Description
Book SynopsisPractical options pricing for better-informed investment decisions. The Heston Model and Its Extensions in VBA is the definitive guide to options pricing using two of the derivatives industry''s most powerful modeling toolsthe Heston model, and VBA. Light on theory, this extremely useful reference focuses on implementation, and can help investors more efficientlyand accuratelyexploit market information to better inform investment decisions. Coverage includes a description of the Heston model, with specific emphasis on equity options pricing and variance modeling, The book focuses not only on the original Heston model, but also on the many enhancements and refinements that have been applied to the model, including methods that use the Fourier transform, numerical integration schemes, simulation, methods for pricing American options, and much more. The companion website offers pricing code in VBA that resides in an extensive set of Excel spreadsheets.
The Heston
Table of Contents
Foreword xi
Preface xiii
Acknowledgments xv
About This Book xvii
VBA Library for Complex Numbers xix
Chapter 1 The Heston Model for European Options 1
Model Dynamics 1
The Heston European Call Price 2
Dividend Yield and the Put Price 8
Consolidating the Integrals 9
Black-Scholes as a Special Case 10
Conclusion 12
Chapter 2 Integration Issues, Parameter Effects, and Variance Modeling 13
Remarks on the Characteristic Functions 14
Problems with the Integrand 16
The Little Heston Trap 18
Effect of the Heston Parameters 20
Variance Modeling in the Heston Model 26
Moment Explosions 38
Bounds on Implied Volatility Slope 40
Conclusion 42
Chapter 3 Derivations Using the Fourier Transform 45
Derivation of Gatheral (2006) 46
Attari (2004) Representation 47
Carr and Madan (1999) Representation 49
Conclusion 61
Chapter 4 The Fundamental Transform for Pricing Options 63
The Payoff Transform 64
Option Prices Using Parseval’s Identity 70
Volatility of Volatility Series Expansion 75
Conclusion 81
Chapter 5 Numerical Integration Schemes 83
The Integrand in Numerical Integration 84
Newton-Cotes Formulas 85
Gaussian Quadrature 90
Integration Limits, Multidomain Integration, and Kahl and Jäckel Transformation 98
Illustration of Numerical Integration 103
Fast Fourier Transform 106
Fractional Fast Fourier Transform 108
Conclusion 114
Chapter 6 Parameter Estimation 115
Estimation Using Loss Functions 116
Speeding Up the Estimation 126
Differential Evolution 128
Maximum Likelihood Estimation 132
Risk-Neutral Density and Arbitrage-Free Volatility Surface 135
Conclusion 140
Chapter 7 Simulation in the Heston Model 143
General Setup 144
Euler Scheme 146
Milstein Scheme 147
Implicit Milstein Scheme 149
Transformed Volatility Scheme 152
Balanced, Pathwise, and IJK Schemes 155
Quadratic-Exponential Scheme 157
Alfonsi Scheme for the Variance 161
Moment-Matching Scheme 165
Conclusion 167
Chapter 8 American Options 169
Least-Squares Monte Carlo 169
The Explicit Method 174
Beliaeva-Nawalkha Bivariate Tree 178
Medvedev-Scaillet Expansion 191
Chiarella and Ziogas American Call 200
Conclusion 208
Chapter 9 Time-Dependent Heston Models 209
Generalization of the Riccati Equation 209
Bivariate Characteristic Function 210
Linking the Bivariate CF and the General Riccati Equation 212
Mikhailov and Nögel Model 214
Elices Model 219
Benhamou-Miri-Gobet Model 223
Black-Scholes Derivatives 231
Conclusion 232
Chapter 10 Methods for Finite Differences 235
The PDE in Terms of an Operator 236
Building Grids 236
Finite Difference Approximation of Derivatives 239
Boundary Conditions for the PDE 240
The Weighted Method 241
Explicit Scheme 248
ADI Schemes 251
Conclusion 256
Chapter 11 The Heston Greeks 257
Analytic Expressions for European Greeks 258
Finite Differences for the Greeks 263
Numerical Implementation of the Greeks 264
Greeks under the Attari and Carr-Madan Formulations 267
Greeks under the Lewis Formulations 273
Greeks Using the FFT and FRFT 276
American Greeks Using Simulation 279
American Greeks Using the Explicit Method 281
American Greeks from Medvedev and Scaillet 284
Conclusion 285
Chapter 12 The Double Heston Model 287
Multidimensional Feynman-Kac Theorem 288
Double Heston Call Price 288
Double Heston Greeks 292
Parameter Estimation 297
Simulation in the Double Heston Model 301
American Options in the Double Heston Model 306
Conclusion 308
Bibliography 309
About the Website 317
Index 319
