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The Heston Model and its Extensions in Matlab and C#


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Foreword ix Preface xi Acknowledgments xiii CHAPTER 1 The Heston Model for European Options 1 Model Dynamics 1 The European Call Price 4 The Heston PDE 5 Obtaining the Heston Characteristic Functions 10 Solving the Heston Riccati Equation 12 Dividend Yield and the Put Price 17 Consolidating the Integrals 18 Black-Scholes as a Special Case 19 Summary of the Call Price 22 Conclusion 23 CHAPTER 2 Integration Issues, Parameter Effects, and Variance Modeling 25 Remarks on the Characteristic Functions 25 Problems With the Integrand 29 The Little Heston Trap 31 Effect of the Heston Parameters 34 Variance Modeling in the Heston Model 43 Moment Explosions 56 Bounds on Implied Volatility Slope 57 Conclusion 61 CHAPTER 3 Derivations Using the Fourier Transform 63 The Fourier Transform 63 Recovery of Probabilities With Gil-Pelaez Fourier Inversion 65 Derivation of Gatheral (2006) 67 Attari (2004) Representation 69 Carr and Madan (1999) Representation 73 Bounds on the Carr-Madan Damping Factor and Optimal Value 76 The Carr-Madan Representation for Puts 82 The Representation for OTM Options 84 Conclusion 89 CHAPTER 4 The Fundamental Transform for Pricing Options 91 The Payoff Transform 91 The Fundamental Transform and the Option Price 92 The Fundamental Transform for the Heston Model 95 Option Prices Using Parseval?s Identity 100 Volatility of Volatility Series Expansion 108 Conclusion 113 CHAPTER 5 Numerical Integration Schemes 115 The Integrand in Numerical Integration 116 Newton-Cotes Formulas 116 Gaussian Quadrature 121 Integration Limits and Kahl and J ackel Transformation 130 Illustration of Numerical Integration 136 Fast Fourier Transform 137 Fractional Fast Fourier Transform 141 Conclusion 145 CHAPTER 6 Parameter Estimation 147 Estimation Using Loss Functions 147 Speeding up the Estimation 158 Differential Evolution 162 Maximum Likelihood Estimation 166 Risk-Neutral Density and Arbitrage-Free Volatility Surface 170 Conclusion 175 CHAPTER 7 Simulation in the Heston Model 177 General Setup 177 Euler Scheme 179 Milstein Scheme 181 Milstein Scheme for the Heston Model 183 Implicit Milstein Scheme 185 Transformed Volatility Scheme 188 Balanced, Pathwise, and IJK Schemes 191 Quadratic-Exponential Scheme 193 Alfonsi Scheme for the Variance 198 Moment Matching Scheme 201 Conclusion 202 CHAPTER 8 American Options 205 Least-Squares Monte Carlo 205 The Explicit Method 213 Beliaeva-Nawalkha Bivariate Tree 217 Medvedev-Scaillet Expansion 228 Chiarella and Ziogas American Call 253 Conclusion 261 CHAPTER 9 Time-Dependent Heston Models 263 Generalization of the Riccati Equation 263 Bivariate Characteristic Function 264 Linking the Bivariate CF and the General Riccati Equation 269 Mikhailov and No gel Model 271 Elices Model 278 Benhamou-Miri-Gobet Model 285 Black-Scholes Derivatives 299 Conclusion 300 CHAPTER 10 Methods for Finite Differences 301 The PDE in Terms of an Operator 301 Building Grids 302 Finite Difference Approximation of Derivatives 303 The Weighted Method 306 Boundary Conditions for the PDE 315 Explicit Scheme 316 ADI Schemes 321 Conclusion 325 CHAPTER 11 The Heston Greeks 327 Analytic Expressions for European Greeks 327 Finite Differences for the Greeks 332 Numerical Implementation of the Greeks 333 Greeks Under the Attari and Carr-Madan Formulations 339 Greeks Under the Lewis Formulations 343 Greeks Using the FFT and FRFT 345 American Greeks Using Simulation 346 American Greeks Using the Explicit Method 349 American Greeks from Medvedev and Scaillet 352 Conclusion 354 CHAPTER 12 The Double Heston Model 357 Multi-Dimensional Feynman-KAC Theorem 357 Double Heston Call Price 358 Double Heston Greeks 363 Parameter Estimation 368 Simulation in the Double Heston Model 373 American Options in the Double Heston Model 380 Conclusion 382 Bibliography 383 About the Website 391 Index 397

About the Author

FABRICE DOUGLAS ROUAH is a quantitative analyst who specializes in financial modeling of derivatives for pricing and risk management at Sapient Global Markets, a global consultancy. Prior to joining Sapient, Rouah worked at State Street Corporation and McGill University. He is the coauthor and/or coeditor of five books on hedge funds, commodity trading advisors, and option pricing. Rouah holds a PhD in finance and an MSc in statistics from McGill University, and a BSc in applied mathematics from Concordia University.

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