The Heston Model and Its Extensions in VBA E-Book

Fabrice D. Rouah

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

Seitenzahl: 470

Veröffentlichungsjahr: 2015

Beschreibung

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.

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The Heston Model and Its Extensions in VBA

FABRICE DOUGLAS ROUAH

Cover image: © Gilles Gheerbrant (1 2 3 4 au hasard) Cover design: © Gilles Gheerbrant

Copyright © 2015 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 VBA + website / Fabrice D. Rouah.       pages cm. – (Wiley finance)    Includes bibliographical references and index. ISBN 978-1-119-00330-4 (paperback) – ISBN 978-1-119-00332-8 (ePDF) – ISBN 978-1-119-00331-1 (ePub)   1. Options (Finance)–Mathematical models.2. Options (Finance)–Prices.   3. Finance–Mathematical models.   4. Visual Basic for Applications (Computer program language)   I. Title.    HG6024.A3R6778 2015    332.64′5302855133–dc23

2014043249

TO GIOVANNA

Contents

Foreword

Preface

Acknowledgments

About This Book

VBA Library for Complex Numbers

NOTE

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

NOTES

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, MULTIDOMAIN 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 NӦGEL MODEL

ELICES MODEL

BENHAMOU-MIRI-GOBET MODEL

BLACK-SCHOLES DERIVATIVES

CONCLUSION

NOTES

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

MULTIDIMENSIONAL 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

EULA

List of Tables

Chapter 2

Table 2.1

Chapter 4

Table 4.1

Chapter 5

Table 5.1

Table 5.2

Chapter 6

Table 6.1

Table 6.2

Chapter 7

Table 7.1

Chapter 8

Table 8.1

Chapter 12

Table 12.1

List of Illustrations

VBA Library for Complex Numbers

Figure 0.0

Quadrants of the Complex Plane

Chapter 1

Figure 1.1

Heston Price Using the Trapezoidal Rule

Figure 1.2

Heston Price Using Gauss-Laguerre Integration

Figure 1.3

Heston Integrand and Maturity

Chapter 2

Figure 2.1

Heston Price Using One or Two Characteristic Functions

Figure 2.2

Discontinuities in the Heston Integrand

Figure 2.3

Oscillations in the Heston Integrand

Figure 2.4

Integrand for

P

1

, Original and Little Trap Formulations

Figure 2.5

Effect of Correlation on Density

Figure 2.6

Effect of Volatility of Variance on Density

Figure 2.7

Effect of Correlation on Heston Prices Relative to Black-Scholes

Figure 2.8

Effect of Volatility of Variance on Heston Prices Relative to Black-Scholes

Figure 2.9

Effect of Heston Parameters on Implied Volatility

Figure 2.10

Heston Local Volatility and Implied Volatility

Figure 2.11

Market Implied Volatility and Heston Volatility, S&P 500 Index

Figure 2.12

Market and Heston Implied Volatilities from Puts on DIA

Figure 2.13

Implied Volatility at Extreme Strikes, DJIA Puts

Chapter 3

Figure 3.1

Attari (2004) and Heston (1993) Integrands

Figure 3.2

Heston (1993) and Carr and Madan (1999) Call Price

Figure 3.3

Roger Lee Bounds and Optimal Damping Factor

Figure 3.4

Lord and Kahl Optimal Alpha Function

Figure 3.5

Carr-Madan Integrands

Figure 3.6

Carr-Madan Integrands for OTM Options

Figure 3.7

OTM Pricing Using the Heston and Carr-Madan Methods

Chapter 4

Figure 4.1

Real Part of the Put Payoff Transform

Figure 4.2

Integration Strip for the Call Option

Figure 4.3

Contours of Integration

Figure 4.4

Comparison of Call Prices

Figure 4.5

Call Prices Using the Volatility Expansion

Figure 4.6

Reproduction of Figure 3.3.1. of Lewis (2000)

Chapter 5

Figure 5.1

Heston Prices Using Newton-Cotes Rules

Figure 5.2

Laguerre Polynomial of Fifth Order

Figure 5.3

Integrand Decay and Maturity

Figure 5.4

Multidomain Integration

Figure 5.5

The Kahl and Jäckel (2005) Integrand

Figure 5.6

Call Prices Using the Fast Fourier Transform

Figure 5.7

Call Prices Using the Fractional Fast Fourier Transform

Figure 5.8

Call Prices Using the FRFT, Narrow Strike Range

Chapter 6

Figure 6.1

S&P 500 Market and Heston Implied Volatilities with RMSE Parameter Estimates.

Figure 6.2

Starting Values for Rho and Sigma

Figure 6.3

RMSE Parameter Estimates Using the SVC Objective Function

Figure 6.4

Market and DE Implied Volatilities

Figure 6.5

Market and Heston Implied Volatilities Using MLEs

Figure 6.6

Risk-Neutral Densities for the S&P Index

Figure 6.7

Three Properties of the RNDs

Figure 6.8

Total Variance for the S&P 500 Data

Chapter 7

Figure 7.1

Stock Price and Variance under Negative Correlation

Figure 7.2

Stock Price and Variance under Positive Correlation

Figure 7.3

Heston Call Prices under the Euler and Milstein Schemes

Figure 7.4

Heston Call Prices under the QE and MM Schemes

Chapter 8

Figure 8.1

Pricing American Puts Using the LSM Algorithm

Figure 8.2

Pricing American Puts Using the Explicit Method

Figure 8.3

Jumps in the Transformed Stock Price

Figure 8.4

Put Prices Using the Beliaeva and Nawalkha Tree

Figure 8.5

American Put Prices under Black-Scholes

Figure 8.6

Barrier Levels for the Medvedev-Scaillet Approximation

Figure 8.7

American Puts Using Medvedev-Scaillet Expansion

Figure 8.8

Boundary Points and American Call Price

Figure 8.9

Early Exercise Boundary

Figure 9.1

Call Prices Using Different Characteristic Functions

Chapter 9

Figure 9.2

Mikhailov and Nögel Option Prices

Figure 9.3

Implied Volatility from the Elices (2009) Model

Figure 9.4

Implied Volatility from the BGM Model

Figure 9.5

Time-Varying Parameters

Chapter 10

Figure 10.1

Nonuniform Grid

Figure 10.2

Value of the European Call along the Stock Price and Variance Grids

Figure 10.3

Call Prices Using the Weighted Method and Uniform Grid

Figure 10.4

Call Prices Using the Weighted Method and Nonuniform Grid

Figure 10.5

Call Prices Using the Explicit Method

Figure 10.6

Call Prices Using ADI Schemes

Chapter 11

Figure 11.1

Black-Scholes and Heston PDEs

Figure 11.2

Heston and Black-Scholes Greeks

Figure 11.3

Gamma from the Heston Model

Figure 11.4

Theta from the Heston Model

Figure 11.5

Delta and Gamma under the Attari Formulation

Figure 11.6

Greeks under the Carr-Madan Formulation

Figure 11.7

Greeks under the Lewis (2000) Formulation

Figure 11.8

Greeks under the Lewis (2001) Formulation

Figure 11.9

Greeks under the Fast Fourier Transform

Figure 11.10

Greeks under the Fractional Fast Fourier Transform

Figure 11.11

American Greeks Using the LSM Algorithm

Figure 11.12

American Greeks Using the Explicit Method

Figure 11.13

Greeks from Medvedev-Scaillet Model

Figure 11.14

Vega from Medvedev-Scaillet Model

Chapter 12

Figure 12.1

The Double Heston Model

Figure 12.2

Greeks from the Double Heston Model

Figure 12.3

Double Heston Model Parameter Estimates

Figure 12.4

Implied Volatilities from Double Heston Model

Figure 12.5

Risk-Neutral Densities from the Double Heston Model

Figure 12.6

Simulation Schemes for the Double Heston Model

Figure 12.7

Simulated Stock Price and Variances with the Double Heston Model

Guide

Cover

Table of Contents

Preface

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Foreword

I am pleased to write this forward for Fabrice Rouah's final book on the “Heston Model." He had previously written an excellent book entitled Option Pricing Models and Volatility Using Excel/VBA, surveying a variety of modern models. He had also written the comprehensive book The Heston Model and Its Extensions in Matlab and C#. While these languages are widely used in large-scale econometric and commercial applications, there is conspicuous gap for people who want to implement stochastic volatility models in VBA. This present book fills that gap.

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. In the spirit of the classic Numerical Recipes books, the present book explains a variety of numerical methods and illustrates them with VBA routines. It includes numerical integration, the calculation of Greeks, American options, many simulation-based methods for valuation, finite difference numerical schemes, and subsequent developments such as the introduction of time-dependent parameters and the double version of the model. There is an old saying: “Teach a man to fish, and you will feed him for a lifetime.” In a similar spirit, this book will support students, researchers, and practitioners who need to understand their computations for a lifetime of implementation.

Professor Steven L. Heston

Robert H. Smith School of Business

University of Maryland

January 15, 2015

Preface

In 20 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 for 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 equity markets. It is now a stylized fact in these markets that returns distribution is 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 can 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 predate 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 not only 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 present 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's (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 VBA.

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 the original book on which this book is based. And to my team at Wiley—Bill Falloon, Meg Freeborn, and Vincent Nordhaus. 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 for both this book and the original work: Amir Atiya, Sébastien Bossu, Carl Chiarella, Elton Daal, Redouane El-Kamhi, Judith and James Farer, Jacqueline Gheerbrant, Emmanuel Gobet, Greg N. Gregoriou, Antoine Jacquier, Roy Horan, Dominique Legros, Pierre Leignadier, Alexey Medvedev, Sanjay K. Nawalkha, Razvan Pascalau, Jean Rouah, Olivier Scaillet, Martin Schmelzle, Giovanna Sestito, Nick Webber, and Estarose Wolfson. Finally, a special mention to Kevin Samborn at Sapient Global Markets for his help and support.

About This Book

This book is a condensed and readapted version of the original work entitled The Heston Model and Its Extensions in Matlab and C#, which was published by John Wiley & Sons in September 2013. It is geared for professionals who make extensive use of VBA in their professional lives. Much of the content in this book is taken directly from the original book, but with edits to condense the material and to present the VBA code with which the models are implemented. Readers of this book who seek theoretical details and mathematical derivations of the models may wish to consult the companion book.

This current work is light on theory but heavy on implementation. As such, it serves as a “VBA cookbook” for the models covered in the original book, rather than a reference guide for the theoretical aspects of the models. Indeed, the mathematical proofs and derivations of model results are omitted—these appear in the original book. Instead, the current book explains the VBA user-defined functions that implement the models; presents examples of how to run the functions; explains model results; and provides screen shots of the Excel files that accompany this book, all of which are available on the Wiley web site.

All the calculations for the implementation of the models covered in this book are done in VBA code that resides in Excel files. The code is used to create the user-defined VBA functions for the implementation. The Excel files serve as an interface between the model inputs and settings, and the VBA user-defined functions that process these inputs to produce model outputs such as prices, volatilities, and graphs. The VBA functions are arranged in separate modules that regroup functions with a common objective.

The VBA code and modules are available in the VBA editor, which can be accessed from Excel by pressing the <ALT> key, followed by <F11>. VBA code snippets are used throughout the book to illustrate how the user-defined VBA functions work. The snippets are for illustrative purposes only, and are therefore not complete—they contain only those portions of the functions that are vital to the calculations. Nonessential portions do not appear in the snippets. The VBA code itself appears as black-colored font in the snippets, comments appear as green-colored font, and text in quotations appears as red-colored font.

NOTE

1

We thank Nick Webber for pointing this out.

CHAPTER 1The Heston Model for European Options

Abstract

Here, we present the European call price under the Heston model. We first present the model and then illustrate 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 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 speed 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.

CIR process, European call, characteristic function, dividend yield, put-call parity, Black-Scholes

In this chapter, we present the European call price under the Heston model. We first present the model and then illustrate 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 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, St, 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),