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).
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.