Table of Contents
Praise
Title Page
Copyright Page
Dedication
Table of Figures
List of Tables
Foreword
I
II
Preface
HOW THIS BOOK IS ORGANIZED
Acknowledgments
CHAPTER 1 - Stochastic Volatility and Local Volatility
STOCHASTIC VOLATILITY
LOCAL VOLATILITY
CHAPTER 2 - The Heston Model
THE PROCESS
THE HESTON SOLUTION FOR EUROPEAN OPTIONS
DERIVATION OF THE HESTON CHARACTERISTIC FUNCTION
SIMULATION OF THE HESTON PROCESS
CHAPTER 3 - The Implied Volatility Surface
GETTING IMPLIED VOLATILITY FROM LOCAL VOLATILITIES
LOCAL VOLATILITY IN THE HESTON MODEL
IMPLIED VOLATILITY IN THE HESTON MODEL
THE SPX IMPLIED VOLATILITY SURFACE
CHAPTER 4 - The Heston-Nandi Model
LOCAL VARIANCE IN THE HESTON-NANDI MODEL
A NUMERICAL EXAMPLE
DISCUSSION OF RESULTS
CHAPTER 5 - Adding Jumps
WHY JUMPS ARE NEEDED
JUMP DIFFUSION
CHARACTERISTIC FUNCTION METHODS
STOCHASTIC VOLATILITY PLUS JUMPS
CHAPTER 6 - Modeling Default Risk
MERTON’S MODEL OF DEFAULT
CAPITAL STRUCTURE ARBITRAGE
LOCAL AND IMPLIED VOLATILITY IN THE JUMP-TO-RUIN MODEL
THE EFFECT OF DEFAULT RISK ON OPTION PRICES
THE CREDITGRADES MODEL
CHAPTER 7 - Volatility Surface Asymptotics
SHORT EXPIRATIONS
THE MEDVEDEV-SCAILLET RESULT
INCLUDING JUMPS
LONG EXPIRATIONS: FOUQUE, PAPANICOLAOU, AND SIRCAR
SMALL VOLATILITY OF VOLATILITY: LEWIS
EXTREME STRIKES: ROGER LEE
ASYMPTOTICS IN SUMMARY
CHAPTER 8 - Dynamics of the Volatility Surface
DYNAMICS OF THE VOLATILITY SKEW UNDER STOCHASTIC VOLATILITY
DYNAMICS OF THE VOLATILITY SKEW UNDER LOCAL VOLATILITY
STOCHASTIC IMPLIED VOLATILITY MODELS
DIGITAL OPTIONS AND DIGITAL CLIQUETS
CHAPTER 9 - Barrier Options
DEFINITIONS
LIMITING CASES
THE REFLECTION PRINCIPLE
THE LOOKBACK HEDGING ARGUMENT
PUT-CALL SYMMETRY
QUASISTATIC HEDGING AND QUALITATIVE VALUATION
ADJUSTING FOR DISCRETE MONITORING
PARISIAN OPTIONS
SOME APPLICATIONS OF BARRIER OPTIONS
CONCLUSION
CHAPTER 10 - Exotic Cliquets
LOCALLY CAPPED GLOBALLY FLOORED CLIQUET
REVERSE CLIQUET
NAPOLEON
CHAPTER 11 - Volatility Derivatives
SPANNING GENERALIZED EUROPEAN PAYOFFS
VARIANCE AND VOLATILITY SWAPS
VALUING VOLATILITY DERIVATIVES
LISTED QUADRATIC-VARIATION BASED SECURITIES
SUMMARY
Postscript
Bibliography
Index
Table of Figures
FIGURE 1.1 SPX daily log returns from December 31, 1984, to December 31, 2004. Note the −22.9% return on October 19, 1987!
FIGURE 1.2 Frequency distribution of (77 years of) SPX daily log returns compared with the normal distribution. Although the −22.9% return on October 19, 1987, is not directly visible, the x-axis has been extended to the left to accommodate it!
FIGURE 1.3 Q-Q plot of SPX daily log returns compared with the normal distribution. Note the extreme tails.
FIGURE 3.1 Graph of the pdf of x conditional on x = log(K) for a 1-year European option, strike 1.3 with current stock price = 1 and 20% volatility.
FIGURE 3.2 Graph of the SPX-implied volatility surface as of the close on September 15, 2005, the day before triple witching.
FIGURE 3.3 Plots of the SVI fits to SPX implied volatilities for each of the eight listed expirations as of the close on September 15, 2005. Strikes are on the x-axes and implied volatilities on the y-axes. The black and grey diamonds represent bid and offer volatilities respectively and the solid line is the SVI fit.
FIGURE 3.4 Graph of SPX ATM skew versus time to expiry. The solid line is a fit of the approximate skew formula () to all empirical skew points except the first; the dashed fit excludes the first three data points.
FIGURE 3.5 Graph of SPX ATM variance versus time to expiry. The solid line is a fit of the approximate ATM variance formula () to the empirical data.
FIGURE 3.6 Comparison of the empirical SPX implied volatility surface with the Heston fit as of September 15, 2005. From the two views presented here, we can see that the Heston fit is pretty good for longer expirations but really not close for short expirations. The paler upper surface is the empirical SPX volatility surface and the darker lower one the Heston fit. The Heston fit surface has been shifted down by five volatility points for ease of visual comparison.
FIGURE 4.1 The probability density for the Heston-Nandi model with our parameters and expiration T = 0.1.
FIGURE 4.2 Comparison of approximate formulas with direct numerical computation of Heston local variance. For each expiration T, the solid line is the numerical computation and the dashed line is the approximate formula.
FIGURE 4.3 Comparison of European implied volatilities from application of the Heston formula () and from a numerical PDE computation using the local volatilities given by the approximate formula (). For each expiration T, the solid line is the numerical computation and the dashed line is the approximate formula.
FIGURE 5.1 Graph of the September 16, 2005, expiration volatility smile as of the close on September 15, 2005. SPX is trading at 1227.73. Triangles represent bids and offers. The solid line is a nonlinear (SVI) fit to the data. The dashed line represents the Heston skew with Sep05 SPX parameters.
FIGURE 5.2 The 3-month volatility smile for various choices of jump diffusion parameters.
FIGURE 5.3 The term structure of ATM variance skew for various choices of jump diffusion parameters.
FIGURE 5.4 As time to expiration increases, the return distribution looks more and more normal. The solid line is the jump diffusion pdf and for comparison, the dashed line is the normal density with the same mean and standard deviation. With the parameters used to generate these plots, the characteristic time T* = 0.67.
FIGURE 5.5 The solid line is a graph of the at-the-money variance skew in the SVJ model with BCC parameters vs. time to expiration. The dashed line represents the sum of at-the-money Heston and jump diffusion skews with the same parameters.
FIGURE 5.6 The solid line is a graph of the at-the-money variance skew in the SVJ model with BCC parameters versus time to expiration. The dashed line represents the at-the-money Heston skew with the same parameters.
FIGURE 5.7 The solid line is a graph of the at-the-money variance skew in the SVJJ model with BCC parameters versus time to expiration. The short-dashed and long-dashed lines are SVJ and Heston skew graphs respectively with the same parameters.
FIGURE 5.8 This graph is a short-expiration detailed view of the graph shown in .
FIGURE 5.9 Comparison of the empirical SPX implied volatility surface with the SVJ fit as of September 15, 2005. From the two views presented here, we can see that in contrast to the Heston case, the major features of the empirical surface are replicated by the SVJ model. The paler upper surface is the empirical SPX volatility surface and the darker lower one the SVJ fit. The SVJ fit surface has again been shifted down by five volatility points for ease of visual comparison.
FIGURE 6.1 Three-month implied volatilities from the Merton model assuming a stock volatility of 20% and credit spreads of 100 bp (solid), 200 bp (dashed) and 300 bp (long-dashed).
FIGURE 6.2 Payoff of the 1 × 2 put spread combination: buy one put with strike 1.0 and sell two puts with strike 0.5.
FIGURE 6.3 Local variance plot with λ = 0.05 and σ = 0.2.
FIGURE 6.4 The triangles represent bid and offer volatilities and the solid line is the Merton model fit.
FIGURE 7.1 For short expirations, the most probable path is approximately a straight line from spot on the valuation date to the strike at expiration. It follows that (k,T) ≈ [ν(0,0) + ν(k,T)]/2 and the implied variance skew is roughly one half of the local variance skew.
FIGURE 8.1 Illustration of a cliquet payoff. This hypothetical SPX cliquet resets at-the-money every year on October 31. The thick solid lines represent nonzero cliquet payoffs. The payoff of a 5-year European option struck at the October 31, 2000, SPX level of 1429.40 would have been zero.
FIGURE 9.1 A realization of the zero log-drift stochastic process and the reflected path.
FIGURE 9.2 The ratio of the value of a one-touch call to the value of a European binary call under stochastic volatility and local volatility assumptions as a function of strike. The solid line is stochastic volatility and the dashed line is local volatility.
FIGURE 9.3 The value of a European binary call under stochastic volatility and local volatility assumptions as a function of strike. The solid line is stochastic volatility and the dashed line is local volatility. The two lines are almost indistinguishable.
FIGURE 9.4 The value of a one-touch call under stochastic volatility and local volatility assumptions as a function of barrier level. The solid line is stochastic volatility and the dashed line is local volatility.
FIGURE 9.5 Values of knock-out call options struck at 1 as a function of barrier level. The solid line is stochastic volatility; the dashed line is local volatility.
FIGURE 9.6 Values of knock-out call options struck at 0.9 as a function of barrier level. The solid line is stochastic volatility; the dashed line is local volatility.
FIGURE 9.7 Values of live-out call options struck at 1 as a function of barrier level. The solid line is stochastic volatility; the dashed line is local volatility.
FIGURE 9.8 Values of lookback call options as a function of strike. The solid line is stochastic volatility; the dashed line is local volatility.
FIGURE 10.1 Value of the “Mediobanca Bond Protection 2002-2005” locally capped and globally floored cliquet (minus guaranteed redemption) as a function of MinCoupon. The solid line is stochastic volatility; the dashed line is local volatility.
FIGURE 10.2 Historical performance of the “Mediobanca Bond Protection 2002-2005” locally capped and globally floored cliquet. The dashed vertical lines represent reset dates, the solid lines coupon setting dates and the solid horizontal lines represent fixings.
FIGURE 10.3 Value of the Mediobanca reverse cliquet (minus guaranteed redemption) as a function of MaxCoupon. The solid line is stochastic volatility; the dashed line is local volatility.
FIGURE 10.4 Historical performance of the “Mediobanca 2000-2005 Reverse Cliquet Telecommunicazioni” reverse cliquet. The vertical lines represent reset dates, the solid horizontal lines represent fixings and the vertical grey bars represent negative contributions to the cliquet payoff.
FIGURE 10.5 Value of (risk-neutral) expected Napoleon coupon as a function of MaxCoupon. The solid line is stochastic volatility; the dashed line is local volatility.
FIGURE 10.6 Historical performance of the STOXX 50 component of the “Mediobanca 2002-2005 World Indices Euro Note Serie 46” Napoleon. The light vertical lines represent reset dates, the heavy vertical lines coupon setting dates, the solid horizontal lines represent fixings and the thick grey bars represent the minimum monthly return of each coupon period.
FIGURE 11.1 Payoff of a variance swap (dashed line) and volatility swap (solid line) as a function of realized volatility Σ. Both swaps are struck at 30% volatility.
FIGURE 11.2 Annualized Heston convexity adjustment as a function of T with Heston-Nandi parameters.
FIGURE 11.3 Annualized Heston convexity adjustment as a function of T with Bakshi, Cao, and Chen parameters.
FIGURE 11.4 Value of 1-year variance call versus variance strike K with the BCC parameters. The solid line is a numerical Heston solution; the dashed line comes from our lognormal approximation.
FIGURE 11.5 The pdf of the log of 1-year quadratic variation with BCC parameters. The solid line comes from an exact numerical Heston computation; the dashed line comes from our lognormal approximation.
FIGURE 11.6 Annualized Heston VXB convexity adjustment as a function of t with Heston parameters from December 8, 2004, SPX fit.
List of Tables
TABLE 3.1 At-the-money SPX variance levels and skews as of the close on September 15, 2005, the day before expiration.
TABLE 3.2 Heston fit to the SPX surface as of the close on September 15, 2005.
TABLE 5.1 September 2005 expiration option prices as of the close on September 15, 2005. Triple witching is the following day. SPX is trading at 1227.73.
TABLE 5.2 Parameters used to generate and .
TABLE 5.3 Interpreting and .
TABLE 5.4 Various fits of jump diffusion style models to SPX data. JD means Jumps Diffusion and SVJ means Stochastic Volatility plus Jumps.
TABLE 5.5 SVJ fit to the SPX surface as of the close on September 15, 2005.
TABLE 6.1 Upper and lower arbitrage bounds for one-year 0.5 strike options for various credit spreads (at-the-money volatility is 20%).
TABLE 6.2 Implied volatilities for January 2005 options on GT as of October 20, 2004 (GT was trading at 9.40). Merton vols are volatilities generated from the Merton model with fitted parameters.
TABLE 10.1 Estimated “Mediobanca Bond Protection 2002-2005” coupons.
TABLE 10.2 Worst monthly returns and estimated Napoleon coupons. Recall that the coupon is computed as 10% plus the worst monthly return averaged over the three underlying indices.
TABLE 11.1 Empirical VXB convexity adjustments as of December 8, 2004.
Further Praise for The Volatility Surface
“As an experienced practitioner, Jim Gatheral succeeds admirably in combining an accessible exposition of the foundations of stochastic volatility modeling with valuable guidance on the calibration and implementation of leading volatility models in practice.”
—Eckhard Platen, Chair in Quantitative Finance, University of Technology, Sydney
“Dr. Jim Gatheral is one of Wall Street’s very best regarding the practical use and understanding of volatility modeling.
The Volatility Surface reflects his in-depth knowledge about local volatility, stochastic volatility, jumps, the dynamic of the volatility surface and how it affects standard options, exotic options, variance and volatility swaps, and much more. If you are interested in volatility and derivatives, you need this book!
—Espen Gaarder Haug, option trader, and author to The Complete Guide to Option Pricing Formulas
“Anybody who is interested in going beyond Black-Scholes should read this book. And anybody who is not interested in going beyond Black-Scholes isn’t going far!”
—Mark Davis, Professor of Mathematics, Imperial College London
“This book provides a comprehensive treatment of subjects essential for anyone working in the field of option pricing. Many technical topics are presented in an elegant and intuitively clear way. It will be indispensable not only at trading desks but also for teaching courses on modern derivatives and will definitely serve as a source of inspiration for new research.”
—Anna Shepeleva, Vice President, ING Group
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To Yukiko and Ayako
Foreword
I
Jim has given round six of these lectures on volatility modeling at the Courant Institute of New York University, slowly purifying these notes. I witnessed and became addicted to their slow maturation from the first time he jotted down these equations during the winter of 2000, to the most recent one in the spring of 2006. It was similar to the progressive distillation of good alcohol: exactly seven times; at every new stage you can see the text gaining in crispness, clarity, and concision. Like Jim’s lectures, these chapters are to the point, with maximal simplicity though never less than warranted by the topic, devoid of fluff and side distractions, delivering the exact subject without any attempt to boast his (extraordinary) technical skills.
The class became popular. By the second year we got yelled at by the university staff because too many nonpaying practitioners showed up to the lecture, depriving the (paying) students of seats. By the third or fourth year, the material of this book became a quite standard text, with Jim G.’s lecture notes circulating among instructors. His treatment of local volatility and stochastic models became the standard.
As colecturers, Jim G. and I agreed to attend each other’s sessions, but as more than just spectators—turning out to be colecturers in the literal sense, that is, synchronously. He and I heckled each other, making sure that not a single point went undisputed, to the point of other members of the faculty coming to attend this strange class with disputatious instructors trying to tear apart each other’s statements, looking for the smallest hole in the arguments. Nor were the arguments always dispassionate: students soon got to learn from Jim my habit of ordering white wine with read meat; in return, I pointed out clear deficiencies in his French, which he pronounces with a sometimes incomprehensible Scottish accent. I realized the value of the course when I started lecturing at other universities. The contrast was such that I had to return very quickly.
II
The difference between Jim Gatheral and other members of the quant community lies in the following: To many, models provide a representation of asset price dynamics, under some constraints. Business school finance professors have a tendency to believe (for some reason) that these provide a top-down statistical mapping of reality. This interpretation is also shared by many of those who have not been exposed to activity of risk-taking, or the constraints of empirical reality.
But not to Jim G. who has both traded and led a career as a quant. To him, these stochastic volatility models cannot make such claims, or should not make such claims. They are not to be deemed a top-down dogmatic representation of reality, rather a tool to insure that all instruments are consistently priced with respect to each other-that is, to satisfy the golden rule of absence of arbitrage. An operator should not be capable of deriving a profit in replicating a financial instrument by using a combination of other ones. A model should do the job of insuring maximal consistency between, say, a European digital option of a given maturity, and a call price of another one. The best model is the one that satisfies such constraints while making minimal claims about the true probability distribution of the world.
I recently discovered the strength of his thinking as follows. When, by the fifth or so lecture series I realized that the world needed Mandelbrot-style power-law or scalable distributions, I found that the models he proposed of fudging the volatility surface was compatible with these models. How? You just need to raise volatilities of out-of-the-money options in a specific way, and the volatility surface becomes consistent with the scalable power laws.
Jim Gatheral is a natural and intuitive mathematician; attending his lecture you can watch this effortless virtuosity that the Italians call sprezzatura. I see more of it in this book, as his awful handwriting on the blackboard is greatly enhanced by the aesthetics of LaTeX.
—Nassim Nicholas Taleb1
June, 2006
Preface
Ever since the advent of the Black-Scholes option pricing formula, the study of implied volatility has become a central preoccupation for both academics and practitioners. As is well known, actual option prices rarely if ever conform to the predictions of the formula because the idealized assumptions required for it to hold don’t apply in the real world. Consequently, implied volatility (the volatility input to the Black-Scholes formula that generates the market price) in general depends on the strike and the expiration of the option. The collection of all such implied volatilities is known as the volatility surface.
This book concerns itself with understanding the volatility surface; that is, why options are priced as they are and what it is that analysis of stock returns can tell us about how options ought to be priced.
Pricing is consistently emphasized over hedging, although hedging and replication arguments are often used to generate results. Partly, that’s because pricing is key: How a claim is hedged affects only the width of the resulting distribution of returns and not the expectation. On average, no amount of clever hedging can make up for an initial mispricing. Partly, it’s because hedging in practice can be complicated and even more of an art than pricing.
Throughout the book, the importance of examining different dynamical assumptions is stressed as is the importance of building intuition in general. The aim of the book is not to just present results but rather to provide the reader with ways of thinking about and solving practical problems that should have many other areas of application. By the end of the book, the reader should have gained substantial intuition for the latest theory underlying options pricing as well as some feel for the history and practice of trading in the equity derivatives markets. With luck, the reader will also be infected with some of the excitement that continues to surround the trading, marketing, pricing, hedging, and risk management of derivatives.
As its title implies, this book is written by a practitioner for practitioners. Amongst other things, it contains a detailed derivation of the Heston model and explanations of many other popular models such as SVJ, SVJJ, SABR, and CreditGrades. The reader will also find explanations of the characteristics of various types of exotic options from the humble barrier option to the super exotic Napoleon. One of the themes of this book is the representation of implied volatility in terms of a weighted average over all possible future volatility scenarios. This representation is not only explained but is applied to help understand the impact of different modeling assumptions on the shape and dynamics of volatility surfaces—a topic of fundamental interest to traders as well as quants. Along the way, various practical results and tricks are presented and explained. Finally, the hot topic of volatility derivatives is exhaustively covered with detailed presentations of the latest research.
Academics may also find the book useful not just as a guide to the current state of research in volatility modeling but also to provide practical context for their work. Practitioners have one huge advantage over academics: They never have to worry about whether or not their work will be interesting to others. This book can thus be viewed as one practitioner’s guide to what is interesting and useful.
In short, my hope is that the book will prove useful to anyone interested in the volatility surface whether academic or practitioner.
Readers familiar with my New York University Courant Institute lecture notes will surely recognize the contents of this book. I hope that even aficionados of the lecture notes will find something of extra value in the book. The material has been expanded; there are more and better figures; and there’s now an index.
The lecture notes on which this book is based were originally targeted at graduate students in the final semester of a three-semester Master’s Program in Financial Mathematics. Students entering the program have undergraduate degrees in quantitative subjects such as mathematics, physics, or engineering. Some are part-time students already working in the industry looking to deepen their understanding of the mathematical aspects of their jobs, others are looking to obtain the necessary mathematical and financial background for a career in the financial industry. By the time they reach the third semester, students have studied financial mathematics, computing and basic probability and stochastic processes.
It follows that to get the most out of this book, the reader should have a level of familiarity with options theory and financial markets that could be obtained from Wilmott (2000), for example. To be able to follow the mathematics, basic knowledge of probability and stochastic calculus such as could be obtained by reading Neftci (2000) or Mikosch (1999) are required. Nevertheless, my hope is that a reader willing to take the mathematical results on trust will still be able to follow the explanations.
HOW THIS BOOK IS ORGANIZED
The first half of the book from Chapters 1 to 5 focuses on setting up the theoretical framework. The latter chapters of the book are more oriented towards practical applications. The split is not rigorous, however, and there are practical applications in the first few chapters and theoretical constructions in the last chapter, reflecting that life, at least the life of a practicing quant, is not split into neat boxes.
Chapter 1 provides an explanation of stochastic and local volatility; local variance is shown to be the risk-neutral expectation of instantaneous variance, a result that is applied repeatedly in later chapters. In Chapter 2, we present the still supremely popular Heston model and derive the Heston European option pricing formula. We also show how to simulate the Heston model.
In Chapter 3, we derive a powerful representation for implied volatility in terms of local volatility. We apply this to build intuition and derive some properties of the implied volatility surface generated by the Heston model and compare with the empirically observed SPX surface. We deduce that stochastic volatility cannot be the whole story.
In Chapter 4, we choose specific numerical values for the parameters of the Heston model, specifically ρ = −1 as originally studied by Heston and Nandi. We demonstrate that an approximate formula for implied volatility derived in Chapter 3 works particularly well in this limit. As a result, we are able to find parameters of local volatility and stochastic volatility models that generate almost identical European option prices. We use these parameters repeatedly in subsequent chapters to illustrate the model-dependence of various claims.
In Chapter 5, we explore the modeling of jumps. First we show why jumps are required. We then introduce characteristic function techniques and apply these to the computation of implied volatilities in models with jumps. We conclude by showing that the SVJ model (stochastic volatility with jumps in the stock price) is capable of generating a volatility surface that has most of the features of the empirical surface. Throughout, we build intuition as to how jumps should affect the shape of the volatility surface.
