# Mathematical Modeling and Computation in Finance: With Exercises and Python and MATLAB Computer Codes 1786348055, 9781786348050

This book discusses the interplay of stochastics (applied probability theory) and numerical analysis in the field of qua

3,079 688 9MB

English Pages 1310 Year 2019

Contents
Preface
Acknowledgment
Using this Book
1.1 Stochastic variables
1.1.1 Density function, expectation, variance
1.1.2 Characteristic function
1.1.3 Cumulants and moments
1.2 Stochastic processes, martingale property
1.2.1 Wiener process
1.2.2 Martingales
1.2.3 Iterated expectations (Tower property)
1.3 Stochastic integration, It ˆo integral
1.3.1 Elementary processes
1.3.2 Ito isometry
1.3.3 Martingale representation theorem
1.4 Exercise set
2 Introduction to Financial Asset Dynamics
2.1 Geometric Brownian motion asset price process
2.1.1 Ito process
2.1.2 Ito’s lemma
2.1.3 Distributions of S(t) and logS(t)
2.2 First generalizations
2.2.1 Proportional dividend model
2.2.2 Volatility variation
2.2.3 Time-dependent volatility
2.3 Martingales and asset prices
2.3.1 P-measure prices
2.3.2 Q-measure prices
2.3.3 Parameter estimation under real-world measure P
2.4 Exercise set
3 The Black-Scholes Option Pricing Equation
3.1 Option contract definitions
3.1.1 Option basics
3.1.2 Derivation of the partial differential equation
3.1.3 Martingale approach and option pricing
3.2 The Feynman-Kac theorem and the Black-Scholes model
3.2.1 Closed-form option prices
3.2.2 Green’s functions and characteristic functions
3.2.3 Volatility variations
3.3 Delta hedging under the Black-Scholes model
3.4 Exercise set
4 Local Volatility Models
4.1 Black-Scholes implied volatility
4.1.1 The concept of implied volatility
4.1.2 Implied volatility; implications
4.1.3 Discussion on alternative asset price models
4.2 Option prices and densities
4.2.1 Market implied volatility smile and the payoff
4.2.2 Variance swaps
4.3 Non-parametric local volatility models
4.3.1 Implied volatility representation of local volatility
4.3.2 Arbitrage-free conditions for option prices
4.3.4 Simulation of local volatility model
4.4 Exercise set
5 Jump Processes
5.1 Jump diffusion processes
5.1.1 Ito’s lemma and jumps
5.1.2 PIDE derivation for jump diffusion process
5.1.3 Special cases for the jump distribution
5.2 Feynman-Kac theorem for jump diffusion process
5.2.1 Analytic option prices
5.2.2 Characteristic function for Merton’s model
5.2.3 Dynamic hedging of jumps with the Black-Scholes model
5.3 Exponential Levy processes
5.3.1 Finite activity exponential L´evy processes
5.3.2 PIDE and the L´evy triplet
5.3.3 Equivalent martingale measure
5.4 Infinite activity exponential Levy processes
5.4.1 Variance Gamma process
5.4.2 CGMY process
5.4.3 Normal inverse Gaussian process
5.5 Discussion on jumps in asset dynamics
5.6 Exercise set
6 The COS Method for European Option Valuation
6.1 Introduction into numerical option valuation
6.1.1 Integrals and Fourier cosine series
6.1.2 Density approximation via Fourier cosine expansion
6.2 Pricing European options by the COS method
6.2.1 Payoff coefficients
6.2.2 The option Greeks
6.2.3 Error analysis COS method
6.2.4 Choice of integration range
6.3 Numerical COS method results
6.3.1 Geometric Brownian Motion
6.3.2 CGMY and VG processes
6.4 Exercise set
7 Multidimensionality, Change of Measure, Affine Processes
7.1 Preliminaries for multi-D SDE systems
7.1.1 The Cholesky decomposition
7.1.2 Multi-D asset price processes
7.1.3 Ito’s lemma for vector processes
7.1.4 Multi-dimensional Feynman-Kac theorem
7.2 Changing measures and the Girsanov theorem
7.2.2 Change of num´eraire examples
7.2.3 From P to Q in the Black-Scholes model
7.3 Affine processes
7.3.1 Affine diffusion processes
7.3.2 Affine jump diffusion processes
7.3.3 Affine jump diffusion process and PIDE
7.4 Exercise set
8 Stochastic Volatility Models
8.1 Introduction into stochastic volatility models
8.1.1 The Sch¨ obel-Zhu stochastic volatility model
8.1.2 The CIR process for the variance
8.2 The Heston stochastic volatility model
8.2.1 The Heston option pricing partial differential equation
8.2.2 Parameter study for implied volatility skew and smile
8.2.3 Heston model calibration
8.3 The Heston SV discounted characteristic function
8.3.1 Stochastic volatility as an affine diffusion process
8.3.2 Derivation of Heston SV characteristic function
8.4 Numerical solution of Heston PDE
8.4.1 The COS method for the Heston model
8.4.2 The Heston model with piecewise constant parameters
8.4.3 The Bates model
8.5 Exercise set
9 Monte Carlo Simulation
9.1 Monte Carlo basics
9.1.1 Monte Carlo integration
9.1.2 Path simulation of stochastic differential equations
9.2 Stochastic Euler and Milstein schemes
9.2.1 Euler scheme
9.2.2 Milstein scheme: detailed derivation
9.3 Simulation of the CIR process
9.3.1 Challenges with standard discretization schemes
9.3.2 Taylor-based simulation of the CIR process
9.3.3 Exact simulation of the CIR model
9.4 Monte Carlo scheme for the Heston model
9.4.1 Example of conditional sampling and integrated variance
9.4.2 The integrated CIR process and conditional sampling
9.4.3 Almost exact simulation of the Heston model
9.4.4 Improvements of Monte Carlo simulation
9.5 Computation of Monte Carlo Greeks
9.5.1 Finite differences
9.5.2 Pathwise sensitivities
9.5.3 Likelihood ratio method
9.6 Exercise set
10 Forward Start Options; Stochastic Local Volatility Model
10.1 Forward start options
10.1.1 Introduction into forward start options
10.1.2 Pricing under the Black-Scholes model
10.1.3 Pricing under the Heston model
10.1.4 Local versus stochastic volatility model
10.2 Introduction into stochastic-local volatility model
10.2.1 Specifying the local volatility
10.2.2 Monte Carlo approximation of SLV expectation
10.2.3 Monte Carlo AES scheme for SLV model
10.3 Exercise set
11 Short-Rate Models
11.1 Introduction to interest rates
11.1.1 Bond securities, notional
11.1.2 Fixed-rate bond
11.2 Interest rates in the Heath-Jarrow-Morton framework
11.2.1 The HJM framework
11.2.2 Short-rate dynamics under the HJM framework
11.2.3 The Hull-White dynamics in the HJM framework
11.3 The Hull-White model
11.3.1 The solution of the Hull-White SDE
11.3.2 The HW model characteristic function
11.3.3 The CIR model under the HJM framework
11.4 The HJM model under the T-forward measure
11.4.1 The Hull-White dynamics under the T-forward measure
11.4.2 Options on zero-coupon bonds under Hull-White model
11.5 Exercise set
12 Interest Rate Derivatives and Valuation Adjustments
12.1 Basic interest rate derivatives and the Libor rate
12.1.1 Libor rate
12.1.2 Forward rate agreement
12.1.3 Floating rate note
12.1.4 Swaps
12.1.5 How to construct a yield curve
12.2 More interest rate derivatives
12.2.1 Caps and floors
12.2.2 European swaptions
12.3 Credit Valuation Adjustment and Risk Management
12.3.2 Approximations in the calculation of CVA
12.3.3 Bilateral Credit Value Adjustment (BCVA)
12.3.4 Exposure reduction by netting
12.4 Exercise set
13 Hybrid Asset Models, Credit Valuation Adjustment
13.1 Introduction to affine hybrid asset models
13.1.1 Black-Scholes Hull-White (BSHW) model
13.1.2 BSHW model and change of measure
13.1.3 Schobel-Zhu Hull-White (SZHW) model
13.1.4 Hybrid derivative product
13.2 Hybrid Heston model
13.2.1 Details of Heston Hull-White hybrid model
13.2.2 Approximation for Heston hybrid models
13.2.3 Monte Carlo simulation of hybrid Heston SDEs
13.2.4 Numerical experiment, HHW versus SZHW model
13.3 CVA exposure profiles and hybrid models
13.3.1 CVA and exposure
13.3.2 European and Bermudan options example
13.4 Exercise set
14 Advanced Interest Rate Models and Generalizations
14.1 Libor market model
14.1.1 General Libor market model specifications
14.1.2 Libor market model under the HJM framework
14.2 Lognormal Libor market model
14.2.1 Change of measure in the LMM
14.2.2 The LMM under the terminal measure
14.2.3 The LMM under the spot measure
14.2.4 Convexity correction
14.3 Parametric local volatility models
14.3.1 Background, motivation
14.3.2 Constant Elasticity of Variance model (CEV)
14.3.3 Displaced diffusion model
14.3.4 Stochastic volatility LMM
14.4 Risk management: The impact of a financial crisis
14.4.1 Valuation in a negative interest rates environment
14.4.2 Multiple curves and the Libor rate
14.4.3 Valuation in a multiple curves setting
14.5 Exercise set
15 Cross-Currency Models
15.1 Introduction into the FX world and trading
15.1.1 FX markets
15.1.2 Forward FX contract
15.1.3 Pricing of FX options, the Black-Scholes case
15.2 Multi-currency FX model with short-rate interest rates
15.2.1 The model with correlated, Gaussian interest rates
15.2.2 Pricing of FX options
15.2.3 Numerical experiment for the FX-HHW model
15.2.4 CVA for FX swaps
15.3 Multi-currency FX model with interest rate smile
15.3.1 Linearization and forward characteristic function
15.3.2 Numerical experiments with the FX-HLMM model
15.4 Exercise set
References
Index