The Heston Model and Its Extensions in VBA

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