Applications of Fourier Transform to Smile Modeling (eBook)

Preface 7
Contents 11
Option Valuation and the Volatility Smile 16
Stochastic Processes for Stocks 16
Brownian Motion 16
Stock Price as Geometric Brownian Motion 18
Itô Process and Itô's Lemma 19
The Black-Scholes Model 20
Options and Dynamic Hedging 20
Risk-Neutral Valuation 22
Self-financing and No Arbitrage 23
Equivalent Martingale Measures 26
Volatility Quotations in Markets 30
Implied Volatilities 30
Market Quotations 31
Special Case: FX Market 32
Characteristic Functions in Option Pricing 35
Constructing Characteristic Functions (CFs) 36
A General Process for Stock Price 36
Valuation of European-style Options via CFs 37
Special Case: FX Options 41
Understanding Characteristic Functions 43
Properties of Characteristic Functions 43
Economic Interpretation of CFs 46
Examination of Existing Option Models 49
Relationship between CF to PDE 53
Advantages of CF and Modular Pricing 56
Stochastic Volatility Models 58
Introduction 58
The Heston Model 61
Model Setup and Properties 61
PDE Approach to Pricing Formula 63
Expectation Approach to Pricing Formula 65
Various Representations of CFs 67
The Schöbel-Zhu Model 68
Model Setup and Properties 68
Derivation of CFs 71
Numerical Examples 73
Double Square Root Model 76
Model Setup and Properties 76
Numerical Examples 82
Other Stochastic Volatility Models 83
Appendices 86
Numerical Issues of Stochastic Volatility Models 90
Alternative Pricing Formulas with CFs 91
The Formula á la Black-Scholes 91
The Carr and Madan Formula 91
The Attari Formula 93
Risk Sensitivities 93
Delta and Gamma 94
Various Vegas 95
Curvature and Slope 97
Volga and Vanna 97
Direct Integration (DI) 99
The Gaussian Integration 99
Multi-Domain Integration 100
Strike Vector Computation 101
Fast Fourier Transform (FFT) 102
Algorithms of FFT 102
Restrictions 104
Direct Integration vs. FFT 105
Computation Speed 106
Computation Accuracy 107
Matching Market Data 107
Calculation of Greeks 108
Implementation 108
Logarithm of Complex Number 112
Definition 112
Three Algorithms Dealing with Branch Cut 114
When Main Argument Is Appropriate 116
Calibration to Market Data 117
General Procedure 117
Fixing Velocity Parameter 118
Fixing Spot Volatility 119
Markovian Projection 121
Simulating Stochastic Volatility Models 125
Simulation Scheme 126
Discretization 126
Moment-Matching 127
Problems in the Heston Model 128
Negative Values in Paths 128
Log-normal Scheme 129
Transformed Volatility Scheme 130
QE Scheme 132
The Broadie-Kaya Scheme 134
Some Other Schemes 136
Simulation Examples 137
Maximum and Minimum 138
Multi-Asset Model 142
Stochastic Interest Models 146
Introduction 146
The Cox-Ingosoll-Ross Model 149
The Zero-Correlation Case 149
The Correlation Case 151
The Vasicek Model 153
The Longstaff Model 155
The Zero-Correlation Case 156
The Correlation Case 157
Correlations with Stock Returns: SI versus SV 159
Poisson Jumps 164
Introduction 164
Simple Jumps 169
Lognormal Jumps 171
Pareto Jumps 174
The Kou Model: An Equivalence to Pareto Jumps 176
Affine Jump-Diffusions 179
Lévy Jumps 184
Introduction 185
Stochastic Clock Models 188
Variance-Gamma Model 190
Normal Inverse Gaussian Model 192
Time-Changed Lévy Process 194
Uncorrelated Time-Change 195
Correlated Time-Change 199
The Barndorff-Nielsen and Shephard Model 204
Alpha Log-Stable Model 205
Empirical Performance of Various Lévy Processes 207
Monte-Carlo Simulation 209
Generating Random Variables 209
Simulation of Lévy Process 212
Integrating Various Stochastic Factors 214
Stochastic Factors as Modules 214
Integration Approaches 217
Modular Approach 217
Time-Change Approach 223
Pricing Kernels for Options and Bonds 228
Criterions for Model Choice 229
Exotic Options with Stochastic Volatilities 233
Forward-Starting Options 234
Barrier Options 236
Introduction 236
Two Special Cases 238
Numerical Examples 243
Lookback Options 245
Introduction 245
Pricing Formulas with Stochastic Factors 248
Asian Options 254
Introduction 254
The Black-Scholes World 255
Asian Options in a Stochastic World 259
Approximations for Arithmetic Average Asian Options 262
A Model for Asian Interest Rate Options 264
Correlation Options 267
Introduction 267
Exchange Options 270
Quotient Options 273
Product Options 274
Other Exotic Options 276
Appendices 277
Libor Market Model with Stochastic Volatilities 282
Introduction 282
Standard Libor Market Model 284
Model Setup 284
Term Structure and Smile of Volatility 289
Swap Market Model 291
Model Setup 291
Correlation Structure 295
Convexity Adjustments for CMS 300
Incorporating Stochastic Volatility 303
The Andersen and Brotherton-Ratcliffe Model 304
The Piterbarg Model 307
The Wu and Zhang Model 311
The Zhu Model 314
The Belomestny, Matthew and Schoenmakers Model 321
Conclusive Remarks 325
References 327
Index 335