The Heston Model and Its Extensions in VBA + Website

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

Foreword

Preface

Acknowledgments

About This Book

VBA Library for Complex Numbers

Chapter 1: The Heston Model for European Options

Model Dynamics

The Heston European Call Price

Dividend Yield and the Put Price

Consolidating the Integrals

Black–Scholes as a Special Case

Conclusion

Chapter 2: Integration Issues, Parameter Effects, and Variance Modeling

Remarks on the Characteristic Functions

Problems With the Integrand

The Little Heston Trap

Effect of the Heston Parameters

Variance Modeling in the Heston Model

Moment Explosions

Bounds on Implied Volatility Slope

Conclusion

Chapter 3: Derivations Using the Fourier Transform

Derivation of Gatheral (2006)

Attari (2004) Representation

Carr and Madan (1999) Representation

Conclusion

Chapter 4: The Fundamental Transform for Pricing Options

The Payoff Transform

Option Prices Using Parseval s Identity

Volatility of Volatility Series Expansion

Conclusion

Chapter 5: Numerical Integration Schemes

The Integrand in Numerical Integration

Newton–Cotes Formulas

Gaussian Quadrature

Integration Limits, Multi–Domain Integration, and Kahl and Jäckel Transformation

Illustration of Numerical Integration

Fast Fourier Transform

Fractional Fast Fourier Transform

Conclusion

Chapter 6: Parameter Estimation

Estimation Using Loss Functions

Speeding up the Estimation

Differential Evolution

Maximum Likelihood Estimation

Risk–Neutral Density and Arbitrage–Free Volatility Surface

Conclusion

Chapter 7: Simulation in the Heston Model

General Setup

Euler Scheme

Milstein Scheme

Implicit Milstein Scheme

Transformed Volatility Scheme

Balanced, Pathwise, and IJK Schemes

Quadratic–Exponential Scheme

Alfonsi Scheme for the Variance

Moment Matching Scheme

Conclusion

Chapter 8: American Options

Least–Squares Monte Carlo

The Explicit Method

Beliaeva–Nawalkha Bivariate Tree

Medvedev–Scaillet Expansion

Chiarella and Ziogas American Call

Conclusion

Chapter 9: Time–Dependent Heston Models

Generalization of the Riccati Equation

Bivariate Characteristic Function

Linking the Bivariate CF and the General Riccati Equation

Mikhailov and  Nögel Model

Elices Model

Benhamou–Miri–Gobet Model

Black–Scholes Derivatives

Conclusion

Chapter 10: Methods for Finite Differences

The PDE in Terms of an Operator

Building Grids

Finite Difference Approximation of Derivatives

Boundary Conditions for the PDE

The Weighted Method

Explicit Scheme

ADI Schemes

Conclusion

Chapter 11: The Heston Greeks

Analytic Expressions for European Greeks

Finite Differences for the Greeks

Numerical Implementation of the Greeks

Greeks Under the Attari and Carr–Madan Formulations

Greeks Under the Lewis Formulations

Greeks Using the FFT and FRFT

American Greeks Using Simulation

American Greeks Using the Explicit Method

American Greeks from Medvedev and Scaillet

Conclusion

Chapter 12: The Double Heston Model

Multi–Dimensional Feynman–Kac Theorem

Double Heston Call Price

Double Heston Greeks

Parameter Estimation

Simulation in the Double Heston Model

American Options in the Double Heston Model

Conclusion

Bibliography

About the Website

Index

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.