The Heston Model and its Extensions in Matlab and C# E-Book

Fabrice D. Rouah

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley Finance Editions
• Sprache: Englisch

Seitenzahl: 520

Veröffentlichungsjahr: 2013

Beschreibung

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.

Leseprobe

Table of Contents

Title Page

Copyright

Foreword

Preface

Acknowledgments

Chapter 1: The Heston Model for European Options

Model Dynamics

The European Call Price

The Heston PDE

Obtaining the Heston Characteristic Functions

Solving the Heston Riccati Equation

Dividend Yield and the Put Price

Consolidating the Integrals

Black-Scholes as a Special Case

Summary of the Call Price

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

The Fourier Transform

Recovery of Probabilities With Gil-Pelaez Fourier Inversion

Derivation of Gatheral (2006)

Attari (2004) Representation

Carr and Madan (1999) Representation

Bounds on the Carr-Madan Damping Factor and Optimal Value

The Carr-Madan Representation for Puts

The Representation for OTM Options

Conclusion

Chapter 4: The Fundamental Transform for Pricing Options

The Payoff Transform

The Fundamental Transform and the Option Price

The Fundamental Transform for the Heston Model

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 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

Conclusion

Chapter 7: Simulation in the Heston Model

General Setup

Euler Scheme

Milstein Scheme

Milstein Scheme for the Heston Model

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

The Weighted Method

Boundary Conditions for the PDE

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

Code Functionality by Chapter

Access to the Code and File Formats

Index

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Cover illustration: Gilles Gheerbrant, “1 2 3 4 au hasard” (1976);

© Gilles Gheerbrant

Cover design: Gilles Gheerbrant

Copyright © 2013 by Fabrice Douglas Rouah. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.

Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Rouah, Fabrice, 1964-

The Heston model and its extensions in Matlab and C# / Fabrice Douglas Rouah.

pages cm. – (Wiley finance series)

Includes bibliographical references and index.

ISBN 978-1-118-54825-7 (paper); ISBN 978-1-118-69518-0 (ebk); ISBN 978-1-118-69517-3 (ebk)

1. Options (Finance)–Mathematical models. 2. Options (Finance)–Prices. 3. Finance–Mathematical models. 4. MATLAB. 5. C# (Computer program language) I. Title.

HG6024.A3R6777 2013

332.64′53028553–dc23

2013019475

Foreword

I am pleased to introduce The Heston Model and Its Extensions in Matlab and C# by Fabrice Rouah. Although I was already familiar with his previous book entitled Option Pricing Models and Volatility Using Excel/VBA, I was pleasantly surprised to discover he had written a book devoted exclusively to the model that I developed in 1993 and to the many enhancements that have been brought to the original model in the twenty years since its introduction. Obviously, this focus makes the book more specialized than his previous work. Indeed, it contains detailed analyses and extensive computer implementations that will appeal to careful, interested readers. This book should interest a broad audience of practitioners and academics, including graduate students, quants on trading desks and in risk management, and researchers in option pricing and financial engineering.

There are existing computer programs for calculating option prices, such as those in Rouah's prior book or those available on Bloomberg systems. But this book offers more. In particular, it contains detailed theoretical analyses in addition to practical Matlab and C# code for implementing not only the original model, but also the many extensions that academics and practitioners have developed specifically for the model. The book analyzes numerical integration, the calculation of Greeks, American options, many simulation-based methods for pricing, finite difference numerical schemes, and recent developments such as the introduction of time-dependent parameters and the double version of the model. The breadth of methods covered in this book provides comprehensive support for implementation by practitioners and empirical researchers who need fast and reliable computations.

The methods covered in this book are not limited to the specific application of option pricing. The techniques apply to many option and financial engineering models. The book also illustrates how implementation of seemingly straightforward mathematical models can raise many questions. For example, one colleague noted that a common question on the Wilmott forums was how to calculate a complex logarithm while still guaranteeing that the option model produces real values. Obviously, an imaginary option value will cause problems in practice! This book resolves many similar difficulties and will reward the dedicated reader with clear answers and practical solutions. I hope you enjoy reading it as much as I did.

Professor Steven L. Heston Robert H. Smith School of Business University of Maryland January 3, 2013

Preface

In the twenty years since its introduction in 1993, the Heston model has become one of the most important models, if not the single most important model, in a then-revolutionary approach to pricing options known as stochastic volatility modeling. To understand why this model has become so important, we must revisit an event that shook financial markets around the world: the stock market crash of October 1987 and its subsequent impact on mathematical models to price options.

The exacerbation of smiles and skews in the implied volatility surface that resulted from the crash brought into question the ability of the Black-Scholes model to provide adequate prices in a new regime of volatility skews, and served to highlight the restrictive assumptions underlying the model. The most tenuous of these assumptions is that of continuously compounded stock returns being normally distributed with constant volatility. An abundance of empirical studies since the 1987 crash have shown that this assumption does not hold in equities markets. It is now a stylized fact in these markets that returns distributions are not normal. Returns exhibit skewness, and kurtosis—fat tails—that normality cannot account for. Volatility is not constant in time, but tends to be inversely related to price, with high stock prices usually showing lower volatility than low stock prices. A number of researchers have sought to eliminate this assumption in their models, by allowing volatility to be time-varying.

One popular approach for allowing time-varying volatility is to specify that volatility be driven by its own stochastic process. The models that use this approach, including the Heston (1993) model, are known as stochastic volatility models. The models of Hull and White (1987), Scott (1987), Wiggins (1987), Chensey and Scott (1989), and Stein and Stein (1991) are among the most significant stochastic volatility models that pre-date Steve Heston's model. The Heston model was not the first stochastic volatility model to be introduced to the problem of pricing options, but it has emerged as the most important and now serves as a benchmark against which many other stochastic volatility models are compared.

Allowing for non-normality can be done by introducing skewness and kurtosis in the option price directly, as done, for example, by Jarrow and Rudd (1982), Corrado and Su (1997), and Backus, Foresi, and Wu (2004). In these models, skewness and kurtosis are specified in Edgeworth expansions or Gram-Charlier expansions. In stochastic volatility models, skewness can be induced by allowing correlation between the processes driving the stock price and the process driving its volatility. Alternatively, skewness can arise by introducing jumps into the stochastic process driving the underlying asset price.

The parameters of the Heston model are able to induce skewness and kurtosis, and produce a smile or skew in implied volatilities extracted from option prices generated by the model. The model easily allows for the inverse relationship between price level and volatility in a manner that is intuitive and easy to understand. Moreover, the call price in the Heston model is available in closed form, up to an integral that must be evaluated numerically. For these reasons, the Heston model has become the most popular stochastic volatility model for pricing equity options.

Another reason the Heston model is so important is that it is the first to exploit characteristic functions in option pricing, by recognizing that the terminal price density need not be known, only its characteristic function. This crucial line of reasoning was the genesis for a new approach for pricing options, known as pricing by characteristic functions. See Zhu (2010) for a discussion.

In this book, we present a treatment of the classical Heston model, but also of the many extensions that researchers from the academic and practitioner communities have contributed to this model since its inception. In Chapter 1, we derive the characteristic function and call price of Heston's (1993) original derivation. Chapter 2 deals with some of the issues around the model such as integrand discontinuities, and also shows how to model implied and local volatility in the model. Chapter 3 presents several Fourier transform methods for the model, and Chapter 4 deals exclusively with Alan Lewis' (2000, 2001) approach to stochastic volatility modeling, as it applies to the Heston model. Chapter 5 presents a variety of numerical integration schemes and explains how integration can be speeded up. Chapter 6 deals with parameter estimation, and Chapter 7 presents classical simulation schemes applied to the model and several simulation schemes designed specifically for the model. Chapter 8 deals with pricing American options in the Heston model. Chapter 9 presents models in which the parameters of the original Heston model are allowed to be piecewise constant. Chapter 10 presents methods for obtaining the call price that rely on solving the Heston partial differential equation with finite differences. Chapter 11 presents the Greeks in the Heston model. Finally, Chapter 12 presents the double Heston model, which introduces an additional stochastic process for variance and thus allows the model to provide a better fit to the volatility surface.

All of the models presented in this book have been coded in Matlab and C#.

Acknowledgments

I would like to thank Steve Heston not only for having bestowed his model to the financial engineering community, but also for contributing the Foreword to this book and to Leif B.G. Andersen, Marco Avellaneda, Peter Christoffersen, Jim Gatheral, Espen Gaarder Haug, Andrew Lesniewski, and Alan Lewis for their generous endorsement. And to my team at Wiley—Bill Falloon, Meg Freeborn, Steven Kyritz, and Tiffany Charbonier—thank you. I am also grateful to Gilles Gheerbrant for his strikingly beautiful cover design.

Special thanks also to a group who offered moral support, advice, and technical reviews of the material in this book: Amir Atiya, Sébastien Bossu, Carl Chiarella, Elton Daal, Redouane El-Kamhi, Judith Farer, Jacqueline Gheerbrant, Emmanuel Gobet, Greg N. Gregoriou, Antoine Jacquier, Dominique Legros, Pierre Leignadier, Alexey Medvedev, Sanjay K. Nawalkha, Razvan Pascalau, Jean Rouah, Olivier Scaillet, Martin Schmelzle, and Giovanna Sestito. Lastly, a special mention to Kevin Samborn at Sapient Global Markets for his help and support.

Chapter 1

The Heston Model for European Options

Abstract

In this chapter, we present a complete derivation of the European call price under the Heston model. We first present the model and obtain the various partial differential equations (PDEs) that arise in the derivation. We show that the call price in the Heston model can be expressed as the sum of two terms that each contains an in-the-money probability, but obtained under a separate measure, a result demonstrated by Bakshi and Madan (2000). We show how to obtain the characteristic function for the Heston model, and how to solve the Riccati equation from which the characteristic function is derived. We then show how to incorporate a continuous dividend yield and how to compute the price of a European put, and demonstrate that the numerical integration can be speeded up by consolidating the two numerical integrals into a single integral. Finally, we derive the Black-Scholes model as a special case of the Heston model.

Model Dynamics

The Heston model assumes that the underlying stock price, , follows a Black-Scholes–type stochastic process, but with a stochastic variance that follows a Cox, Ingersoll, and Ross (1985) process. Hence, the Heston model is represented by the bivariate system of stochastic differential equations (SDEs)

1.1

where .

We will sometimes drop the time index and write , , and for notational convenience. The parameters of the model are

the drift of the process for the stock;

the mean reversion speed for the variance;

the mean reversion level for the variance;

the volatility of the variance;

the initial (time zero) level of the variance;

the correlation between the two Brownian motions and ; and

the volatility risk parameter. We define this parameter in the next section and explain why we set this parameter to zero.

We will see in Chapter 2 that these parameters affect the distribution of the terminal stock price in a manner that is intuitive. Some authors refer to as an unobserved initial state variable, rather than a parameter. Because volatility cannot be observed, only estimated, and because represents this state variable at time zero, this characterization is sensible. For the purposes of estimation, however, many authors treat as a parameter like any other. Parameter estimation is covered in Chapter 6.

It is important to note that the volatility is not modeled directly in the Heston model, but rather through the variance . The process for the variance arises from the Ornstein-Uhlenbeck process for the volatility given by

1.2

Applying It's lemma, follows the process

1.3

Defining , , and expresses from Equation (1.1) as (1.3).

The stock price and variance follow the processes in Equation (1.1) under the historical measure , also called the physical measure. For pricing purposes, however, we need the processes for under the risk-neutral measure . In the Heston model, this is done by modifying each SDE in Equation (1.1) separately by an application of Girsanov's theorem. The risk-neutral process for the stock price is

1.4

where

It is sometimes convenient to express the price process in terms of the log price instead of the price itself. By an application of It's lemma, the log price process is

The risk-neutral process for the log price is

1.5

If the stock pays a continuous dividend yield, , then in Equations (1.4) and (1.5) we replace by .

The risk-neutral process for the variance is obtained by introducing a function into the drift of in Equation (1.1), as follows

1.6

where

1.7

The function is called the volatility risk premium. As explained in Heston (1993), Breeden's (1979) consumption model yields a premium proportional to the variance, so that , where is a constant. Substituting for in Equation (1.6), the risk-neutral version of the variance process is

1.8

where and are the risk-neutral parameters of the variance process.

To summarize, the risk-neutral process is

1.9

where and with the risk-neutral measure.

Note that, when , we have and so that these parameters under the physical and risk-neutral measures are the same. Throughout this book, we set , but this is not always needed. Indeed, is embedded in the risk-neutral parameters and . Hence, when we estimate the risk-neutral parameters to price options we do not need to estimate . Estimation of is the subject of its own research, such as that by Bollerslev et al. (2011). For notational simplicity, throughout this book we will drop the asterisk on the parameters and the tilde on the Brownian motion when it is obvious that we are dealing with the risk-neutral measure.

Properties of the Variance Process

The properties of are described by Cox, Ingersoll, and Ross (1985) and Brigo and Mercurio (2006), among others. It is well-known that conditional on a realized value of , the random variable (for ) follows a non-central chi-square distribution with degrees of freedom and non-centrality parameter , where

1.10

and with . The mean and variance of , conditional on the value are, respectively

1.11

The effect of the mean reversion speed on the moments is intuitive and explained in Cox, Ingersoll, and Ross (1985). When the mean approaches the mean reversion rate and the variance approaches zero. As the mean approaches the current level of variance, , and the variance approaches .

If the condition holds, then the drift is sufficiently large for the variance process to be guaranteed positive and not reach zero. This condition is known as the Feller condition.

The European Call Price

In this section, we show that the call price in the Heston model can be expressed in a manner which resembles the call price in the Black-Scholes model, which we present in Equation (). Authors sometimes refer to this characterization of the call price as “Black-Scholes–like” or “à la Black-Scholes.” The time- price of a European call on a non-dividend paying stock with spot price , when the strike is and the time to maturity is , is the discounted expected value of the payoff under the risk-neutral measure