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The Heston Model and its Extensions in Matlab and C#

ISBN: 978-1-118-69517-3
432 pages
August 2013
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Description

Tap into the power of the most popular stochastic volatility model for pricing equity derivatives

Since its introduction in 1993, the Heston model has become a popular model for pricing equity derivatives, and the most popular stochastic volatility model in financial engineering. This vital resource provides a thorough derivation of the original model, and includes the most important extensions and refinements that have allowed the model to produce option prices that are more accurate and volatility surfaces that better reflect market conditions. The book's material is drawn from research papers and many of the models covered and the computer codes are unavailable from other sources.

The book is light on theory and instead highlights the implementation of the models. All of the models found here have been coded in Matlab and C#. This reliable resource offers an understanding of how the original model was derived from Ricatti equations, and shows how to implement implied and local volatility, Fourier methods applied to the model, numerical integration schemes, parameter estimation, simulation schemes, American options, the Heston model with time-dependent parameters, finite difference methods for the Heston PDE, the Greeks, and the double Heston model.

  • A groundbreaking book dedicated to the exploration of the Heston model—a popular model for pricing equity derivatives
  • Includes a companion website, which explores the Heston model and its extensions all coded in Matlab and C#
  • Written by Fabrice Douglas Rouah a quantitative analyst who specializes in financial modeling for derivatives for pricing and risk management

Engaging and informative, this is the first book to deal exclusively with the Heston Model and includes code in Matlab and C# for pricing under the model, as well as code for parameter estimation, simulation, finite difference methods, American options, and more.

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Table of Contents

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

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Author Information

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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