − DF)

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where DF represents the default rate. For the 3-rated bond, this is: 0.3 √ (0.0015)(1 − 0.0015) = 1.2%. This number shows the range of variation of loss outcome around the expectation for a given loan. Example 3. CreditMetrics approach This approach is more complex and accurate for the computation of default risk. It allows for the possibility of changes in value of the bond due to credit quality changes, even in the absence of default. In the case of non-default, there are different events corresponding to better credit quality, the same credit quality or a worse credit quality. Using the product of the probability of each event by the resulting gain or loss, allows the computation of the expected loss. The product of the probability of the event by the square of the deviation of loss over all states allows the computation of the variance of loss, and the unexpected loss (as the square root of the loss variance).

7.2. Problems with Characterizing Default Probabilities with Discrete Categories The computation of default probabilities is based on the fact that the default rate for assets within a given rating grade is the same (the historical average default rate). The results in Kealhofer et al. (1998) show that: — the historical average default rate and the historical transition probabilities are different from the actual rate and actual transition probabilities for loans within a given grade, — there are differences of default rate within a bond rating grade. In practice, the variations in default rates within a grade can impact the pricing guidelines based upon rating. Risk measurement systems based upon rating group default rates and rating transition probabilities are subject to measurement errors.

8. Managing Default Risk in Portfolios of Derivatives The growth in the derivatives industry justifies the desire to monitor the expected default, to price default risk and to preserve capital adequacy. The default risk measures are used to implement a mean–variance portfolio analysis. This context allows the determination of the overall portfolio risk.

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8.1. Default Risk Versus Market Risk It is well known from portfolio theory that market risk cannot be managed through portfolio diversification. For a portfolio of assets, diversification works quite well when the correlation between the assets is very small. Default risks have very low correlations and depend heavily on the firm’s business. If you consider the correlation of default risk between two counterparties, then the probability that they default at the same time are closely related. Hence, default risk can be managed through diversification. However, the default risk of a given firm is closely related to its future cashflows and not to the performance of other companies. Default risk can be hedged by means of a financial instrument which is related to a specific counterparty. The simplest way is to get a guarantee of the counterparty’s performance from a third party. Firms selling a third-party credit guarantees are specializing in credit risk and use diversification to manage this risk.

8.2. Portfolio Default Risk The analysis suggests that an investor managing a portfolio of derivatives must minimise default risk by maximizing diversification. A portfolio manager must enhance performance in order to attain the highest return for an acceptable level of risk. The management of default risk needs the knowledge of the three critical quantities: — the probability of default for each counterparty, — the joint probability between each pair of counterparties, — the correlation between the quantities exposed to loss in the event of default, for each pair of contracts.

8.3. Expected Exposure In the absence of recovery, the loss for a counterparty in the event of default is the replacement cost of the original contract. The exposure created by a derivative contract corresponds to the value of the contract at a future date when there is no default risk. The assumption 1, A1 in Kealhofer et al. (1998) stipulates that X = max(0, a + bM) where X is the amount of the exposure, M is a random market variable with a distribution function F(m) and a and b are known constants. The exposure amount can then be written as an option payoff where the constants (a > 0, b > 0) corresponds to a call and (a >< 0, b < 0) corresponds to a put.

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The assumption 2, A2 stipulates that the risk of default of the counterparty is independent of the size of the exposure generated by a given deal. Expected payoff The payoff to a deal corresponds to its value above “break-even” pricing on a given transaction. This latter corresponds to a pricing such that the value of the cash inflows less the flows given up, cover the cost of doing the deal in the absence of default risk. The expected loss under the deal depends on the expected exposure (EX), the EDF and the LGD. Using these assumptions, Kealhofer shows that the expected exposure for a single contract is: E(X) = (a)F(−a/b) + (b)H(−a/b) where H(z) is the truncated mean of M calculated from the lower bound to z. The expected payoff for a single contract is: E(pay of fi ) = pay of fi − Eli and Eli = E(Xi )(EDFi )LGDi . The expected payoff for a portfolio is given by the sum of the expected payoffs of its components:  E(Pp ) = E(Pi ). The expected return can be defined as the expected payoff divided by the notional amount of the deal: E(Pp )  E(Rp ) =  = wi E(Ri ) Ni where: Ni : notional amount of contract i, wi : notional amount of contract i as a fraction of the total notional amount, and E(Ri ): expected payoff to contract i as a fraction of the notional amount of exposure i.

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Unexpected loss Unexpected loss or standard deviation of loss measures the range of possible losses and is given by:  UL = (EDF) var(X) + EDF(1 − EDF)(E2 (X))LGD/N where the variance of exposure is given by: var(X) = a2 F(1 − F ) + 2abH(1 − F ) + b2 (J 2 − H 2 ) and J 2 is the truncated second moment of F . The quantities H, F and J are functions. The portfolio unexpected loss is given by:  ULp = wi wj (Cov(Xi , Xj )(EDFi × EDFj + JDFij )  + E(Xi )E(Xj )JDFij ))LGDi LGDj

(Ni Nj )

with: Cov(Xi , Xj ) = ai aj (1 − Fj ) + bi aj Hi (1 − Fj ) + ai bj (Hi − Fi Hj ) + bi bj (Ji2 − Hi Hj (1 + 2Fj − 2Fi )). This covariance corresponds to the case of two put-type exposures. This formula shows that the portfolio unexpected loss is a function of the following parameters: — — — —

the magnitude of the expected exposures, the covariance between exposures, the EDF and the LGD for each counterparty, the joint default frequency (JDF) between each pair of counterparties.

The above formulas can be applied to conventional loan portfolios and can be extended to derivatives portfolios.

9. Bond Pricing Models, Default Risk and theTerm Structure of Credit Risk 9.1. Mathematical Default Price Model Models for the valuation of credit exposure correspond in general to transaction-based models or portfolio models. Transaction-based models

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specify the nature of the loss exposure (static or non-static), the default risk process, the recovery rate and the correlations between the default risk, the exposure process and the recovery rate. These different parameters are modelled as distributions of the value of the cashflows through price over a given period. The exposure is estimated using the distribution of loss exposures, the expected and unexpected losses as well as the expected recovery rate. Monte Carlo methods are often used to model the distributions. Several other models are proposed in the literature. The reader can refer to Das (1995) and Hughston (1996). Hughston (1996) model assumes that bonds are represented by a portfolio of zero coupon bonds plus a credit spread. He used the Heath, Jarrow and Morton’s context to value risky bonds by incorporating a poisson process to refelect the risk of default. Jarrow, Lando and Turnbull (1994) assume that credit ratings follow a Markov chain and use a matrix of ratings transitions to model default. This model uses a fixed recovery rate. Das, Sanjiv and Tufano (1996) extend the analysis to a context where the recovery rate is a random variable. The portfolio models recognize the correlation between default events or the joint default frequency (JDF) as in KMV’s EDF model. The model incoporates equity correlations.

9.2. Using Default Rates to Model theTerm Structure of Credit Risk A term structure of credit risk is defined by the behavior of credit spreads over time. Rosenfeld (1984) and Ramaswamy and Sundaresan (1993) use contingent claims models to study credit risk. Fong and Casella (1998) develop a yield spread model using the standard context of bond pricing. Recall that the price of a coupon bond with an annual coupon c and N years to maturity is given by: Price =

N  t=1

c 1 + t (1 + y) (1 + y)N

where y is the yield to maturity. The model assumes that the coupon rate is equal to the promised y, investors are risk neutral and bonds are held until maturity or default. If we denote by i the yield on a comparable default-risk free bond, then the credit spread is (y − i). If the bond is priced at par, the credit spread is (c − i).

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Fong and Casella (1998) give the following expression for the certaintyequivalent version of the bond price: Price =

N  St c + St−1 dt µ(c + 1) t=1

(1 + i)t

+

SN (1 + i)N

where: St : likelihood or probability that a payment due in t years will be honored, St−1 : probability that the issuer survives through year (t − 1), dt : probability of default during year t, µ(c + 1): fraction µ (the recovery rate) of the coupon plus principal (c + 1) to be received in the event of default. This equation can be used to determine the risky coupon rate. The credit spread (c − i) compensates a buy-and-hold risk-neutral investor for different marginal default rates dt and an expected recovery rate µ. Fong and Casella (1998) use historical default rates and a recovery rate estimate from Moody’s long-term default study to simulate this formula. Credit spreads generated by this formula are very sensitive to the estimation of the recovery rate.

10. A Comment on Market versus Accounting Based Measures of Default Risk Default appears when the firm does not honor its debt obligations. The default probability for the borrowing firm can be measured using data and models. Data can be used from financial statements and market prices for equity and debt.

10.1. The Problem of Default The anticipation of default by KMV’s model is represented by a probability distribution. The loss in the event of default or the loss given default depends on the security and its seniority. The role of loss given default is to transform the default distribution into a loss function. The KMV model uses equity prices and financial statements.

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10.2. A Market Based Measure of Default Risk Expected Default Frequency (EDF) in the KMV’s credit monitor system gives the probability that default could occur over a given period. The expected loss corresponds to the EDF times the loss in the event of default. The EDF is based on the market value of assets, the volatility of future asset values and the shape of its distribution as well as the face value of the liabilities. The market value of assets (variable 1) is estimated from the observed equity market value. The issuer’s underlying asset riskiness (variable 2) is measured using a generalization of the option pricing theory in the lines of Black, Scholes and Merton. The shape of the distribution of financial asset values (variable 3) in the region of distress is computed from data on historical default. The face value of liabilities (variable 4) corresponds to the level of current liabilities for short horizons and to the debt servicing requirement for longer horizons.

10.3. Ordinal versus Cardinal In general, credit analysis uses an “ordinal” default rank or a letter to appreciate default. KMV uses besides a number for each credit letter, elaborating a linkage from ordinal to cardinal using historical default probabilities. The EDFs in credit monitor are cardinal. Therefore, the KMV default output models can be used in valuation and portfolio analyses. Using statement data, the private firm model estimates the firm’s equity value and risk. When compared to the public company model, the private firm model in KMV provides default prediction power. The way to manage the risk of bank portfolios lies in maximizing diversification with respect to return opportunities.

Summary A credit derivative can be seen as “any instrument that enables the trading/management of credit risk in isolation from the other types of risk associated with an underlying asset.” These instruments may include: credit default products, credit spread products, total return products, basket products and credit linked notes. Banks are the principal actors in the market of credit risk. They represent the buyers, sellers, and intermediaries of credit derivatives. Bankers are the principal buyers of credit protection. However, any firm or institution with a portfolio subject to credit risk can use credit derivatives. This is the case for firms with concentrated portfolios who can use credit derivatives to manage credit lines. Sellers of credit protection 12:16:11.

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are motivated by the desire to enhance their risk adjusted return on capital. Banks searching for higher yields are using credit derivatives written on lower rated assets. Straight credit default swaps seem to be the most widely used of all credit derivative products. Total return swaps and credit derivatives written into collateralised obligations seem to have larger notional values than other types of credit derivatives. Any type of sovereign or non-sovereign, bond, loan, or senior claim of any credit rating can be an underlying asset. Bonds seems to be the most used assets, due to their relatively liquid secondary market and generic form. Loans have less generic structures and are used in this market. The strength of the default relationship between two borrowers can be appreciated by default correlation. When this correlation is zero, the probability that they default simultaneously corresponds to the product of their individual default probabilities. When the correlation varies between zero and one, the higher the correlation, the higher the probability that the two borrowers default at the same time. The academic research on credit risk estimation can be classified into three classes. The first concerns the estimation of expected default frequencies and recovery rates in the event of default. The second is focused on the estimation of volatility or unexpected losses using an assumption regarding the bond market level diversification. The third class concerns the estimation of volatility of value for a given portfolio which is not perfectly diversified. Expected losses can be explained by the expected probability of default and the recovery rate in the same context. In practice, rating agencies use an accounting analytic approach. Market participants use either statistical methods or apply the main concepts in option pricing theory. Default pricing models provide a unified framework to quantify the credit risk of an individual transaction and to value a credit derivative using the concepts of loss exposures, recovery rates and default probabilities. A basic default pricing model ignores default correlations within a portfolio of credit exposures. The basic model focuses on the risk of default and ignores the changes in credit risk or rating migrations. More complicated models as the approach in CreditMetrics accounts for default correlations and rating migrations.

Questions 1. What is the definition of credit risk analysis? 2. What is credit exposure? 3. What is the average shortfall? 12:16:11.

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4. 5. 6. 7. 8.

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What is credit scoring? What is the current exposure? What is the default probability? What is credit quality migration? What is migration analysis?

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Ramaswamy, K and S Sundaresan (1993). Does default risk in coupons affect the valuation of corporate bonds: a contingent claims model. Financial Management, Autumn. Rosenfeld, E (1984). Contingent claims analysis of corporate capital structures: an empirical investigation. Journal of Finance, 39, 611–625. Stevenson, B and W Fadhil (1995). Modern portfolio theory: can it work for commercial loans. Commercial Lending Review, 10(2), 4–12. Vasicek, O (1984). Credit valuation. Document N: 999-0000-021. Wagner, H (1996). The pricing of bonds in bankruptcy and financial restructuring. Journal of Fixed Income, June, 40–47.

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Credit Derivatives: The Basic Concepts

This chapter is organized as follows: 1. Section 2 defines credit derivatives. It presents also the main structures of credit derivatives products. 2. Section 3 presents the main uses and applications of credit derivatives. 3. Section 4 presents the general context for the pricing of credit derivatives and default risk. 4. Section 5 provides the main approaches used in the rating agencies models and the proprietary models. 5. Section 6 studies default risk and the term structure of credit risk. 6. Section 7 develops a model for default probabilities within a context of incomplete information.

1. Introduction everal over-the-counter credit derivatives are negotiated since 1991. Credit derivatives are financial instruments which isolate credit risk. They facilitate the trading of credit risk, its transfer and hedging. There are different categories of credit derivatives: forwards, swaps, options and some building blocks combining some of these main products. Credit derivatives are often presented in the form of three classes of instruments: total return swaps (total rate of return swaps, loan swaps or credit swaps), credit-default instruments and credit-spread instruments. Total return swaps are conceived to transfer the credit risk to the counterparty. Credit-default instruments give a certain payoff upon the occurence of a default event. They are often in the form of a credit-default swap or

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default options. Credit spreads instruments are often in the form of forward or option contracts on credit-sensitive assets. Credit derivatives allow a restructuring of the risk/return profiles of credits and permit investors to access new markets. The investors have the possibility to structure and to optimize the risk adjusted performance of their liabilities by diversifying them among several markets and instruments. Credit derivatives serve at least three goals. They improve risk management, optimize exposure and facilitate access to markets. The pricing of credit derivatives accounts for the exposure, the default probability and the expected recovery rate. The approach is then adapted in different contexts using replication portfolios. The main difficulty concerns the measurement of default risk.

2. Credit Derivatives: Definitions and Main Concepts 2.1. Forward Contracts Forward contracts on bonds can be either cash-settled or physical-settled. The cash-flows for this forward agreement commits the buyer to buy a given bond at a specified future date at a predetermined price specified at contract origination (time t = 0). The agreement can specify that instead of using the price, the bond’s spread over a treasury asset or a benchmark will be used. In this operation there are two maturity dates: the maturity of the forward agreement and the maturity of the reference bond. The maturity of the forward agreement is in general shorter than that of the reference bond. Since, in general, the default risk is borne by the buyer of the forward contract on the spread, he will pay at the maturity date the following quantity: (spread in forward agreement — spread at maturity) duration (notional amount). When there is a credit event, the transaction is marked to market and unwound.

2.2. The Structure of Credit Default Instruments Credit default instruments dissociate the risk of default on credit obligations. They are often presented in the form of credit-default swaps or credit-default options. In general, when there is a credit event by the reference credit, then according to the terms of a credit-default swap, the bank pays the counterparty an agreed default payment. The counterparty pays a periodic fee and benefits from the protection of the risk of default of the reference credit. This credit derivative structure is based on the replication of the

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total performance of the underlying credit asset (a reference bond, a loan). The swap is done between an investor and the bank. The investor assumes the risks of the reference bond. The bank passes through all payments of the bond and in return, the investor makes a payment akin to a funding cost. The transaction is based on a notional amount. The current bond price is used to compute the settlement value under the transaction. The investor receives interest payments and pays a money-market interest rate plus or minus a specified margin.

2.2.1. Total Return Swaps In a total return swap, the rate payer makes periodic payments to another party (the total return payer). He receives the total return less the principal and interest payments plus or minus price changes of the reference asset. The total return swap is often used in the swap structures on default risk. The two parties in a total return swap define at origination the initial value P0 of the reference asset and agree on the reference rate. At any time between t = 0 and the maturity date corresponding to settlement dates, the asset receiver obtains the cash-flows from the reference asset. He pays a certain amount fixed with reference to the reference rate. At the maturity date of the contract, the value of the reference asset PT is used. If this value is greater than P0 , the asset receiver gets the difference (PT − P0 ), otherwise, he pays the difference (P0 − PT ). The credit swap can also be based directly on a spread. In this case, the asset receiver pays at maturity the difference in the spread of the reference asset over a treasury security with a comparable maturity at origination and at maturity.

2.2.2. Credit Default Swaps This derivative contract allows one party (the protection seller) to receive fixed periodic payments from the protection buyer. The payments are in return of making a single contingent payment covering losses with respect to a reference asset following a default or an other specified “credit event”. The main idea behind credit default swaps is that they strip off the default risk of some reference assets. This risk will be traded separately. By implementing a credit default swap, the protection seller earns investment income and the protection buyer hedges the risk of default on the reference asset. This contract allows investors to hedge credits without implementing costly strategies consisting of buying and selling cash securities and loans. 12:16:18.

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The following example is adapted from a sponsor’s statement by Barclays Capital. Example Consider an investor A who gains customised access through a bank B to a corporate bond by selling three-year default protection to the bank B on the bond. The investor A receives a fixed premium of 120 bp per annum and agrees to make a credit event payment if the borrower defaults on the bond. In the credit event, the swap terminates and the investor A pays the bank B the notional times the percentage fall from par of the bond. The investor A can also settle the swap by buying the bond from the bank B at par.

2.2.3. Basket Default Swaps An investor can sell default protection on several assets. In a first-to-default basket default swap, the protection seller assumes the default risk on a basket of bonds by agreeing to compensate market losses on the first asset in the basket to default.

2.2.4. Credit Default Exchange Swap It is possible to swap a default risk on an asset for that of an other asset. In a credit default exchange swap, both parties act simultaneously as protection buyers and protection sellers. Example Consider two institutions A and B. A trades the default risk of a loan it holds for that of a complementary loan held by B. A lays off the default risk on loan A in return for assuming default risk on loan B. If a reference credit experiences a credit event than the protection seller must make a credit event payment to the protection buyer. This can terminate the trade. The trade could continue with the protection buyer in the rest of transaction paying an agreed rate. Parties do not make periodic payments and swap just the contingent payments when default risks are perfectly matched.

2.2.5. Credit Linked Notes (CLNs) CLNs are associated to the credit performance of underlying assets. A principal protected note protects a preset portion of principal. A principal linked structure pays enhanced fixed coupons and can redeem principal at a rate associated with the credit performance of reference assets.

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Credit default notes allow investors to buy or sell default protection on reference credits.

2.2.6. Basket Default Notes It is possible to earn additional yield on sovereign bonds by buying CLNs associated with more than one reference obligation.

2.2.7. Levered Portfolio Notes These notes are basket default notes giving leveraged exposure to a portfolio of reference bonds. They allow investors to gain significant returns on higher grade assets by receiving an enhanced coupon for selling default protection on a certain number of investment grade bonds.

2.3. The Structure of Credit Spread Instruments Credit spreads refer to a difference with respect to the riskless interest rate. This difference compensates the investor for the risk of default. Credit spread derivatives can be defined using the riskless rate as a benchmark (absolute spread) or using two credit assets (relative spread). In a creditspread swap between a bank and a counterparty, the latter gains (loses) if the spread decreases (increases). Call and put options on credit spreads give, respectively, the right to the buyer to buy (to sell) the spread and benefit from a decrease (an increase) in the spread. For further readings about the credit derivatives and their different forms, the reader can refer to Das (1998), Brown (1996) and Howard (1995).

2.4. Credit Spread Derivatives (CSDs) CSDs may be presented in the form of options, forwards or swaps associated with a credit spread difference determined between the current yield of a given reference asset and a risk free asset. The dissociation between credit spread risk, market risk and interest rate risk allows the investors to hedge credit spread risk. Investors using credit spread derivatives can implement long or short exposures to reference assets. These derivatives protect investors from credit deterioration. Credit spread puts are used to hedge against rising credit spreads. The default put is widely used in operations involving options on default risk. The buyer of a default put pays a premium to the seller who assumes the default risk for the reference

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asset. When there is a credit event, the seller pays the buyer a default premium. Local currency derivatives combine currency derivatives with credit derivatives. They permit foreign investors to manage currency and convertibility risk.

3. Using Credit Derivatives Credit derivatives allow a restructuring of the risk/return profiles of credits and permit investors to access new markets. The investors have the possibility to structure and to optimize the risk adjusted performance of their liabilities by diversifying them among several markets and instruments. Credit derivatives serve at least three goals. They improve risk management, optimize exposure and facilitate access to markets. Smithson (1995) advances two main reasons at least for the use of credit derivatives. The first reason is that credit derivatives represent a mean to implement the main prescriptions of modern portfolio theory. The second reason is that they can be used in the management of interest rate risks.

3.1. Constructing Efficient Portfolios Credit derivatives can reduce transaction costs and provide managers with a way to “short sell” loans or bonds. They offer to investors the possibility to leverage bond or loan positions. These products complete also the market since they provide a mean to synthesize assets which are not available in the market.

3.2. Implied Forecasts of the Probability of Default When constructing a treasury zero-coupon curve, the “bootstrap” methodology is often used. The same methodology can be applied to construct a corporate zero-coupon curve using the available corporate bond curve. Using this information, it is possible to construct a credit spread zero curve by computing the difference between the corporate zero rate and the treasury zero rate. The credit spread zero curve can then be used to compute the forward credit spread curve. This latter curve embodies the market’s forecasts regarding future default probabilities. The implied future default probabilities are at the origin of two main uses of credit derivative transactions. Sometimes, the future default probability implied by the forward credit spread curve is greater than the expected

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default rate computed from historical data. This result is confirmed in the study of Spahr, Sunderman and Amalu (1991). These authors observed that third-party insurers could have offered corporate bond insurance at a cost significantly lower than the prevailing market default risk premium. In this sense, a credit derivative transaction can be initiated to implement this view. It is also possible to make a credit derivative transaction to implement arbitrage using a credit which is traded in different debt markets. This is possible if the instrument in the two credit markets has different implied future default probabilities. The market is important for the management of credit risk. Credit derivatives consider credit risk as a disaggregated commodity distinct from other financial risks. Trading in credit risk facilitates the diversification of credit portfolios. Credit derivatives offer to investors a mean to access credit risks. The applications of credit derivatives in risk management allow the creation of methods to transfer credit risk. These instruments allow a dissociation between the term for which the credit risk is assumed and the term of the underlying credit obligation. They also allow some flexibility in the creation of different kinds of credit exposure. These instuments provide also some flexibility in terms of the definition of the credit event which leads to the default pay-out. Credit derivatives are also used in the management of concentration risk in credit portfolios of financial institutions. The concept of concentration risk refers to the additional risk of credit losses when the portfolios of credit risks are not well diversified. For detailed applications of credit derivatives, the reader can refer to Das (1998) and Reoch (1996) among others.

4. Valuation and Pricing of Credit Derivatives The pricing of credit derivatives accounts for the exposure, the default probability and the expected recovery rate. The approach is then adapted in different contexts using replication portfolios. The main difficulty concerns the measurement of default risk. There are two main approaches for the pricing of credit securities. The first approach traced back to the Black and Scholes (1973) seminal paper. In this approach a credit security is regarded as a contingent claim on the value of the issuing firm. The option pricing theory is applied to the pricing of the claims on the firm’s assets. In this approach, default is modeled as the first time the firm’s value hits a prespecified boundary. Models of this category are based on a continuous process for which the time of default is a predictable stopping time. The payoff in

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the case of default is represented by a fixed fraction of the firm’s value in the event of bankruptcy. This approach is adopted in Merton (1974), Black and Cox (1976), Geske (1977), Leland (1994), among others. The second approach models the time of default as the time of the first jump of a Poisson process with random intensity, referred to as a Cox process. This approach is used by Jarrow and Turnbull (1995) and Schonbucher (1997) among others.

4.1. The Pricing of Total Return Credit Swaps Total return credit swaps are in general priced off the risk-free return in combination with the pricing of a default swap. The method of replication reproduces the loan characteristics using surrogate instruments. In general, the correlation between the bonds and equity is used to implement a “composite” hedge. The derivation of default risk is based on the firm’s model developed in Merton (1974), Black and Scholes (1973) and Geske (1977).

4.2. The Pricing of Credit-Spread Derivatives The credit spread is a key element in the valuation of credit derivatives because it represents a compensation for the assumed credit risk. The knowledge of the credit spread and the use of the classic option pricing methodologies allow the computation of forward and option prices on the credit spread. The credit spread is defined as the difference between the yield of a security and the yield of a corresponding risk free asset. The concept of a risk-neutral yield credit spread is often used in practice and is defined as the difference in the yield of a risky bond and the yield on a risk-free bond. The determination of risk-neutral credit spreads allows the modelling of the term structure of credit spreads. The reader can refer to the study of Litterman and Iben (1988). The pricing of forwards on credit spreads uses the classic methodology which comprises three steps. The first step uses the spot price of an asset and the risk free rate. The second step determines the forward prices of the securities. The third step determines the forward credit spread by the difference between the forward asset yield and the forward riskless rate. The pricing of credit-spread options depends on the volatility parameter of the underlying asset which is the forward credit spread of the nominated assets. The forward spread corresponds

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to the differential between the forward rates of the assets at the option’s maturity date.

4.2.1. Pricing Spread Options Garman (1992) and Ravindran (1993) present different approaches for the pricing of credit spread options. McDermott (1993) applies Black’s (1976) model to the pricing of spread options. In this case, the call option formula is given by: √ c = e−rT [(S − K)N(h) + σ TN  (h)] √ with h = 1/σ T (S − K) where: S: K: σ: r: T:

the forward yield spread, the strike yield spread, the standard deviation of the yield spread, the risk free rate, the option’s time to maturity. In the same context, the value of the put option is given by: √ p = e−rT [(K − S)(1 − N(h)) + σ TN  (h)].

Credit spread options can also be priced using the approach in Margrabe (1978) for the option to exchange one asset for an other. The option value is given by: c = e−q2 TS2 N(d1 ) + e−q1 TS1 N(d2 ) √ √ 2 with
d1 = [ln(S2 /S1 ) + (q1 − q2 (1/2)σ )T ]/σ T , d2 = d1 − σ T , σ = σ12 + σ22 − 2ρσ1 σ2 with:

S1 , S2 : the spot prices of assets 1 and 2, q1 , q2 : the yields on assets 1 and 2, σ1 , σ2 : the volatilities of assets 1 and 2, ρ: the correlation coefficient between the assets 1 and 2, T : the time to maturity.

4.2.2. Credit Spread Puts There are several specifications of credit spread puts. A credit spread put ˜ T2 ) with maturity T1 < T2 gives the holder the on a defaultable bond b(t,

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right to sell defaultable bonds at time T1 at a price that corresponds to a yield spread s¯ above the yield of an otherwise identical default free bond B(T1 , T2 ). The exchange ratio is defined by S¯ = e−¯s(T −t) .

4.2.3. Exchange Credit Options Exchange credit options are credit derivatives which give the holder the ˜ 1 , T2 ) for S¯ default free bonds right to exchange one defaultable bond b(T ˜ 1 , T2 ) is expressed as B(T1 , T2 ) at time T1 . The defaultable bond b(T QT1 b(T1 , T2 ) where QT1 is a factor representing the default reductions in the face value and b(T1 , T2 ) is a default adjusted price. The payoff of this credit derivative is: ˜ ¯ max[SB(T 1 , T2 ) − b(T1 , T2 ), 0]. This payoff shows that the credit put can be viewed as an exchange option. This contract protects against any losses worse than a final credit spread s¯ as well as all defaults.

4.2.4. Credit Spread Puts with Face Value Protection These credit derivatives give the holder the right to exchange n defaultable ˜ 1 , T2 ) for S¯ default free bonds at a time T1 where the n defaultable bonds b(T bonds have a face value of one. The payoff of this credit derivative is: ¯ max[SB(T 1 , T2 ) − b(T1 , T2 ), 0]. This contract insures the holder against final credit spreads greater than s¯ . The protection in this case covers a portfolio of a given face value instead of a fixed number of defaultable bonds. The price of the underlying bond is adjusted for the default less quota Q.

4.2.5. Credit Spread Puts with Security Protection These credit derivatives give the holder the right to exchange one default˜ 1 , T2 ) for S¯ default free bond portfolios yB(T1 , T2 ) havable bond b(T ˜ 1 , T2 ). The payoff of this credit derivative ing the same face value as b(T ¯ is y max[SB(T1 , T2 ) − b(T1 , T2 ), 0]. This credit spread put protects one defaultable bond against final yield spreads which are greater than s¯ . Since Q appears in this payoff in a linear fashion, the default effects may be accounted for into a higher discount rate. The default risk is borne by

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the holder of this contract referred to as a credit spread put with security protection.

4.3. An Introduction to the Pricing Approaches for Credit Derivatives The pricing of individual transactions or single transactions concerns the loss exposure, the default probability, the recovery rate and the correlations between these features in order to predict future credit losses. The pricing in a portfolio approach is focused on the correlations between individual exposures, which means that we must account for the joint default probabilities (correlations between default) and the correlation between loss exposure and recovery rates. The main focus of these approaches concerns the modelling of default. Default models are based on rating or rating migration to account for value changes in a security due to changes in credit risk. Ratings models infer from historical data the probability of a change in value of a security due to changes in credit-spread as a consequence of ratings migration or default. Default models use also creditspread approaches based on the term structure of credit spreads to model the default risk.

4.3.1. Modelling Credit (Loss) Exposure Credit or loss exposure is the amount exposed to risk of loss in the event of default. The loss exposure can be static in the sense that is independent of changes in market variables. It can also be dynamic when it depends of changes in market variables. The loss exposure is often appreciated in terms of expected or unexpected credit loss. The expected credit loss is often determined as an average expected credit exposure over the life of the transaction. The unexpected credit loss is determined as a loss in the worst case or the maximum exposure at a given time.

4.3.2. Modelling Recovery Rates and Default Risk The recovery refers to the amount of probable loss exposure that can be recovered after a default event. The recovery rate corresponds in general to the percentage of par value of the security recovered in the event of default. In general, this rate is published by major rating agencies like Moody’s Investor Services.

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5. The Rating Agencies Models and the Proprietary Models 5.1. The Rating Agencies Models The models of default risk developed by rating agencies are based on the current rating and the time to maturity of the obligation. These models determine the cumulative risk of default (the total default probability over a given period) and/or the marginal risk of default (the change in default probability over some periods). In a first special report by Standard & Poor’s (1999), we find that corporate defaults rise sharply in 1998. Their database contains a vast collection of statistics on default and rating migration behavior on CD-ROM under the trade name CreditProTM . The data used correspond to the issuer credit ratings that reflect S&P’s opinion of a company’s overall capacity to pay its obligations. This opinion is based on the obligor’s ability to meet its financial commitments on a timely based. This indicates in general the likelihood of default regarding the firm’s financial obligations. The definition of default corresponds to the first occurrence of a payment default on any financial obligation. Preferred stocks are not considered as financial obligations since a missed preferred stock dividend cannot be equated to a default. The studies of default by S&P are based on groupings called static pools. A static pool is constructed at the first day of each year covered by the study and followed from that point on. All obligors are followed year to year within each pool. All of Standard & Poor’s default studies reveal a well defined correlation between credit quality and default remoteness. The rating letters in the study are: AAA, AA, A, BBB, BB, B and CCC. In general, the higher the rating, the lower the default probability and vice versa. Besides, the lower an obligor’s original rating, the shorter the time it takes to observe a default. Using default ratios, the study shows that default rates over a one-year horizon exhibit a high degree of volatility. The default patterns seem to share broad similarities across all pools. This result suggests that S&P rating standards are consistent over time. Using transition analysis, each one year transition matrix shows all rating movements between letter categories from the beginning through the end of the year. Rating transition ratios give useful information to investors and credit professionals. The one-year rating transition ratios by rating category reveal that higher ratings are long lived. The S&P study assumes that the rating transition rates follow a first-order Markov process. This allows to model cumulative default rates over several horizons. Rating transtion matrices are constructed to produce stressed default rates. Multi-year 12:16:18.

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transitions are also constructed for periods of two through 15 years using the same methodology as for single year transitions. This allows to compute average transition matrices whose ratios represent the historical incidence of the ratings. The study reveals also that, for example, 10-year transition ratios are less reliable than their one-year counterparts. The analysis reveals an increase in corporate defaults in 1998. In a second special report by Standard & Poor’s, Brand and Bahar (1999) study the recoveries on defaulted bonds tied to seniority rankings. While the first study is concerned with the probability that a given issuer might default, this second research concerns the expected recovery in the event of default. These two factors represent the credit risk implicit in holding a bond. In general, recoveries are estimated on the basis of the prices the defaulted securities fatch at some time after the default event. The data correspond to 533 S&P’s rated straight-debt issues that defaulted in the period 1981–1997. We denote by: Pd : the prices at the end of the default month, (referring to a default event), Pe : prices (just) preceding liquidation or emergence from out-of-court settlement or Chapter 11 reorganization, (referring to an emergence event). The methodology uses the fact that recoveries are based on the ultimate values yielded by the completion of the bankruptcy process. Table 1 reproduced from the study of S&P summarizes the main findings. Table 1:

Recoveries shortly after default.

Seniority-ranking

Nobs

SA, Pd

SD, Pd

WA, Pd

CV, Pd

Senior-secured Senior-unsecured Subordinated Junior-subordinated Total

65 180 144 144 533

58.52 49.60 38.29 35.30 43.77

22.27 26.51 25.23 22.29 25.81

58.11 53.66 36.86 36.41 43.93

0.38 0.53 0.66 0.63 0.59

with: Nobs : SA: SD: WA: CV:

the number of observations, the simple average, the standard deviation, the weighted average by issue size in dollars, the coefficient of variation.

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Table 1 shows that investors who liquidate a position in defaulted securities (shortly after default) expect to recover, on average, about 44 cents on the dollar. This means that on average, creditors receive about 40 cents on the dollar. Table 1 shows also a certain predictable degree of variation across seniority classes. The study shows that average recoveries at default and emergency are higher the seniority rankings of the issues used. It also reveals that excess returns exhibit much uncertainty at default and emergence and that uncertainty is in general higher, the lower the seniority of the debt used.

5.2. The Proprietary Models 5.2.1. The Default Probability Problem The default probability can be calculated as the probability that asset values will be less than the value of the claims on the firm’s assets. In this spirit, KMV Corporation (Kealhofer, McQuown and Vasicek) develop the expected default frequency (EDF) model. The model needs an estimation of the market asset values, the volatility of the assets and the market value of the liabilities. The volatility can be implied from an option pricing model. Using the asset value, the volatility and the cumulative liabilities, it is possible to calculate the default risk of the firm. The model determines the default probability using the distance in volatility between the asset value and the point at which the asset value will be less than the liabilities. The EDF model uses large databases of firms in the computation of historical default frequencies. This approach allows the estimation of expected and unexpected default losses and derives default within a volatility framework. Default risk corresponds to the uncertainty surrounding the firm’s ability to service its obligations. The default probability of a firm depends on the market value of its assets, on the risk of the assets and on the firm’s liabilities. The risk of the assets is given by the standard deviation of the annual percentage change in the asset value. The methodology of KMV Corporation as shown in the work of Crosbie (1998), looks for a default point or the asset value at which the firm will default. The firm defaults when its market net worth is zero. A ratio of default risk referred to as distance default, compares the market net worth to the size of a standard deviation move in the asset value: Distance default =

(market value assets − default point) . (market value assets − asset volatility)

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The ratio says that a firm is n-standard deviation away from default. When the probability distribution is known, the default probability can be computed directly. KMV Corporation proposes a model of default probability, CreditMonitor which computes the expected default frequency (EDF). EDF is the probability of default during some coming years. The determination of the defaut probability of a firm is done in three steps. The first step: the asset value and its volatility are estimated using the asset market value, the volatility of equity and the book value of liabilities. Using an option pricing based approach where equity is viewed as a call on the underlying assets with a strike price equal to the book value of the firm’s liabilities, it is possible to estimate the value of the firm and the volatility of its assets. The second step: the distance to default is calculated using the asset value and its volatility and the book value of liabilities. The default probability depends on the current value and the distribution of the asset value, the volatility, the expected rate of growth in the asset value, the horizon and the level of the default point. If the future distribution of asset values is known, the default probability and the EDF correspond to the likelihood that the final asset value be below the default point. The third step: the default probability is computed using historical data on default and bankruptcy frequencies. A frequency table is generated to link the likelihood of default to different levels of distance default. CreditMetrics models the process of value changes resulting from changes in credit quality. It gives risk management information which is tailored to the name, industry and sector concentrations. The details and data are freely available on the Internet. See also Gupton, Finger and Bhatia (1997). Miller (1998) develops a framework to determine whether a particular quantitative method of measuring credit risk using option valuation, represents a refinement over broader categorisation methods. They develop a methodology for testing whether a quantitative credit rating system is a statistically significant refinement of a broader rating system. Other studies are interested in KMV Corporation’s Credit Monitor system (an optionsbased credit rating system) and how it compared with S&P’s ratings for US companies. S&P ratings provide indications of credit worthiness. Miller (1998) uses the S&P’s senior unsecured debt ratings as reported by S&P Compustat database service. The KMV’s ratings take the form of expected

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default frequencies (EDFs). As the likelihood of default varies over the business cycle, there is no direct mapping betwen KMV and S&P ratings. The study uses also the one-year EDF. Miller (1998) tests the KMV’s predictive power relative to an S&P rating using nonparametric methods. He uses the Kolmogorov–Smirnoff test to quantify the degree to which EDFs provide information which is not contained in S&P ratings. The study shows that KMV credit Monitor has very strong predictive value for up to 18 months prior to default.

5.2.2. Modelling Expected Versus Unexpected Losses The expected loss or the average expected credit loss in the absence of correlation between the different factors is given by: expected loss = loss exposure (default probability)(1 − recovery rate). The expected loss in the presence of correlation between the factors is given by: expected loss = loss exposure (default probability)(1 − recovery rate) × (multivariate probability density function). The unexpected loss or the worst case credit loss is based on the worst case probability of default. It corresponds to the expected loss plus the volatility of this loss given a certain level of confidence. The models for assessing credit portfolio risks of JPMorgan (1997), CreditMetrics, and the one of Credit Suisse Financial Products (1997), CreditRisk+ , present several similarities. The models in CreditRisk+ seem appropriate for illiquid loan portfolios. The models in CreditMetrics seem appropriate for liquid bond portfolios. Rolfes and Broeker (1998) develop an “easy to implement” procedure to integrate ratings migrations into CreditRisk+ . The latter system ignores initially ratings migrations. Their model combines the rating migration concept of CreditMetrics with the actuarial approach of CreditRisk+ . They show that this approach can be used instead of computationally inefficient Monte-Carlo simulations. The calculation of credit loss distributions Arvanitis et al. (1998) present some models for the calculation of credit loss distributions. The authors deal also with the problems of modelling default correlations. They introduce a method for modelling stochastic recovery rates. There are two classes of models in the computation of credit loss

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distributions. The first class corresponds to the analytical approach. The accuracy of the results obtained in this context depends on the portfolio characteristics. The second class of models use simulation techniques and Monte-Carlo methods. The unexpected loss The unexpected loss measure in risk management defines the worst loss at a specified level of confidence for a given horizon. This concept is applied to measure the economic capital for some years. The level of confidence is defined as (100 − x) percentile of the loss distribution. In this context, x is defined with respect to the credit rating of the institution. The confidence level is often determined with respect to a normal distribution as a multiple of the standard deviation. The error in the estimation of the unexpected loss using a normal distribution is: (ULσ − ULp )/ULσ , where ULσ is the appropriate multiple of the portfolio’s standard deviation and ULp is the true percentile. In the presence of n assets, the portfolio loss due to defaults for a period T can be written as: n  L(T ) = (1 − δi )Xi (T)di (T) i=1

where: δi : recovery rate, Xi (T): a stochastic exposure given by the greater of the present value of the asset and zero or max(PVi (.), 0), di (T): a binary default function taking the value 1 for default and 0 otherwise. The expression of L(T) cannot be calculated explicitly for several assets. The saddle-point approximation The moment generating function (mgf) of a random variable V is given by:  ∞ sV MV (s) = E[e ] = estPV(t)dt. −∞

If we consider a weighted sum of binary variables with weights wi and default probabilities pi , then the mgf is given by: n  ML (s) = (1 − pi + pi ewi s ). i=1

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The method of the steepest descents or saddle-point method allows to obtain analytical approximations to the tail of the distribution without any prior assumption regarding the shape of the loss distribution. This method seems to give good results in the determination of tail probabilities of credit loss distributions. These analytical expressions can be used in computing loss distributions, for unequal exposures and unequal default probabilities. Simulation methods The use of stochastic exposures and recovery rates, credit migration, etc. needs simulation methods. Arvanitis et al. (1998) assume that the entire correlation structure is determined from a multivariate normal distribution, for which the higher order correlations are defined by the pairewise ones. The multivariate normal distribution represents in this case the standardized asset returns of the counterparties. The correlation structure is defined by mapping the binomial default probabilities pi on to thresholds ki = −1 (pi ) of a normal distribution. The correlations between the normal variates ij are related to the correlations between binary variables ρij by equating the expectation of pairs of joint Binary events to the joint probability of a correlated bivariate normal distribution: E[di dj ] = (ij , ki , kj ). The correlation coefficient is given by the following expression relating ρij and ij : ρij =

(ij , ki , kj ) − pi pj . pi (1 − pi )pj (1 − pj )

This expression can be solved numerically. For a Monte-Carlo simulation, correlated binary (default) events must be simulated from a correlated multivariate normal distribution. The covariance matrix can be factorized as C = AAT for some matrix A. For a multivariate process u, the vector Au presents the required covariance matrix C. The correlation between the normal random variables can lead to the determination of the binary default variable di which is given by the indicator function I(Yi < ki ). Yi is the ith element of the vector Au. This simulation method which allows for stochastic exposures can be compared with the loan equivalent approach. This approach asumes that exposures are constant since there are no covariance between the exposures of the assets. The treatment of stochastic recovery rates in Arvanitis et al. (1998) follows from the default and migration model. In the event of default, the

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recovery rate is defined by a series of further thresholds below the default threshold. The position of the recovery rate thresholds is determined by matching the historical mean and variance. The exposure distributions are estimated by simulating underlyings over many paths. The assets are then revalued at a number of discrete time points on these paths.

6. How to Measure and Appreciate Credit Risk 6.1. The Calculation of the Counterparty Credit Risk Shimko and Humphreys (1998) propose a simple calculation to value counterparty credit risk. The method ensures that deals are priced to reflect differential credit risks. The concept of the “loan-equivalent” considers the counterparty default exposure and determines what size of loan would have the same default characteristics. This method is based on the following idea: if a counterparty has a credit rating, the cost of credit associated with a loan corresponds to the credit spread. For example, if a swap exposure has a loan equivalent, then the credit costs of a swap are similar to those of its loan equivalent. The knowledge of the credit costs of a swap determines the profitability of the swap. Example Shimko and Humphreys (1998) provide an example for the determination of the loan-equivalent and credit charges in the comparison of two profitable trades: a 100,000 dollar value trade with an A-rated counterparty and a 300,000 dollar trade with a B-rated counterparty. The method is implemented in four steps. First, they determine the potential “maximum” exposure of each transaction using the VaR over the life of the deal. Second, the recovery rate is estimated for each counterparty. This is a difficult task. Third, the loanequivalent for each potential counterparty is calculated. This is realized by multiplying the maximum exposure by the ratio of the recovery rates. Fourth, the loan-equivalent is regarded as a letter of credit. This allows the determination of the appropriate charge for the trade. In this step, the loanequivalent is multiplied by the amount the potential counterparty would need to pay for a letter of credit of the same magnitude. The cost must be equal to the spread between a bond for that company and the short term riskless rate. The appropriate credit charge can be obtained by multiplying the spread to the risk-free rate by the loan-equivalent.

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6.2. How to Measure Credit Risk 6.2.1. The Basic Risk Premium Concept The risk premium concept in credit risk has different meanings. It can refer to a payment compensating for both the expected value of losses and the volatility that the instrument contributes to a diversified portfolio. In this case, the risk premium is considered as an expected loss risk premium (EL). It can also refer just to the uncertain component of credit loss. In this case, the risk premium is considered as an unexpected loss risk premium (UL). For a commercial insurance, a “premium” (covering the expected and unexpected loss) represents a cost paid by the policy-holder to transfer the risk of loss to the issuer of the policy. The analogy with respect to insurance is simple. The counterparty who acquires risk in a credit derivative transaction is in a position similar to the sale of a credit insurance. The risk premium is represented by the cost of the insurance. For a credit default swap, the party ceding the risk is purchasing a constant-premium term insurance policy. This insurance covers credit loss on a specific loan. A financial institution can purchase a put option on the credit loss making an upfront payment for the protection (the option premium). Belkin et al. (1998) define the risk premium for a credit instrument in a way that focuses on the cost of swapping unexpected total return (UTR) as opposed to the cost of insuring against unexpected default loss (UDL).

6.2.2. Natural and Risk-Neutral Measures The probability measure that governs the migration of the borrower’s risk rating over time is considered as the natural process measure. The arbitrage pricing theory can be applied under some assumptions to credit markets. This allows to compute the market price or credit derivative from the expected net present value of the cashflows. Ginzberg, Maloney and Willner (1994) and Belkin, Forest and Suchower (1998) develop a formal procedure for constructing the risk-neutral migration measure. The procedure can be described as follows. First, the risk-neutral measure is used for one period term loans. These loans are priced through observed market credit spreads. Second, arbitrage methods are used to price two-state “reference loans” by treating such instruments as contingent claims on one period loans. Third, the prices of multi-period loans are obtained in a recursive procedure by approximating their one period cash flows in terms of the reference loans.

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6.2.3. Defining Credit Risk Premiums Belkin et al. (1998) define the unexpected total return (UTRrp ) risk premium ˜ ] with: as UTRrp = E[V ] − E[V UTRrp : unexpected total return risk premium, E[V ]: expectation of the net present value V of the credit instrument cashflows to the lender with respect to the natural migration measure, ˜ E[V ]: expectation of the net present value V with respect to the risk-neutral measure. The discounting applied to the computation of V involves the riskless interest rate, r. The definition of UTRrp consists in the difference between the net present value that the lender expects to realize by holding the credit instrument and the price that it would get by selling the instrument at its fair market price. The UTRrp corresponds for example to the cost of a credit swap. The UDL risk premium corresponds to the premium paid on a default loss “insurance policy”. The financial institution transfers in a swap, both the upside and downside risks of any unexpected deviation in total return. The financial institution insures against the one-sided risk of default loss in excess of expected loss in a default-loss insurance case.

6.2.4. Properties of Credit Risk Premiums Credit risk premiums present several properties. First, when the cash flows related to a credit instrument are deterministic, the associated risk premium must be zero. The risk premium can be positive or negative. The NPV of two pooled cashflows tied to the same borrower rating migration process corresponds to the sum of the NPVs of the component cashflows. Since the expectation is a linear operator, it is possible to calculate the risk premium for a hedged position relative to a single credit exposure as the sum of the risk premiums of the separate instruments in the hedge. The above definition of the risk premium can generalize the concept of marginal risk premiums. Consider a borrower whose rating grade at a one-year loan origination is i. We denote by: pD (i): probability of default, LIED: loss in the event of default or the fraction of the loan balance uncovered in the event of default,

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(i): the par credit spread on the loan for a given i for the borrower risk grade. When loss in the event of default or the fraction of the loan balance uncovered in the event of default LIED is deterministic, the marginal risk premium u(i) for a borrower with risk grade i is u(i) = (i) − pD (i) LIED where u(i) is the excess of the par credit over the expected default loss. The latter is calculated with respect to the natural migration measure. Under the risk-neutral measure, the default probability for one year as defined in Ginzburg, Maloney and Willner (1994) is πD (i) = (i)/LIED. In this case, the difference: ˜ ] = (1 − pD (i) LIED) − (1 − πD (i) LIED) E[V ] −E[V = (i) − pD (i) LIED = u(i). This expression shows that in the case of a one-year loan, the UTR risk premium is equivalent to the marginal risk premium. The UDL risk premium for the same one-year loan is πD (i)(1 − pD (i)) LIED = (i)(1 − pD (i)). Examination of the two last expressions shows that the UDL risk premium differs from the UTR risk premium by the quantity pD (i)(LIED − (i)). This term is positive since the par credit spread (i) is less than LIED. For multi-period loans, the UDL risk premium is not always larger than the UTR risk premium.

6.3. Estimating Credit Spread Risk Using Extreme Value Theory The widening in credit spreads in the third quarter of 1998, leads to trading losses which exceeded largely reported value at risk estimates. A quantile plot of daily shifts in swap spreads reveals that the empirical distribution is leptokurtic. The question is to choose the theoretical distribution to model daily shifts. The approach proposed by Phoa (1999) is based on extreme value theory and the Fisher–Tripett theorem. This theorem allows the approximation of the probability distribution of the maxima by one of three standard distribution functions. When the individual observations indicate a fat-tailed distribution, the maxima will have a Frechet distribution of the form: fα (x) = exp(−x−α ).

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When the observations have a bounded distribution, the maxima will have a Weinbull distribution of the form: gα (x) = exp(−xα ). These two distribution functions use a tail index α. When the observations are normally distributed, the maxima will have a Gumbel distribution of the form h(x) = exp(−e−x ). Using the Australian dollar swap data, Phoa (1999) shows that the Frechet distribution works well. His model needs an estimation of the shifting and scaling parameters and the tail index. The tail index is estimated using the Hill estimator (see for details Muller et al. (1998)). This model seems to give approximately good results.

7. Measuring Implied Default Probability: Information Costs Do Matter In a recent paper, Duffie–Lando (2001) considers the implications of imperfect information for term structures of credit spreads on corporate bonds. Pointing out that bond investors cannot observe the issuer’s assets directly and receive instead only imperfect accounting reports from which it is difficult to make inference, they demonstrate there must exist a default arrival intensity process related to the information perfection. Then they focus on the shapes of the term structure of credit spreads and demonstrate these can have many different behaviour. This section takes another viewpoint and considers that the whole information on the underlying firm assets behavior may be found. But, as there is no free lunch, necessary investigation induce a cost. This information cost is shown to change the perceived default probability. The risk neutral price process of the underlying firm asset, V , is known to be well described by the following stochastic differential equation: d ln V = (r + λV − 0.5σ 2 )dt + σ dz where z is a standard Brownian motion and σ the constant volatility. In the following, it serves as the state variable. To make the framework as simple as possible, let us assume that the firm asset value process is used as a signalling variable and that the default is declared at maturity of the bond if this state variable is not above a threshold K. Following Longstaff– Schwartz (1995), one further assumes that the recovery m is constant. As a result, any default risky corporate bond may be priced by: Gd (t, T, K) = p0 (t, T )(1 − (1 − m)Qk (τV < T )).

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Here p0 represents the price of an equivalent risk free bond, m denotes the recovery rate and Qk is the forward risk neutral default probability. Due to the behavior of the state variable, both p0 and Qk depend on the riskless interest rate r. In addition, the default probability depends on the way the default event may be declared. Because of the simple assumption we make, one has N(−d2 ) where       √ 1 2 V + r + λV − σ T σ T. d2 = ln K 2 Suppose now that one observes a term structure of anticipated default probability q(T˜ )T . This term structure must be treated with care and in particular, the unbiased risk neutral default probability √ (corrected from the informaλ −1 ˜ tional costs) is equal to: N(N (q(T )) + σ T ). First, one observes that the unbiased default probability is larger than the observed one. Second, it is a function of the square root of the term considered and the firm volatility. More interesting the correction is independent of the default thresholds. The base case considers that an underlying firm value with a 40% volatility and supposes that the horizon is 5 years. Figure 1 in Bellalah and Moreaux (2003) plots default probabilities obtained when it is assumed that there are information costs. The observed 5-years default probability is supposed to be either 0, 0.5, 10 or 25. The underlying firm value has a 40% volatility. For this first experiment, information costs are allowed to be worth 60%. It appears that even if the observed default probability is null, the unbiased (real) default probability is non-negligible! For further analysis, we refer to Bellalah and Moreaux (2002).

Summary The pricing of credit derivatives and credit instruments depends on the default probability of the reference asset, the expected recovery rate and the nature of the exposure. The pricing approaches concern individual transactions and portfolios. The pricing of individual transactions or single transactions concerns the loss exposure, the default probability, the recovery rate and the correlations between these features in order to predict future credit losses. The pricing in a portfolio approach is focused on the correlations between individual exposures, which means that we must account for the joint default probabilities (correlations between default) and the correlation between loss exposure and recovery rates.

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The proprietary models are based on an option approach to default in which the equity of a levered firm is equivalent to a call on the net asset value of the firm. This approach is initiated by Black and Scholes (1973), Merton (1974), Black and Cox (1976), Geske (1977), etc. This approach considers the position of debtholders as a combination of a long position in a bond plus a short position in a put on the firm’s assets. Belkin et al. (1998) develop a direct approach to measuring credit risk at the transaction level. They identify credit risk as the “cost” of buying credit “insurance” in the spirit of the models of Merton and Bodie (1992) and Merton and Perold (1993). According to this approach, each credit instrument has an associated risk premium that reflects the competitively determined cost of a form of credit insurance.

Questions 1. 2. 3. 4. 5. 6. 7. 8.

What are credit derivatives? What are the main uses of credit derivatives? How are credit derivatives valued? What are the specific features of rating agencies models? What are the specific features of the proprietary models? How to appreciate credit risk? How to measure credit risk? How to estimate credit spread risk using extreme value theory?

Bibliography Arvanitis, A, C Browne, J Gregory and M Richard (1998). A credit risk tool box. Risk, December, 50–55. Belkin, B, L Forest, S Aguais and S Suchower (1998). Expect the unexpected. Risk, November, 34–39. Belkin, B, L Forest and S Suchower (1998). The effect of systematic credit risk on loan portfolio value-at-risk and loan pricing. CreditMetrics Monitor, First quarter, 17–28. Bellalah, M and F Moreaux (2003). Measuring implied probabilities: information costs do matter. Working Paper. Black, F (1976). Studies of stock price volatility changes. Proceedings of the American Statistical Association, 177–181. Black, F and J Cox (1976). Valuing corporate securities: some effects of bond indenture provisions. Journal of Finance, 31, 351–367. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659.

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Brown, C (1996). Credit derivatives. Financial Derivatives and Risk Management, 5 (March), 21–33. Credit Suisse Financial Products (1997). CreditRisk, a credit risk management framework (http://www.csfb.csh.com). Crosbie, P (1998). Modelling Default Risk. KMV Corporation, In Credit Derivatives, Das, 1998. Das, S (1998). Credit Derivatives: Trading & Management of Credit & Default Risk, John Wiley & Sons. Duffie, D and D Lando (2001). Term structure of credit spreads with incomplete accounting information. Econometrica, 69, 633–664. Garman, M (1992). Spread the load. Risk, 5(11), 68–84. Geske, R (1977). The valuation of corporate liabilities as compound options. Journal of Financial and Quantitative Analysis, November, 541–552. Ginzburg, A, KJ Maloney and R Willner (1994). Debt rating migration and the valuation of commercial loans. Citibank Portfolio Strategies Group Report, December. Gupton, G, C Finger and M Bhatia (1997). CreditMetrics — Technical Document, JPMorgan, April. Howard, K (1995). An introduction to credit derivatives. Derivatives Quarterly, Winter, 28–37. Jarrow, R and S Turnbull (1995). Pricing derivatives on financial securities subject to credit risk. Journal of Finance, L(1), March, 53–85. JPMorgan (1997). CreditMetrics, Technical document (http://www.jpmorgan.com/ RiskManagement/CreditMetrics). Leland, H (1994). Corporate debt value, bond covenants, and optimal capital structure. Journal of Finance, 4, 1213–1252. Litterman, R and T Iben (1988). Corporate Bond Valuation and the Term Structure of Credit Spreads. Goldman Sachs Financial Strategies Group, New York. Longstaff, F and E Schwartz (1995). A simple approach to valuing risky fixed and floating rate debt. Journal of Finance, 5, 789–820. Margrabe, W (1978). The value of an option to exchange one asset for another. Journal of Finance, 33, 177–186. McDermott, S (1993).A survey of spread options for fixed income investors. In: Handbook of Derivatives & Synthetics (eds. AR Klein and J Lederman), Probus Publishing, Chicago, IL, Chapter 4. Merton, RC (1974). On the pricing of corporate debt. Journal of Finance, 29, 449–470. Merton, R and Z Bodie (1992). On the management of financial guarantees. Financial Management, Winter. Merton, R and A Perold (1993). Management of risk capital in financial firms. In: Financial Services: Perspectives and Challenges (ed. Samuel Hayes), Harvard Business School Press. Miller, R (1998). Refining ratings. Risk, August, 97–99. Muller, U, M Dacorogna and O Pictet (1998). Heavy tails in high frequency financial data. In: A Practical Guide to Heavy Tails: Statistical Techniques and Applications (eds. Adler, R, R Feldman and M Taqqu), Basle, Birkhauser. Phoa, W (1999). Estimating credit spread risk using extreme value theory. Journal of Portfolio Management, Spring, 69–73.

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Ravindran, M (1993). Low fat spreads. Risk, 6(10), 66–67. Reoch, R (1996). Credit derivatives and applications. Financial Derivatives and Risk Management, 5, 4–10. Rolfes, B and F Broeker (1998). Good migrations. Risk, November, 72–73. Schonbucher, P. Pricing Credit Risk Derivatives. Working Paper, Financial Markets Group, London School of Economics. Shimko, D and B Humphreys (1998). I want to be a loan. Risk, November, 95. Smithson, C (1995). Credit derivatives. Risk, 8(12), December, 38–39. Spahr, RW, MA Sunderman and C Amalu (1991). Corporate bond insurance: feasibility and insurer risk assessment. Journal of Risk and Insurance, 58 (September), 418–437.

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Chapter 14

Default Risk and the Pricing of Corporate Bonds, Swaps and Options

This chapter is organized as follows: 1. Section 2 develops the standard context for the pricing of corporate bonds and default risk. It presents the traditional contingent claims-modeling of default risk and studies the pricing of corporate spreads. 2. Section 3 develops a general context for the analysis and valuation of credit risk in the presence of stochastic interest rates, stochastic risk of default and information uncertainty. 3. Section 4 presents other recent contributions in the analysis of credit risk and credit derivatives. 4. Section 5 studies the pricing of risky debt and compares the Longstaff and Schwartz (1995) and Merton (1974) models. 5. Section 6 shows how to construct a credit curve. It studies the time until default, the survival function and the hazard rate function by accounting for the effects of incomplete information. 6. Section 7 develops the pricing of credit risk derivatives in the context of multifactor models, stochastic interest rates and credit risk.

1. Introduction n the traditional contingent claims analysis of Black–Scholes (1973) and Merton (1974), default risk is shown to be equivalent to a European put option on the corporate assets. This standard approach has some weaknesses. In particular, it does not account for the interest rate uncertainty, the bankruptcy-triggering mechanism and the deviations from the strict priority rule. For corporate bondholders, default by the issuer cannot be

I

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neglected. Future possible losses are reflected in current bond yields. The extra yield or the corporate default spread rewards bondholders for carrying the risk of not being repaid. Corporate bonds are more difficult to price than equivalent Treasury bonds. Black–Scholes (1973) and Merton (1974) developed the general context for pricing corporate bonds using the contingent claim framework. Default risk is seen as a European put option on the corporate assets. This pioneering approach considers corporate liabilities as options on the total value of the firm. Black and Cox (1976) examined the effects of some indenture provisions. Geske (1977) explained how coupon bonds can be seen as portfolios of compound options. Recent contributions in this field account for interest rate uncertainty, for default to appear before the maturity of the bond and allow for deviations from the absolute priority rule. Guoming and Guo (1997) study the applicability and the performance of Longstaff and Schwartz (1995) and Merton (1974) models for the pricing of corporate contingent liabilities. Corporate debt and its derivatives depend on interest rate risk and default or credit risk. Default risk of risky debt can be measured in terms of a default premium. This premium is defined as the spread between the yield to maturity of risky debt and equivalent riskless debt for the same maturity. Default term structure of interest rates corresponds to the yield spread as a function of time to maturity. Default risk is specified using a default probability and a recovery rate. Default probability corresponds to the likelihood of occurrence of a default event between current time and a future date. Recovery rates indicate the proportion of payment of the bond in the event of default. The credit curve is studied and constructed in a discrete and in a continuous time setting. In the first case, the credit curve corresponds to a sequence of default probabilities over each period referred to as conditional martingale default probabilities. In the second case, the credit curve can be expressed in terms of a hazard rate function defining the instantaneous default probability in the future. The rate of return of risky bonds turns to be equal to the risk-free rate plus the mean loss due to default and information costs. Das (1995) develops some models for the pricing of derivatives on credit risk. These derivatives are proposed in 1992 by the International Swap Dealers Association (ISDA). He shows that the price of the credit risk option is given by the expected forward value of a put option on a risky bond with a credit level adjusted strike price. The model uses a compound option approach in a world with stochastic interest rates where the strike price is stochastic. Credit risk derivatives show often a pay-off related to

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the credit risk characteristics of a particular asset. Even if the concept of credit derivatives appears in 1992, bond insurance and financial guarantees represent a substantial market in the municipal bond area.

2. Pricing of Corporate Bonds and Default Risk 2.1. Traditional Contingent Claims-Modeling of Default Risk Consider a firm issuing at time t = 0 two classes of securities: a single class of zero-coupon bonds and equity. The face value of debt is F and it matures in T years. The total value of assets in the balance sheet is At = Et + Dt , where E refers to equity and D refers to debt. The traditional Black–Scholes (1973), Merton (1974) analysis of corporate debt is based on four assumptions. The first uses the traditional assumption of complete markets. The second assumption specifies the corporate asset process. It can be written under the risk-neutral probability as dAT = r dt + σA dWt At with r: instantaneously interest rate, σA : instantaneous volatility of the return on corporate assets, Wt : a standard Brownian motion. The third assumption refers to the Modigliani–Miller theorem, namely, the value of the firm is independent of its capital structure. The last assumption concerns the limited liability of shareholders.

2.2. Pricing Corporate Debt The option approach captures the shareholders’ option to walk away when things go wrong. In the case of solvency, assets generate enough value to match the face value of debt. Hence, bondholders receive the promised payment F and DT = F, if AT ≥ F . In the case of insolvency, the net worth of the firm assets is less than the face value of debt. The firm is declared bankrupt. Hence DT = AT , if

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AT < F . The holders of residual claims (shareholders) have the following pay-off: ET = AT − F if AT ≥ F or zero otherwise. Hence, at the debt’s maturity, the value of equity and debt can be written as ET = max[AT − F, 0] and DT = F − max[F − AT , 0]. In this context, the value of equity at time t before the maturity of the debt is given by Et = CE (At , F) where CE (At , F) corresponds to the value of a standard European call with a strike price F . The value of this call is given by the Black–Scholes formula: CE (At , F) = At N(d1 ) − F e−rT [N(d2 )] where
At 


 + r + 21 σA2 (T − t) d1 = √ σA T − t √ d1 = d2 + σA T − t ln

F

where N(.) is the cumulative normal distribution. The final pay-off of liabilities shows that the value of debt is given by Dt = F e−r(T −t) − PE (At , F), where PE (At , F) corresponds to the shareholders’ put-to-default. The put value is given by PE (At , F ) = −At N(−d1 ) + F e−rTN(−d2 ) where ln( AFt ) + (r + 21 σA2 )(T − t) , d1 = √ σA T − t

√ d1 = d2 + σA T − t.

This analysis reveals that a corporate debt can be seen as a portfolio comprising a long position on a risk-free zero-coupon bond F e−r(T −t) and a short position on a put-to-default PE (At , F ). The put reflects the effect of the limited liability.

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2.3. Pricing Corporate Spreads For a risky debt, bondholders ask for a risk premium to compensate them for the default risk. The term structure of default spreads can be easily obtained in this context. At time t = 0, the yield Y0 of a corporate zero-coupon bond D0 with a face value F and a maturity T is Y0 = −

1 D0 . ln F T

Using the previous bond’s price formula for D0 gives   1 1 N(−d1 ) + N(d2 ) Y0 = r − ln T l0 where l0 denotes the quasi-debt ratio because of a discounting factor. The corporate spread S0 corresponds to the difference between the yield Y0 and the yield on an otherwise equivalent riskless bond. It is given by S0 = Y0 − r = −

  1 1 ln N(−d1 ) + N(d2 ) . T l0

The spread S0 captures the risk premium of the corporate debt.

2.4. Limits of the Traditional Approach The traditional modeling of default risk of Black–Scholes (1973) and Merton (1974) considers a flat structure of interest rates, a simple bankruptcy mechanism and applies the strict priority rule. These points are considered as the main three weaknesses of these pioneering papers. First, in practice, interest rates are stochastic and the assumption of constant interest rates is untenable for interest rate sensitive instruments. Second, the bankruptcytriggering mechanism plays a crucial role in the pricing of corporate debt. The analysis in Merton (1974) ignores the possibility of early default. Third, the deviations from the strict priority rule or the violations of the strict priority rule should be modeled to better reflect the complexity of the bankruptcy process.

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3. Credit Risk, Stochastic Interest Rates, Stochastic Risk of Default and Information Uncertainty 3.1. The Value of the Firm as a Random Variable in the Presence of Stochastic Interest Rates and Shadow Costs The dynamics of the firm value are governed by the following stochastic differential equation: dA = µA dt + σA dX where A refers to the firm value. The firm defaults if its value attains a critical level Ab . It can be shown in the standard Black–Scholes–Merton context, that the bond value V satisfies the following equation:  dV =

∂V ∂2 V 1 + σ 2 A2 2 ∂A 2 ∂t

 dt +

∂V dA. ∂A

Using the fact that the expected instantaneous return must be equal to the riskless rate plus the information costs on V , we have ∂V 1 ∂2 V ∂V + σ 2 A2 2 + µA − (r + λV )V = 0. ∂t 2 ∂A ∂A The bond value at maturity must be equal to its nominal value: V(A, T) = D. If the firm’s value is less than the debt value and default occurs then V(Ab , t) = 0. Stochastic interest rates When the interest rate follows the dynamics dr = u(r, t)dt + w(r, t)dX1 and the correlation coefficient between the two processes is ρ, then the debt value is a function of A, r, t and can be written as V(A, r, t). Using the same dynamics for A, it is possible to construct a portfolio comprising a long position in the firm’s corporate bond and a short position in  units of a

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risk-less zero-coupon bond Z(r, t):  = V(A, r, t) − Z(r, t). It is possible to compute the change in the portfolio’s value as follows:   1 2 2 ∂2 V ∂2 V 1 2 ∂2 V ∂V + σ A dt + ρσwA + w d = 2 ∂r 2 ∂r∂A 2 ∂A2 ∂t     ∂V ∂Z ∂Z 1 2 ∂2 Z ∂V dA + − dr −  + w dt. + ∂A ∂r ∂r ∂t 2 ∂r 2 The appropriate choice of the  allows the elimination of the terms is dr. If we set the return on this portfolio to be equal to the risk-free rate plus information costs, the following equation is obtained: ∂V 1 ∂2 V 1 ∂2 V ∂2 V ∂V + w2 2 + ρσwA + σ 2 A2 2 + (u − γw) ∂t 2 ∂r ∂r∂A 2 ∂A ∂r ∂V + µA − (r + λV )V = 0. ∂A This equation results from hedging the diffusive component resulting from the interest rate risk. The term γ corresponds to the price of the interest rate risk. However, it is important to note that the firm’s value is unhedged. This equation must be solved using the following terminal condition V(A, r, T ) = D under the following boundary condition V(Ab , r, t) = 0 corresponding to the value of the bond when the firm’s value reaches the critical level. In the special case of a zero correlation between the firm value and interest rates, the solution for the bond’s value is V(A, r, t) = Z(r, t; T )H(A, t). The function H satisfies the following equation: ∂H ∂2 H ∂H 1 =0 + σ 2 A2 2 + µA 2 ∂A ∂A ∂t under the following conditions: H(Ab , t) = 0, H(A, T) = 1, A > Ab . This model appears in Longstaff and Schwartz (1995) and Wilmott (1998). In this model, default risk appears in H. The variables in this model are difficult to measure.

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3.2. Modeling Default Risk with Measurable Parameters and Firm Valuation Following the analysis in Wilmott (1998), consider the dynamics for the firm’s gross annualized earnings: dE = µE dt + σE dX. Let us denote by E∗ the fixed costs and by kE the floating costs. The firm’s profit can be computed as the difference between the gross earnings, the fixed costs and the variable costs or E − E∗ − kE = (1 − k)E − E∗ . This profit can be put in a bank to yield the riskless rate. Hence, cash in the bank account is accumulated at the risk-less rate:  t C= ((1 − k)E(τ) − E∗ )er(t−τ) dτ. 0

Differentiating gives dC = ((1 − k)E − E∗ + rC)dt. The firm has a debt D maturing in T years. The debt is repaid at T if there is “money” in the account. Hence, the repayment can be written as max(min(C, D), 0). We denote by V(E, C, t) the present value of the expected amount, which satisfies the following differential equation: ∂V 1 2 2 ∂2 V ∂V ∂V + σ E + ((1 − k)E − E∗ + rC) − (r + λV )V = 0 + µE 2 ∂t 2 ∂E ∂E ∂C under the condition V(E, C, T) = max(min(C, D), 0). The value of the firm It is possible to use this model for firm valuation. Let the value of the business correspond to the present value of the expected cash in the bank at some time T0 in the future. Hence, the firm’s value satisfies the following equation: ∂V 1 2 2 ∂2 V ∂V ∂V + ((1 − k)E − E∗ + rC) − (r + λV )V = 0. + µE + σ E 2 2 ∂E ∂E ∂C ∂t When the business has no liability, then the value at T0 is V(E, C, T0 ) = max(C, 0). When the liability is limited, the value satisfies V(E, C, T0 ) = C.

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3.3. The Poisson Process and the Instantaneous Default Risk within Information Uncertainty We denote by p the instantaneous default risk and by p dt the probability of default between times t and t + dt. Let us denote by P(t; T) the probability that the company does not default before time T given that default has not occurred at t. The probability of default between future times t  and t  + dt  corresponds to p dt times the probability that default has not appeared until time t  . Hence, the rate of change in the required probability can be written as ∂P/∂t  = pP(t  ; T). In the absence of default at the beginning, P(T, T) = 1 and the solution is e−p(T −t) . In this context, the expected value of a zerocoupon bond Z paying one at T is given by e−p(T −t) Z. The bond’s yield to maturity is given by log Z log(e−p(T −t) Z) =− + p. T −t T −t This analysis shows that the effect of the default risk on the yield appears in the addition of a spread p. Now, we consider the pricing of derivatives in the absence of a correlation between the diffusive change in interest rate and the Poisson process. In this context, it is possible to construct a portfolio with a long position in the security and a short position in the riskles asset:  = V(r, p, t) − Z(r, t). Since the probability that the bond does not default is (1 − p dt), then the change in the portfolio value can be written as   ∂V 1 2 ∂2 V ∂V d = + w dr dt + 2 ∂t 2 ∂r ∂r    ∂Z ∂Z 1 2 ∂2 Z + w dr . dt + − ∂t 2 ∂r 2 ∂r −

It is possible to eliminate the risky term dr by choosing  to eliminate the interest rate risk. In the event of default with a probability p dt, the change in the portfolio value is d = −V + O(dt 1/2 ). Taking expectations, and using the bond equation in the absence of default, the risky bond must satisfy the following equation: ∂V 1 ∂2 V ∂V + w2 2 + (u − γw) − (r + λV + p)V = 0. ∂t 2 ∂r ∂r 12:16:26.

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This model is very simple and gives the relationship between a risky bond and a risk-free bond in the presence of default and incomplete information.

3.4. Time-Dependent Intensity, the Term Structure of Default Under Stochastic Risk of Default and Information Uncertainty Consider a firm for which the capital structure comprises equity and zerocoupon bonds. When the default risk depends on time, p(t) and there is no correlation between this risk and interest rate risk, then the real expected value of the bond is Ze−

T t

p(τ) t);

t ≥ 0.

The subscript A is omitted for simplicity of exposition. S(t) provides the probability that an asset will attain age t. When F(0) = 0, S(0) = 1. He defines the probability density function f(t) as: f(t) = F  (t) = −S  (t) = lim+ →0

Pr[t < T ≤ t + ] 

and denotes by T − x > t | T > x the probability of future survival time for a security having survived x years. In this context, the conditional

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probability that the asset A will default within the next t years conditional on the survival in x years is given by: tqx = Pr[T − x ≤ t | T > x]; t ≥ 0 tpx = 1 − tqx = Pr[T − x > t | T > x]; t ≥ 0.

(1)

When x = 0: tp0 = S(t);

x≥0

if t = 1 we have: px = Pr[T − x > 1 | T > x] qx = Pr[T − x ≤ 1 | T > x] where qx refers to the marginal probability. In a discrete setting, the credit curve is given by the sequence of q0 , q1 , . . . , qn .

6.2. The Hazard Rate Function The hazard rate function gives the instantaneous default probability for an asset that has attained age x: Pr[x < T ≤ x + x | T > x] =

f(x)x F(x + x) − F(x) = . 1 − F(x) 1 − F(x)

The hazard rate function, h(x) = f(x)/(1 − F(x)). The relation between the hazard rate function, the distribution and the survival function is: h(x) =

f(x) S  (x) =− . 1 − F(x) S(x)

It is possible to express the survival function in terms of the hazard rate t function using S(t) = e− 0 h(s)ds . It is also possible to express tqx and tpx in terms of the hazard rate function as: tpx = e−

t 0

h(s+x)ds

= e−

 x+t x

h(s)ds

.

Besides, the distribution function for T is: F(t) = 1 − S(t) = 1 − e−

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t 0

h(s)ds

(2)

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and the density function for T is: f(t) = S(t) × h(t) = h(t)e−

t 0

h(s)ds

.

When the hazard rate is constant, the density function follows an exponential distribution with parameter h: f(t) = he−ht ;

t > 0,

h > 0.

In this case, the probability of survival over the [0, t] is: tpx = e−

t 0

h(x+s)ds

= e−ht = (px )t .

6.3. The Discrete Version Consider the valuation of corporate bond subject to default. The bond promises the cash flows C1 to Cn at different dates t1 to tn . The discount factor computed from the Treasury yield curve is P(t0 , t). We denote by V(ti ) the bond’s market value at ti immediately after the payment of Ci . The current bond price can be calculated by a recursive formula between the bond values at times ti and ti+1 . If the absence of default, the terminal value of the bond in the period [ti , ti+1 ] is Ci+1 + Vi+1 . In the presence of default, the recoverable value at time ti+1 is R(ti+1 )[Ci+1 + Vi+1 ]. In the presence of a survival probability pi , the recursive formula is given by: P(t0 , ti+1 ) [Pi [Ci+1 + V(ti+1 )] + (1 − pi )R(ti+1 )[Ci+1 + V(ti+1 )]] P(t0 , ti ) P(t0 , ti+1 ) = [pi + (1 − pi )R(ti+1 )[Ci+1 + V(ti+1 )]] P(t0 , ti )

V(ti ) =

where i = 0, 1, 2, . . . , n − 1. The boundary condition is: V(tn ) = 0. By substitution in this last equation, we obtain:   n i−1  V(t0 ) = P(t0 , ti )  [pj + (1 − pj )R(ti+1 )] Ci . i=1

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(3)

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If we denote the credit discount factor by:   i−1  DC(ti ) =  [pj + (1 − pj )R(ti+1 )]

529

(4)

j=0

and the credit risk adjusted discount factor by: Q(t0 , Ti ) = P(t0 , ti )DC(ti ) then, the initial bond value is: V(0) =

n

Q(t0 , ti ) · Ci .

(5)

i=1

This equation gives the value of a defaultable bond by adjusting the discount factor for default. The total discount factor is the product of the risk free discount factor and the pure credit discount factor when the underlying factors affecting default and the interest rate are independent. The marginal default probabilities are implied using (3) and pi = e−

 ti+1 ti

h(s)ds

.

ˆ ∈ (ti−1 , ti )] the In the presence of a constant hazard rate: h(s) = hi , E[s −hi+1 (ti+1 −ti ) marginal default probabilities are pi = e .

6.4. The Continuous Model Equation (3) can be used to obtain the continuous version of the model in Duffie and Singleton (1997). Using Eq. (2): pj = e

−

 tj+1 tj

h(s)ds

for a small t = tj+1 − tj , it is possible to write that: pj + (1 − pj )R(tj+1 ) = e =e

12:16:26.

−

 tj+1 tj

h(s)ds

−(1−R(tj+1 ))

 tj+1
h(s)ds  − + 1 − e tj R(tj+1 )

 tj+1 tj

h(s)ds

.

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The credit discount factor can be written as follows: i−1 

pj + (1 − pj )R(tj+1 ) =

j=0

i−1 

e

−(1−R(tj+1 ))

 tj+1 tj

h(s)ds

j=0

 i−1   = exp j=0

 tj+1

(1 − R(ti+1 ))h(s)ds .

tj

When the interval (0, ti ] is divided into nt small intervals, then:   i−1  tj+1 (1 − R(ti+1 ))h(s)ds lim [pj + (1 − pj )R] = lim exp − n→∞

n→∞

= e−

 ti 0

j=0 [1−R(s)]h(s)ds

tj

.

Assuming that: P(t0 , ti ) = e−

 ti 0

(r(s)+λ)ds

where r(s) is the short rate and λ is a shadow cost of incomplete information, then Eq. (3) can be written as: V(t0 ) =

n

Ci × e−

 ti 0

[r(s)+λ+(1−R(s))h(s)]ds

.

(6)

i=1

This result generalizes the one in Duffie and Singleton (1997) for a zero coupon bond to the case of a coupon bearing bond in the presence of incomplete information.

6.5. A Numerical Example Li (1998) provides a numerical example to illustrate the construction of the credit curve using a series of outstanding corporate bonds with different maturities. He assumes that h(t) = hi where t is between ti and ti−1 . For the first bond maturing in one year with an annual coupon of 10.95%, he uses Eq. (6) as follows: 100 =

10.95  0.5 [r(s)+λ+(1−R(s))h(s)]ds e0 2  1 10.95 + 100 + × e− 0 [r(s)+λ+(1−R(s))h(s)]ds 2

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or: 100 =

10.95 × P(0, 0.5)e−[(1−R)h(1)]0.5 2   10.95 × P(0, 1.0)e−[(1−R)h(1)]10 . + 100 + 2

Assuming a constant recovery rate R, it is possible to solve for h(1). The same method applies to solve for h(2) for the second bond and so on until the hazard rate is determined for the last bond. This approach is useful in the valuation of several credit derivatives.

7. The Pricing of Credit Risk Derivatives Das (1995) uses the contingent claims approach for the pricing of derivatives credit risk options (denoted CROs). The writer of a CRO (on any underlying risky debt instrument) agrees to compensate the buyer for a prespecified fall in credit standing of the issuer of the underlying credit instrument. The buyer of a CRO is protected against the loss in value of a bond when its yield rises above, or its rating falls below, a “strike” level. This level corresponds to the acceptable default spread on the bond. When the CRO is exercised, the payoff corresponds to the price of the bond at the prevailing default-free interest rate plus the strike default spread. Hence, the pay-off of a CRO corresponds to the amount by which the strike price exceeds the then-prevailing market price of the bond.

7.1. The Model for Credit Risk Derivatives Following Das (1995), we denote, respectively, by: TB : debt’s maturity, TW : maturity of the CRO with TW < TB , S: stock value, F : face value of outstanding debt, B: market value of outstanding debt, V : value of the firm at time 0, VW = V(TW ): firm value at time TW , VB = V(TB ): firm value at time TB , σ: volatility of return on V , σS : volatility of stock return,

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r: risk free interest rate, r ∗ : credit spread for a given debt rating, rd : continuous dividend rate, rc : continuous coupon rate on debt, dz: a standard Wiener process, λV : an information cost on the asset V . We denote by: C(V, K, TW , σ, r, λ): Black and Scholes (1973) call price on V with strike K, maturity TW , volatility σ, interest rate r and information cost λ, P(V, K, TW , σ, r, λ): Black and Scholes put price in the same context, K(.): CRO strike price which is stochastic, R = R(r, rd , r ∗ , rc ): interest rates set. Consider the following dynamics for the value of the firm: dV = (αV − rc F )dt + σV dz

(7)

with α = r − rd + λV . The solution to Eq. (7) for V(0) is:       σ2 σ2 V(t) = V(0) exp α − t + σZ(t) − rc F ×exp α − s + σZ(t) 2 2      t σ2 × s + σZ(s) ds. exp − α − 2 0 Credit options are options on debt where the underlying asset is the firm’s value. Valuation of the CRO requires computation of the strike price which corresponds to the value of the bond at time TW if the credit rating is at the ∗ exercise level. Hence, the effective strike price is K = F e−(r+r +λ)(TH −TW ) . This corresponds to the discounted bond price (at the riskless rate) plus the strike credit spread for the remaining maturity of the bond, (TB − TW ). When the interest rate is constant, the CRO’s payoff is max(0, K − BW ) where BW corresponds to the bond’s value at time TW . Using the put–call parity relationship, the bond value can be written as the difference between its present value PV (F) and the Black–Scholes put price. It can also be written as the difference between the firm value VW and the call price as follows: BW = PV(F) − P(VW , F, TB − TW , σ, r) = VW − C(VW , F, TB − TW , σ, r).

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The value of the firm V ∗ below which the CRO will be exercised can be computed using: K − VW + C(VW , F, TB − TW ) = 0. When rc = rd = 0, the value of the CRO is computed as:  x[V ∗ ] √ 2 −(r+λ)TW ∗ CRO(V, σ, r, r , TB , TW , F ) = e [K − V e(r+λ−σ /2)TW +xσ TW −∞ √ (r+λ−σ 2 /2)TW +xσ TW

− C(V e − TW , σ, r)]φ(x)dx

, F, TB

where K = F exp[−(r + r ∗ + λ)(TB − TW )] log(V ∗ /V) − (r + λ − σ 2 /2)TW x[V ∗ ] = √ σ TW

and

where φ(.) is the probability density function of the standard normal distribution. The term λ indicates the information cost on the option CRO. This ordinary integral expression in x can be easily computed by numerical integration. The simulations done in Das (1995) show that the value of the CRO falls with a decrease in the debt equity ratio. A higher value of r ∗ reduces the CRO value.

7.2. Multifactor Models, Stochastic Interest Rates and Credit Risk It is possible to extend the previous model by accounting for stochastic interest rates. In this context, Das (1995) uses the following dynamics for the firm’s value: dV = µV dt + δ1 V dz1 + δ2 V dz2 where Z = (z1 , z2 ) is a vector Wiener process with variance (δ1 , δ2 ). The Heath, Jarrow and Morton (1992) is used for the dynamics of forward rates: df(t, T ) = α(t, T )dt + σ(t, T )dz1 (t) where f(t, T) denotes the instantaneous forward interest rates for date T observed at time t. The spot rate r(t) is expressed in terms of forward rates as r(t) = f(t, t).

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The price of a default-free bond at time t for maturity T is given by:    T f(t, y)dy . P(t, T ) = exp − t

Using Ito’s Lemma gives: dP(t, T ) = [r(t) + b(t, T )]P(t, T ) dt + a(t, T )P(t, T ) dz1 (t) where



T

a(t, T ) = −

σ(t, y)dy t

 b(t, T ) = −

t

T

1 α(t, y)dy + 2



2

T

σ(t, y)dy

.

t

Since equity is seen as a call on firm value maturing at time TB , the value of risky debt in the firm can be written as:   1 max(0, VB − F) . B(0, TB ) = V(0) − S(0) = EQ V(0) − P(0, TB ) The solution is: B(0, TB ) = V(0)[1 − (k)] − FP(0, TB )(k − ) log[V(0)/FP(0, TB )] + 1/22   TB  2 2 2 a(t, TB )dt +  = (δ1 + δ2 )TB 2δ1 ds k=

0

TB

a(t, TB )2 dt.

0

7.3. Implementation of the Model Das (1995) makes some simplifying assumptions for the sake of model implementation. First, he considers a volatility σ(t, T) = σ. This gives a term structure model as the one in Ho and Lee (1986). Hence:  T  T a(t, T ) = − σ(t, u)du = − σ du = −σ(T − t) 0

and

 0

T

σT 2 a(t, T )dt = 2

12:16:26.

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T

a(t, T )2 dt =

and 0

σ2T 3 . 3

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Second, in the presence of continuous dividend payouts, the value of the risky debt is computed using the following formula: B(0, TB ) = V(0)[1 − (k)e−rd TB ] − FP(0, TB )(k − ) log[V(0)/FP(0, TB )] − rd TB − 1/22  σ 2 TB3 . 2 = (δ21 + δ22 )TB − δ1 σTB2 + 3

k=

(8)

Example Das (1995) provides an example in which riskless discount bond prices are given by: P(0, T ) = e−(0.06+0.05T −0.0001T

2 )T

.

The polynomial structure allows to generate different realistic term structures. The riskless term structure embodying the yield Y(T ) for different maturities is given by: 1 Y(T ) = − log[P(0, T )]. T The term structure of risky debt with stochastic interest rates YB (TB ) is represented by: 1 YB (TB ) = − log[B(0, TB )]. TB Using two curves YB (.) and Y(.), it is possible to generate different shapes of the credit spread term structure.

7.4. Derivatives on Risky Debt: Pricing CROs Since risky debt contains an option on the firm value, the pricing of options on risky debt is equivalent to the pricing of a compound option. Since CROs are written on the firm risk component of the corporate bond stripped of the interest rate risk, the pricing of CROs needs to adjust the strike price to reflect this feature. For a constant firm value volatility, the date 0 forward value of the riskless bond as of time TW is: PW = P(TW , TB , Z1 )  2   σ TW TB (TW − TB ) P(0, TB ) exp − σ(TB − TW ) TW Z1 . = p(0, TW ) 2 12:16:26.

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The date 0 forward value for the firm value as of time TW is: VW = V(TW , Z1 , Z2 )



1 = V(0)M(TW , Z1 ) × exp − (δ21 + δ22 )TW + δ1 Z1 + δ2 Z2 2 where

 M(TW , Z1 ) =



 2

TW3

1 σ × exp  6 P(0, TW )

− Z1



σ 2 TW3 3

.

Equation (8) gives the date TW price of risky debt for each realization of (Z1 , Z2 ) where the terms V(0) and P(0, TW ) are replaced by VW and PW from the above equations. The maturity TB is replaced by TB − TW . This risky debt price is denoted by: B[VW , PW ] = B[TW , TB ; Z1 , Z2 ]. Consider the following example, for the computation of the strike price. When the current default spread for AA bonds over the riskless rate for a maturity (TB − TW ) is 100 basis points, the buyer of a CRO receives compensation when the price of the risky bond falls below the price of a bond at the riskless yield plus 100 basis points at date TW . The strike credit spread is r ∗ = 0.01. A range of strikes K(.) corresponding to each outcome of Z1 is generated using a different riskless bond price at TW . The CRO stochastic strike price is given by: K(TW , TB , Z1 , F, r ∗ )     1 ∗ = F exp − − log PW + r (TB − TW ) . (TB − TW )

(9)

Hence, for each outcome of Z1 , the price P(.) is inverted to get the riskless yield. The spread r ∗ is added to compute K(.). The discounting factor is (9) corresponds to the riskless yield plus the strike credit spread, r ∗ . The price of the CRO is given by: CRO = CRO[δ1 , δ2 , σ, TW , TB , V(0), P(0, .), F, rd , r ∗ ] = EQ [max[K(TW , TB , Z1 , F, r ∗ ) − B(TW , TB , 0)]]   max[K(TW , TB , Z1 , F, r ∗ ) = Z1

Z2

− B(TW , TB ), 0]φ(Z1 )φ(Z2 )dZ2 dZ1 .

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Equation (10) shows an integral which is difficult to calculate. However, it can be approximated using a binomial distribution. The following scheme is used for any number n of discrete points. The√ith outcome of the random variable Z takes the value: Zi = ((n − 2i)/ n), ∀i = 0, . . . , n with
n
n probability Probi = i 21n , ∀i = 0, . . . , n where i is the number of combinations of n things taken i at a time.

Summary Corporate debt and its derivatives depend on interest rate risk and default or credit risk. Default risk of risky debt can be measured in terms of a default premium. This premium is defined as the spread between the yield to maturity of risky debt and equivalent riskless debt for the same maturity. Default term structure of interest rates corresponds to the yield spread as a function of time to maturity. Default risk is specified using a default probability and a recovery rate. Default probability corresponds to the likelihood of occurrence of a default event between current time and a future date. Recovery rates indicate the proportion of payment of the bond in the event of default. Merton (1974) assumes that a firm has only one class of pure discount bonds and that default occurs only at the maturity date of risky debt. This is the case when the firm value is less than the promised payment since bondholders take over the firm at that date. In this setting, equity can be seen as a call on the firm value and the bond value corresponds to the difference between the values of the firm and equity. Hence, risky bond prices can be obtained by applying the Black– Scholes model. Merton (1974) provides a simple framework for the pricing of risky debt. Geske (1977) extends Merton (1974) model to the case of risky coupon bonds with a finite time to maturity and discrete coupon payments. This analysis is extended by Black and Cox (1977) to the analysis of safety covenants and subordinated debt, etc. Guoming and Guo (1997) study the applicability and the performance of Longstaff and Schwartz (1995) and Merton (1974) models for the pricing of corporate contingent liabilities. The emergence of new credit derivatives in addition to the standard products such as credit default swaps, total return swaps, credit spread options, collateralized bonds and loan obligations, credit default swaption, etc., raises the question of their pricing.

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The key element in the pricing of standard and complex credit derivatives is the building of a credit curve. This curve gives the instantaneous default probability of a party at any future time. This curve is important in the pricing of credit derivatives as the yield curve is fundamental in the pricing of interest rate sensitive derivatives. Li (1998) develops a method for the construction of a credit curve using the yield spread curve of a corporation. Using observable information and not historical default experience information, he assumes that there exists a series of bonds maturing in n years issued by the same company and having the same seniority. The yield spread curve is obtained using the yield to maturity of corresponding Treasury bonds. The credit curve is generated using the yield spread curve and an exogenous assumption about the recovery rate. In the case of default, the bondholder receives a fixed percentage of the bond price (the recovery rate) immediately prior to default as in Duffie and Singleton (1997). A credit risk option, CRO, is a compound option with a stochastic exercise price since it is an option on risky debt, which is itself an option on the firm’s assets. This is the same concept as that introduced in Geske (1977). Das (1995) extends Merton’s model to account for stochastic interest rates in the pricing of CROs. The pricing of credit risk can also be done using the instantaneous default risk approach where a bond defaults with Poisson probability. The pricing can also be done with reference to the process for credit spreads. Jarrow, Lando and Turnbull (1994) value bonds off the riskless interest rate process and provide for credit rating to follow a Markov process. Hull and White (1995) develop a model for the pricing of derivative securities subject to default risk in which the holder of the security can recover a proportion of its no-default value in the event of default. Their model is based on the assumption that both the probability of default and the size of the proportional recovery are random. They refer to a derivative security subject to default risk as a vulnerable derivative security. Jarrow and Turnbull (1997) use their 1995 model to price swap contracts by considering the default characteristics of both counterparties. Their approach allocates the counterparties to a credit risk class and implies from the credit class the martingale default probabilities. The value of the swap is calculated using a recursive valuation procedure. Their model is applied for interest rate swaps and foreign currency swaps. For more details about the models in this chapter, we refer the reader to Hull and White (1995), Jarrow and Turnbull (1995, 1997), etc.

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Questions 1. 2. 3. 4. 5. 6. 7. 8. 9.

How can corporate bonds be valued? Describe the traditional contingent claims-modeling of default risk. How are corporate spreads priced? Describe the Longstaff–Schwartz model. Can you compare the Longstaff and Schwartz and Merton models for the pricing of risky debt? How to construct a credit curve? What is a hazard rate function? What is a CRO? How to value interest rate swaps?

Bibliography Black, F and J Cox (1976). Valuing corporate securities: some effects of bond indenture provisions. Journal of Finance, 31, 351–367. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659. Brown, SJ and P Dybvig (1986). The empirical implications of the Cox, Ingersoll, Ross theory of the term structure of interest rates. Journal of Finance, 41, 1209–1227. Das, S (1995). Credit risk derivatives. Journal of Derivatives, Spring, 7–23. Duffie, D and K Singleton (1997). Modeling Term Structure of Defaultable Bonds, Graduate School of Business, Stanford University, 40–44. Geske, R (1977). The valuation of corporate liabilities as compound options. Journal of Financial and Quantitative Analysis, November, 541–552. Guoming Wei, D and D Guo (1997). Pricing risky debt: an empirical comparison of the Longstaff and Schwartz and Merton models. Journal of Fixed Income, September, 8–28. Heath, D, R Jarrow and A Morton (1992). Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation. Econometrica, 60, 77–105. Ho, TSY and SB Lee (1986). Term structure movements and pricing interest rate contingent claims. Journal of Finance, 41, 1011–1029. Hull, J and A White (1995). The impact of default risk on the prices of options and other derivative securities. Journal of Banking and Finance, 19, 299–322. Jarrow, R, D Lando and S Turnbull (1994). A Markov model for the term structure of credit risk spreads. Working Paper, Cornell University. Jarrow, R and S Turnbull (1995). Pricing derivatives on financial securities subject to credit risk. Journal of Finance, L(1), March. Jarrow, R and S Turnbull (1997). When swaps are dropped. Risk, May, 70–77. Leland, H (1994). Corporate debt value, bond covenants, and optimal capital structure. Journal of Finance, 4, 1213–1252. Li, D (1998). Constructing a credit curve. Risk, November, 40–44.

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Longstaff, F and E Schwartz (1995). A simple approach to valuing risky fixed and floating rate debt. Journal of Finance, 5, 789–820. Merton, RC (1974). On the pricing of corporate debt. Journal of Finance, 29, 449–470. Vasicek, O (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5, 177–188. Willmott, P (1998). Derivatives, John Wiley and Sons.

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Chapter 15

Contingent Claims Analysis and Its Applications in Corporate Finance: The Case of Real Options This chapter is organized as follows: 1. Section 2 presents the main concepts regarding the identification of corporate liabilities as compound options. 2. Section 3 explains the applications of the Black–Scholes model in this context. 3. Section 4 extends the option pricing analysis to the pricing of several corporate liabilities.

1. Introduction

S

ince the path-breaking contribution by Black–Scholes (1973) was published, many papers have studied the contingent claims analysis (CCA). It is a technique for computing the price of a security for which the payoffs depend upon the prices of one or more other securities. This technique can be applied to a number of tactical and strategic corporate financial decision problems. According to Merton (1998): “The most influential development in terms of impact on finance practice was the Black–Scholes model for option pricing. . . This success in turn increased the speed of adoption for quantitative financial models to help value options and assess risk exposures” (p. 324). Corporate liabilities can be viewed as combinations of simple option contracts. This chapter presents an overview of CCA and its applications to several corporate financial problems. It presents some applications of contingent claims analysis (CCA) to a variety of corporate financial problems. The problems include capital budgeting decisions and the characterization Bellalah, Mondher. Exotic Derivatives and Risk: Theory, Extensions and541 Applications, World Scientific Publishing Co 12:16:34.

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of the strategic value of a project as a series of “operating options” in the presence of incomplete information. The analysis can be extended to account for the effects of shadow costs of incomplete information.

2. Corporate Liabilities as Options: The Basic Concepts We denote by: C(S, T, X), (c(S, T, X)), the American (European) call option on the asset S, with a strike price X and time to maturity T , P(S, T, X), (p(S, T, X)), the American (European) put option on the asset S, with a strike price X and time to maturity T . We recall that the value of a call option at maturity is given by: C(S, 0, X) = Max(S − X, 0).

(1)

This expression is true forAmerican and European calls. In the same context, the value of the American put is given by: P(S, 0, X) = Max(X − S, 0).

(2)

This expression is true for American and European puts. There is an important relationship between European call and put prices. Consider an investment position: a long call and a short put on the same underlying asset and strike price. This portfolio can be written as: I1 = c(S, T, X) − p(S, T, X).

(3)

Now, consider a portfolio with the following composition: buy one unit of the underlying asset S and borrow on a discount basis X dollars for T periods at rate r. This portfolio can be written as: I2 = S − Xe−rt .

(4)

In T periods, the position is worth I2 = S − X. Since the two portfolios have the same final value, the initial net investment in the two portfolios must be the same. This gives the put–call parity relationship: c(S, T, X) − p(S, T, X) = S − Xe−rt .

(5)

Consider the economic balance sheet of a simple firm with two liabilities: equity E and a single issue of a zero-coupon debt D. The left side of the balance sheet is the economic value of the firm. The right side shows the

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economic value of all the liabilities of the firm. If on the debt’s maturity date, the value of the firm V is higher than the promised principal B, the debt will be paid off, D = B. In this case, the value of equity is V − B. However, if V < B, the value of equity is zero, E = 0. Hence, on the debt’s maturity, T = 0, the value of equity can be represented by E(V, 0, B) = Max[V − B, 0].

(6)

The value of debt at T = 0, is given by D(V, 0, B) = Min(V, B).

(7)

Equation (6) shows that equity is analogous to a European call option written on the firm value, V , with a strike price, B: E(V, T, B) = c(V, T, B).

(8)

The put–call parity can be written as: V = c(V, T, B) + BerT − p(V, T, B).

(9)

V =E+D

(10)

But, since

and since the value of equity is given by (8), then D(V, T, B) = BerT − p(V, T, B).

(11)

The value of risky debt corresponds to the price of a risk-free bond minus the price of put written on the value of the firm. Expression (11) shows that risky debt plus a loan guarantee has the same value as risk free debt. The loan guarantee is like insurance. If at T = 0, V > B, the guarantee will pay nothing since the firm is valuable to retire the debt. If V < B, the guarantor must pay B − V , in order that the debt be fully repaid. Hence, the value of the loan guarantee can be written as: G(V, 0, B) = Max(B − V, 0).

(12)

This payoff is analogous to a European put option on the value of the firm, i.e., G(V, T, B) = p(V, T, B).

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3. The Black–Scholes Model and Its Applications The model is based on the following six assumptions: A1. A2. A3. A4. A5. A6.

There are no transaction costs or differential taxes. Borrowing and lending at the same rate. The short-term risk-free rate r is constant. Short sales are unrestricted. Trading takes place continuously. The dynamics of the stock price are given by dS = αS dt + σS dz.

Assumptions (A1)–(A5) are mainly institutional assumptions. Assumption (A6) implies that the distribution of unanticipated stock price changes is lognormal. Using the Black–Scholes approach, the call option C(S; T, X) satisfies the following PDE: 1 2 2 σ S CSS + rSCS − CT − rC = 0 (13) 2 which must be solved under the following conditions: C(0, T ) = 0.

(13a)

This condition shows that the underlying asset becomes worthless when the call becomes worthless. C(S, T )/S → 1

as S → ∞.

(13b)

This condition shows that the value of the call approaches that of the underlying asset for very large values of the underlying asset. C(S, 0) = Max(S − X, 0).

(13c)

This condition indicates the value of the call at expiration. The call value is given by: C(S, T, X) = SN(d1 ) − Xe−rtN(d2 )
 log(S/X) + r + 21 σ 2 T d1 = √ σ T 
log(S/X) + r − 21 σ 2 T . d2 = √ σ T

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This analysis can be extended to account for the effects of information uncertainty. In this context, we can use the model in Bellalah (1999) instead of the Black–Scholes model.

4. Contingent Claim Analysis and the Pricing of Corporate Liabilities The CCA approach to the pricing of corporate liabilities uses the firm’s total capital structure and prices the different components of the capital structure. Consider the following assumption that must be added to the Black–Scholes A1–A6 assumptions: The movements of the firm value, V , through time can be described by the diffusion-type equation: ¯ dV = (αV − P)dt + σV dz where α is the instantaneous expected rate of return to the firm per unit time, P¯ is the net total payout by the firm per unit time, and σ 2 is the variance of return on the firm per unit time. Merton (1977) derives a dynamic portfolio strategy, which involves mixing positions in the firm and the riskless asset. The strategy allows to produce a pattern of returns that exactly replicates the return to any given corporate liability of the firm. The portfolio must be adjusted in response to changes in time and the value of the firm. This replication argument leads to a fundamental partial differential equation which must be satisfied by the prices of corporate liabilities. The equity E(V, T, B) satisfies the following equation: 1 2 2 ¯ V − ET − rE + P¯ = 0 σ V EVV + (rV − P)E (15) 2 where p¯ is the payout per unit time from the firm to the equity. The equation must be solved under the following boundary and terminal conditions: E(0, T ) = 0.

(15a)

Condition (15a) shows that the equity is worthless when the firm is worthless. E(V, T )/V → 1,

as V → ∞.

(15b)

Condition (15b) shows that for very high values of the firm, the value of the equity approaches the value of the firm. E(V, 0) = Max(V − B, 0).

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Condition (15c) gives the value of equity on the debt’s maturity date. When the firm does not pay dividends, p¯ = 0, the debt corresponds to a zerocoupon bond, P¯ = 0, the equity valuation problem in Eq. (15) is equivalent to the valuation of the Black–Scholes call in Eq. (13). The value of equity is given by Eq. (14) where S is replaced by V , X by B, T corresponds to the maturity of the debt and σ 2 is the variance of the return to the firm. Using the same replication argument as in Merton (1974), it is possible to show that the value of risky debt must satisfy the following PDE: 1 2 2 σ V DVV + rVDV − DT − rD = 0 2

(16)

since the firm does not make payouts P¯ = 0, and the debt is a zero-coupon bond. The boundary and terminal conditions are as follows: D(0, T ) = 0.

(16a)

Condition (16a) shows that the debt is worthless when the firm is worthless. D(V, T )/V → Be−rt

as V → ∞.

(16b)

Condition (16b) shows that for very high values of the firm, the value of the risky debt approaches that of a riskless bond, Be−rT . D(V, 0) = Min(V, B).

(16c)

Condition (16c) corresponds to the value of debt at its maturity date. Merton (1974) gives the following solution for the debt valuation problem given by Eqs. (16), (16a), (16b) and (16c): D(V, T, B) = Be−rtN(h1 ) + VN(h2 )

(17)


 log(V/B) + r − 21 σ 2 T h1 = √ σ T
 log(B/V) + r − 21 σ 2 T . h2 = √ σ T This solution shows the following properties: — The higher the value of the firm V , the higher the value of the risky debt D.

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— The higher the promised principal, B, the higher the value of the risky debt D. — The later the maturity of the debt T , the lower the value of the risky debt D. — The higher the volatility of the firm value σ 2 , the lower the value of the risky debt D. — The higher the risk-free rate, r, the lower the value of the risky debt D. The CCA approach allows to handle more complex problems. For example, consider a firm financed by equity (receiving dividends) and a callable coupon bond. Let us denote the callable coupon bond by F(V, T, B). As in Merton (1974), the value of this callable bond must satisfy the following system: 1 2 2 σ V FVV + (rV − c¯ − d)FV − FT − rF + c¯ = 0 2

(18)

F(0, T ) = 0

(18a)

F(V¯ (T ), T ) = K(T )

(18b)

F(V, 0) = Min(V, B)

(18c)

where P¯ = c¯ + d, p¯ = c¯ , where c¯ is the coupon on the debt and d stands for the dollar dividend to the equity. Condition (18b) shows the existence of a schedule of firm values, V¯ (T ) at or above which it is optimal for the firm to call the debt using a call price K(T ) with K(0) = B. The solution gives the value of the debt and the value V¯ (T ). Another application of CCA to complex securities concerns the valuation of callable convertible debt, H(V, T, X). The analysis in Ingersoll (1976) and Brennan and Schwartz (1977) shows that the partial differential equation is similar to (18), (18a), (18b), (18c) except that conditions (18b) and (18c) are replaced by: H(V¯ (T )T ) = γ V¯ (T )

(18b )

H(V, 0 ) = Min(V, Max)(B, γV)

(18c )

where γ corresponds to the fraction of the equity to be held by owners of convertible debt if all of the bonds are converted. The solution gives the value of the convertible bond and the firm schedule V¯ (T ) at or above which

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it is optimal to call the convertibles and force conversion. Ingersoll (1976) shows that when d = 0, CCA implies that: V¯ (T ) = K(T )/γ.

(19)

This shows that a convertible bond should be called at the moment the bond’s common value equals the call price. A major obstacle in applying CCA to corporate financial problems concerns the observability of the firm’s value. If all of a firm’s liabilities do not trade and there is a traded asset which moves closely with the unobserved firm value, then the variance can be estimated from this traded asset. The approach needs that only one asset of the firm is traded. Otherwise, if the firm value is unobservable, there are no traded assets correlated with the firm value, but the firm has at least two traded liabilities, then a firm’s value and variance rate can be inferred from the model. Traditional capital budgeting procedures assume that a project will operate each year during its lifetime. However, for some projects, it may not be optimal to operate each year because for some time revenue is not expected to cover variable cost. In addition to the option to temporarily shut down the project, management has the option to terminate it. Myers and Majd (1983) present a quantitative analysis of the option to abandon. Myers (1977) shows the importance of CCA in the evaluation of a firm’s “growth opportunities”. Again, the analysis in the previous sections can be extended to account for the effects of information uncertainty.

Summary Discounted cash flow methods seem to cause systematic undervaluation of projects since the strategic value of projects is ignored. The flexibility of a project seems to be a description of the options made available to management as part of the project. These options are “operating” options. An example is given by the choice between building a power plant that burns only oil and one that can burn either oil or coal. The latter solution provides more flexibility and costs more. It also provides management with the option to select which fuel to use and can switch back and forth in response to energy market conditions. The valuation equations of CCA are derived from arbitrage arguments using traded securities. The main question is the validity of these equations for capital budgeting projects which are not traded. The objective of capital budgeting is the estimation of the price

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of a project (or an asset) if it were traded. Since the absence of arbitrage is a necessary condition for equilibrium prices, the no arbitrage price of an option on a traded asset must be the equilibrium price of an option on a corresponding nontraded project. According to Merton (1998): “CCA will become as important and common place a tool for capital budgeting decisions in the future as it is for financial market decisions in the present”.

Questions 1. 2. 3. 4.

What are corporate liabilities? How can corporate liabilities be identified as options? How can corporate liabilities be valued? What are the main difficulties in the applications of CCA?

Bibliography Bellalah, M (1999). The valuation of futures and commodity options with information costs. Journal of Futures Markets, September. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659. Brennan, MJ and E Schwartz (1977). Saving bonds, retractable bonds, and callable bonds. Journal of Financial Economics, 5, 67–88. Ingersoll, JE (1976). A theoretical and empirical investigation of the dual purpose funds: an application of contingent claims analysis. Journal of Financial Economics, 3(1/2), 83–123. Merton, RC (1974). On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance, 19(2), 449–470. Merton, RC (1977). On the pricing of contingent claims and the Modigliani–Miller Theorem. Journal of Financial Economics, 5(2), 241–250. Merton, RC (1998). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141–183. Myers, SC (1977). Determinants of corporate borrowing. Journal of Financial Economics, 5(2), 147–175. Myers, SC and S Majd (1983). Calculating abandonment value using option pricing theory. Working Paper 1462-83, Massachusetts Institute of Technology, Sloan School of Management.

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Chapter 16

Extended Discounted Cash Flow Techniques and Real Options Analysis within Information Uncertainty This chapter is organized as follows: 1. Section 2 presents a framework for the valuation of the firm in the presence of information costs. Using Merton’s (1987) model, we extend the analysis in Modigliani–Miller (1958, 1963) to account for the effects of incomplete information. We extend the concepts of economic value added and standard DCF analysis. 2. Section 3 uses the main results in the real option literature to make the standard analogy between financial and real options. This allows the presentation of the main applications of the real option pricing theory. 3. Section 4 develops a context for the pricing of real options in a continuous-time setting. We provide some analytic formulas for the pricing of standard and complex European and American commodity options in the presence of information costs. We extend the results in Lint and Pennings (1998) for the pricing of the option on market introduction. 4. Section 5 develops some models for the pricing of real options in a discrete-time setting by accounting for the role of shadow costs of incomplete information. We first extend the Cox, Ross and Rubinstein (1979) model to account for managerial flexibility and the option to abandon. Then, we use the generalization in Trigeorgis (1990) for the pricing of several complex investment opportunities with embedded real options to account for the effects of information costs. Most of the models presented in this chapter can be applied to the valuation of biotechnology projects and investments with several stages. Bellalah, Mondher. Exotic Derivatives and Risk: Theory, Extensions and550 Applications, World Scientific Publishing Co 12:16:42.

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1. Introduction

T

he standard literature on capital budgeting techniques uses the net present value as a reference criteria in investment decisions. The analysis is mainly based on the use of the cost of capital in the discounting of future cash flows. A project is accepted if its extended Net Present Value, NPV is positive, otherwise it is rejected. The extended NPV corresponds to the standard NPV plus the flexibility in investment decisions. The standard technique for calculating the NPV has not changed much since Fisher (1907) by discounting the expected cash flow at an appropriate discount rate. The research in this area is based on the specification and estimation of the discount rate. Over the last two decades, a body of academic research takes the methodology used in financial option pricing and applies it to real options in what is well known as real options theory. This approach recognizes the importance of flexibility in business activities. Today, options are worth more than ever because of the new realities of the actual economy: information intensity, instantaneous communications, high volatility, etc. The literature on real options and discounted cash flow techniques ignores the role of information uncertainty. However, these costs play a central role in financial markets and capital budgeting decisions. Financial models based on complete information might be inadequate to capture the complexity of rationality in action. Some factors and constraints, like entry into a business are not costless and may influence the short run behavior of asset prices. The treatment of information and its associated costs play a central role in capital markets. If an investor does not know about a trading opportunity, he will not act to implement an appropriate strategy to benefit from it. However, the investor must determine if potential gains are sufficient to warrant the costs of implementing the strategy. These costs include time and expenses required to create data base to support the strategy, to build models and to get informed about the technology. This argument applies in varying degrees to the adoption in practice of new structural models of evaluation. This reasoning holds not only for individual investors but also for professional managers who spend resources and time in the same spirit. It is also valid for the elaboration and implementation of option pricing models. Hence, recognition of information costs might be important in asset valuation and has the potential to explain empirical biases exhibited by prices computed from complete information models. As shown in Merton (1987), the “true” discounting rate for future risky cash flows must be coherent with his simple model of capital market equilibrium with incomplete information. This

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model can be used in the valuation of real assets. Nowadays, a rich set of criteria is used to recognize the companies real options. Consultants look beyond traditional financial analysis techniques to get reasonable guidelines in investment practices. Actual decision making in firms resort to real options. The value of the firm can have two components: the value of the existing projects and the value of the options hold by the firm to do other things. The use of standard option valuation techniques in the valuation of real assets is based on some important assumptions. Managers are interested not only in real options, but also in the latest outgrowth in DCF analysis; the Economic Value Added. EVA simply means that the company is earning more than its cost of capital on its projects. EVA is powerful in focusing senior management attention on shareholder value. Its main message concerns whether the company is earning more than the cost of capital. It says nothing about the future and on the way the companies can capitalize on different contingencies. Hence, a useful criterion must account for both the DCF analysis and real options. The NPV and the EP (economic profit) ignore the complex decision process in capital investment. In fact, business decisions are in general implemented through deferral, abandonment, expansion or in series of stages. This chapter presents a survey of some results regarding the standard discounted cash flow techniques, the economic value added and real options. Since the standard literature ignores the role of market frictions and the effect of incomplete information, we rely on Merton’s (1987) model of capital market equilibrium with incomplete information to introduce information costs in the pricing of real assets. Using this model instead of the standard CAPM allows a new definition of the weighted average cost of capital and offers some new tools to compute the value of the firm and its assets in the presence of information uncertainty. Using the methodology in Bellalah (2001a,b) for the pricing of real options, we propose some new results by extending the standard models to account for shadow costs of incomplete information. The extended models can be used for the valuation of R&D projects as well as projects with several stages like joint ventures.

2. The Cost of Capital, the Value of the Firm and Its Investment Opportunities in the Presence of Shadow Costs of Incomplete Information The standard analysis in corporate investments needs the projection of the project’s cash flows and then to perform an NPV analysis. The discount 12:16:42.

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rate is set with regard to the risk of the project. The riskier the project, the higher the manager sets the discount rate. This approach leads to a real bias toward projects that produce return in the short run. In fact, the more distant the payoff horizon, the more uncertainty enters the game so that even huge pay back opportunities, if long term, tend to be discounted away. The NPV analysis obliges managers to compute present values of their investments as if they have engaged all the costs. This standard approach ignores the presence of information costs. However, information plays a central role in the valuation of financial assets and must be accounted for in the valuation process. Merton (1987) presents a simple context to account for information costs.

2.1. The Cost of Capital and the Value of the Firm Value within Information Uncertainty The cost of capital or the weighted average cost of capital, (WACC), is a central concept in corporate finance. It is used in the computation of the net present value, NPV, and in the discounting of future risky streams. The standard analysis in Modigliani–Miller (1958, 1963) ignores the presence of market frictions and assumes that information is costless. Or, as it is well known in practice, information costs represent a significant component in the determination of returns from investments in financial and real assets. Merton (1987) provides a simple context to account for these costs by discounting future risky cash flows at a rate that accounts for these costs. In this context, the cost of capital and the firm’s value can be computed in an economy similar to that in Merton (1987). We denote, respectively, by: D: B: S: O: τ: Vu : V: kd : kb : ke :

the face value of debt, the market value of debt, the market value of equity, perpetual operating earnings, the corporate tax rate, the value of the unlevered firm, the value of the levered firm, the cost of debt, the current market yield on the debt, the cost of equity or the required return for levered equity,

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ko : the market value-weighted of these components known as the WACC, ρ: the market cost of equity for an unlevered firm in the presence of incomplete information. Using the main results in the Modigliani and Miller analysis and Merton’s λ, it is clear that discounting factors must account for the shadow cost of information regarding the firm and its assets. By adding Merton’s λ in the analysis of Modigliani–Miller in the discounting of the different streams of cash-flows for levered and unlevered firms, similar very simple formulas can be derived in an extended Modigliani–Miller–Merton context. The formulas follow directly from the analysis in Modigliani–Miller and the fact that future risky streams must be discounted at a rate that accounts for Merton’s λ. Table 1 presents the main results regarding the components of the costs of capital and the values of the levered and unlevered firms with information costs. These results show the components of cost of capital and the values of the firms in the presence of information costs. When these costs are equal to zero, this table is equivalent to the results in the Modigliani–Miller analysis. Table 1: Summary of the main results regarding the components of the costs of capital and the values of the levered and unlevered firms with information uncertainty with kb
= kb + λd and kd
= kd + λd . No tax ρ = SO + λu

Corporate tax
 ρ = SO + λu (1 − τ)

k
B = D kd
b [O−kd
D] ke = S

k
B = D kd
b [(O−kd
D)(1−τ)] ke = S

u


ke = ρ + B S (ρ − kb )

u

 
ρ+ B S (ρ − kb ) (1 − τ)

Vu = O ρ = Su V = Vu

Vu = (1 − τ) O ρ

ko = ke VS + kb
VB ko = O V

ko = ke VS + kb
(1 − τ) VB ko = O V (1 − τ)
 ko = ρ 1 − τ VB

ko = ρ

V = Vu + τB

The term λd indicates the information cost for the debt and the term λu corresponds to the information cost for the unlevered firm.

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Table 2: Summary of the main results regarding the components of the costs of capital and the values of the levered and unlevered firms with information costs: the standard case. No tax

Corporate tax

ρ = 10% B = 10,000 ke = 15% ke = 15% Vu = 20,000 V = 20,000 S = 10,000 ko = 10% ko = 10% ko = 10%

ρ = 10% B = 10,000 ke = 15% ke = 15% Vu = 12,000 V = 16,000 S = 6,000 ko = 7.5% ko = 7.50% ko = 7.5%

O = 2,000, D = 10,000, B = 10,000, S = 10,000, V = 20,000, τ = 40%, ρ = 10% and kd = 5%, λu = 0%, λd = 0%.

The results show how to calculate the firm’s value, the weighted average cost of capital, and the net present value of future risky cash flows in the presence of information costs. The above formulas are simulated for an illustrative purpose using: O = 2,000, D = 10,000, B = 10,000, S = 10,000, V = 20,000, τ = 40%, ρ = 10% and kd = 5%, λu = 0%, λd = 0%. These figures represent the standard benchmark case. The simulations allow to appreciate the impact of information costs on the computation of the different values of the levered and unlevered firm and the costs of capital with corporate taxes. The fact that ke is equal to 15% in this case is consistent with the MM assumptions. The effect of incomplete information on the firm value and the cost of capital is simulated using the following data: O = 2,000, D = 10,000, B = 10,000, S = 10,000, V = 20,000, τ = 40%, ρ = 10%, kd = 5%, λu = 3% and λd = 1%. The value of ke is equal to 26% in this case. Every scenario is consistent with the Modigliani–Miller assumptions and the Merton’s shadow cost (λ). When compared to the benchmark case with no information costs, we see that information costs increase significantly ke . These shadow costs reduce the value of the firm in the two cases: with no tax and with corporate tax.

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Table 3: The main results for the cost of capital and the values of the levered and unlevered firms with information costs. No tax

Corporate tax

ρ = 13% B = 10,000 ke = 26% ke = 26% Vu = 15384.62 V = 15384.62 S = 5384.62 ko = 13% ko = 13% ko = 13%

ρ = 13% B = 10,000 ke = 26% ke = 26% Vu = 9230.77 V = 13230.77 S = 3230.77 ko = 9.07% ko = 9.07% ko = 9.07%

O = 2,000, D = 10,000, B = 10,000, S = 10,000, V = 20,000, τ = 40%, ρ = 10%, kd = 5%, λu = 3% and λd = 1%.

2.2. Application to a Biotechnology Firm For a biotechnology firm, the development of a drug needs several stages: discovery, pre-clinical, Phase I clinical trials, Phase II clinical trials, Phase III clinical trials, submission for review and post approval. We show how to apply Merton’s (1987) model of capital market equilibrium with incomplete information for the computation of the cost of capital, the expected net present value (ENPV) in the decision tree method. Following the analysis in Kellogg and Charnes (1999), we will generalize their decision-tree method and the application of the binomial model to account for shadow costs of incomplete information. A model is constructed to compute the expected net present value (ENPV) without accounting for growth options. The ENPV can be computed in the presence of information costs. In the decision tree method, the ENPV is computed as: ENPV =

7 

ρi

i=1

T 5   qj CCFj,t DCFit + ρ 7 t (1 + rd ) (1 + rc )t t=1 j=1

where: i = 1, . . . , 7: an index of the 7 stages in the project, ρi : the probability that stage i is the end stage for product i, T : the time at which all future cash flows become zero,

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DCFit : the expected development stage cash flow at time t given that stage i is the end stage, rd : the discount rate for development cash flows, j = 1 to 5: an index of quality for the product, qj : the probability that the product is of quality j, and rc : the discount rate for commercialization cash flows. The discounting rates rd and rc can be estimated using Merton’s CAPMI as in Bellalah (2000, 2001b). This method is easy to implement and accounts for the effects of information costs in project valuation.

2.3. Economic Value Added, EVA, and Information Costs In standard financial theory, every company’s ultimate aim is to maximize shareholders’ wealth. The maximization of value is equivalent to the maximization of long-term yield on shareholders’ investment. Currently, EVA is the most popular value based measure. A manager accepts a project with positive NPV; i.e., for which the internal rate of return (IRR) is higher than the cost of capital. With practical performance measuring, the rate of return to capital is used because the IRR cannot be measured. However, the accounting rate of return is not an accurate estimate of the true rate of return. As shown in several studies, ROI underestimates the IRR in the beginning of the period and overestimates it at the end. This phenomenon is known as wrong periodizing. The EVA valuation technique provides the true value of the firm regardless of how the accounting is done. The EVA is simply a modified version of the standard DCF analysis in a context where all of the adjustments in the EVA to the DCF must result net to zero. EVA can be superior to accounting profits in the measurement of value creation. In fact, EVA recognizes the cost of capital and, the riskiness of the company. Maximizing EVA can be set as a target while maximizing an accounting profit or accounting rate of return can lead to an undesired outcome. The weighted average cost of capital, WACC, is computed using Merton’s (1987) model of capital market equilibrium with incomplete information for the cost of equity component. The WACC is computed as in Table 1. Stewart (1990) defines the EVA as the difference between the net operating profit after taxes (NOPAT) and the cost of capital. EVA gives the 12:16:42.

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same results as the discounted cash flow techniques or the net present value (NPV). It can be described by one of the three equivalent formulas: EVA = NOPAT − cost of capital × (capital employed) or EVA = rate of return − cost of capital × (capital employed) or EVA = (ROI − WACC) capital employed with: Rate of return = NOPAT/capital, Capital = total balance sheet − non-interest bearing debt at the beginning of the year, ROI = the return on investment after taxes, i.e., an accounting rate of return. The cost of capital is the WACC as in the Modigliani–Miller analysis where the cost of equity is defined with respect to the CAPM of Sharpe (1964), Lintner (1965) and Mossin (1966). In the presence of information costs, the cost of capital can be determined in the context of Merton’s model of capital market equilibrium as described above. In this case, the above formulas must be used. Hence, the analysis in Stewart (1990) can be extended using the CAPMI of Merton (1987) rather than the standard CAPM in the computation of EVA. In the presence of taxes, EVA can also be calculated as: EVA = [NOP − ((NOP − excess depreciation − other increase in reserves) × (tax rate))] − WACC × (capital) where NOP is the net operating profit. Stewart (1990) defines the market value added, MVA, as the difference between a company’s market and book values: MVA = total market asset value − capital invested. 12:16:42.

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When the book and the market values of debt are equal, MVA can be written as: MVA = market value of equity − book value of equity. The MVA can also be defined as: MVA = the present value of all future EVA. Using the above definitions, it is evident that: market value of equity = book value of equity + present value of all future EVA. In this context, this formula is always equivalent to discounted cash flow and net present value.Again, the cost of capital with information costs represents an appropriate rate for the discounting of all the future EVA. Hence, the main concepts in Stewart (1990) can be extended without difficulties to account for the shadow costs of incomplete information in the spirit of Merton’s model.

3. From Financial Options to Real Options: Some Standard Applications Managers recognize that the NPV analysis is incomplete and shortsighted. This analysis ensures in theory perpetual profitability for a company. The NPV fails because it assumes the decision to invest in a project is all or nothing. Hence, it ignores the presence of many incremental points in a project where the option exists to go forward or abort. For a survey of the literature on real options, the reader can refer to Trigeorgis (1990, 1993a,b,c, 1995, 1996), Pindyck (1991), Paddock, Siegel and Smith (1988), Newton (1996), Myers (1984), McDonald and Siegel (1984, 1986), Myers and Majd (1990) among others. Realistic view of the capital budgeting process portrays projects as a sequence of options. For a review of the main results in this literature, the reader can refer to Luehrman (1997, 1998), Baghay et al. (1996), Carr (1988), etc. Real option valuation maps out the possibilities available to a company, including those not readily apparent in the decision tree. By varying the discount rate through the tree, it accounts for the relative level of risk for different cash flows. Real option valuation can also identify the optimal course of the company at each stage in the process.

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3.1. The Standard Analogy Between Financial and Real Options There is a well established analogy between financial options and corporate investments that lead to future opportunities. It is evident for a manager why investing today in research and development or in a new marketing program can lead to a possibility of new markets in the future. Dixit (1992, 1995) and Dixit and Pindyck (1994) suggest that option theory provides helpful explanations since the goal of the investments is to reveal information about technological possibilities, production costs or market potential. Consider, for example, a generic investment opportunity or a capital budgeting project to see the analogy with financial options. The difficult task lies in mapping a project onto an option. A corporate investment opportunity looks like a call because the firm has the right but not the obligation to acquire a given underlying (the operating assets of a project or a new business). If the manager finds a call option in the market similar to the investment opportunity, then the value of that option can give him information about the value of the investment opportunity. Using this analogy between financial options and real options allows one to know more about the project. This approach is more interesting than the standard discounting cash flow techniques DCT. For a review of the main results in this literature, the reader can refer to Baldwin and Ruback (1986), Dentskevich and Salkin (1991), Ingersoll and Ross (1992), etc. The option implicit in the project (the real option) and the NPV without the option are easily compared when the project can no longer be delayed. Opportunities may be thought of as possible future operations. When a manager decides how much to spend on R&D, or on which kind of research and development R&D, he is valuing real opportunities. The crucial decision to invest or not will be made in general after some uncertainty is resolved or when time runs out. An opportunity is analogous to an option. Option pricing models contain parameters to capture information about cash, time, and risk. The theory handles simple contingencies better than standard DCF models. The reader can refer to Kogut (1991), Kogut and Kulatilaka (1994a,b), McDonald and Siegel (1984, 1986) among others. A real option confers flexibilities to its holder and can be economically important. Paddock, Siegel and Smith (1988) and Berger, Ofek and Swary (1996) show that the value of a firm is the combined value of the assets already in use and the present value of the future investment opportunities. There are several situations that lead to real options in different sectors in the

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economy. Most of these options appear in Dentskevich and Salkin (1991), Dixit (1992, 1995), Faulkner (1996) among others.

3.2. Standard and Complex Real Options and Their Applications: Some Examples If you consider the example of high-tech start-up companies, these firms are valued mainly for their real options rather than their existing projects. The market recognizes today the value of these options. While standard options are easily identified, it is more difficult to identify compound and learning options. Compound options generate other options among exercise. These options involve sequenced or staged investments. When a manager makes an initial investment, he has the right to make a second investment, which in turn gives the right to make a third investment, and so on. Learning options allow the manager to pay to learn about an uncertain technology or system. Staged investments give managers the right to abandon or scale up projects, to expand into new geographic areas and investing in research and development. The main element in the determination of profitability in certain cyclical activities is the ability of timing a business cycle to build for example a new factory. The manager does not have to commit himself outright to a new factory. He has the option of staging the investment over a given period by paying a certain amount up front for design, another amount in a period for pre-construction work and another outlay to complete construction at the end of the year. This gives him the flexibility to walk away if profit projections fall below a given level or to abandon at the end of the initial construction phase and save a given additional outlay. The factory is designed to convert an input into an output and its profitability would be a function of the spread between these prices. The manager can invest in new factories only when the input/output spread is higher than its long-term average. The NPV assumes that the factory is built and operated, ignoring the flexibility offered to managers. A first example of compound options can be found in a staged investment, which may be assimilated to a sequence of stages where each stage is contingent on the completion of its predecessor. This is the case for a company seeking to expand in foreign markets. The firm might start in a single territory. It can then learn and modify the specific features of its products. The first experience enables the firm to expand into similar overseas markets. However, the manager must weigh the value of

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the option to expand cautiously against the potential costs of coming second in some or all of these markets. This situation corresponds also to joint ventures and the valuation of joint ventures and biotechnology products where each stage is contingent on the subsequent stages. A second example is given in the market for corporate control and acquisitions. A sequence of acquisitions represents a staged series of investments and can be assimilated to compound options. Real options can be used in this context to value all possible contingencies. In this case, the literature regarding exotic options can be applied to value the different real options. A third example corresponds to mining companies. Mining companies must often give an answer to the following question: when to develop the properties they own and how much to bid for the right to implement additional properties. These decisions refer to a combination of options: the option to learn about the quantity of ore and the option to defer the development waiting for favorable prices. In general, learning options appear when a company has the possibility to speed up the arrival of information by making an investment. Real option theory can be used to determine the optimal time to exercise the option. When the company does not know the quantity of ore in its mine, it has a learning option: to pay money to find out. Here also, the main models for the pricing of exotic options can be applied. A fourth example is given for the development of a natural gas field (compound rainbow options) Combinations of learning options and rainbow options can arise for some firms.a A fifth example is given by R&D in pharmaceuticals (Rainbow options) projects in R&D combine learning and compound options. R&D projects contain both technological and product uncertainties. Consider a pharmaceutical company ranking different R&D projects in order of priority. The real option approach handles both uncertainties. R&D projects can be classified as compound rainbow options, each contingent on the preceding options and on multiple sources of uncertainty (rainbow options and multi factor options). In this context, the models for the pricing of exotic options can be applied.

a Consider a company deciding on how much production capacity to install in an undeveloped natural

gas field. The company can create a decision tree for a real option valuation (ROV) model to weigh up the various decisions in view of the uncertainty regarding the price and quantity. Using the information regarding the volatility of gas prices and quantity, the ROV model can estimate the total value of the different courses open to the company. The reader can refer to the work of Brennan (1991), Brennan and Schwartz (1985), Pickles and Smith (1993), etc.

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4. The Valuation of Real Options with Information Costs in a Continuous-Time Setting Several models in financial economics are proposed to deal with the ability to delay an irreversible investment expenditure. These models undermine the theoretical foundation of standard neoclassical investment models and invalidate the net present value criteria in investment choice under uncertainty. For a survey of this literature, the reader can refer to Pindyck (1991) and the references therein. Before presenting some models for the valuation of real options in a continuous time setting, we present the general context for the valuation of financial options with information costs. We first present the valuation of simple options then a formula for the valuation of compound options.

4.1. The Valuation of Simple European and American Commodity Options with Information Costs Following Black (1976), we assume that all the parameters of the Merton’s (1987) CAPMI are constant through time. Under these assumptions, the value of the commodity option, C(S, t), can be written as a function of the underlying price and time. In this context, the valuation equation is given by 1 2 2 σ S CSS + (b + λS )SCS − (r + λC )C + Ct = 0. 2

(1)

This equation appears in Bellalah (1999) for the pricing of commodity options. When the information costs regarding the underlying asset and the option (λS , λC ) are set equal to zero, this equation collapses to that in Barone-Adesi and Whaley (1987). The value of a European commodity call is: C(S, T ) = Se((b−r−(λC −λS ))T)N(d1 ) − Ke−(r+λC )TN(d2 ) with:



    √ 1 2 S + b + σ + λS T σ T, d1 = ln K 2 √ d2 = d1 − σ T

and where N(.) is the cumulative normal density function.

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When λS and λC are equal to zero and b = r, this formula is the same as that in Black and Scholes. A direct application of the approach in BaroneAdesi and Whaley (1987), allows to write down immediately the formulas for American commodity options with information costs. In this context, the American commodity option value CA (S, T ) is given by: CA (S, T ) = C(S, T ) + A2 (S/S ∗ )q2

when S < S ∗

CA (S, T ) = S − K

when S ≥ S ∗

with: S∗ (1 − e(b+λS −r−λC )TN(d1 (S ∗ ))) (3) q2  1 = 2 (−(N − 1) + (N − 1)2 + 4 Mk ), N = 2(r + λC )/σ 2 , M = A2 =

q2

2(b + λS )/σ 2 , k = 1 − e−(r+λC )T . The critical underlying commodity price is given by an iterative procedure from the following equation:   ∗ S (1 − e(b+λS −r−λC )TN(d1 (S ∗ ))) S ∗ − K = C(S ∗ , T ) + . (4) q2 In the same context, the American commodity option put value PA (S, T ) is given by: PA (S, T ) = P(S, T ) + A1 (S/S ∗ )q1 PA (S, T ) = K − S

when S > S ∗∗ when S ≤ S ∗∗

with: −S ∗∗ (1 − e(b+λS −r−λC )TN(−d1 (S ∗∗ ))) (5) q1  = 21 (−(N − 1) − (N − 1)2 + 4 Mk ), N = 2(r + λC )/σ 2 , M = A1 =

q1

2(b + λS )/σ 2 , k = 1 − e−(r+λC )T . The critical underlying commodity price is given by an iterative procedure from the following equation: K − S ∗ = P(S ∗ , T ) −

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(6)

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A similar algorithm as the one developed in Barone-Adesi and Whaley (1987) can be used to determine the critical underlying asset price. The above formulas can be applied to the valuation of several real options embedded in project valuation. In particular, the formulas can be applied for the pricing of European and American call and put options in the presence of a continuous dividend stream. The advantages of these formulas over many formulas for American options is the speed of computation since this analytic approximation is faster than numerical methods and the lattice approaches. These formulas can be used in the valuation of complex projects as those described in Trigeorgis (1990).

4.2. The Valuation of Compound Options within Information Costs Several projects are often valued using the concept of compound options introduced by Geske (1979). For example, the development process for a new product requires several stages where the manager resorts to the new information revealed up to that point to decide whether to abandon or to continue the project. This is particularly the case for a biotechnology firm for which the development of a drug needs several stages. The idea is that engaging in the development phase is equivalent to buying a call on the value of a subsequent product. Hence, there is the initial option and the growth option. In the presence of only two stages a formula for a call on a call can be used. We show how to value compound options in the presence of information costs. For the sake of simplicity, we use the general context proposed by Geske (1979). Investors suffer sunk costs to obtain information about the equity and the assets of the firm. The costs regarding the equity and the firm’s cash-flows reflect the agency costs and the asymmetric information costs. These costs characterize also joint ventures. In this situation, the formula is given by:

√

√

C0 = V0 e−(λc −λv )TN2 h + σv t, k + σv T , − Me−(r+λc )TN2 h, k,

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t T

t T



 − Ke−(r+λc )tN(h).

(7)

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The value V¯ is determined by the following equation: √ St − K = V¯ e−(λc −λv )(T −t) N(k + σv T − t) − Me−(r+λc )(T −t) N(k) − K = 0 with:

    

√ V 1 2 h = ln + r + λv − σv t σv t ¯ 2 V 

    √ 1 2 V + r + λv − σv T σv T . k = ln M 2

If the information cost is zero, this compound option pricing formula becomes that in Geske (1979). This formula is also useful for the valuation of real options in the presence of information costs. Table 4 provides the simulation results for the compound option formula with information costs and the Geske’s compound call formula using the following parameters: K = 20, M = 100, r = 0.08, T = 0.25, t = 0.125, σv = 0.4. The parameters used for information costs are: case a: case b: case c: case d:

(λc (λc (λc (λc

= λv = 0%), = λv = 2%), = 1%, λv = 2%), = 1%, λv = 2%).

In case (a), we have exactly the same values as those generated by the formula in Geske (1979). The table shows that the compound option price is an increasing function of the firm’s or the project’s assets. This result is Table 4: Simulation of equity values as compound options in the presence of information costs using our model for the following parameters. The following parameters are used: K = 20, M = 100, r = 0.08, T = 0.25, t = 0.125, σv = 0.4. C0

λc = 0% λv = 0%

λc = 2% λv = 2%

λc = 1% λv = 2%

λc = 1% λv = 2%

110 120 130

6.82 15.17 26.52

7.13 15.65 27.16

7.16 15.70 27.25

7.14 15.67 27.20

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independent of the values attributed to information costs. The compound option price is an increasing function of the information costs regarding the firm’s assets, λv . When λv is fixed, this allows the study of the effects of the other information costs on the option value. In this case, the option price seems to be a decreasing function of the information cost λc . We intend to test this model on real data.

4.3. Investment Timing, Project Valuation and the Pricing of Real Assets with Compound Options within Information Uncertainty The timing option gives the right to the manager to choose the most advantageous moment to implement the investment project and allows him to pull out of the project when the economic environment turns out to be unfavorable. Several standard models are proposed in the literature for the pricing of these options. Lee (1988) proposes a model for the valuation of the timing option arising from the uncertainty of the project value and for the detection of the optimal timing. He considers three cases: the optimal timing of plant and equipment replacement, the real estate development and the marketing of a new product. The investment project is interpreted as the replacement of a capital asset, the inauguration of a new product and the development of real estate. The manager has the option to implement the project in the time interval [0, T ] where T is the option’s maturity. The possibility to implement an investment project in [0, T ] can be seen as an American call option on a security with no dividend payments. In the presence of information costs, our formula for the valuation of American options can be used to price options in this context. Let us denote by: V : the present value of the project implemented, S: the present value of the project not yet implemented, I: the cost of the project, D: a known anticipated jump in the project’s value, C(S, 0, T, I): an American call without dividend where 0 refers to the starting time, c(S, 0, T, I): a European call option, and PTi (0, T ): the value of timing option.

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The value of PTi (0, T ) corresponds to the difference between the value of the deferrable investment opportunity when the timing option is “alive” and when the timing option is “dead”. The project’s value if it is implemented now is: C(S, 0, 0, I ) = Max[V − I, 0] where the NPV of the implemented investment opportunity is (V − I). In this case, the timing option value is given by: PTi (0, T ) = C(S, 0, T, I ) − C(S, 0, 0, I ) PTi (0, T ) = Min[C(S, 0, T, I ), C(S, 0, T, I ) − (V − I )] ≥ 0.

(8)

This equation shows that it is profitable to implement the project now (V − I > 0) when the value of the timing option is equal to the value of the deferrable investment opportunity minus NPV. The cost of waiting D can be seen as a dividend in the pricing of American call options. It is possible to study three different specifications. Specification 1 (i) The present value changes of the not-yet-implemented project is: dS/S = µ dt + σ dz. (ii) If the project is implemented before t ∗ , it generates an extra cash-flow at t ∗ : Vt∗ = St ∗ + D.

(9)

This specification corresponds, for example, to the real estate development. In fact, leaving property vacant can be seen as holding a timing option on the real estate development. The cost of development is I. Specification 2 Same as (i) of specification 1. The cost of the project increases by D when implemented after t ∗ : It∗+h = I, Xt ∗ −h = I − D,

for all h > 0 for h > 0.

(10)

It is possible to use the formula in Whaley (1981) to compute the value of the optimal timing option and the optimal timing of project implementation.

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It is possible to show that the value of an American call in the presence of a cash discrete dividend and information costs is given by: 

 t∗ ∗ ∗ C = S e((b−r−(λC −λS ))t )N(b1 ) + e((b−r−(λC −λS ))t )N2 a1 , −b1 , − T 

 t∗ −(r+λC )t ∗ −(r+λC )t ∗ −I e N(b2 ) + e N2 a2 , −b2 , − T ∗

+ De−(r+λC )t N(b2 ) with:

(11)



  √ 1 a1 = ln(S/I) + b + σ 2 + λS t ∗ σ t∗ 2 √ a2 = a1 − σ t ∗   

√ 1 2 b1 = ln(S/Scr,t ∗ ) + b + σ + λS ti σ t∗ 2 √ b2 = b1 − σ t ∗

where Scr,t ∗ corresponds to the trigger point present value, N(.) stands for the cumulative normal distribution and N2 (., ., ) is the bivariate cumulative normal density function with upper integral limits a and b and a correlation coefficient ρ. The “trigger point” for specification 1 is given by: PTi (t ∗ , T ) = c(Scr,t ∗ , t ∗ , T, I ) − (Scr,t ∗ + D − I ) = 0.

(12)

The trigger-point value for specification 2 is given by a formula identical to Eq. (17). This case fits well with the replacement of plant and equipment. If we denote by S, I and T the present value, the cost of replacement and the remaining life, then a firm keeping the equipment in operation will face expenditures at time t ∗ of amount D. In this case, formula (11) can be applied to compute the value of the timing option and trigger point present value. These two specifications allow a single occurrence of discrete cash flow at time t ∗ . It is possible to generalize the results using specification 3. Formula (11) is simulated in Tables 5–7. The parameters are S = 175, D = 1.5, r = 0.1 and the constant “carrying cost” is 0.6. We use different values for the information costs λS and λC . The option has a maturity date of one month. The volatility is σ = 0.32 and the “dividend” is paid in 24 days. Table 5 uses these parameters with no information costs. It gives the

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Table 5: Simulations of option values for the continuous-time model using the following parameters: S = 175, r = 0.1, D = 1.5, T = 30, t = 24, σ = 0.32, λc = 0, λs = 0. Strike 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 240

Call

ca

cb

cc

ca + cb − cc

S∗

76.03 71.06 66.09 61.13 56.16 51.19 46.22 41.26 36.30 31.37 26.50 21.78 17.31 13.25 9.72 6.82 4.56 2.91 1.77 1.03 0.57 0.00

74.42 69.46 64.51 59.55 54.59 49.63 44.67 39.72 34.79 29.90 25.11 20.52 16.23 12.38 9.07 6.37 4.27 2.74 1.68 0.98 0.55 0.00

74.25 69.11 63.96 59.07 53.65 48.48 43.32 38.49 32.99 27.87 22.85 18.05 13.64 9.78 6.63 4.22 2.52 1.41 0.74 0.36 0.16 0.00

72.65 67.51 62.37 57.49 52.07 46.92 41.76 36.94 31.47 26.39 21.46 16.80 12.56 8.92 5.98 3.77 2.23 1.24 0.64 0.31 0.14 0.00

76.03 71.06 66.09 61.13 556.16 51.19 46.23 41.26 36.30 31.37 26.50 21.78 17.31 13.25 9.72 6.82 4.56 2.91 1.77 1.03 0.57 0.00

100.02 105.00 110.00 115.00 120.00 125.00 130.00 135.00 140.00 145.00 150.00 154.99 159.99 164.99 169.99 174.99 179.99 184.99 189.99 194.99 200.00 240.00

computation of the American call value referred to as call, the option ca , the option cb , the option cc , the algebraic sum of the three options (ca + cb − cc ) and the critical underlying asset price. The results are given for different “strike prices” varying from 100 to 240. Table 5 shows that the algebraic sum of the three options is equal to the American call price. The “critical asset price” corresponding to an early exercise is an increasing function of the strike price. Table 6 uses the same data except for information costs. Information costs are set equal to λS = 0.01 and λC = 0.001. The reader can check that the algebraic sum of the three options is exactly equal to the American call price. With these costs, the call price is slightly higher than in Table 5. Table 7 uses the same parameters except for the information costs which are set equal to λS = 0.1 and λC = 0.05.

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Table 6: Simulations of option values for the continuous-time model using the following parameters: S = 175, r = 0.1, D = 1.5, T = 30, t = 24, σ = 0.32, λc = 0.001, λs = 0.01. Strike 100 120 140 145 150 155 160 165 170 175 180 185 190 195 200 240

Call

ca

cb

cc

ca + cb − cc

S∗

76.14 56.427 36.41 31.48 26.61 21.89 17.42 13.34 9.80 6.89 4.62 2.95 1.80 1.05 0.58 0.00

74.56 54.73 34.93 30.04 25.25 20.65 16.35 12.49 9.16 6.44 4.33 2.78 1.71 1.00 0.56 0.00

74.34 53.73 33.06 27.94 22.92 18.11 13.67 9.83 6.66 4.24 2.53 1.42 0.74 0.36 0.17 0.00

72.76 52.19 31.57 26.49 21.55 16.87 12.61 8.97 6.02 3.80 2.25 1.25 0.65 0.32 0.14 0.00

76.14 56.27 36.41 31.48 26.61 21.89 17.42 13.34 9.80 6.89 4.62 2.95 1.80 1.05 0.58 0.00

100.02 120.00 140.01 145.00 150.00 154.99 159.99 164.99 169.99 174.99 179.99 184.99 189.99 194.99 200.00 240.00

Specification 3 (i) The present value of the implemented project V follows the equation: dV/V = µ dt + σ dz. (ii) If the project is not implemented immediately, its value will fall by a known amount Di at time ti where i = 1, 2, . . . , n. (iii) If the project is implemented at time tk , its present value is given by: Sk = V0 −

k−1 

Di e−(r+λs )ti ,

0 ≤ ti < tk ≤ T.

i=1

In this expression, Sk corresponds to the present value at time 0 for the project to be implemented at tk . V0 corresponds to the present value of the project to be implemented now. The cost of waiting is given by the difference between the two present values. In this case, an extended version of the Black’s (1975a,b) approximation with information costs can be used: C(S, 0, T, I) = max[c(Sk , 0, tk , I ) | k = 1, 2, . . . , n].

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Table 7: Simulations of option values for the continuous-time model using the following parameters: S = 175, r = 0.1, D = 1.5, T = 30, t = 24, σ = 0.32, λc = 0.05, λs = 0.1. Strike 100 120 140 145 150 155 160 165 170 175 180 185 190 195 200 240

Call

ca

cb

cc

ca + cb − cc

S∗

76.92 57.12 37.32 32.40 27.54 22.80 18.29 14.15 10.52 7.49 5.09 3.31 2.05 1.22 0.69 0.00

75.55 55.79 36.06 31.18 26.38 21.75 17.39 13.43 9.97 7.11 4.85 3.16 1.97 1.18 0.67 0.00

74.95 54.31 33.59 28.45 23.40 18.55 14.08 10.15 6.90 4.41 2.65 1.49 0.78 0.38 0.18 0.00

73.58 52.98 32.33 27.23 22.25 17.51 13.18 9.42 6.36 4.04 2.41 1.34 0.70 0.34 0.16 0.00

76.92 57.12 37.32 32.40 27.54 22.80 18.29 14.15 10.52 7.49 5.09 3.31 2.05 1.22 0.69 0.00

100.02 120.00 140.01 145.00 150.00 154.99 159.99 164.99 169.99 174.99 179.99 184.99 189.99 194.99 200.00 240.00

At each instant th , just before the known present value decline, Dh , it is possible to compute the trigger point project value, Vcr,h as in Lee (1988) using the following equation: Sk = Vcr,h −

k 

Di e−(r+λs )(ti −th ) ,

k = h, h + 1, . . . , n

(14)

i=1

where k∗ is the argument of k at which [c(Sk , th , I ) | n ≥ k ≥ h] is a maximum and: c(Sk∗ , th , tk∗ , I ) = Vcr,h − I.

(15)

In this expression: tk∗ : the planned optimal timing when the manager decides to wait, Sk∗ : the present value at tk of the project when it is implemented at the optimal planned time. A firm has a timing option on the introduction of a product with a cost I for a time horizon T . If a new product is introduced at time 0, its present value

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V can be described by the above dynamics. Before a given firm introduces the product, the introduction by the competitor at time tk can reduce the value of a given firm new product by Dk . Each episode of innovation at time i can reduce the value of the new planned product line by Di . This fits with specification 3. In this case, Eqs. (38) and (39) can be used for an optimal timing decision.

4.4. Research and Development and the Option on Market Introduction in the Presence of Information Costs Several companies face the difficulty of selecting an optimal portfolio of research projects. As it appears in the analysis of Lint and Pennings (1998), the standard DCF techniques for capital budgeting can distort the process of selecting a portfolio of research projects. When managers have the option to abandon a project, it is possible to think of the cost of R&D as an option on major follow-on investments. Newton and Pearson (1994) provide an option pricing framework for R&D investments. Lint and Pennings (1998) report the application of an option pricing model for setting the budget of R&D projects. Their model captures a discontinuous arrival of new information that affects the project’s value. R&D options can be viewed as European when two conditions hold.b In the Lint and Pennings’s (1998) model, the variance of the underlying value σ 2 is given by the product of a parameter representing the number of annual business shifts η and a parameter γ for the expected absolute change in the underlying value at every business shift: σ 2 = ηγ 2 . Applying asymptotic theory, the option value can be approximated with the Black and Scholes (1973) formula where σ 2 is replaced by ηγ 2 , or:  (16) C(S, T ) = S(t)N(d + η(T − t)γ) − Ie−r(T −t) N(d )   

   1 2 S(t) + r − ηγ (T − t) σ η(T − t)γ d = ln I 2 b Lint and Pennings (1998) assume that the costs associated with the irreversible investment, required

for market introduction, and the time for completing R&D are given with reasonable accuracy. By ignoring dividends, they propose a simple model which is an extension for R&D option pricing in practice. The approach in Lint and Pennings (1998) is based on a discontinuous arrival of information affecting the project.

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where S(t), I, r and T − t stand, respectively, for the underlying value at present, the costs for market introduction, the risk free rate and the option’s time to maturity. Lint and Pennings (1998) use their model in Philips and show that the option value is largely determined by the opportunity to make a final decision on market introduction with more technological and market information. They show that the option value must compensate the R&D costs necessary to create the option. Their estimation of the option value of the potential benefits to market new products based on R&D goes beyond myopic use of DCF analysis. In the conclusion of their paper, they suggest to classify a variety of past and current R&D projects into sets of similar risks and returns. This can allow the estimation of the value of future idiosyncratic R&D projects by option analysis as in Newton and Pearson (1994). This line of research imposes an information cost in the spirit of the costs in Merton’s (1987) model of capital market equilibrium with incomplete information.c It is possible to use the methodology in Lint and Pennings (1998) and in Bellalah (1999) to account for the role of information costs. In this case, the option value is given by:  C(S, T ) = S(t)e−(λC −λS )(T −t)N(d + η(T − t)γ) − Ie−(r+λC )(T −t)N(d ) (17)     

 S(t) 1 2 d = ln + r + λS − ηγ (T − t) σ η(T − t)γ I 2 where λS and λc denote, respectively, the information costs relative to S and C.

5. The Valuation of Real Options and R&D Projects within Information Costs in a Discrete-Time Setting The majority of the papers concerned with the pricing of real assets in a discrete-time setting derive from the models for financial options pioneered by Cox, Ross and Rubinstein (1979). c These costs appear also in the models of Bellalah and Jacquillat (1995) and Bellalah (1999) for the

pricing of financial options in the presence of incomplete information.

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5.1. The Valuation of Real Assets in a Simple Discrete-Time Framework Salkin (1991) extends the basic binomial option pricing methodology to derive a consistent technique for the pricing of real hydrocarbon reserves. We extend this analysis to account for the effect of information costs. In the classic binomial model of Cox, Ross and Rubinstein (1979), the price of the underlying asset goes up (u) or down (d ) with a probability p and (1 − p). The use of this model is based on the presence of a “twin security” which exactly mimics the structure of the project. Consider an investor who can either trade a commodity or invest in a project which supplies the commodity. The use of the dynamics of prices of the commodity must provide a good foundation for the examination of the structure of the cash flows of the project. By introducing information costs, the probability of an upward movement in the underlying asset price can be shown to be equal to p = (r + λc − d )/(u − d ). The price uncertainty is described by a lattice: Si,t = S0,0 ui d i−t , where S0,0 is the price of the underlying commodity. Let us denote by: Pt : Ft : Vt : τ:

the production of a commodity at time t, the fixed costs of production at time t, the variable costs of production per unit of commodity at time t, and corporation tax rate on positive cash flows at time t.

These profiles can be used to construct gross revenue, net revenue and post-tax cash flows. Using a lattice of post-tax cash flows, it is possible to calculate the expected NPV of the project (ENPV). The lattice gross revenue Gi,t corresponds to the spot lattice Si,t times the production profile Pt for all time and states t. Gi,t = Si,t Pt . The net revenue lattice Ni,t pre-taxation corresponds to the gross revenue less the cost profiles Ft and Vt : Ni,t = Gi,t − Ft − Pt Vt . The application of a taxation rate to all positive cash flows, gives a lattice that describes the cash flows of the project: i,t = Ni,t ≥ 0, Ni,t (1 − τ) i,t = Ni,t < 0, Ni,t . The resulting lattice describes the post-tax cash flows of the project. The added value to the project resulting from the ability to implement any decision contingent on the cash flows, i,t . In general, a decision rule is used to decide on the abandonment of a project, the contraction of its scale, the 12:16:42.

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expansion of its scale, capacity, etc. For example, the decision to abandon is taken when both the post-tax cash flows in the current period are negative, and the expected future post-cash flows from the current time t and state i is negative. The expected value of all future post-tax cash flows from current time t can be calculated by beginning at the end for T = N. If we denote by i,t the expected value of all future post-tax cash flows for the current time t and state i, then: i,t =

1 [p(i+1,t+1 + i+1,t+1 ) + (1 − p)(i,t+1 + i,t+1 )] R + λc

(18)

where R refers to one plus the riskless rate of interest. Now, it is possible to get a structure of cash flows that accounts for the abandonment decision i,t = max[i,t ; i,t ]. Repeating this procedure for all states at each period gives the project’s value 0,0 with the embedded option to abandon the production. The process by which 0,0 is calculated is denoted by: = Fn (Pt , Ft , Vt , τ, σ, r, λs , λc , S0,0 ).

5.2. The Valuation of a Biotechnology Firm Using a Discrete-Time Framework within Information Costs Following the analysis in Kellogg and Charnes (1999), the value of the firm can be found also using the binomial lattice with the addition of a growth option. The growth option is represented by a second binomial lattice for a research phase. The current value of the asset S (or S0,0 ) is computed using the discounted value of the expected commercialization cash flows to time zero as: S0,0 = S =

5  j=1

qj

T  CCFjt (1 + rc )t t=1

where the discount rate is estimated using Merton’s CAPMI. The number of stages can be arbitrarily any number. It is possible to construct an n-period binomial lattice of asset values. The value of the underlying asset S goes up by u or down by d. This multiplicative process is continued for n period until the nth lattice. Kellogg and Charnes (1999) use the fact that u = eσ and d = e−σ and impose that h = Sul = Seσl where l corresponds to a given number of years. They used an example in which the periods are supposed to have a length of one 12:16:42.

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year. The next step is to add in the value of the growth option. The idea is that engaging in the development phase is equivalent to buying a call on the value of a subsequent product. Hence, there is the initial option and the growth option. The value of the growth option at the time of the launch of the first product is added to each of the Ek values of the first NME. Once the binomial tree of asset values is completed, it is possible to compute the possible payoffs and roll back the values using the risk neutral probabilities. The different payoffs are computed as Pk = max[Ek (θt ) − DCFt , 0] where θt is the probability of continuation to the next year in t and DCFt , the R&D payment in year t. The Pk values are rolled back by multiplying the adjacent values, such as P1 and P2 (denoted by Vt+1,k and Vt+1,k+1) by the risk neutral probabilities p and (1−p), the probability of continuation to the next year and a discount factor to obtain Vt,k . The risk neutral probabilities are calculated as: p=

e(r+λS )t − d . (u − d)

As the option values are rolled back, they are adjusted for the probability of success at that phase of development and for the cost of development that year. The option values can be obtained at each node as: √ t

Vt,k = max[(pVt+1,k + (1 − p)Vt+1,k+1 )θt e−(r+λV )

− DCFt , 0].

5.3. The Generalization of Discrete-Time Models for the Pricing of Projects and Real Assets within Information Uncertainty Trigeorgis (1990) proposed a log-transformed binomial model for the pricing of several complex investment opportunities with embedded real options. The model can be extended to account for information costs. The value of the expected cash flows or the underlying asset V satisfies the following dynamics: dV = α dt + σ dz. V Consider the variable X = log V and K = σ 2 dt. If we divide the project’s life T into N discrete intervals of length τ, then K can be approximated from σ 2 (T/N ).

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Within each interval, X moves up by an amount X = H with probability π or down by the same amount X = −H with probability (1 − π). The mean of the process is E(dX) = µK; and its variance is Var(dX) = K with µ = ((r + λS )/σ 2 ) − 1/2. The mean and the variance of the discrete process are: E(X) = 2πH − H

and

Var(X) = H 2 − [E(X)]2 .

The discrete-time process is consistent with the continuous diffusion process when: 2πH − H = µK with (r + λs ) 1 − µ= σ2 2

  1 µK so π = 1+ 2 H

and H 2 − (µK)2 = K

so that H =

 K + (µK)2 .

The model can be implemented in four steps. In the first step, the cash flows CF are specified. In the second step, the model determines the following key variables: the time-step K from (σ 2 T/N), the drift µ from (r + λS )/(σ 2 − 21 ), the  state-step H from K + (µK)2 and the probability π from 21 (1+(µK/H )). Let j be the integer of time steps (each of length K), i the integer index for the state variable X (for the net number of ups less downs). Let R(i) be the total investment opportunity value (the project plus its embedded options). In the third step, for each state i, the project’s values are V(i) = e(X0 +iH) . The total investment opportunity values are given by the terminal condition R(i) = max[V(i), 0]. The fourth step follows an iterative procedure. Between two periods, the value of the opportunity in the earlier period j at state i, R
(i) is given by:





R (i) = e

−(r+λc )

K σ2



[πR(i + 1) + (1 − π)R(i − 1)].

In this setting, the values of the different real options can be calculated by specifying their payoffs. The payoff of the option to switch or abandon for salvage value S is: R
= max(R, S).

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The payoff of the option to expand by e by investing an amount I4 is: R
= R + max(eV − I4 , 0). The payoff of the option to contract the project scale by c saving an amount I3
is R
= R + max(I3
− cV, 0). The payoff of the option to abandon by defaulting on investment I2 is: R
= max(R − I2 , 0). The payoff of the option to defer (until next period) is: R
= max(e−(r+λc )TE(Rj+1 ), Rj ). When a real option is encountered in the backward procedure, then the total opportunity value is revised to reflect the asymmetry introduced by that flexibility or real option. This general procedure can be applied for the valuation of several projects and firms in the presence of information costs.

Summary This chapter provides some of the main results in the literature regarding the valuation of the firm and its assets using the real option theory when we account for the effects of information uncertainty. We propose some simple models for the analysis of the investment decision under uncertainty, irreversibility and sunk costs. First, we use Merton (1987) model of capital market equilibrium with incomplete information to determine the appropriate rate for the discounting of future risky cash flows under incomplete information. The use of this model allows the computation of the weighted average cost of capital under incomplete information. This cost can be used to reformulate the Modigliani–Miller (1958, 1963). It allows the extension of the standard DCF analysis, the EVA and the theory of firm valuation under incomplete information. Second, we review the main possible and potential applications of option pricing theory to the valuation of simple and complex real options. Third, we develop some simple models for the pricing of European and American commodity options in the presence of information costs. We propose also simple analytic formulas for the pricing of compound options in the presence of information costs. These formulas are useful in the study of the main results in the literature regarding the investment timing and the

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pricing of real assets using standard and complex options in the presence of incomplete information. The analysis is extended to the valuation of research and development and the option on market introduction. It is also applied to the valuation of flexibility as a compound option in the same context. Fourth, a general context is proposed for the valuation of real options and the pricing of real assets in a discrete-time setting. Salkin (1991) shows how to apply the Cox, Ross and Rubinstein (1979) model for the valuation of complex capital budgeting decisions. The methodology is applied to a hypothetical case of a marginal natural resource project. The real benefit of this technique arises in its ability to value more realistically situations in which traditional techniques attributed little or no worth. Following the analysis in Salkin (1991), we develop a simple context for the valuation of real options using option pricing techniques in the presence of information costs. Then, using the Trigeorgis (1990) general log-transformed binomial model for the pricing of complex investment opportunities, we provide a context for the valuation of these options under incomplete information. Trigeorgis (1990) proposed a log-transformed binomial model for the pricing of several complex real options. We use that generalization to account for information costs in the pricing of complex investment opportunities. Our approach can be extended to price most well-known real options in the presence of information costs. While the estimation of the magnitude of these costs is done in Bellalah and Jacquillat (1995) for financial options, it is possible to look for a convenient approach to estimate these costs for real options. We let this point for a future research. The analogy between standard and exotic financial options facilitates considerably the valuation of real options. It is possible to use the main results in exotic options to value different real options. However, it is important to note that real options can be sometimes more difficult to value in the presence of information costs and a dependency between different real options in the same project.

Questions 1. Define the cost of capital. 2. How can one compute the value of the firm in standard analysis? 3. What are real options?

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4. Define the concept of economic value added (EVA). 5. What are the main differences between financial options and real options? 6. What is the standard analogy between financial and real options? 7. How can one value real options with information costs in a continuoustime setting? 8. How can one value real simple European and American commodity options with information costs? 9. How can one value compound options within information costs? 10. How can one appreciate research and development and the option on market introduction in the presence of information costs? 11. How can one value real options and R&D projects within information costs in a discrete-time setting? 12. How can one value generalized discrete-time models for the pricing of projects and real assets within information uncertainty?

Bibliography Baghay, M, C Coley, D White, C Conn and R McLean (1996). Staircases to growth. The McKinsey Quarterly, 4, 38–61. Baldwin, C and R Ruback (1986). Inflation, uncertainty and investment. Journal of Finance, 41, 657–669. Barone-Adesi, G and RE Whaley (1987). Efficient analytic approximation of American option values. Journal of Finance, XLII(2), 81, 303–320. Bellalah, M (1999). The valuation of futures and commodity options with information costs. Journal of Futures Markets, 19 (September), 645–664. Bellalah, M (2000). Strategies d’investissements technologiques, options relles et information. In: Gestion des risques dans un cadre International. Economica. Bellalah, M (2001a). Irreversibility, sunk costs and investment under incomplete information. R&D Management Journal, April. Bellalah, M (2001b). A re-examination of corporate risks under incomplete information. International Journal of Finance and Economics, 6, 59–67. Bellalah, M and B Jacquillat (1995). Option valuation with information costs: theory and tests. Financial Review, 30, 617–635. Berger, PG, E Ofek and I Swary (1996). Investor valuation of the abandonment option. Journal of Financial Economics, 42, 257–287. Black, F (1975a). The Pricing of Complex Options and Corporate Liabilities. Graduate School of Business, University of Chicago, Chicago, IL. Black, F (1975b). Fact and fantasy in the use of options. Financial Analyst Journal, 31, July–August. Black, F (1976). The pricing of commodity contracts. Journal of Financial Economics, 79(3), 167–179.

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Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659. Brennan, MJ (1991). The price of convenience and the valuation of commodity contingent claims. In Lund, D and B Oksendal (eds.) Stochastic Models and Option Values. North Holland, New York. Brennan, MJ and E Schwartz (1985). Evaluating natural resource investments. Journal of Business, 58, 135–157. Carr, P (1988). The valuation of sequential exchange opportunities. Journal of Finance, XLIII(2), December. Cox, JC, SA Ross and M Rubinstein (1979). Option pricing: a simplified approach. Journal of Financial Economics, 7(3), 229–263. Dentskevich, P and G Salkin (1991). Valuation of real projects using option pricing techniques. OMEGA International Journal of Management Science, 19(4), 207–222. Dixit, AK (1992). Investment and hysteresis. Journal of Economic Perspectives, 6(1), 107– 132. Dixit, AK (1995). Irreversible investment with uncertainty and scale economies. Journal of Economic Dynamics and Control, 19, 327–350. Dixit, A and RS Pindyck (1994). Investment Under Uncertainty. Princeton University Press. Faulkner, TW (1996). Applying options thinking to R&D valuation. Research and Technology Management, May–June. Fisher, I (1907). The Rate of Interest: Its Nature, Determination and Relation to Economic Phenomena. Macmillan Co., New York. Geske, R (1979). The valuation of compound options. Journal of Financial Economics, 7, 375–380. Ingersoll, JE and SA Ross (1992). Waiting to invest: investment and uncertainty. Journal of Business, 65(I), 1–30. Kellogg, D and JC Charnes (1999). Using real-options valuation methods for a biotechnology firm. Working Paper, University of Kansas, School of Business. Kogut, B (1991). Joint ventures and the option to expand and acquire. Management Science, 37(1), 19–33. Kogut, B and N Kulatilaka (1994a). Operating flexibility, global manufacturing, and the option value of a multinational network. Management Science, 40(1), 123–139. Kogut, B and N Kulatilaka (1994b). Options thinking and platform investments: investing in opportunity. California Management Review, 36(2), Winter. Lee, CJ (1988). Capital budgeting under uncertainty: the issue of optimal timing. Journal of Business Finance and Accounting, 15(2), Summer. Lint, O and E Pennings (1998). R&D as an option on market introduction. R&D Management, 28(4), 279–287. Lintner, J (1965). Security prices, risk and maximal gains from diversification. Journal of Finance, 20, 587–615. Luehrman, T (1997). A general guide to valuation. Harvard Business Review, May–June, 132–142. Luehrman, T (1998). Investment opportunities as real options: getting started on the numbers. Harvard Business Review, July–August. McDonald, R and D Siegel (1984). Option pricing when the underlying asset earns a belowequilibrium rate of return: a note. Journal of Finance, 39, 261–265.

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McDonald, R and D Siegel (1986). The value of waiting to invest. Quarterly Journal of Economics, 101, 707–728. Merton, RC (1987). A simple model of capital market equilibrium with incomplete information. Journal of Finance, 42, 483–510. Modigliani, F and M Miller (1958). The cost of capital, corporation finance and the theory of investment. American Economic Review, June, 261–279. Modigliani, F and M Miller (1963). Corporate income taxes and the cost of capital. American Economic Review, June, 433–443. Mossin, J (1966). Equilibrium in a capital asset market. Econometrica, 34, 768–783. Myers, SC (1984). Finance theory and financial strategy. Interfaces, 14(1), 126–137. Myers, S and S Majd (1990). Abandonment value and project life. Fabozzi, F (ed.), Advances in Futures and Options Research, 4, 1–21. Newton, D (1996). Opting for the right value for R&D. The Financial Times, June 28. Newton, D and A Pearson (1994). Application of option pricing theory to R&D. Management, 24, 83–89. Paddock, J, D Siegel and J Smith (1988). Option valuation of claims on real assets: the case of offshore petroleum leases. Quarterly Journal of Economics, 103, 479–508. Pickles, E and JL Smith (1993). Petroleum property evaluation: a binomial lattice implementation of option pricing theory. The Energy Journal, 14(2), 1–26. Pindyck, RS (1991). Irreversibility, uncertainty, and investment. Journal of Economic Literature, September, 1100–1148. Salkin, G (1991). Valuation of real projects using option pricing techniques. OMEGA International Journal of Management Science, 19, 207–222. Sharpe, WF (1964). Capital asset prices: a theory of market equilibrium under conditions of risk. Journal of Finance, 19, 425–442. Stewart, GB (1990). The Quest for Value. Harper Business, New York. Trigeorgis, L (1990). A real option application in natural-resource investments. Advances in Futures and Options Research, 4, 153–156. Trigeorgis, L (1993a). The nature of options interactions and the valuation of investments with multiple real options. Journal of Financial and Quantitative Analysis, 28(1), 1–20. Trigeorgis, L (1993b). Real Options in Capital Investments, Models, Strategies, and Applications. Praeger Publishers, Westport. Trigeorgis, L (1993c). Real options and interactions with financial flexibility. Financial Management, 22 (Autumn), 202–224. Trigeorgis, L (1995). Real Options in Capital Investment: Models, Strategies, and Applications. Praeger, Westport, Conn. Trigeorgis, L (1996). Real Options, Managerial Flexibility and Strategy in Resource Allocation. The MIT Press. Whaley, RE (1981). On the valuation of American call options on stocks with known dividends. Journal of Financial Economics, 9, 375–380.

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Chapter 17

Option Pricing When the Underlying Asset is Nonobservable

This chapter is organized as follows: 1. Section 2 shows how to extend the Black–Scholes theory to account for the effects of nontradability and nonobservability. 2. Section 3 extends the analysis to account for the effects of incomplete information.

1. Introduction According to Merton (1998): “

T

he most influential development in terms of impact on finance practice was the Black–Scholes model for option pricing. . . This success in turn increased the speed of adoption for quantitative financial models to help value options and assess risk exposures” (p. 324). In fact, the conceptual framework used to derive the option formula is used to price the risk in financial and nonfinancial applications. The extra-ordinary growth in the use of derivatives can be explained by the vast saving in transactions costs derived from their use. These costs can be further reduced with further improved technology and experience in the use of derivatives. New financial products and market designs improved computer and telecommunications technology. Innovations have improved efficiency by expanding opportunities for risk sharing, lowering transaction costs, and reducing agency costs and information costs. Merton (1998) and Perold (1992) show that the cost of implementing financial strategies for institutions using derivatives can be one-tenth to one-twentieth of the cost of executing them in the underlying cash market Bellalah, Mondher. Exotic Derivatives and Risk: Theory, Extensions and584 Applications, World Scientific Publishing Co 12:16:53.

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securities. Hence, roughly speaking, if the information cost in using derivatives for a given asset is 5%, then the information cost for the derivative is about one-tenth of 5%. In a more general level, Merton (1998) shows that “implementation of derivative-security technology and markets within smaller and emerging countries may help form important gateways of access to world capital markets and global risk sharing”.

2. General Derivation of Derivative-Security Pricing The derivation of the option pricing formula in Black–Scholes (1973) and Merton (1973, 1998) is based on the following five assumptions. Assumption 1: Concerns frictionless and continuous markets with no transactions costs or differential taxes. Besides, borrowing and lending rates are equal. Assumption 2: Concerns the dynamics of the underlying asset where instantaneous returns are described by dV = [αV − D1 (V, t)]dt + σV dZ where: α: σ2: dZ: D1 :

the instantaneous expected rate of return on the security, instantaneous variance rate, a Wiener process, dividend payment flow rate.

Assumption 3: process.

Stipulates that default-free bond returns follow an Ito

Assumption 4: Concerns investor preferences and expectations. Investors agree on σ 2 and it is not assumed that they agree on the expected rate of return, α. Assumption 5: Says that the option price is twice-continuously differentiable of the asset price. This is rather a derived consequence of the analysis. When D1 = 0, the Black–Scholes call formula is √ (C[S, t]) = VN(d ) − L · exp(−r[T − t])N(d − σ T − t) with

√ d = (ln[V/L] + [r + σ 2 /2])[T − t]/σ T − t

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where N(.) is the cumulative density function for the standard normal distribution and L stands for the strike price. As it appears in Merton’s (1998) paper, his main contribution to the Black–Scholes option pricing theory was to demonstrate the following result: in the limit of continuous trading, the Black–Scholes dynamic trading strategy designed to offset the risk exposure of an option would provide a perfect hedge. Hence, when trading is done without cost, the Black–Scholes dynamic strategy using the option’s underlying asset and a risk-free bond would exactly replicate the option’s payoff. In the absence of a continuous trading, which represents an idealized prospect, replication with discrete trading intervals is at best only approximate. Merton (1976) studies a mixture of jump and diffusion processes to capture the prospect of nonlocal movements in the underlying asset’s return process. Replication is not possible when the sample path of the underlying asset is not continuous. In this case, the derivation of an option pricing model is completed by using an equilibrium asset pricing model. This approach is used in the original Black–Scholes model who derived their formula using the standard CAPM. As it appears in the work of Black (1989), Scholes (1998) and as Merton (1998) asserts: “Fisher Black always maintained with me that the CAPM-version of the option model derivation was more robust because continuous trading is not feasible and there are transaction costs”. As it is well-known, in all standard asset pricing models, assets showing only diversifiable risk or nonsystematic risk are valued to yield an expected return equal to the riskless rate. Merton (1998) reexamines the imperfect replication problem for a derivative security when the underlying asset is not continuously available for trading. Consider, for example, a derivative security W(t) at time t for which payouts are described by functions of observable asset prices and time. The terms are expressed as: If V(t) ≥ V¯ (t), If V(t) ≤ V _ (t),

for 0 ≤ t < T for 0 ≤ t < T If t = T

then W(t) = f [V(t), t] then W(t) = g[V(t), t] then W(t) = h[V(t), t].

(2)

This payoff shows that the first time V(t) ≥ V¯ (t) or V(t) ≤ V _ (t), the owner of the derivative asset must exchange it for cash. Otherwise, the security is redeemed at t = T for cash according to (2). The derivative receives a

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payment flow rate D2 (V, t). The relevant region for analysis V _ (t) ≤ V(t) ≤ ¯ V (t) for 0 ≤ t ≤ T . After specifiying the derivative-security characteristics, it is possible to study the fundamental production technology for hedging the risk of issuing a derivative and for evaluating the cost of its production. Consider a hypothetical financial intermediary that creates derivative securities in principal transactions for its customers by selling them contracts. When the intermediary cannot perfectly replicate the payoffs to the issued derivative, it can adopt one of two strategies. The first strategy is to obtain adequate equity to bear the residual risks of the imperfectly hedged positions. The second strategy is to securitize the positions by bundling them into a portfolio for a special-purpose financial vehicule. The objective is to find a trading portfolio P(t) strategy using the traded assets and the riskless asset that satisfies ‘closely’ the following four properties: (i) (ii) (iii) (iv)

at t, if V(t) = V _ (t), then P(t) = g[V _ (t), t]; at t, if V(t) = V¯ (t), then P(t) = f [V¯ (t), t]; for each t, the payout rate on the portfolio is D2 (V, t) dt; at t = T , P(T) = h[V(T)].

This portfolio is referred to as the hedging portfolio for the derivative security in (2), (portfolio*). When the portfolio meets exactly the above conditions, it becomes a ‘replicating portfolio’ for the derivative security. Merton (1998) determines the optimal hedging portfolio in two steps. In the first step, he finds a portfolio strategy using traded assets with the smallest ‘tracking error’ in replicating the returns on the underlying asset. This portfolio refers to a ‘V -fund’. In the second step, the hedging portfolio for the derivative asset is obtained as a dynamic portfolio strategy mixing the V -fund and the riskless asset. Consider the dynamics for the price of asset i at time t: dSi = αi Si dt + σi Si dZi

(3)

where i = 1, . . . , n. In this equation, αi is the instantaneous expected rate of return, dZi a Wiener process, σi,j is the instantaneous covariance between the returns on i dSj ) = σij dt and σii = σi2 . i and j or ( dS Si Sj Let ηi be the correlation between dZi and dZ in assumption 2 with dZi dZ = ηi dt. We denote by S(t) the value of the V -fund portfolio and by

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wi (t) the fraction of that portfolio invested in asset i at time t. The balance is invested in riskless assets. The dynamics of S are given by dS = [µS − D1 (V, t)]dt + δS dq

(4)

where µ=r+

n


wi (t)[αi − r]

i=1 n n



δ = 2

wi (t)wj (t)σij

i=1 j=1

 n


dq =



wi (t)σi dZi

δ.

i=1

The wi ’s are chosen so as to minimize the unanticipated part of the difference between the return on the underlying asset and the traded portfolio’s return, i.e., to minimize the variance of [dS/S − dV/V ]. Merton (1992, Theorem 15.3, p. 502) shows that wi (t) = σ

n


vki σk ηk ,

i = 1, . . . , n

(5)

k=1

where νki corresponds to the kth–ith element of the inverse variance– covriance matrix of the returns on the n risky continuously traded assets. Merton (1992, p. 502) shows that the correlation between the V -fund and the underlying asset dZ dq = ρ dt is  n n 1/2

ρ= vki σk ηk ηi (6) k=1 i=1

and δ = ρσ.

(7)

dS/S − dV/V = (µ − α)dt + θ db

(8)

The tracking error dynamics are where θ 2 = (1 − ρ2 )σ 2 and the Wiener process db =

(ρ da q − dZ)  (1 − ρ2 )

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dSi = 0, Si db

i = 1, . . . , n.

(9)

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This shows that the tracking error in Eq. (8) is uncorrelated with the returns on all traded assets. This is the consequence of picking the portfolio that minimizes the tracking error. The derivation of the production process for the hypothetical financial intermediary to best hedge the cash flows of the derivative security begins with the use of (5) to construct the V -fund portfolio by the quantitative analysis department. The department must solve the following linear parabolic partial differential equation for the function F(V, t]: 1 2 σ (V, t)V 2 F11 [V, t] + [rV − D1 (V, t)]F1 [V, t] 2 − rF [V, t] + F2 [V, t] + D2 (V, t)

(10)

under the boundary conditions F [V¯ (t), t] = f [V¯ (t), t]  0

(11)

F [bV(t), t] = g[bV(t), t]  0

(12)

F [V, t] = h[V ]  0

(13)

where the subscripts for F(., .) refer to partial derivatives. This gives a unique solution for F . The quant department has to do the following actions at t, (0 ≤ t ≤ T): (i) ask the trading desk for the prices of traded assets to compute the price S(t) for the V -fund and to estimate V(t); (ii) compute from the solution to (10)–(13) M(t) = F1 [V(t), t]V(t) (iii) this amount M(t) must be invested in the V -fund for the period t, t +dt; (iv) compute Y(t) = F [V(t), t] and store Y(t) for further analysis. The prescription for the trading-desk activities is as follows. At t = 0, give the trading desk P(0) as an initial funding for portfolio (*) which contains the V -fund and the riskless asset. The value of portfolio (*) at t is denoted by P(t). The trading desk must: (a) determine the prices of the underlying asset V(t), (b) pay the cash distribution D2 [V(t), t]dt to the customer holding the derivative security, (c) determine the value of the balance of P(t),

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(d) readjust the portfolio by investing M(t) in the V -fund and [P(t) − M(t)] in the riskless asset. Hence, the dynamics of portfolio (*) are given by dP = M(t)

D1 (V, t) dS + M(t) dt + [P − M(t)]r dt − D2 (V, t)dt S S

(14)

where M(t)(dS/S ) is the price appreciation, M(t) D1 (V,t) dt is the dividend S payments received into the portfolio, [P − M(t)]r dt is the interest earned by the portfolio, D2 (V, t) is the cash distribution to customer. Since M(t) = F1 [V(t), t]V(t), the dynamics of P can be obtained by substitution from (4) into (14): dP = F1 [V, t]V dS/S + F1 [V, t]VD1 [V, t]/S + (P − F1 [V, t]V )r dt − D2 (V ) dt = [F1 V(µ − r) + rP − D2 ]dt + F1 Vδ dq.

(15)

From (iv), Y(t) = F [V, t] and V(t) = V and since F is solution to (10)–(13), applying Ito’s lemma, we have 1 dY = F1 [V, t] dV + F2 [V, t]dt + F11 [V, t](dV)2 2  1 2 2 σ V F11 + F1 (αV − D1 ) + F2 dt + F1 Vσ dZ = 2

(16)

because (dV)2 = σ 2 V 2 dt. Since F [V, t] satisfies (10), one has 1 2 2 σ V F11 − D1 F1 + F2 = rF − rVF1 − D2 . 2

(17)

Substituting (17) into (16), it is possible to write (16) as dY = [F1 (α − r)V + F − D2 ]dt + F1 Vσ dZ.

(18)

Let Q(t) = P(t) − Y(t) so that dQ = dP − dY . Substituting for dP from (15) and for dY from (18) and using (8) gives dQ = rQ dt + F1 V(dS/S − dV/V) = (rQ + F1 V [µ − α])dt + F1 Vθ db. (19)

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The case of perfect replication For the case of perfect replication, ρ = 1. When there is no tracking error, Eq. (19) reduces to an ordinary differential equation with solution Q(t) = Q(0) exp(rt)

(20)

where Q(0) = P(0) − Y(0) = P(0) − F [V(t), t]. If the initial funding is P(0) = F [V(0), 0] then from (20), Q(t) = 0 for all t and P(t) = F [V(t), t].

(21)

When Eqs. (11)–(13) are compared with (2), using (21), the (*)-portfolio strategy generates the same flows as the derivative security. Hence, F [V(0), 0] corresponds to the cost to the intermediary for producing the derivative. If the derivative security is traded, then W(t) = P(t) = F [V(t), t].

(22)

Hence, equilibrium prices must satisfy Eq. (22) as in Black–Scholes where the V -fund corresponds to a single underlying asset. In this analysis, the cost of creating the security is F [V(t), t]. The case of imperfect replication In this case, the cumulative arithmetic tracking error for the hedging portfolio is Q = P − Y . Since in (19) the tracking error for the derivative security is perfectly correlated with the tracking error of the V -fund, and in (9) the tracking error for the hedging portfolio is uncorrelated with the returns on traded assets, Merton (1998) shows that the replication-based valuation can be used for the pricing of the derivative security even though replication is not feasible. He shows that the tracking-error component of the hedging portfolio satisfies a stronger no correlation condition than either a zero-beta asset in the CAPM, a zero multibeta asset of the Intertemporal CAPM, or a zero factor-risk asset of the Arbitrage Pricing Theory. By any of these theories, the equilibrium condition from either Eqs. (8) or (19) is that: µ = α.

(23)

If (23) obtains, then the equilibrium price for the derivative security F [V(t), t] is the same “as if” the underlying asset is traded. Hence, the Black–Scholes can be used even when the underlying asset is not traded. In the literature on incomplete markets, Eq. (23) need not obtain because

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of the cost of creating, for example, the securities necessary to span a risk exceeds the benefits or because of adverse selection problems, etc. In systems with today’s financial technology, the tracking error variations are likely to be specific to the underlying project, person, firm and (23) must obtain. How the valuation formula is modified if the underlying asset price is not continuously observable? Assume that the price is observable at time t = 0 and at maturity t = T . In between, there is no direct observation, D1 (V, t) = 0. The dynamics of V in assumption (2) allow the construction of the V -fund in (5). Consider the random variable X(t) = V(t)/S(t), the cumulative tracking error with X(0) = 1. Applying Ito’s lemma, one has from (8), (9), and (23) that the dynamics for X are: dX = θX db.

(24)

This equation shows that the distribution for X(t), conditional on X(0) = 1 is lognormal. The expected value of X(t) is 1 and the variance of ln(X(t)) is θ 2 t. The PDE for F , corresponding to (10) uses the best estimate of V(t) which is S(t) and is written as 1 2 2 δ S F11 [S, t] + rSF1 [S, t] − rF [S, t] + F2 [S, t] 2 subject to the terminal condition for S(T −) = S, 0=

F [S, T ] = E[h(SX)]

(25)

(26)

where h is defined as in (13), X follows a log-normal distribution, E(X) = 1, and the variance of ln(X) = θ 2 t. At t = T , V(T) is revealed and the value of S “jumps” by the total cumulative tracking error of X(t) from its value S at t = T − to S(T) = V(T). The solution to (25) and (26) with h(V) = max[0, V − L] is given by, for 0 < t < T , √ F [S, t] = SN(u) − L · exp(−r[T − t]) × N(u − γ) with √ u = (ln[S/L] + r[T − t] + γ/2)/ γ and γ = δ2 (T − t) + θ 2 T

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where N(.) is the cumulative density function for the standard normal distribution. This equation can be compared with that in Black–Scholes (1973). This allows to study the effect of the underlying asset price when it is not observable. The main difference is that the variance over the remaining life of the option does not go to zero as t approaches T . This is due to the “jump” event at the expiration date corresponding to the cumulative effect of tracking error. This section studied the conditions under which the Black–Scholes model can be applied even when the underlying asset-equivalent is neither continuously traded nor continuously observable. In most examples of real options (if not all), the underlying “asset” is rarely traded in anything approximating a continuous market and therefore its price is not continuously observable either.

3. The General Derivation of Derivative-Security Pricing in the Presence of Information Costs We can use Merton’s (1987) simple model of capital market equilibrium with incomplete information to provide a simple model that accounts for the effects of nontradability and nonobservability. Recall that the value of a standard call in the presence of information costs is given by: C(V, T ) = V e(−(λC −λv )T )N(d1 ) − Le−(r+λC )TN(d2 ) with:

 

 √ V 1 2 d1 = ln + r + σ + λv T σ T L 2 √ d2 = d1 − σ T

where λC and λv indicate, respectively, the information costs regarding C and V . The previous analysis applies until Eq. (9). When Merton’s (1987) model is used, the return on V must be (r + λv ) and the return on F must be (r + λF ). In this context, Eq. (10) becomes for the function F(V, t]: 1 2 σ (V, t)V 2 F11 [V, t] + [(r + λv )V − D1 (V, t)]F1 [V, t] 2 − (r + λF )F [V, t] + F2 [V, t] + D2 (V, t)

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under the same boundary conditions F [V¯ (t), t] = f [V¯ (t), t]  0

(11 )

F [V _ (t), t] = g[V¯ (t), t]  0

(12 )

F [V, t] = h[V ]  0

(13 )

where the subscripts for F(., .) refer to partial derivatives. This gives a unique solution for F . As in Merton (1998), it is possible to use any equilibrium asset pricing model, like the standard CAPM or the CAPMI of Merton (1987) to show the following equilibrium condition from (8) or (19): µ = α.

(23)

Using Eq. (24) and the partial differential equation corresponding to (10 ) that determines the hedging strategy, we have an equation similar to (25): 1 2 2 δ S F11 [S, t]+(r+λs )SF1 [S, t]−(r+λF )F [S, t]+F2 [S, t] 2 subject to the terminal condition for S(T −) = S, 0=

F [S, T ] = E[h(SX)]

(25 )

(26 )

where h is defined as in (13), X follows a log-normal distribution, E(X) = 1, and the variance of ln(X) = θ 2 t. At t = T , V(T ) is revealed and the value of S “jumps” by the total cumulative tracking error of X(t) from its value S at t = T − to S(T ) = V(T ). The solution to (25 ) and (26 ) with h(V) = max[0, V − L] is given by, for 0 < t < T , √ F [S, t] = Se(−(λF −λs )[T −t]) N(u) − L · e−(r+λC )[T −t] × N(u − γ) with

√ u = (ln[S/L] + (r + λv )[T − t] + γ/2)/ γ

(27 )

and γ = δ2 (T − t) + θ 2 T where N(.) is the cumulative density function for the standard normal distribution. This equation can be compared with that in Black–Scholes (1973) in a context of incomplete information.

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ch17

Option Pricing When the Underlying Asset is Nonobservable

595

Summary This chapter deals with the valuation of options in the case of nontradability and nonobservability of the underlying asset. We show how to extend the Black–Scholes theory to account for the effects of nontradability and nonobservability. The analysis is generalized to account for the effects of incomplete information. Our analysis is useful for the analysis and valuation of real options.

Bibliography Black, F (1989). How we came up with the option formula. Journal of Portfolio Management, 15, 4–8. Black, F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659. Merton, RC (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183. Merton, RC (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3, 125–144. Merton, RC (1987). A simple model of capital market equilibrium with incomplete information. Journal of Finance, 42, 483–510. Merton, RC (1992). Continuous-Time Finance. Blackwell, Cambridge, MA. Merton, RC (1998). Applications of option pricing theory: twenty-five years later. American Economic Review, 2, 323–348. Perold, AF (1992). BEA Associates: Enhanced Equity Index Funds. Harvard Business School, Case 292–024. Scholes, M (1998). Derivatives in a dynamic environment. American Economic Review, 2, 350–370.

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Index

Index calls on puts, 131 capped contracts, 226 capped equity, 226, 259, 260, 264 classic barrier options, 297, 372, 374, 401 collars, 226, 265 commodity futures, 74, 75, 77, 113, 116, 117, 227 commodity futures options, 74, 116 complex binaries, 267, 269, 315 complex chooser options, 131, 173 complex investment opportunities, 550, 577, 580 compound option approach, 131, 168, 512 compound options, 131, 133, 158, 170, 172, 173, 212, 512, 541, 561–563, 565–567, 579, 581 contingent claims-modeling, 511, 513, 539 continuous dividend yield, 74, 77, 106, 107, 114, 116, 117, 213–215, 219, 307, 380 continuous strike options, 267, 315 corporate bonds, 212, 447, 452, 455, 467, 506, 511–513, 530, 539 corporate liabilities, 462, 512, 541, 542, 545, 549 corridor options, 267, 304 credit curve, 456–459, 511, 512, 526, 527, 530, 538, 539 credit derivatives, 447, 480, 481, 484, 485, 488–491, 493, 494, 507, 508, 511, 513, 531, 537, 538 credit quality migration, 446, 447, 449, 451, 452, 454, 482 credit risk literature, 446, 471 credit valuation, 446, 447, 460–463

American commodity, 76, 550, 563, 564, 579, 581 arithmetic Asian options, 348, 368, 370 Asian options, 348–350, 359, 367–370 asset pricing, 1, 2, 4, 29, 34, 36, 37, 70–72, 75, 80, 89, 104, 115, 121–123, 139, 463, 586, 594 barrier options, 262, 264, 267–273, 281, 282, 287, 289, 290, 292–294, 297, 300, 306–308, 310, 314–316, 343, 372–375, 377, 379, 380, 385, 386, 394, 395, 401, 402 basket options, 176, 186, 191, 199, 348, 350, 370 binary barrier options, 267, 273, 281, 282, 287, 294, 316 binary options, 267–269, 274, 275, 281, 301, 302, 305, 315, 316 bivariate normal density, 75, 127, 205 bond default rates, 446, 473 bonds, 3, 79, 80, 94, 132, 133, 158, 159, 172, 175, 176, 189–191, 200, 202, 212, 225, 226, 250, 251, 264, 265, 318, 395, 400, 447, 448, 452, 455–460, 463, 467, 472, 473, 478, 481, 485, 487–489, 491, 493, 496, 506, 511–513, 520, 523, 530, 536–539, 547 calls, 1, 9, 24, 27, 29, 71, 72, 79, 131, 144, 205, 207, 208, 210–213, 225, 260, 268, 271, 299, 301, 307, 308, 318, 320, 326, 328, 341, 343–346, 361, 391, 392, 394–399, 468, 542 calls on calls, 131 597

October 23, 2008

598

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B-613

9in x 6in

Index

Exotic Derivatives and Risk: Theory, Extensions and Applications

CreditMetrics, 446–452, 454–459, 471, 472, 474, 481, 498, 499 cross-currency two-way equity, 226, 258 cumulative normal distribution, 75, 126, 161, 162, 178, 194, 464, 514, 569 cumulative Parisian options, 372, 402 currency options, 74, 91, 226, 227, 231, 349, 374

exotic options, 267, 268, 301, 302, 305, 349, 350, 372, 373, 393, 401, 562, 580 exotic timing option, 317, 318, 333–336, 345–347 extendible bonds, 202, 212, 225 extendible options, 202, 204, 210, 212, 225 extendible warrants, 225

default risk, 3, 446, 447, 467, 468, 470, 473–480, 484–488, 490, 491, 493–495, 497, 506, 511–513, 515, 517–520, 537–539 delayed barrier options, 372, 375, 379, 380, 385, 386, 402 delta approximation, 403, 413, 431, 432 delta-gamma approximation, 403, 414 derivative assets, 2, 5, 72, 74, 76, 105, 116, 218, 302, 433 derivative security, 74, 76, 98, 106–109, 112, 116, 117, 123, 126, 359, 360, 538, 585–587, 589, 591, 593 derivatives products, 484 discrete-time, 1, 4, 5, 29, 38, 70, 72, 75, 574–577, 580, 581 distributions, 1, 58, 72, 80, 219, 340, 404, 408, 416, 419, 421, 424, 428, 432–434, 439, 442, 443, 463, 478, 499, 501, 502 double lookback, 317–319, 336, 338, 339, 341, 342, 347 down-and-out call options, 267, 270, 302 dual strike options, 175, 176, 195

flexible arithmetic, 348, 367–370, flexible Asian options, 348, 350, 367, 370 foreign stock investments, 202, 218, 221, 225 forward-start options, 131 futures market, 3, 73, 74, 76, 176, 191, 200

economic value added, 550, 552, 557, 581 efficient frontier, 1, 31, 32, 36, 37, 72 embedded real options, 550, 577 equity options, 1, 38, 72, 91, 228, 263 equity swaps, 226, 254, 255, 257, 258, 263, 264 equity swaptions, 226, 262 equity-linked foreign exchange options, 226, 228, 234 European commodity, 74, 115, 116, 563 European lookback options, 317 European options, 75, 76, 135, 192, 200, 227, 307, 351, 372

general differential equation, 74, 77, 110, 116, 117 geometric Asian options, 348, 370 guaranteed exchange-rate contracts, 202, 203, 218, 221, 224, 225 hazard rate, 511, 512, 520, 521, 527–529, 531, 539 hedging parameters, 75, 97, 119, 120 hybrid instruments, 226 incomplete information, 1, 2, 5, 37, 70–75, 77, 104, 107, 110, 112, 113, 116, 117, 120–123, 131, 137, 147, 151, 152, 157, 183, 208, 210–212, 234, 266, 293, 317, 325, 335, 348, 360–362, 484, 511, 520, 530, 542, 550–552, 554–557, 559, 574, 579, 580, 584, 593–595 indexed notes, 226, 247, 265 information uncertainty, 5, 75, 175, 192, 200, 222, 226, 250–254, 267, 511, 516, 519, 520, 526, 545, 548, 550–554, 567, 577, 579, 581 inside and outside barrier options, 267 Ito’s lemma, 74, 76, 82, 98–102, 107, 109, 110, 117, 124, 159, 177, 534, 590, 592 KMV approach, 446, 463, 470 knock-out options, 267, 304, 373, 374, 378

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B-613

9in x 6in

Index

Index

limited risk options, 317, 323, 331, 347 linear step options, 372, 377, 401 lookback options, 317–321, 324, 325, 332, 333, 336, 338, 339, 341, 342, 346, 347 margin account, 131, 132, 137, 172 market frictions, 202, 552, 553 non-path dependent options, 175, 186, 187 nonlinear positions, 403, 416, 438, 440, 441 notional principal, 226, 250, 255, 257–260, 264 occupation time derivatives, 372, 375, 379, 401, 402 option price sensitivities, 74, 76, 95, 117, 346, 405 option pricing, 1, 38, 40, 71, 73–78, 103, 104, 106, 115, 121–123, 263, 299, 352, 413, 431, 438, 439, 461, 462, 465, 467–469, 471, 473, 480, 481, 490, 491, 497, 498, 541, 550, 551, 560, 566, 573, 575, 579, 580, 584–586 option pricing theory, 74, 76, 106, 123, 263, 461, 462, 465, 467–469, 473, 480, 481, 490, 550, 579, 586 option strategies, 1, 6, 75 option to exchange, 131, 132, 137, 139, 172, 179, 229, 492 options on several assets, 175, 186 options on the minimum, 172, 175, 176, 179, 199, 200, 330 outperformance options, 202, 215, 224, 225 Parisian options, 372, 387–389, 402 passport options, 202, 222, 225 pay floating, 226, 257, 264 pay later options, 131, 133, 143, 145, 147, 172, 173 pay-on-exercise options, 202, 213, 214, 225 performance incentive fee contract, 131 portfolio options, 175, 176, 195, 200

599

probability, 29, 30, 40, 53, 127, 169, 181, 183, 184, 205, 219, 274, 275, 299, 318, 334, 336–338, 340, 342, 353, 356, 358, 366, 368, 370, 375, 388, 389, 403, 404, 406–409, 412, 416, 419, 420, 427, 430, 433, 435, 438, 439, 442, 445, 447, 448, 451, 455, 461–465, 467–470, 472–475, 479–481, 485, 489, 490, 494–499, 501, 503–507, 512, 513, 519–524, 526–528, 533, 537, 538, 556, 557, 575, 577, 578 puts, 1, 25, 71, 72, 131, 142, 196, 209–212, 225, 268, 299, 301, 318, 320, 327, 329, 334, 341, 343–345, 362, 393–396, 398, 460, 467, 488, 492, 493, 542 puts on calls, 131 puts on puts, 131 quantos, 226–228, 234, 240, 265 rainbow options, 175, 176, 189, 199, 200, 562 range forward contract, 226, 227, 244–246, 263, 265 ratchet options, 131, 141, 173 real options, 121, 541, 550–552, 559–563, 565, 566, 574, 577–581, 593, 595 replication argument, 74, 76, 101, 117, 545, 546 reporting management system, 403, 436 risk measures, 75, 117, 144, 403, 414, 427, 436, 439, 440, 449, 460, 474 risk-neutral argument, 75, 77, 116, 117 risk-neutral valuation, 75, 113, 125, 380 RiskMetrics, 403–406, 408, 412, 413, 432, 433, 435, 436, 440, 471 risky asset, 71, 76, 131–133, 139, 172, 176, 177 risky debt, 511, 512, 515, 523, 531, 534–539, 543, 546, 547 semilookbacks, 317, 342 several combined strategies, 1, 11 simple arithmetic, 372, 377, 401 simple chooser options, 131, 148, 149, 151, 173

October 23, 2008

600

16:48

B-613

9in x 6in

Index

Exotic Derivatives and Risk: Theory, Extensions and Applications

simple writer extendible options, 202, 210 soft barrier options, 267, 268, 306, 308, 310, 315, 316 soft binary options, 267, 301, 316 spot asset, 6, 74, 76, 106, 115–117, 317 standard DCF analysis, 550, 557, 579 standard lookbacks, 317 static hedges, 372, 374 stochastic interest rates, 511, 512, 516, 533, 535, 538 stochastic risk, 511, 516, 520 structured Monte-Carlo methods, 403 survival function, 511, 526, 527 swaps, 2, 226, 250, 254, 255, 257, 258, 262–264, 410, 449, 450, 481, 484–488, 491, 511, 537–539

swaptions, 226, 254, 262, 264 switch options, 267, 301, 302, 304, 315, 380 Taylor series, 74, 76, 82, 98, 100–102, 117, 348, 416, 424 uncertain strike price, 131, 139, 140, 172, 173 underlying assets, 1, 6, 58, 72, 135, 137, 176, 181, 191, 192, 298, 319, 346, 431, 487, 498 value at risk, 403–406, 411, 440, 505 wildcard options, 128, 175, 177, 199, 200