Option Pricing Models and Volatility Using Excel-VBA E-Book

Fabrice D. Rouah

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

Seitenzahl: 621

Veröffentlichungsjahr: 2007

Beschreibung

This comprehensive guide offers traders, quants, and students the tools and techniques for using advanced models for pricing options. The accompanying website includes data files, such as options prices, stock prices, or index prices, as well as all of the codes needed to use the option and volatility models described in the book. Praise for Option Pricing Models & Volatility Using Excel-VBA "Excel is already a great pedagogical tool for teaching option valuation and risk management. But the VBA routines in this book elevate Excel to an industrial-strength financial engineering toolbox. I have no doubt that it will become hugely successful as a reference for option traders and risk managers." --Peter Christoffersen, Associate Professor of Finance, Desautels Faculty of Management, McGill University "This book is filled with methodology and techniques on how to implement option pricing and volatility models in VBA. The book takes an in-depth look into how to implement the Heston and Heston and Nandi models and includes an entire chapter on parameter estimation, but this is just the tip of the iceberg. Everyone interested in derivatives should have this book in their personal library." --Espen Gaarder Haug, option trader, philosopher, and author of Derivatives Models on Models "I am impressed. This is an important book because it is the first book to cover the modern generation of option models, including stochastic volatility and GARCH." --Steven L. Heston, Assistant Professor of Finance, R.H. Smith School of Business, University of Maryland

Leseprobe

Contents

Preface

Chapter 1: Mathematical Preliminaries

Introduction

Complex Numbers

Finding Roots of Functions

OLS and WLS

Nelder-Mead Algorithm

Maximum Likelihood Estimation

Cubic Spline Interpolation

Summary

Exercises

Solutions to Exercises

Chapter 2: Numerical Integration

Introduction

Newton-Coates Formulas

Implementing Newton-Cotes Formulas in VBA

Gaussian Quadratures

Summary

Exercises

Solution to Exercises

Appendix

Chapter 3: Tree-Based Methods

Introduction

CRR Binomial Tree

Leisen-Reimer Binomial Tree

Edgeworth Binomial Tree

Flexible Binomial Tree

Trinomial Tree

Adaptive Mesh Method

Comparing Trees

Implied Volatility Trees

Allowing for Dividends and The Cost-of-Carry

Summary

Exercises

Solutions to Exercises

Chapter 4: The Black-Scholes, Practitioner Black-Scholes, and Gram-Charlier Models

Introduction

The Black-Scholes Model

Implied Volatility and The DVF

The Practitioner Black-Scholes Model

The Gram-Charlier Model

Summary

Exercises

Solutions to Exercises

Chapter 5: The Heston (1993) Stochastic Volatility Model

Introduction

The Heston (1993) Model

Increasing Integration Accuracy

The Fundamental Transform

Sensitivity Analysis

Summary

Exercises

Solutions to Exercises

Appendix

Chapter 6: The Heston and Nandi (2000) GARCH Model

Introduction

Persistent Volatility in Asset Returns

Garch Variance Modeling

The Heston and Nandi (2000) Model

Summary

Exercises

Solutions to Exercises

Chapter 7: The Greeks

Introduction

Black-Scholes Greeks

Greeks From The Trees

Greeks From The Gram-Charlier Model

Greeks From The Heston (1993) Model

Greeks From The Heston and Nandi (2000) Model

Greeks By Finite Differences

Summary

Exercises

Solutions to Exercises

Appendix

Chapter 8: Exotic Options

Introduction

Single-Barrier Options

Digital Options

Asian Options

Floating-Strike Lookback Options

Summary

Exercises

Solutions to Exercises

Chapter 9: Parameter Estimation

Introduction

Unconditional Moments

Maximum Likelihood for Garch Models

Estimation by Loss Functions

Other Estimation Methods

Summary

Exercises

Solutions to Exercises

Chapter 10: Implied Volatility

Introduction

Obtaining Implied Volatility

Explaining Smiles and Smirks

Summary

Exercises

Solutions to Exercises

Chapter 11: Model-Free Implied Volatility

Introduction

Theoretical Foundation

Implementation

Interpolation-Extrapolation Method

Model-Free Implied Forward Volatility

The Vix Index

Summary

Exercises

Chapter 12: Model-Free Higher Moments

Introduction

Theoretical Foundation

Implementation

Verifying Implied Moments

Gram-Charlier Implied Moments

Summary

Exercises

Solutions to Exercises

Chapter 13: Volatility Returns

Introduction

Straddle Returns

Delta-Hedged Gains

Volatility Exposure

Variance Swaps

Summary

Exercises

Solutions to Exercises

Appendix A: A VBA Primer

References

About the CD-ROM

About the Authors

Index

Founded in 1807, John Wiley & Sons is the oldest independent publishing company in the United States. With offices in North America, Europe, Australia, and Asia, Wiley is globally committed to developing and marketing print and electronic products and services for our customers’ professional and personal knowledge and understanding.

The Wiley Finance series contains books written specifically for finance and investment professionals as well as sophisticated individual investors and their financial advisors. Book topics range from portfolio management to e-commerce, risk management, financial engineering, valuation, and financial instrument analysis, as well as much more.

For a list of available titles, visit our Web site at www.WileyFinance.com.

Copyright © 2007 by Fabrice Douglas Rouah and Gregory Vainberg. All rights reserved.

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

Published simultaneously in Canada.

Wiley Bicentennial Logo: Richard J. Pacifico

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, [phone], fax [phone], or on the Web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, [phone], fax [phone], or online at www.wiley.com/go/permissions.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at [phone], outside the United States at [phone] or fax [phone].

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. For more information about Wiley products, visit our Web site at www.wiley.com.

Designations used by companies to distinguish their products are often claimed as trademarks. In all instances where John Wiley & Sons, Inc. is aware of a claim, the product names appear in initial capital or all capital letters. Readers, however, should contact the appropriate companies for more complete information regarding trademarks and registration.

Library of Congress Cataloging-in-Publication Data:

Rouah, Fabrice, 1964-

Option pricing models and volatility using Excel®-VBA / Fabrice Douglas Rouah, Gregory Vainberg.

p. cm. –(Wiley finance series)

Includes bibliographical references and index.

ISBN: 978-0-471-79464-6 (paper/cd-rom)

1. Options (Finance)–Prices. 2. Capital investments–Mathematical–Mathematical models. 3. Options (Finance)–Mathematical models. 4. Microsoft Excel (Computer file) 5. Microsoft Visual Basic for applications. I. Vainberg, Gregory, 1978-II. Title.

HG6024.A3 R678 2007

332.64′53–dc22

2006031250

If this e-book refers to media such as a CD or DVD, you may download this material at http://booksupport.wiley.com. For more information about Wiley products, visit www.wiley.com

To Jacqueline, Jean, and Gilles

—Fabrice

To Irina, Bryanne, and Stephannie

—Greg

Preface

This book constitutes a guide for implementing advanced option pricing models and volatility in Excel/VBA. It can be used by MBA students specializing in finance and risk management, by practitioners, and by undergraduate students in their final year. Emphasis has been placed on implementing the models in VBA, rather than on the theoretical developments underlying the models. We have made every effort to explain the models and their coding in VBA as simply as possible. Every model covered in this book includes one or more VBA functions that can be accessed on the CD-ROM. We have focused our attention on equity options, and we have chosen not to include interest rate options. The particularities of interest rate options place them in a separate class of derivatives.

The first part of the book covers mathematical preliminaries that are used throughout the book. In Chapter 1 we explain complex numbers and how to implement them in VBA. We also explain how to write VBA functions for finding roots of functions, the Nelder-Mead algorithm for finding the minimum of a multivariate function, and cubic spline interpolation. All of these methods are used extensively throughout the book. Chapter 2 covers numerical integration. Many of option pricing and volatility models require that an integral be evaluated for which no closed-form solution exists, which requires a numerical approximation to the integral. In Chapter 2 we present various methods that have proven to be extremely accurate and efficient for numerical integration.

The second part of this book covers option pricing formulas. In Chapter 3 we cover lattice methods. These include the well-known binomial and trinomial trees, but also refinements such as the implied binomial and trinomial trees, the flexible binomial tree, the Leisen-Reimer tree, the Edgeworth binomial tree, and the adapted mesh method. Most of these methods approximate the Black-Scholes model in discrete time. One advantage they have over the Black-Scholes model, however, is that they can be used to price American options. In Chapter 4 we cover the Black-Scholes, Gram-Charlier, and Practitioner Black-Scholes models, and introduce implied volatility. The Black-Scholes model is presented as a platform upon which other models are built. The Gram-Charlier model is an extension of the Black-Scholes model that allows for skewness and excess kurtosis in the distribution of the return on the underlying asset. The Practitioner Black-Scholes model uses implied volatility fitted from a deterministic volatility function (DVF) regression, as an input to the Black-Scholes model. It can be thought of as an ad hoc method that adapts the Black-Scholes model to account for the volatility smile in option prices. In Chapter 5 we cover the Heston (1993) model, which is an extension of the Black-Scholes model that allows for stochastic volatility, while in Chapter 6 we cover the Heston and Nandi (2000) GARCH model, which in its simplest form is a discrete-time version of the model in Chapter 5. The call price in each model is available in closed form, up to a complex integral that must be evaluated numerically. In Chapter 6 we also show how to identify the correlation and dependence in asset returns, which the GARCH model attempts to incorporate. We also show how to implement the GARCH(1,1) model in VBA, and how GARCH volatilities can be used for long-run volatility forecasting and for constructing the term structure of volatility. Chapter 7 covers the option sensitivities, or Greeks, from the option pricing models covered in this book. The Greeks for the Black-Scholes and Gram-Charlier models are available in closed form. The Greeks from Heston (1993), and Heston and Nandi (2000) models are available in closed form also, but require a numerical approximation to a complex integral. The Greeks from tree-based methods can be approximated from option and asset prices at the beginning nodes of the tree. In Chapter 7 we also show how to use finite differences to approximate the Greeks, and we show that these approximations are all close to their closed-form values. In Chapter 8 we cover exotic options. Most of the methods we present for valuing exotic options are tree-based. Particular emphasis is placed on single-barrier options, and the various methods that have been proposed to deal with the difficulties that arise when tree-based methods are adapted to barrier options. In Chapter 8 we also cover Asian options, floating-strike lookback options, and digital options. Finally, in Chapter 9 we cover basic estimation methods for parameters that are used as inputs to the option pricing models covered in this book. Particular emphasis is placed on loss function estimation, which estimates parameters by minimizing the difference between market and model prices.

The third part of this book deals with volatility and higher moments. In Chapter 10 we present a thorough treatment of implied volatility and show how the root-finding methods covered in Chapter 1 can be used to obtain implied volatilities from market prices. We explain how the implied volatility curve can shed information on the distribution of the underlying asset return, and we show how option prices generated from the Heston (1993) and Gram-Charlier models lead to implied volatility curves that account for the smile and skew in option prices. Chapter 11 deals with model-free implied volatility. Unlike Black-Scholes implied volatility, model-free implied volatility does not require the restrictive assumption of a particular parametric form for the underlying price dynamics. Moreover, unlike Black-Scholes implied volatilities, which are usually computed using at-the-money or near-the-money options only, model-free volatilities are computed using the whole cross-section of option prices. In Chapter 11 we also present methods that mitigate the discretization and truncation bias brought on by using market prices that do not include a continuum of strike prices, and that are available only over a bounded interval of strike prices. We also show how to construct the Chicago Board Options Exchange® volatility index, the VIX, which is now based on model-free implied volatility. Chapter 12 extends the model-free methods of Chapter 11, and deals with model-free skewness and kurtosis. We show how applying interpolation-extrapolation to these methods leads to much more accurate approximations to the integrals that are used to estimate model-free higher moments. In Chapter 13 we treat volatility returns, which are returns on strategies designed to profit from volatility. We cover simple straddles, which are constructed using a single call and put. Zero-beta straddles are slightly more complex, but have the advantage that they are hedged against market movements. We also introduce a simple model to value straddle options, and introduce delta-hedged gains. Similar to zero-beta straddles, delta-hedged gains are portfolios in which all risks except volatility risk have been hedged away, so that the only remaining risk to the portfolio is volatility risk. Finally, we cover variance swaps, which are an application of model-free volatility for constructing a call option on volatility.

This book also contains a CD-ROM that contains Excel spreadsheets and VBA functions to implement all of the option pricing and volatility models presented in this book. The CD-ROM also includes solutions to all the chapter exercises, and option data for IBM Corporation and Intel Corporation downloaded from Yahoo! (finance.yahoo.com).

ACKNOWLEDGMENTS

We have several people to thank for their valuable help and comments during the course of writing this book. We thank Peter Christoffersen, Susan Christoffersen, and Kris Jacobs. We also thank Steven Figlewski, John Hull, Yue Kuen Kwok, Dai Min, Mark Rubinstein, and our colleagues Vadim Di Pietro, Greg N. Gregoriou, and especially Redouane El-Kamhi. Working with the staff at John Wiley & Sons has been a pleasure. We extend special thanks to Bill Falloon, Emilie Herman, Laura Walsh, and Todd Tedesco. We are indebted to Polina Ialamova at OptionMetrics. We thank our families for their continual support and personal encouragement. Finally, we thank Peter Christoffersen, Steven L. Heston, and Espen Gaarder, for kindly providing the endorsements.

Chapter 1

Mathematical Preliminaries

INTRODUCTION

In this chapter we introduce some of the mathematical concepts that will be needed to deal with the option pricing and stochastic volatility models introduced in this book, and to help readers implement these concepts as functions and routines in VBA. First, we introduce complex numbers, which are needed to evaluate characteristic functions of distributions driving option prices. These are required to evaluate the option pricing models of Heston (1993) and Heston and Nandi (2000) covered in Chapters 5 and 6, respectively. Next, we review and implement Newton’s method and the bisection method, two popular and simple algorithms for finding zeros of functions. These methods are needed to find volatility implied from option prices, which we introduce in Chapter 4 and deal with in Chapter 10. We show how to implement multiple linear regression with ordinary least squares (OLS) and weighted least squares (WLS) in VBA. These methods are needed to obtain the deterministic volatility functions of Chapter 4. Next, we show how to find maximum likelihood estimators, which are needed to estimate the parameters that are used in option pricing models. We also implement the Nelder-Mead algorithm, which is used to find the minimum values of multivariate functions and which will be used throughout this book. Finally, we implement cubic splines in VBA. Cubic splines will be used to obtain model-free implied volatility in Chapter 11, and model-free skewness and kurtosis in Chapter 12.

COMPLEX NUMBERS

Operations on Complex Numbers

Many of the operations on complex numbers are done by isolating the real and imaginary parts. Other operations require simple tricks, such as rewriting the complex number in a different form or using its complex conjugate. Krantz (1999) is a good reference for this section.

Addition and subtraction of complex numbers is performed by separate operation on the real and imaginary parts. It requires adding and subtracting, respectively, the real and imaginary parts of the two complex numbers:

Multiplying two complex numbers is done by applying the distributive axiom to the product, and regrouping the real and imaginary parts:

The complex conjugate of a complex number is defined as and is useful for dividing complex numbers. Since , we can express division of any two complex numbers as the ratio

Exponentiation of a complex number is done by applying Euler’s formula, which produces

(1.1)

so that the real and imaginary parts of , respectively.

Operations Using VBA

In this section we describe how to define complex numbers in VBA and how to construct functions for operations on complex numbers. Note that it is possible to use the built-in complex number functions in Excel directly, without having to construct them in VBA. However, we will see in later chapters that using the built-in functions increases substantially the computation time required for convergence of option prices. Constructing complex numbers in VBA, therefore, makes computation of option prices more efficient. Moreover, it is sometimes preferable to have control over how certain operations on complex numbers are defined. There are other definitions of the square root of a complex number, for example, than that given by applying DeMoivre’s Theorem. Finally, learning how to construct complex numbers in VBA is a good learning exercise.