Numerical Solution Of The American Option Pricing Problem, The: Finite Difference And Transform Approaches

by Carl Chiarella, Boda Kang, Gunter H Meyer

Overview

The early exercise opportunity of an American option makes it challenging to price and an array of approaches have been proposed in the vast literature on this topic. In The Numerical Solution of the American Option Pricing Problem, Carl Chiarella, Boda Kang and Gunter Meyer focus on two numerical approaches that have proved useful for finding all prices, hedge ratios and early exercise boundaries of an American option. One is a finite difference approach which is based on the numerical solution of the partial differential equations with the free boundary problem arising in American option pricing, including the method of lines, the component wise splitting and the finite difference with PSOR. The other approach is the integral transform approach which includes Fourier or Fourier Cosine transforms. Written in a concise and systematic manner, Chiarella, Kang and Meyer explain and demonstrate the advantages and limitations of each of them based on their and their co-workers' experiences with these approaches over the years.

Product Details

ISBN-13:	9789814452618
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	12/02/2014
Pages:	224
Product dimensions:	6.00(w) x 9.00(h) x 0.56(d)

Table of Contents

Preface v

1 Introduction 1

2 The Merton and Heston Model for a Call 3

2.1 The Model 3

3 American Call Options under Jump-Diffusion Processes 11

3.1 Introduction 11

3.2 The Problem Statement - Merton's Model 13

3.3 Jamshidian's Representation 15

3.4 Limit of the Early Exercise Boundary at Expiry 27

3.5 The American Call with Log-Normal Jumps 29

3.5.1 Delta for the American call 33

3.6 Properties of the Free Boundary at Expiry 34

3.7 Numerical Implementation 36

3.8 Numerical Results 38

Appendix 41

3.A Proof of Proposition 3.3 41

3.B Proof of Proposition 3.4 44

3.C Proof of Proposition 3.9 46

4 American Option Prices under Stochastic Volatility and Jump-Diffusion Dynamics - The Transform Approach 49

4.1 Introduction 49

4.2 The Problem Statement The Merton-Heston Model 50

4.3 The Integral Transform Solution 53

4.4 The Martingale Representation 62

4.5 Conclusion 69

Appendix 69

4.A Deriving the Inhomogeneous PIDE 69

4.B Verifying Duhamel's Principle 70

4.C Proof of Proposition 4.3 - Fourier Transform of the PIDE 72

4.D Proof of Proposition 4.4 - Laplace Transform of the PDE (4.14) 73

4.E Proof of Proposition 4.5 - Solving the PDE (4.17) 74

4.F Proof of Proposition 4.6 - Inverting the Laplace Transform 83

4.G Proof of Proposition 4.7 Inverting the Fourier Transform 86

4.H Proof of Proposition 4.8 Deriving the Price for a European Call 87

4.1 Deriving the Early Exercise Premium 89

5 Representation and Numerical Approximation of American Option Prices under Heston 93

5.1 Introduction 93

5.2 Problem Statement The Heston Model 97

5.3 Finding the Density Function using Integral Transforms 100

5.4 Solution for the American Call Option 105

5.5 Numerical Scheme for the Free Surface 109

5.6 Conclusion 119

Appendix 120

5.A Proof of Proposition 5.8 - The European Option Price 120

5.B Evaluation of Common Integral Terms in the Heston Model 124

5.C Calculation of the Deltas 126

5.D Proof of Proposition 5.12 131

5.E Moments for the Heston Model 134

5.F Method of Lines for the Heston PDE 135

6 Fourier Cosine Expansion Approach 141

6.1 Heston Model 141

6.1.1 Transformation to the log-variance process 142

6.2 The Pricing Method for European Options 144

6.3 The Pricing Method for American Options 147

6.3.1 The pricing equations 147

6.3.2 Density recovery by Fourier cosine expansions 148

6.3.3 Discrete Fourier-based pricing formula 150

6.4 Two (Higher) Dimensional COS Methods 157

6.4.1 American max option 161

6.4.2 Recursion formula for coefficients V_k1,_k2 (t_m) 161

6.4.3 Approximation methods for the coefficients V(T) and E{ε _v) 165

6.4.4 3D-COS formula 166

6.5 Numerical Results 167

7 A Numerical Approach to Pricing American Call Options under SVJD 169

7.1 The PDE Formulation 170

7.2 Numerical Solution using Finite Differences with Projected Over relaxation (PSOR) 175

7.3 Componentwise Splitting for the SVJD Call 178

7.4 The Method of Lines for the SVJD Call 184

7.5 Numerical Results 188

8 Conclusion 199

Bibliography 201

Index 207

About the Authors 211