The Heston Model and Its Extensions in VBA

Practical 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 tools—the Heston model, and VBA. Light on theory, this extremely useful reference focuses on implementation, and can help investors more efficiently—and accurately—exploit 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 model is the derivatives industry's most popular stochastic volatility model for pricing equity derivatives. This book provides complete guidance toward the successful implementation of this valuable model using the industry's ubiquitous financial modeling software, giving users the understanding—and VBA code—they need to produce option prices that are more accurate, and volatility surfaces that more closely reflect market conditions.

Derivatives pricing is often the hinge on which profit is made or lost in financial institutions, making accuracy of utmost importance. This book will help risk managers, traders, portfolio managers, quants, academics and other professionals better understand the Heston model and its extensions, in a writing style that is clear, concise, transparent and easy to understand. For better pricing accuracy, The Heston Model and Its Extensions in VBA is a crucial resource for producing more accurate model outputs such as prices, hedge ratios, volatilities, and graphs.

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

FABRICE DOUGLAS ROUAH was a quantitative analyst who specialized 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.

Praise for The Heston Model and Its Extensions in VBA

"In his excellent new book, Fabrice Rouah provides a careful presentation of all aspects of the Heston model, with a strong emphasis on getting the model up and running in practice. This highly practical and useful book is recommended for anyone working with stochastic volatility models."
—Leif B. G. Andersen, Bank of America Merrill Lynch

"Without a doubt, Fabrice provides a very valuable contribution to quantitative analysts interested in pricing options with state-of-the art techniques."
—Marco Avellaneda, New York University

"The Heston model is one of the great success stories of academic finance. Rouah's impressive book provides users with all the tools required to implement the Heston model, and wonderfully bridges the gap between academia and practice."
—Peter Christoffersen, University of Toronto

"In this encyclopedic work, the author takes delight in exploring every aspect of the Heston model. Together with its code, this book will prove invaluable to anyone interested in option pricing. I highly recommend it."
—Jim Gatheral, Baruch College, author of The Volatility Surface: A Practitioner's Guide

"This is the most extensive work on the Heston model I have seen: derivations, implementations, and discussions. For anyone interested in the Heston model and its variations, this is an important book to have!"
—Espen Gaarder Haug, Norwegian University of Life Sciences, author of Derivatives Models on Models

"Rouah offers a unique and much needed synthesis of the literature regarding Heston's model of stochastic volatility. The author has accomplished the formidable task of presenting a large body of published academic and industrial research in a coherent, thorough, and very reader-friendly manner."
—Andrew Lesniewski, DTCC

"Beyond Black-Scholes, the Heston model is arguably the most important model in quantitative finance and certainly deserves its own book. Rouah provides here a comprehensive treatment—clearly discussing all the major issues, later extensions, and subtle traps."
—Alan L. Lewis, PhD, author of Option Valuation Under Stochastic Volatility: With Mathematica Code