The VolatilitySurfaceA Practitioners GuideJIM GATHERALForeword by Nassim Nicholas TalebJohn Wiley & Sons, Inc.Further Praise for The Volatility SurfaceAs an experienced practitioner, Jim Gatheral succeeds admirably in com-bining an accessible exposition of the foundations of stochastic volatilitymodeling with valuable guidance on the calibration and implementation ofleading volatility models in practice.Eckhard Platen, Chair in Quantitative Finance, University ofTechnology, SydneyDr. Jim Gatheral is one of Wall Streets very best regarding the practicaluse and understanding of volatility modeling. The Volatility Surface reectshis 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 areinterested in volatility and derivatives, you need this book!Espen Gaarder Haug, option trader, and author to The CompleteGuide to Option Pricing FormulasAnybody who is interested in going beyond Black-Scholes should read thisbook. And anybody who is not interested in going beyond Black-Scholesisnt going far!Mark Davis, Professor of Mathematics, Imperial College LondonThis book provides a comprehensive treatment of subjects essential foranyone working in the eld of option pricing. Many technical topics arepresented in an elegant and intuitively clear way. It will be indispensable notonly at trading desks but also for teaching courses on modern derivativesand will denitely serve as a source of inspiration for new research.Anna Shepeleva, Vice President, ING GroupFounded in 1807, John Wiley & Sons is the oldest independent publishingcompany in the United States. With ofces in North America, Europe,Australia, and Asia, Wiley is globally committed to developing and market-ing print and electronic products and services for our customers professionaland personal knowledge and understanding.The Wiley Finance series contains books written specically for nanceand investment professionals as well as sophisticated individual investorsand their nancial advisors. Book topics range from portfolio managementto e-commerce, risk management, nancial engineering, valuation, andnancial instrument analysis, as well as much more.For a list of available titles, please visit our Web site at www.WileyFinance.com.The VolatilitySurfaceA Practitioners GuideJIM GATHERALForeword by Nassim Nicholas TalebJohn Wiley & Sons, Inc.Copyright c 2006 by Jim Gatheral. All rights reserved.Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.No part of this publication may be reproduced, stored in a retrieval system, or transmitted inany form or by any means, electronic, mechanical, photocopying, recording, scanning, orotherwise, except as permitted under Section 107 or 108 of the 1976 United States CopyrightAct, without either the prior written permission of the Publisher, or authorization throughpayment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the Webat www.copyright.com. Requests to the Publisher for permission should be addressed to thePermissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030,(201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission.Limit of Liability/Disclaimer of Warranty: While the publisher and author have used theirbest efforts in preparing this book, they make no representations or warranties with respect tothe accuracy or completeness of the contents of this book and specically disclaim any impliedwarranties of merchantability or tness for a particular purpose. No warranty may be createdor extended by sales representatives or written sales materials. The advice and strategiescontained herein may not be suitable for your situation. You should consult with aprofessional where appropriate. Neither the publisher nor author shall be liable for any loss ofprot 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, pleasecontact our Customer Care Department within the United States at (800) 762-2974, outsidethe United States at (317) 572-3993 or fax (317) 572-4002.Wiley also publishes its books in a variety of electronic formats. Some content that appears inprint may not be available in electronic formats. For more information about Wiley products,visit our Web site at www.wiley.com.ISBN-13 978-0-471-79251-2ISBN-10 0-471-79251-9Library of Congress Cataloging-in-Publication Data:Gatheral, Jim, 1957The volatility surface : a practitioners guide / by Jim Gatheral ; forewordby Nassim Nicholas Taleb.p. cm. (Wiley nance series)Includes index.ISBN-13: 978-0-471-79251-2 (cloth)ISBN-10: 0-471-79251-9 (cloth)1. Options (Finance)PricesMathematical models. 2.StocksPricesMathematical models. I. Title. II. Series.HG6024. A3G38 2006332.632220151922dc222006009977Printed in the United States of America.10 9 8 7 6 5 4 3 2 1To Yukiko and AyakoContentsLi st of Fi gures xi i iList of Tables xixForeword xxiPreface xxiiiAcknowledgments xxviiCHAPTER 1Stochastic Volatility and Local Volatility 1Stochastic Volatility 1Derivation of the Valuation Equation 4Local Volatility 7History 7A Brief Review of Dupires Work 8Derivation of the Dupire Equation 9Local Volatility in Terms of Implied Volatility 11Special Case: No Skew 13Local Variance as a Conditional Expectationof Instantaneous Variance 13CHAPTER 2The Heston Model 15The Process 15The Heston Solution for European Options 16A Digression: The Complex Logarithmin the Integration (2.13) 19Derivation of the Heston Characteristic Function 20Simulation of the Heston Process 21Milstein Discretization 22Sampling from the Exact Transition Law 23Why the Heston Model Is so Popular 24viiviii CONTENTSCHAPTER 3The Implied Volatility Surface 25Getting Implied Volatility from Local Volatilities 25Model Calibration 25Understanding Implied Volatility 26Local Volatility in the Heston Model 31Ansatz 32Implied Volatility in the Heston Model 33The Term Structure of Black-Scholes Implied Volatilityin the Heston Model 34The Black-Scholes Implied Volatility Skewin the Heston Model 35The SPX Implied Volatility Surface 36Another Digression: The SVI Parameterization 37A Heston Fit to the Data 40Final Remarks on SV Models and Fittingthe Volatility Surface 42CHAPTER 4The Heston-Nandi Model 43Local Variance in the Heston-Nandi Model 43A Numerical Example 44The Heston-Nandi Density 45Computation of Local Volatilities 45Computation of Implied Volatilities 46Discussion of Results 49CHAPTER 5Adding Jumps 50Why Jumps are Needed 50Jump Diffusion 52Derivation of the Valuation Equation 52Uncertain Jump Size 54Characteristic Function Methods 56L evy Processes 56Examples of Characteristic Functionsfor Specic Processes 57Computing Option Prices from theCharacteristic Function 58Proof of (5.6) 58Contents ixComputing Implied Volatility 60Computing the At-the-Money Volatility Skew 60How Jumps Impact the Volatility Skew 61Stochastic Volatility Plus Jumps 65Stochastic Volatility Plus Jumps in the UnderlyingOnly (SVJ) 65Some Empirical Fits to the SPX Volatility Surface 66Stochastic Volatility with Simultaneous Jumpsin Stock Price and Volatility (SVJJ) 68SVJ Fit to the September 15, 2005, SPX Option Data 71Why the SVJ Model Wins 73CHAPTER 6Modeling Default Risk 74Mertons Model of Default 74Intuition 75Implications for the Volatility Skew 76Capital Structure Arbitrage 77Put-Call Parity 77The Arbitrage 78Local and Implied Volatility in the Jump-to-Ruin Model 79The Effect of Default Risk on Option Prices 82The CreditGrades Model 84Model Setup 84Survival Probability 85Equity Volatility 86Model Calibration 86CHAPTER 7Volatility Surface Asymptotics 87Short Expirations 87The Medvedev-Scaillet Result 89The SABR Model 91Including Jumps 93Corollaries 94Long Expirations: Fouque, Papanicolaou, and Sircar 95Small Volatility of Volatility: Lewis 96Extreme Strikes: Roger Lee 97Example: Black-Scholes 99Stochastic Volatility Models 99Asymptotics in Summary 100x CONTENTSCHAPTER 8Dynamics of the Volatility Surface 101Dynamics of the Volatility Skew under Stochastic Volatility 101Dynamics of the Volatility Skew under Local Volatility 102Stochastic Implied Volatility Models 103Digital Options and Digital Cliquets 103Valuing Digital Options 104Digital Cliquets 104CHAPTER 9Barrier Options 107Denitions 107Limiting Cases 108Limit Orders 108European Capped Calls 109The Reection Principle 109The Lookback Hedging Argument 112One-Touch Options Again 113Put-Call Symmetry 113QuasiStatic Hedging and Qualitative Valuation 114Out-of-the-Money Barrier Options 114One-Touch Options 115Live-Out Options 116Lookback Options 117Adjusting for Discrete Monitoring 117Discretely Monitored Lookback Options 119Parisian Options 120Some Applications of Barrier Options 120Ladders 120Ranges 120Conclusion 121CHAPTER 10Exotic Cliquets 122Locally Capped Globally Floored Cliquet 122Valuation under Heston and LocalVolatility Assumptions 123Performance 124Reverse Cliquet 125Contents xiValuation under Heston and LocalVolatility Assumptions 126Performance 127Napoleon 127Valuation under Heston and LocalVolatility Assumptions 128Performance 130Investor Motivation 130More on Napoleons 131CHAPTER 11Volatility Derivatives 133Spanning Generalized European Payoffs 133Example: European Options 134Example: Amortizing Options 135The Log Contract 135Variance and Volatility Swaps 136Variance Swaps 137Variance Swaps in the Heston Model 138Dependence on Skew and Curvature 138The Effect of Jumps 140Volatility Swaps 143Convexity Adjustment in the Heston Model 144Valuing Volatility Derivatives 146Fair Value of the Power Payoff 146The Laplace Transform of Quadratic Variation underZero Correlation 147The Fair Value of Volatility under Zero Correlation 149A Simple Lognormal Model 151Options on Volatility: More on Model Independence 154Listed Quadratic-Variation Based Securities 156The VIX Index 156VXB Futures 158Knock-on Benets 160Summary 161Postscript 162Bibliography 163Index 169Figures1.1 SPX daily log returns from December 31, 1984, to December31, 2004. Note the 22.9% return on October 19, 1987! 21.2 Frequency distribution of (77 years of) SPX daily log returnscompared with the normal distribution. Although the 22.9%return on October 19, 1987, is not directly visible, the x-axishas been extended to the left to accommodate it! 31.3 Q-Q plot of SPX daily log returns compared with the normaldistribution. Note the extreme tails. 33.1 Graph of the pdf of xt conditional on xT = log(K) for a 1-yearEuropean option, strike 1.3 with current stock price = 1 and20% volatility. 313.2 Graph of the SPX-implied volatility surface as of the close onSeptember 15, 2005, the day before triple witching. 363.3 Plots of the SVI ts to SPX implied volatilities for each of theeight 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 volatilitiesrespectively and the solid line is the SVI t. 383.4 Graph of SPX ATM skew versus time to expiry. The solid lineis a t of the approximate skew formula (3.21) to all empiricalskewpoints except the rst; the dashed t excludes the rst threedata points. 393.5 Graph of SPX ATM variance versus time to expiry. The solidline is a t of the approximate ATM variance formula (3.18) tothe empirical data. 403.6 Comparison of the empirical SPX implied volatility surface withthe Heston t as of September 15, 2005. From the two viewspresented here, we can see that the Heston t is pretty goodxiiixiv FIGURESfor longer expirations but really not close for short expirations.The paler upper surface is the empirical SPX volatility surfaceand the darker lower one the Heston t. The Heston t surfacehas been shifted down by ve volatility points for ease of visualcomparison. 414.1 The probability density for the Heston-Nandi model with ourparameters and expiration T = 0.1. 454.2 Comparison of approximate formulas with direct numericalcomputation of Heston local variance. For each expiration T,the solid line is the numerical computation and the dashed lineis the approximate formula. 474.3 Comparison of European implied volatilities fromapplication ofthe Heston formula (2.13) and from a numerical PDE computa-tion using the local volatilities given by the approximate formula(4.1). For each expiration T, the solid line is the numericalcomputation and the dashed line is the approximate formula. 485.1 Graph of the September 16, 2005, expiration volatility smile asof the close on September 15, 2005. SPX is trading at 1227.73.Triangles represent bids and offers. The solid line is a nonlinear(SVI) t to the data. The dashed line represents the Heston skewwith Sep05 SPX parameters. 525.2 The 3-month volatility smile for various choices of jump diffu-sion parameters. 635.3 The term structure of ATM variance skew for various choices ofjump diffusion parameters. 645.4 As time to expiration increases, the return distribution looksmore and more normal. The solid line is the jump diffusion pdfand for comparison, the dashed line is the normal density withthe same mean and standard deviation. With the parametersused to generate these plots, the characteristic time T = 0.67. 655.5 The solid line is a graph of the at-the-money variance skewin the SVJ model with BCC parameters vs. time to expiration.The dashed line represents the sum of at-the-money Heston andjump diffusion skews with the same parameters. 675.6 The solid line is a graph of the at-the-money variance skew inthe SVJ model with BCC parameters versus time to expiration.The dashed line represents the at-the-money Heston skew withthe same parameters. 67Figures xv5.7 The solid line is a graph of the at-the-money variance skewin theSVJJ model with BCC parameters versus time to expiration. Theshort-dashed and long-dashed lines are SVJ and Heston skewgraphs respectively with the same parameters. 705.8 This graph is a short-expiration detailed viewof the graph shownin Figure 5.7. 715.9 Comparison of the empirical SPX implied volatility surface withthe SVJ t as of September 15, 2005. From the two viewspresented here, we can see that in contrast to the Heston case,the major features of the empirical surface are replicated bythe SVJ model. The paler upper surface is the empirical SPXvolatility surface and the darker lower one the SVJ t. The SVJt surface has again been shifted down by ve volatility pointsfor ease of visual comparison. 726.1 Three-month implied volatilities from the Merton model assum-ing a stock volatility of 20%and credit spreads of 100 bp (solid),200 bp (dashed) and 300 bp (long-dashed). 766.2 Payoff of the 1 2 put spread combination: buy one put withstrike 1.0 and sell two puts with strike 0.5. 796.3 Local variance plot with = 0.05 and = 0.2. 816.4 The triangles represent bid and offer volatilities and the solidline is the Merton model t. 837.1 For short expirations, the most probable path is approximatelya straight line from spot on the valuation date to the strike atexpiration. It follows that 2BS

k, T

vloc(0, 0) +vloc(k, T)

/2and the implied variance skew is roughly one half of the localvariance skew. 898.1 Illustration of a cliquet payoff. This hypothetical SPX cliquetresets at-the-money every year on October 31. The thick solidlines represent nonzero cliquet payoffs. The payoff of a 5-yearEuropean option struck at the October 31, 2000, SPX level of1429.40 would have been zero. 1059.1 A realization of the zero log-drift stochastic process and thereected path. 1109.2 The ratio of the value of a one-touch call to the value ofa European binary call under stochastic volatility and localxvi FIGURESvolatility assumptions as a function of strike. The solid line isstochastic volatility and the dashed line is local volatility. 1119.3 The value of a European binary call under stochastic volatilityand local volatility assumptions as a function of strike. The solidline is stochastic volatility and the dashed line is local volatility.The two lines are almost indistinguishable. 1119.4 The value of a one-touch call under stochastic volatility and localvolatility assumptions as a function of barrier level. The solidline is stochastic volatility and the dashed line is local volatility. 1129.5 Values of knock-out call options struck at 1 as a function ofbarrier level. The solid line is stochastic volatility; the dashedline is local volatility. 1159.6 Values of knock-out call options struck at 0.9 as a function ofbarrier level. The solid line is stochastic volatility; the dashedline is local volatility. 1169.7 Values of live-out call options struck at 1 as a function of barrierlevel. The solid line is stochastic volatility; the dashed line islocal volatility. 1179.8 Values of lookback call options as a function of strike. The solidline is stochastic volatility; the dashed line is local volatility. 11810.1 Value of the Mediobanca Bond Protection 20022005 locallycapped and globally oored cliquet (minus guaranteed redemp-tion) as a function of MinCoupon. The solid line is stochasticvolatility; the dashed line is local volatility. 12410.2 Historical performance of the Mediobanca Bond Protection20022005 locally capped and globally oored cliquet. Thedashed vertical lines represent reset dates, the solid lines couponsetting dates and the solid horizontal lines represent xings. 12510.3 Value of the Mediobanca reverse cliquet (minus guaranteedredemption) as a function of MaxCoupon. The solid line isstochastic volatility; the dashed line is local volatility. 12710.4 Historical performance of the Mediobanca 20002005 ReverseCliquet Telecommunicazioni reverse cliquet. The vertical linesrepresent reset dates, the solid horizontal lines represent xingsand the vertical grey bars represent negative contributions to thecliquet payoff. 12810.5 Value of (risk-neutral) expected Napoleon coupon as a functionof MaxCoupon. The solid line is stochastic volatility; the dashedline is local volatility. 129Figures xvii10.6 Historical performance of the STOXX 50 component of theMediobanca 20022005 World Indices Euro Note Serie 46Napoleon. The light vertical lines represent reset dates, theheavy vertical lines coupon setting dates, the solid horizontallines represent xings and the thick grey bars represent theminimum monthly return of each coupon period. 13011.1 Payoff of a variance swap (dashed line) and volatility swap(solid line) as a function of realized volatility T. Both swapsare struck at 30% volatility. 14311.2 Annualized Heston convexity adjustment as a function of T withHeston-Nandi parameters. 14511.3 Annualized Heston convexity adjustment as a function of T withBakshi, Cao, and Chen parameters. 14511.4 Value of 1-year variance call versus variance strike K with theBCC parameters. The solid line is a numerical Heston solution;the dashed line comes from our lognormal approximation. 15311.5 The pdf of the log of 1-year quadratic variation with BCCparameters. The solid line comes from an exact numericalHeston computation; the dashed line comes from our lognormalapproximation. 15411.6 Annualized Heston VXB convexity adjustment as a function oft with Heston parameters from December 8, 2004, SPX t. 160Tables3.1 At-the-money SPX variance levels and skews as of the close onSeptember 15, 2005, the day before expiration. 393.2 Heston t to the SPX surface as of the close on September 15,2005. 405.1 September 2005 expiration option prices as of the close onSeptember 15, 2005. Triple witching is the following day. SPXis trading at 1227.73. 515.2 Parameters used to generate Figures 5.2 and 5.3. 635.3 Interpreting Figures 5.2 and 5.3. 645.4 Various ts of jump diffusion style models to SPX data. JDmeans Jump Diffusion and SVJ means Stochastic Volatility plusJumps. 695.5 SVJ t to the SPX surface as of the close on September 15, 2005. 716.1 Upper and lower arbitrage bounds for one-year 0.5 strike optionsfor various credit spreads (at-the-money volatility is 20%). 796.2 Implied volatilities for January 2005 options on GT as ofOctober 20, 2004 (GT was trading at 9.40). Merton volsare volatilities generated from the Merton model with ttedparameters. 8210.1 Estimated Mediobanca Bond Protection 20022005 coupons. 12510.2 Worst monthly returns and estimated Napoleon coupons. Recallthat the coupon is computed as 10% plus the worst monthlyreturn averaged over the three underlying indices. 13111.1 Empirical VXB convexity adjustments as of December 8, 2004. 159xixForewordIJim has given round six of these lectures on volatility modeling at theCourant Institute of New York University, slowly purifying these notes. Iwitnessed and became addicted to their slow maturation from the rst timehe jotted down these equations during the winter of 2000, to the most recentone in the spring of 2006. It was similar to the progressive distillation ofgood alcohol: exactly seven times; at every new stage you can see the textgaining in crispness, clarity, and concision. Like Jims lectures, these chaptersare to the point, with maximal simplicity though never less than warrantedby the topic, devoid of uff and side distractions, delivering the exact subjectwithout any attempt to boast his (extraordinary) technical skills.The class became popular. By the second year we got yelled at by theuniversity staff because too many nonpaying practitioners showed up to thelecture, 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 lecturenotes circulating among instructors. His treatment of local volatility andstochastic models became the standard.As colecturers, Jim G. and I agreed to attend each others sessions, butas more than just spectatorsturning out to be colecturers in the literalsense, that is, synchronously. He and I heckled each other, making surethat not a single point went undisputed, to the point of other members ofthe faculty coming to attend this strange class with disputatious instructorstrying to tear apart each others statements, looking for the smallest hole inthe arguments. Nor were the arguments always dispassionate: students soongot to learn from Jim my habit of ordering white wine with read meat; inreturn, I pointed out clear deciencies in his French, which he pronounceswith a sometimes incomprehensible Scottish accent. I realized the value ofthe course when I started lecturing at other universities. The contrast wassuch that I had to return very quickly.IIThe difference between Jim Gatheral and other members of the quantcommunity lies in the following: To many, models provide a representationxxixxii FOREWORDof asset price dynamics, under some constraints. Business school nanceprofessors have a tendency to believe (for some reason) that these providea top-down statistical mapping of reality. This interpretation is also sharedby many of those who have not been exposed to activity of risk-taking, orthe constraints of empirical reality.But not to Jim G. who has both traded and led a career as a quant. Tohim, these stochastic volatility models cannot make such claims, or shouldnot make such claims. They are not to be deemed a top-down dogmaticrepresentation of reality, rather a tool to insure that all instruments areconsistently priced with respect to each otherthat is, to satisfy the goldenrule of absence of arbitrage. An operator should not be capable of derivinga prot in replicating a nancial instrument by using a combination of otherones. A model should do the job of insuring maximal consistency between,say, a European digital option of a given maturity, and a call price ofanother one. The best model is the one that satises such constraints whilemaking minimal claims about the true probability distribution of the world.I recently discovered the strength of his thinking as follows. When, bythe fth or so lecture series I realized that the world needed Mandelbrot-stylepower-law or scalable distributions, I found that the models he proposed offudging the volatility surface was compatible with these models. How? Youjust need to raise volatilities of out-of-the-money options in a specic way,and the volatility surface becomes consistent with the scalable power laws.Jim Gatheral is a natural and intuitive mathematician; attending his lec-ture 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 isgreatly enhanced by the aesthetics of LaTeX.Nassim Nicholas Taleb1June, 20061Author, Dynamic Hedging and Fooled by Randomness.PrefaceEver since the advent of the Black-Scholes option pricing formula, thestudy of implied volatility has become a central preoccupation for bothacademics and practitioners. As is well known, actual option prices rarelyif ever conform to the predictions of the formula because the idealizedassumptions required for it to hold dont apply in the real world. Conse-quently, implied volatility (the volatility input to the Black-Scholes formulathat generates the market price) in general depends on the strike and theexpiration of the option. The collection of all such implied volatilities isknown as the volatility surface.This book concerns itself with understanding the volatility surface; thatis, why options are priced as they are and what it is that analysis of stockreturns can tell as about how options ought to be priced.Pricing is consistently emphasized over hedging, although hedging andreplication arguments are often used to generate results. Partly, thatsbecause pricing is key: How a claim is hedged affects only the width of theresulting distribution of returns and not the expectation. On average, noamount of clever hedging can make up for an initial mispricing. Partly, itsbecause hedging in practice can be complicated and even more of an artthan pricing.Throughout the book, the importance of examining different dynamicalassumptions 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 providethe reader with ways of thinking about and solving practical problemsthat should have many other areas of application. By the end of the book,the reader should have gained substantial intuition for the latest theoryunderlying options pricing as well as some feel for the history and practiceof trading in the equity derivatives markets. With luck, the reader will alsobe infected with some of the excitement that continues to surround thetrading, 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 Hestonmodel and explanations of many other popular models such as SVJ, SVJJ,SABR, and CreditGrades. The reader will also nd explanations of thecharacteristics of various types of exotic options from the humble barrierxxiiixxiv PREFACEoption to the super exotic Napoleon. One of the themes of this bookis the representation of implied volatility in terms of a weighted averageover all possible future volatility scenarios. This representation is not onlyexplained but is applied to help understand the impact of different modelingassumptions on the shape and dynamics of volatility surfacesa topic offundamental interest to traders as well as quants. Along the way, variouspractical results and tricks are presented and explained. Finally, the hot topicof volatility derivatives is exhaustively covered with detailed presentationsof the latest research.Academics may also nd the book useful not just as a guide to the currentstate of research in volatility modeling but also to provide practical contextfor their work. Practitioners have one huge advantage over academics: Theynever have to worry about whether or not their work will be interesting toothers. This book can thus be viewed as one practitioners guide to what isinteresting and useful.In short, my hope is that the book will prove useful to anyone interestedin the volatility surface whether academic or practitioner.Readers familiar with my NewYork University Courant Institute lecturenotes will surely recognize the contents of this book. I hope that evenacionados of the lecture notes will nd something of extra value in thebook. The material has been expanded; there are more and better gures;and theres now an index.The lecture notes on which this book is based were originally targetedat graduate students in the nal semester of a three-semester MastersProgram in Financial Mathematics. Students entering the program haveundergraduate degrees in quantitative subjects such as mathematics, physics,or engineering. Some are part-time students already working in the industrylooking to deepen their understanding of the mathematical aspects of theirjobs, others are looking to obtain the necessary mathematical and nancialbackground for a career in the nancial industry. By the time they reach thethird semester, students have studied nancial mathematics, computing andbasic probability and stochastic processes.It follows that to get the most out of this book, the reader should havea level of familiarity with options theory and nancial markets that couldbe obtained from Wilmott (2000), for example. To be able to follow themathematics, basic knowledge of probability and stochastic calculus such ascould be obtained by reading Neftci (2000) or Mikosch (1999) are required.Nevertheless, my hope is that a reader willing to take the mathematicalresults on trust will still be able to follow the explanations.Preface xxvHOW THIS BOOK IS ORGANIZEDThe rst half of the book from Chapters 1 to 5 focuses on setting up thetheoretical framework. The latter chapters of the book are more orientedtowards practical applications. The split is not rigorous, however, andthere are practical applications in the rst few chapters and theoreticalconstructions in the last chapter, reecting that life, at least the life of apracticing 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 instantaneousvariance, a result that is applied repeatedly in later chapters. In Chapter 2,we present the still supremely popular Heston model and derive the HestonEuropean option pricing formula. We also showhowto simulate the Hestonmodel.In Chapter 3, we derive a powerful representation for implied volatilityin terms of local volatility. We apply this to build intuition and derive someproperties of the implied volatility surface generated by the Heston modeland compare with the empirically observed SPX surface. We deduce thatstochastic volatility cannot be the whole story.In Chapter 4, we choose specic numerical values for the parametersof the Heston model, specically = 1 as originally studied by Hestonand Nandi. We demonstrate that an approximate formula for impliedvolatility derived in Chapter 3 works particularly well in this limit. Asa result, we are able to nd parameters of local volatility and stochasticvolatility models that generate almost identical European option prices.We use these parameters repeatedly in subsequent chapters to illustrate themodel-dependence of various claims.In Chapter 5, we explore the modeling of jumps. First we show whyjumps are required. We then introduce characteristic function techniquesand apply these to the computation of implied volatilities in models withjumps. We conclude by showing that the SVJ model (stochastic volatilitywith jumps in the stock price) is capable of generating a volatility surfacethat has most of the features of the empirical surface. Throughout, we buildintuition as to how jumps should affect the shape of the volatility surface.In Chapter 6, we apply our work on jumps to Mertons jump-to-ruinmodel of default. We also explain the CreditGrades model. In passing, wetouch on capital structure arbitrage and offer the rst glimpse into the lessthan ideal world of real trading, explaining how large losses were incurredby market makers.In Chapter 7, we examine the asymptotic properties of the volatilitysurface showing that all models with stochastic volatility and jumps generatevolatility surfaces that are roughly the same shape. In Chapter 8, we showxxvi PREFACEhowthe dynamics of volatility can be deduced fromthe time series propertiesof volatility surfaces. We also show why it is that the dynamics of thevolatility surfaces generated by local volatility models are highly unrealistic.In Chapter 9, we present various types of barrier option and show howintuition may be developed for these by studying two simple limiting cases.We test our intuition (successfully) by applying it to the relative valuation ofbarrier options under stochastic and local volatility. The reection principleand the concepts of quasi-static hedging and put-call symmetry are presentedand applied.In Chapter 10, we study in detail three actual exotic cliquet transactionsthat happen to have matured so that we can explore both pricing andex post performance. Specically, we study a locally capped and globallyoored cliquet, a reverse cliquet, and a Napoleon. Followers of the nancialpress no doubt already recognize these deal types as having been the causeof substantial pain to some dealers.Finally, in Chapter 11, the longest of all, we focus on the pricingand hedging of claims whose underlying is quadratic variation. In sodoing, we will present some of the most elegant and robust results innancial mathematics, thereby explaining in part why the market in volatilityderivatives is surprisingly active and liquid.Jim GatheralAcknowledgmentsIam grateful to more people than I could possibly list here for theirhelp, support and encouragement over the years. First of all, I owe adebt of gratitude to my present and former colleagues, in particular tomy Merrill Lynch quant colleagues Jining Han, Chiyan Luo and YonathanEpelbaum. Second, like all practitioners, my education is partly thanks tothose academics and practitioners who openly published their work. Sincethe bibliography is not meant to be a complete list of references but ratherjust a list of sources for the present text, there are many people who havemade great contributions to the eld and strongly inuenced my work thatare not explicitly mentioned or referenced. To these people, please be sure Iam grateful to all of you.There are a few people who had a much more direct hand in this projectto whom explicit thanks are due here: to Nassim Taleb, my co-lecturer atCourant who through good-natured heckling helped shape the contents ofmy lectures, to Peter Carr, Bruno Dupire and Marco Avellaneda for helpfuland insightful conversations and nally to Neil Chriss for sharing somegood writing tips and for inviting me to lecture at Courant in the rst place.I am absolutely indebted to Peter Friz, my one-time teaching assistant atNYU and now lecturer at the Statistical Laboratory in Cambridge; Peterpainstakingly read my lectures notes, correcting them often and suggestingimprovements. Without him, there is no doubt that there would have beenno book. My thanks are also due to him and to Bruno Dupire for readinga late draft of the manuscript and making useful suggestions. I also wish tothank my editors at Wiley: Pamela Van Giessen, Jennifer MacDonald andTodd Tedesco for their help. Remaining errors are of course mine.Last but by no means least, I am deeply grateful to Yukiko and Ayakofor putting up with me.xxviiCHAPTER1Stochastic Volatilityand Local VolatilityIn this chapter, we begin our exploration of the volatility surface by intro-ducing stochastic volatilitythe notion that volatility varies in a randomfashion. Local variance is then shown to be a conditional expectation ofthe instantaneous variance so that various quantities of interest (such asoption prices) may sometimes be computed as though future volatility weredeterministic rather than stochastic.STOCHASTIC VOLATILITYThat it might make sense to model volatility as a random variable shouldbe clear to the most casual observer of equity markets. To be convinced,one need only recall the stock market crash of October 1987. Nevertheless,given the success of the Black-Scholes model in parsimoniously describingmarket options prices, its not immediately obvious what the benets ofmaking such a modeling choice might be.Stochastic volatility (SV) models are useful because they explain in aself-consistent way why options with different strikes and expirations havedifferent Black-Scholes implied volatilitiesthat is, the volatility smile.Moreover, unlike alternative models that can t the smile (such as localvolatility models, for example), SV models assume realistic dynamics forthe underlying. Although SV price processes are sometimes accused of beingad hoc, on the contrary, they can be viewed as arising from Brownianmotion subordinated to a random clock. This clock time, often referred toas trading time, may be identied with the volume of trades or the frequencyof trading (Clark 1973); the idea is that as trading activity uctuates, sodoes volatility.12 THE VOLATILITY SURFACE0.20.10.00.1FIGURE 1.1 SPX daily log returns from December 31, 1984, to December 31,2004. Note the 22.9% return on October 19, 1987!From a hedging perspective, traders who use the Black-Scholes modelmust continuously change the volatility assumption in order to matchmarket prices. Their hedge ratios change accordingly in an uncontrolledway: SV models bring some order into this chaos.A practical point that is more pertinent to a recurring theme of thisbook is that the prices of exotic options given by models based on Black-Scholes assumptions can be wildly wrong and dealers in such options aremotivated to nd models that can take the volatility smile into accountwhen pricing these.In Figure 1.1, we plot the log returns of SPX over a 15-year period;we see that large moves follow large moves and small moves follow smallmoves (so-called volatility clustering). In Figure 1.2, we plot the frequencydistribution of SPX log returns over the 77-year period from 1928 to 2005.We see that this distribution is highly peaked and fat-tailed relative to thenormal distribution. The Q-Q plot in Figure 1.3 shows just how extremethe tails of the empirical distribution of returns are relative to the normaldistribution. (This plot would be a straight line if the empirical distributionwere normal.)Fat tails and the high central peak are characteristics of mixtures ofdistributions with different variances. This motivates us to model varianceas a random variable. The volatility clustering feature implies that volatility(or variance) is auto-correlated. In the model, this is a consequence of themean reversion of volatility.Note that simple jump-diffusion models do not have this property. After a jump,the stock price volatility does not change.Stochastic Volatility and Local Volatility 30.20 0.15 0.10 0.05 0.00 0.05 0.10010203040506070FIGURE 1.2 Frequency distribution of (77 years of) SPX daily log returns comparedwith the normal distribution. Although the 22.9% return on October 19, 1987, isnot 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 normaldistribution. Note the extreme tails.4 THE VOLATILITY SURFACEThere is a simple economic argument that justies the mean reversionof volatility. (The same argument is used to justify the mean reversion ofinterest rates.) Consider the distribution of the volatility of IBM in 100 yearstime. If volatility were not mean reverting (i.e., if the distribution of volatilitywere not stable), the probability of the volatility of IBM being between 1%and 100% would be rather low. Since we believe that it is overwhelminglylikely that the volatility of IBM would in fact lie in that range, we deducethat volatility must be mean reverting.Having motivated the description of variance as a mean revertingrandom variable, we are now ready to derive the valuation equation.Derivation of the Valuation EquationIn this section, we follow Wilmott (2000) closely. Suppose that the stockprice S and its variance v satisfy the following SDEs:dSt = tSt dt + vtSt dZ1 (1.1)dvt = (St, vt, t) dt + (St, vt, t)vtdZ2 (1.2)with_dZ1dZ2_ = dtwhere t is the (deterministic) instantaneous drift of stock price returns, is the volatility of volatility and is the correlation between random stockprice returns and changes in vt. dZ1 and dZ2 are Wiener processes.The stochastic process (1.1) followed by the stock price is equivalentto the one assumed in the derivation of Black and Scholes (1973). Thisensures that the standard time-dependent volatility version of the Black-Scholes formula (as derived in Section 8.6 of Wilmott (2000) for example)may be retrieved in the limit 0. In practical applications, this is a keyrequirement of a stochastic volatility option pricing model as practitionersintuition for the behavior of option prices is invariably expressed within theframework of the Black-Scholes formula.In contrast, the stochastic process (1.2) followed by the variance is verygeneral. We dont assume anything about the functional forms of () and(). In particular, we dont assume a square root process for variance.In the Black-Scholes case, there is only one source of randomness, thestock price, which can be hedged with stock. In the present case, randomchanges in volatility also need to be hedged in order to form a risklessportfolio. So we set up a portfolio containing the option being priced,whose value we denote by V(S, v, t), a quantity of the stock andStochastic Volatility and Local Volatility 5a quantity 1 of another asset whose value V1 depends on volatility.We have = V S 1 V1The change in this portfolio in a time dt is given byd =_Vt + 12 v S2 2VS2 + v S 2Vv S + 122v22Vv2_ dt 1_V1t + 12 v S2 2V1S2 + v S 2V1v S + 12 2v 2 2V1v2_ dt+_VS 1V1S _dS+_Vv 1V1v_ dvwhere, for clarity, we have eliminated the explicit dependence on t of thestate variables St and vt and the dependence of and on the state variables.To make the portfolio instantaneously risk-free, we must chooseVS 1V1S = 0to eliminate dS terms, andVv 1V1v = 0to eliminate dv terms. This leaves us withd =_Vt + 12v S22VS2 + v S 2VvS + 122v22Vv2_dt 1_V1t + 12v S22V1S2 + v S2V1vS + 122v 22V1v2_dt= r dt= r(V S 1V1) dtwhere we have used the fact that the return on a risk-free portfolio mustequal the risk-free rate r, which we will assume to be deterministic for ourpurposes. Collecting all V terms on the left-hand side and all V1 terms on6 THE VOLATILITY SURFACEthe right-hand side, we getVt + 12v S2 2VS2 + v S2VvS + 122v2 2Vv2 +rSVS rVVv=V1t + 12v S2 2V1S2 + v S2V1vS + 122v2 2V1v2 +rSV1S rV1V1vThe left-hand side is a function of V only and the right-hand side is afunction of V1 only. The only way that this can be is for both sides tobe equal to some function f of the independent variables S, v and t. Wededuce thatVt + 12 v S2 2VS2 + v S 2Vv S + 12 2v 2 2Vv2 +r S VS r V= _ v_ Vv (1.3)where, without loss of generality, we have written the arbitrary function fof S, v and t as _ v_, where and are the drift and volatilityfunctions from the SDE (1.2) for instantaneous variance.The Market Pri ce of Vol ati l i ty Ri sk (S, v, t) is called the market price ofvolatility risk. To see why, we again follow Wilmotts argument.Consider the portfolio 1 consisting of a delta-hedged (but not vega-hedged) option V. Then

1 = V VS Sand again applying It os lemma,d1 =_Vt + 12 v S2 2VS2 + v S 2Vv S + 122v 2 2Vv2_ dt+_VS _ dS +_Vv_ dvStochastic Volatility and Local Volatility 7Because the option is delta-hedged, the coefcient of dS is zero and we areleft withd1 r 1dt=_Vt + 12vS22VS2 + vS 2VvS + 122v22Vv2 rSVS r V_dt+ Vv dv= v Vv_(S, v, t) dt +dZ2_where we have used both the valuation equation (1.3) and the SDE (1.2)for v. We see that the extra return per unit of volatility risk dZ2 is givenby (S, v, t) dt and so in analogy with the Capital Asset Pricing Model, isknown as the market price of volatility risk.Now, dening the risk-neutral drift as

= v we see that, as far as pricing of options is concerned, we could have startedwith the risk-neutral SDE for v,dv =

dt + v dZ2and got identical results with no explicit price of risk term because we arein the risk-neutral world.In what follows, we always assume that the SDEs for S and v are in risk-neutral terms because we are invariably interested in tting models to optionprices. Effectively, we assume that we are imputing the risk-neutral measuredirectly by tting the parameters of the process that we are imposing.Were we interested in the connection between the pricing of optionsand the behavior of the time series of historical returns of the underlying, wewould need to understand the connection between the statistical measureunder which the drift of the variance process v is and the risk-neutralprocess under which the drift of the variance process is

. From now on,we act as if we are risk-neutral and drop the prime.LOCAL VOLATILITYHistoryGiven the computational complexity of stochastic volatility models andthe difculty of tting parameters to the current prices of vanilla options,8 THE VOLATILITY SURFACEpractitioners sought a simpler way of pricing exotic options consistentlywith the volatility skew. Since before Breeden and Litzenberger (1978), itwas understood (at least by oor traders) that the risk-neutral density couldbe derived from the market prices of European options. The breakthroughcame when Dupire (1994) and Derman and Kani (1994) noted thatunder risk neutrality, there was a unique diffusion process consistent withthese distributions. The corresponding unique state-dependent diffusioncoefcient L(S, t), consistent with current European option prices, is knownas the local volatility function.It is unlikely that Dupire, Derman, and Kani ever thought of localvolatility as representing a model of how volatilities actually evolve. Rather,it is likely that they thought of local volatilities as representing some kind ofaverage over all possible instantaneous volatilities in a stochastic volatilityworld (an effective theory). Local volatility models do not therefore reallyrepresent a separate class of models; the idea is more to make a simplifyingassumption that allows practitioners to price exotic options consistentlywith the known prices of vanilla options.As if any proof were needed, Dumas, Fleming, and Whaley (1998) per-formed an empirical analysis that conrmed that the dynamics of the impliedvolatility surface were not consistent with the assumption of constant localvolatilities.Later on, we show that local volatility is indeed an average over instan-taneous volatilities, formalizing the intuition of those practitioners who rstintroduced the concept.A Brief Review of Dupires WorkFor a given expiration T and current stock price S0, the collection{C(S0, K, T)} of undiscounted option prices of different strikes yields therisk-neutral density function of the nal spot ST through the relationshipC(S0, K, T) =_ KdST (ST, T; S0) (ST K)Differentiate this twice with respect to K to obtain (K, T; S0) = 2CK2Dupire published the continuous time theory and Derman and Kani, a discrete timebinomial tree version.Stochastic Volatility and Local Volatility 9so the Arrow-Debreu prices for each expiration may be recovered by twicedifferentiating the undiscounted option price with respect to K. This processis familiar to any option trader as the construction of an (innite size)innitesimally tight buttery around the strike whose maximum payoffis one.Given the distribution of nal spot prices ST for each time T conditionalon some starting spot price S0, Dupire shows that there is a unique riskneutral diffusion process which generates these distributions. That is, giventhe set of all European option prices, we may determine the functionalform of the diffusion parameter (local volatility) of the unique risk neutraldiffusion process which generates these prices. Noting that the local volatilitywill in general be a function of the current stock price S0, we write thisprocess asdSS = tdt + (St, t; S0) dZApplication of It os lemma together with risk neutrality, gives rise to a partialdifferential equation for functions of the stock price, which is a straightfor-ward generalization of Black-Scholes. In particular, the pseudo-probabilitydensities (K, T; S0) = 2CK2 must satisfy the Fokker-Planck equation. Thisleads to the following equation for the undiscounted option price C in termsof the strike price K:CT = 2K222CK2 + (rt Dt)_C K CK_ (1.4)where rt is the risk-free rate, Dt is the dividend yield and C is short forC(S0, K, T).Derivation of the Dupire EquationSuppose the stock price diffuses with risk-neutral drift t(= rt Dt) andlocal volatility (S, t) according to the equation:dSS = t dt + (St, t) dZThe undiscounted risk-neutral value C(S0, K, T) of a European option withstrike K and expiration T is given byC(S0, K, T) =_ KdST (ST, T; S0) (ST K) (1.5)10 THE VOLATILITY SURFACEHere (ST, T; S0) is the pseudo-probability density of the nal spot at timeT. It evolves according to the Fokker-Planck equation:122S2T_2S2T _ S ST(ST ) = TDifferentiating with respect to K givesCK = _ KdST (ST, T; S0)2CK2 = (K, T; S0)Now, differentiating (1.5) with respect to time givesCT =_ KdST_ T (ST, T; S0)_(ST K)=_ KdST_122S2T_2S2T_ ST(ST )_ (ST K)Integrating by parts twice gives:CT = 2K22 +_ KdST ST = 2K222CK2 + (T)_KCK_which is the Dupire equation when the underlying stock has risk-neutraldrift . That is, the forward price of the stock at time T is given byFT = S0 exp__ T0dt t_Were we to express the option price as a function of the forward priceFT = S0exp__ T0 (t)dt_, we would get the same expression minus the driftterm. That is,CT = 12 2K2 2CK2From now on, (T) represents the risk-neutral drift of the stock price process,which is the risk-free rate r(T) minus the dividend yield D(T).Stochastic Volatility and Local Volatility 11where C now represents C(FT, K, T). Inverting this gives2(K, T, S0) =CT12 K2 2CK2(1.6)The right-hand side of equation (1.6) can be computed from known Euro-pean option prices. So, given a complete set of European option pricesfor all strikes and expirations, local volatilities are given uniquely byequation (1.6).We can viewequation (1.6) as a denition of the local volatility functionregardless of what kind of process (stochastic volatility for example) actuallygoverns the evolution of volatility.Local Volatility in Terms of Implied VolatilityMarket prices of options are quoted in terms of Black-Scholes impliedvolatility BS(K, T; S0). In other words, we may writeC(S0, K, T) = CBS(S0, K, BS(S0, K, T) , T)It will be more convenient for us to work in terms of two dimensionlessvariables: the Black-Scholes implied total variance w dened byw(S0, K, T) := 2BS (S0, K, T) Tand the log-strike y dened byy = log_ KFT_where FT = S0exp__ T0 dt (t)_ gives the forward price of the stock at time0. In terms of these variables, the Black-Scholes formula for the future valueof the option price becomesCBS (FT, y, w) = FT_N_d1_ eyN_d2__= FT_N_ yw +w2_ eyN_ yw w2__ (1.7)and the Dupire equation (1.4) becomesCT = vL2_2Cy2 Cy_ + (T) C (1.8)12 THE VOLATILITY SURFACEwith vL = 2(S0, K, T) representing the local variance. Now, by takingderivatives of the Black-Scholes formula, we obtain2CBSw2 =_18 12w + y22w2_ CBSw2CBSyw =_12 yw_ CBSw2CBSy2 CBSy = 2 CBSw (1.9)We may transform equation (1.8) into an equation in terms of impliedvariance by making the substitutionsCy = CBSy + CBSwwy2Cy2 = 2CBSy2 +22CBSywwy + 2CBSw2_wy_2+ CBSw2wy2CT = CBST + CBSwwT = CBSwwT + (T) CBSwhere the last equality follows fromthe fact that the only explicit dependenceof the option price on T in equation (1.7) is through the forward priceFT = S0exp__ T0 dt (t)_. Equation (1.4) now becomes (cancelling (T) Cterms on each side)CBSwwT= vL2_CBSy + 2CBSy2 CBSwwy +22CBSywwy+ 2CBSw2_wy_2+ CBSw2wy2_= vL2CBSw_2 wy + 2_12 yw_ wy+_18 12w + y22w2__wy_2+ 2wy2_Stochastic Volatility and Local Volatility 13Then, taking out a factor of CBSw and simplifying, we getwT = vL_1 ywwy + 14_14 1w + y2w2__wy_2+ 122wy2_Inverting this gives our nal result:vL =wT1 ywwy + 14_14 1w + y2w2_ _wy_2+ 122wy2(1.10)Special Case: No SkewIf the skew wy is zero, we must havevL = wTSo the local variance in this case reduces to the forward Black-Scholesimplied variance. The solution to this is, of course,w(T) =_ T0vL(t) dtLocal Variance as a Conditional Expectationof Instantaneous VarianceThis result was originally independently derived by Dupire (1996) andDerman and Kani (1998). Following now the elegant derivation by Dermanand Kani, assume the same stochastic process for the stock price as in equa-tion (1.1) but write it in terms of the forward price Ft,T = St exp__ Tt ds s_:dFt,T = vtFt,TdZ (1.11)Note that dFT,T = dST. The undiscounted value of a European option withstrike K expiring at time T is given byC(S0, K, T) = E_(ST K)+_Note that this implies that KBS (S0, K, T) is zero.14 THE VOLATILITY SURFACEDifferentiating once with respect to K givesCK = E[ (ST K)]where () is the Heaviside function. Differentiating again with respect toK gives2CK2 = E[ (ST K)]where () is the Dirac function.Now a formal application of It os lemma to the terminal payoff of theoption (and using dFT,T = dST) gives the identityd (ST K)+ = (ST K) dST + 12 vT S2T (ST K) dTTaking conditional expectations of each side, and using the fact that Ft,T isa martingale, we getdC = dE_(ST K)+_ = 12 E_vT S2T (ST K)_ dTAlso, we can writeE_vTS2T (ST K)_ = E[vT |ST = K] 12K2E[ (ST K)]= E[vT |ST = K] 12K2 2CK2Putting this together, we getCT = E[vT |ST = K] 12K2 2CK2Comparing this with the denition of local volatility (equation (1.6)), wesee that2(K, T, S0) = E[vT |ST = K] (1.12)That is, local variance is the risk-neutral expectation of the instantaneousvariance conditional on the nal stock price ST being equal to the strikeprice K.CHAPTER2The Heston ModelIn this chapter, we present the most well-known and popular of all stochas-tic volatility models, the Heston model, and provide a detailed derivationof the Heston European option valuation formula, implementation of whichfollows straightforwardly from the derivation. We also show how to dis-cretize the Heston process for Monte Carlo simulation and with someappreciation for the complexity and expense of numerical computations,suggest a main reason for the Heston models popularity.THE PROCESSThe Heston model (Heston (1993)) corresponds to choosing (S, vt, t) = (vtv) and (S, v, t) = 1 in equations (1.1) and (1.2). These equationsthen becomedSt = t Stdt +vt StdZ1 (2.1)anddvt = (vtv) dt +vtdZ2 (2.2)with_dZ1dZ2_= dtwhere is the speed of reversion of vt to its long-term mean v.The process followed by the instantaneous variance vt may be recognizedas a version of the square root process described by Cox, Ingersoll, andRoss (1985). It is a (jump-free) special case of a so-called afne jumpdiffusion (AJD) that is roughly speaking a jump-diffusion process for whichthe drifts and covariances and jump intensities are linear in the state vector,1516 THE VOLATILITY SURFACEwhich is {x, v} in this case with x = log(S). Dufe, Pan, and Singleton(2000) show that AJD processes are analytically tractable in general. Thesolution technique involves computing an extended transform, which inthe Heston case is a conventional Fourier transform.We now substitute the above values for (S, v, t) and (S, v, t) into thegeneral valuation equation (1.3). We obtainVt + 12 v S22VS2 + v S 2Vv S + 12 2v 2Vv2 +r SVS r V= (v v) Vv (2.3)In Hestons original paper, the price of risk is assumed to be linear in theinstantaneous variance v in order to retain the form of the equation underthe transformation from the statistical (or real) measure to the risk-neutralmeasure. In contrast, as in Chapter 1, we assume that the Heston process,with parameters tted to option prices, generates the risk-neutral measureso the market price of volatility risk in the general valuation equation (1.3)is set to zero in equation (2.3). Since we are only interested in pricing, andwe assume that the pricing measure is recoverable from European optionprices, we are indifferent to the statistical measure.THE HESTON SOLUTION FOR EUROPEAN OPTIONSThis section follows the original derivation of the Heston formula for thevalue of a European-style option in Heston (1993) pretty closely but withsome changes in intermediate denitions as explained later on.Before solving equation (2.3) with the appropriate boundary conditions,we can simplify it by making some suitable changes of variable. Let K bethe strike price of the option, T time to expiration, Ft,T the time T forwardprice of the stock index and x := log_Ft,T/K_.Further, suppose that we consider only the future value to expirationC of the European option price rather than its value today and dene = T t. Then equation (2.3) simplies toC+ 12 v C11 12 v C1+ 12 2v C22+ v C12(v v) C2 = 0(2.4)where the subscripts 1 and 2 refer to differentiation with respect to x and vrespectively.The Heston Model 17According to Dufe, Pan, and Singleton (2000), the solution of equa-tion (2.4) has the formC(x, v, ) = K _exP1(x, v, ) P0(x, v, )_ (2.5)where, exactly as in the Black-Scholes formula, the rst term in the brack-ets represents the pseudo-expectation of the nal index level given thatthe option is in-the-money and the second term represents the pseudo-probability of exercise.Substituting the proposed solution (2.5) into equation (2.4) implies thatP0 and P1 must satisfy the equationPj+ 12v2Pjx2 _12 j_vPjx + 122v2Pjv2 +v 2Pjxv+(a bjv)Pjv = 0 (2.6)for j = 0, 1 wherea = v, bj = j subject to the terminal conditionlim0Pj(x, v, ) =_ 1 if x > 00 if x 0:= (x) (2.7)We solve equation (2.6) subject to the condition (2.7) using a Fouriertransform technique. To this end dene the Fourier transform of Pj throughP(u, v, ) =_ dx ei uxP(x, v, )ThenP(u, v, 0) =_ dx ei ux(x) = 1i uThe inverse transform is given byP(x, v, ) =_ du2ei ux P(u, v, ) (2.8)18 THE VOLATILITY SURFACESubstituting this into equation (2.6) givesPj 12 u2v Pj_12 j_ i u v Pj+ 12 2v 2Pjv2 + i u v Pjv +(a bjv) Pjv = 0 (2.9)Now dene = u22 i u2 +i j u = j i u = 22Then equation (2.9) becomesv_ PjPjv +2 Pjv2_+a Pjv Pj= 0 (2.10)Now substitutePj(u, v, ) = exp{C(u, ) v +D(u, ) v} Pj(u, v, 0)= 1i u exp{C(u, ) v +D(u, ) v}It follows thatPj=_v C+v D_ PjPjv = D Pj2Pjv2 = D2 PjThen equation (2.10) is satised ifC= DD= D+ D2= (Dr+)(Dr) (2.11)The Heston Model 19where we dener = _242=: d2Integrating (2.11) with the terminal conditions C(u, 0) = 0 and D(u, 0)= 0 givesD(u, ) = r1 ed 1 g edC(u, ) = _r 22 log_1 g ed 1 g__ (2.12)where we deneg := rr+Taking the inverse transform using equation (2.8) and performing thecomplex integration carefully gives the nal formof the pseudo-probabilitiesPj in the form of an integral of a real-valued function.Pj(x, v, ) = 12 + 1_ 0du Re_exp{Cj(u, ) v +Dj(u, ) v +i u x}i u_(2.13)This integration may be performed using standard numerical methods.It is worth noting that taking derivatives of the Heston formula withrespect to x or v in order to compute delta and vega is extremely straight-forward because the functions C(u, ) and D(u, ) are independent of xand v.A Digression: The Complex Logarithm in the Integration (2.13)In Hestons original paper and in most other papers on the subject, C(u, )is written (almost) equivalently asC(u, ) = _r+ 22 log_e+d g1 g__ (2.14)20 THE VOLATILITY SURFACEThe reason for the qualication almost is that this denition coincideswith our previous one only if the imaginary part of the complex logarithmis chosen so that C(u, ) is continuous with respect to u. It turns outthat taking the principal value of the logarithm in (2.14) causes C(u, ) tojump discontinuously each time the imaginary part of the argument of thelogarithm crosses the negative real axis. The conventional resolution is tokeep careful track of the winding number in the integration (2.13) so as toremain on the same Riemann sheet. This leads to practical implementationproblems because standard numerical integration routines cannot be used.The paper of Kahl and J ackel (2005) concerns itself with this problem andprovides an ingenious resolution.With our denition (2.12) of C(u, ), however, it seems that wheneverthe imaginary part of the argument of the logarithm is zero, the real partis positive; plotted in the complex plane, it seems that the argument of thelogarithmnever cuts the negative real axis. It follows that with our denitionof C(u, ), taking the principal value of the logarithm seems to lead to acontinuous integrand over the full range of integration. Unfortunately,a proof of this result remains elusive so it must retain the status of aconjecture.DERIVATION OF THE HESTONCHARACTERISTIC FUNCTIONTo anyone other than an option trader, it may seem perverse to rst derivethe option pricing formula and then impute the characteristic function: Thereverse might appear more natural. However, in the context of understand-ing the volatility surface, option prices really are primary and it makes justas much sense for us to deduce the characteristic function from the optionpricing formula as it does for us to deduce the risk-neutral density fromoption prices.By denition, the characteristic function is given byT(u) := E[eiuxT|xt = 0]The probability of the nal log-stock price xT being greater than the strikeprice is given byPr(xT > x) = P0(x, v, )= 12 + 1_ 0du Re_exp{C(u, ) v +D(u, ) v +i u x}iu_The Heston Model 21with x = log(St/K) and = T t. Let the log-strike k be dened by k =log(K/St) = x. Then, the probability density function p(k) must be given byp(k) = P0k= 12_ du

exp{C(u

, ) v +D(u

, ) v i u

k}ThenT(u) =_ dk p(k) ei uk= 12_ du

exp{C(u

, ) v +D(u

, ) v}_ du ei(uu

)k=_ du

exp{C(u

, ) v +D(u

, ) v} (u u

)= exp{C(u, ) v +D(u, ) v} (2.15)SIMULATION OF THE HESTON PROCESSRecall the Heston processdS = Sdt +v S dZ1dv = (v v) dt +v dZ2 (2.16)with_dZ1dZ2_= dtA simple Euler discretization of the variance processvi+1 = vi (viv) t +vit Z (2.17)with Z N(0, 1) may give rise to a negative variance. To deal with thisproblem, practitioners generally adopt one of two approaches: Either theabsorbing assumption: if v < 0 then v = 0, or the reecting assumption: ifv < 0 then v = v. In practice, with the parameter values that are requiredto t equity index option prices, a huge number of time steps is required toachieve convergence with this discretization.22 THE VOLATILITY SURFACEMilstein Discretizati onIt turns out to be possible to substantially alleviate the negative varianceproblem by implementing a Milstein discretization scheme.Specically, by going to one higher order in the It o-Taylor expansionof v(t +t), we arrive at the following discretization of the variance process:vi+1 = vi (viv) t +vit Z + 24 t_Z21_ (2.18)This can be rewritten asvi+1 =_vi+ 2t Z_2 (viv) t 24 tWe note that if vi = 0 and 4 v/2> 1, vi+1 > 0 indicating that the fre-quency of occurrence of negative variances should be substantially reduced.In practice, with typical parameters, even if 4 v/2< 1, the frequencywith which the process goes negative is substantially reduced relative to theEuler case.As it is no more computationally expensive to implement the Milsteindiscretization (2.18) than it is to implement the Euler discretization (2.17),the Milstein discretization is always to be preferred. Also, the stock processshould be discretized asxi+1 = xi vi2 t +_vit Wwith xi := log(Si/S0) and W N(0, 1), E[ZW] = ; if we discretize theequation for the log-stock price x rather than the equation for the stockprice S, there are no higher order corrections to the Euler discretization.An I mpl i ci t Scheme We follow Alfonsi (2005) and considervi+1 = vi (viv) t +vit Z= vi (vi+1v) t +vi+1t Z_vi+1vi_ t Z +higher order termsWe note thatvi+1vi = 2t Z +higher order termsSee Chapter 5 of Kloeden and Platen (1992) for a discussion of It o-Taylor expan-sions.The Heston Model 23and substitute (noting that E[Z2] = 1) to obtain the implicit discretizationvi+1 = vi (vi+1v) t +vi+1t Z 2 t (2.19)Then vi+1 may be obtained as a root of the quadratic equation (2.19).Explicitly,vi+1 =_4vi+t _( v 2/2) (1 + t) +2Z2_+t Z2(1 + t)If 2 v/2> 1, there is guaranteed to be a real root of this expression sovariance is guaranteed to be positive. Otherwise, theres no guarantee andthis discretization doesnt work.Given that Heston parameters in practice often dont satisfy2 v/2> 1, we are led to prefer the Milstein discretization, which isin any case simpler.Sampling from the Exact Transition LawAs Paul Glasserman (2004) points out in his excellent book on Monte Carlomethods, the problem of negative variances may be avoided altogether bysampling from the exact transition law of the process. Broadie and Kaya(2004) show in detail how this may be done for the Heston process but theirmethod turns out also to be very time consuming as it involves integrationof a characteristic function expressed in terms of Bessel functions.It is nevertheless instructive to followtheir argument. The exact solutionof (2.16) may be written asSt = S0 exp_12_ t0vsds +_ t0vsdZs+_1 2_ t0vsdZs_vt = v0+ v t _ t0vsds +_ t0vsdZswith_dZsdZs_= 0The Broadie-Kaya simulation procedure is as follows: Generate a sample from the distribution of vt given v0. Generate a sample from the distribution of _t0 vsds given vt and v0.24 THE VOLATILITY SURFACE Recover _t0vsdZs given _t0 vsds, vt and v0. Generate a sample from the distribution of St given _t0vsdZs and_t0 vsds.Note that in the nal step, the distribution of _t0vsdZs is normal withvariance _t0 vsds because dZs and vs are independent by construction.Andersen and Brotherton-Ratcliffe (2001) suggest that processes likethe square root variance process should be simulated by sampling froma distribution that is similar to the true distribution but not necessarilythe same; this approximate distribution should have the same mean andvariance as the true distribution.Applying their suggested approach to simulating the Heston process,we would have to nd the means and variances of _t0vsdZs, _t0 vsds, vtand v0.Why the Heston Model Is so PopularFrom the above remarks on Monte Carlo simulation, the reader can geta sense of how computationally expensive it can be to get accurate valuesof options in a stochastic volatility model; numerical solution of the PDEis not much easier. The great difference between the Heston model andother (potentially more realistic) stochastic volatility models is the existenceof a fast and easily implemented quasi-closed form solution for Europeanoptions. This computational efciency in the valuation of European optionsbecomes critical when calibrating the model to known option prices.As we shall see in subsequent chapters, although the dynamics of theHeston model are not realistic, with appropriate choices of parameters,all stochastic volatility models generate roughly the same shape of impliedvolatility surface and have roughly the same implications for the valuationof nonvanilla derivatives in the sense that they are all models of the jointprocess of the stock price and instantaneous variance. Given the relativecheapness of Heston computations, its easy to see why the model is sopopular.CHAPTER3The Implied Volatility SurfaceIn Chapter 1, we showed how to compute local volatilities from impliedvolatilities. In this chapter, we show how to get implied volatilitiesfrom local volatilities. Using the fact that local variance is a conditionalexpectation of instantaneous variance, we can estimate local volatilitiesgenerated by a given stochastic volatility model; implied volatilities thenfollow. Given a stochastic volatility model, we can then approximate theshape of the implied volatility surface.Conversely, given the shape of an actual implied volatility surface, wend we can deduce some characteristics of the underlying process.GETTING IMPLIED VOLATILITYFROM LOCAL VOLATILITIESModel CalibrationFor a model to be useful in practice, it needs to return (at least approximately)the current market prices of European options. That implies that we needto t the parameters of our model (whether stochastic or local volatilitymodel) to market implied volatilities. It is clearly easier to calibrate a modelif we have a fast and accurate method for computing the prices of Europeanoptions as a function of the model parameters. In the case of stochasticvolatility, this consideration clearly favors models such as Heston that havesuch a solution; Mikhailov and N ogel (2003), for example, explain how tocalibrate the Heston model to market data.In the case of local volatility models, numerical methods are usuallyrequired to compute European option prices and that is one of the potentialproblems associated with their implementation. Brigo and Mercurio (2003)circumvent this problem by parameterizing the local volatility in such away that the prices of European options are known in closed-form assuperpositions of Black-Scholes-like solutions.2526 THE VOLATILITY SURFACEYet again, we could work with the European option prices directly ina trinomial tree framework as in Derman, Kani, and Chriss (1996) or wecould maximize relative entropy (of missing information) as in Avellaneda,Friedman, Holmes, and Samperi (1997). These methods are nonparametric(assuming actual option prices are used, not interpolated or extrapolatedvalues); they may fail because of noise in the prices and the bid/offer spread.Finally, we could parameterize the risk-neutral distributions as inRubinstein (1998) or parameterize the implied volatility surface directlyas in Shimko (1993) or Gatheral (2004). Although these approaches lookstraightforward given that we know from Chapter 1 how to get localvolatility in terms of implied volatility, they are very difcult to implementin practice. The problem is that we dont have a complete implied volatilitysurface, we only have a few bids and offers per expiration. To apply a para-metric method, we need to interpolate and extrapolate the known impliedvolatilities. It is very hard to do this without introducing arbitrage. Thearbitrages to avoid are roughly speaking, negative vertical spreads, negativebutteries and negative calendar spreads (where the latter are carefullydened).In what follows, we concentrate on the implied volatility structure ofstochastic volatility models so as not to worry about the possibility ofarbitrage, which is excluded from the outset.First, we derive an expression for implied volatility in terms of localvolatilities. In principle, this should allow us to investigate the shape ofthe implied volatility surface for any local volatility or stochastic volatilitymodel because we know from equation (1.12) how to express local varianceas an expectation of instantaneous variance in a stochastic volatility model.Understanding Implied VolatilityIn Chapter 1, we derived an expression (1.10) for local volatility in termsof implied volatility. An obvious direct approach might be to invert thatexpression and express implied volatility in terms of local volatility. How-ever, this kind of direct attack on the problem doesnt yield any easy resultsin general although Berestycki, Busca, and Florent (2002) were able to invert(1.10) in the limit of zero time to expiration.Instead, by exploiting the work of Dupire (1998), we derive a generalpath-integral representation of Black-Scholes implied variance. We start byassuming that the stock price St satises the SDEdStSt= tdt +tdZtwhere the volatility t may be random.The Implied Volatility Surface 27For xed K and T, dene the Black-Scholes gamma

BS(St, (t)) := 2S2tCBS(St, K, (t), T t)and further dene the Black-Scholes forward implied variance functionvK,T (t) = E_2t S2t BS(St, (t)) |F0_E_S2t BS(St, (t)) |F0_ (3.1)where2(t) := 1T t_ TtvK,T(u) du (3.2)Path-by-path, for any suitably smooth function f (St, t) of the randomstock price St and for any given realization {t} of the volatility process, thedifference between the initial value and the nal value of the function f (St, t)is obtained by antidifferentiation. Then, applying It os lemma, we getf (ST, T) f (S0, 0) =_ T0df=_ T0_ fStdSt+ ft dt + 2t2 S2t2fS2tdt_ (3.3)Under the usual assumptions, the nondiscounted value C(S0, K, T) ofa call option is given by the expectation of the nal payoff under therisk-neutral measure. Then, applying (3.3), we obtain:C(S0, K, T) = E_(STK)+|F0_= E[CBS (ST, K, (T), 0) |F0]= CBS (S0, K, (0), T)+E__ T0_CBSStdSt+ CBSt dt + 122t S2t2CBSS2tdt_F0_Now, of course, CBS (St, K, (t), T t) must satisfy the Black-Scholesequation (assuming zero interest rates and dividends) and fromthe denitionof (t), we obtain:CBSt = 12vK,T (t) S2t2CBSS2t28 THE VOLATILITY SURFACEUsing this equation to substitute for the time derivative CBSt , we obtain:C(S0, K, T) = CBS (S0, K, (0), T)+E__ T0_CBSStdSt+ 12_2t vK,T (t)_S2t2CBSS2tdt_F0_= CBS (S0, K, (0), T)+E__ T012_2t vK,T (t)_S2t2CBSS2tdtF0_ (3.4)where the second equality uses the fact that St is a martingale.In words, the last term in equation (3.4) gives the expected realizedprot on a sale of a call option at an implied volatility of , delta-hedgedusing the deterministic forward variance function vK,T when the actualrealized volatility is t.From the denition (3.2) of vK,T(t), we have thatE_S2t BS(St, (t)) |F0_vK,T (t) = E_2t S2t BS(St, (t)) |F0_so the second term in equation (3.4) vanishes and from the denition ofimplied volatility, (0) is the Black-Scholes implied volatility at time 0 ofthe option with strike K and expiration T (i.e., the Black-Scholes formulamust give the market price of the option).Explicitly,BS(K, T)2= (0)2= 1T_ T0E_2t S2t BS(St) |F0_E_S2t BS(St) |F0_ dt (3.5)Equation (3.5) expresses implied variance as the time-integral of expectedinstantaneous variance 2t under some probability measure.The interpretation of equation (3.5) is that to compute the Black-Scholes implied volatility of an option, we need to average the possiblerealized volatilities over all possible scenarios, in particular over all possiblepaths of the underlying stock. Each such scenario is weighted by the gammaof the option; the protability of the delta hedger in any time interval isdirectly proportional to the gamma and the difference between expectedinstantaneous variance (or local variance) and realized instantaneous vari-ance. In particular, at inception of the delta hedge, there is only one possiblestock price (the then stock price) and only paths that end at the strike priceneed be included in the average because gamma elsewhere is precisely zero.The Implied Volatility Surface 29Following Lee (2005), we may rewrite (3.5) more elegantly asBS(K, T)2= (0)2= 1T_ T0EGt[2t ] dt (3.6)thus interpreting the denition (3.1) of v(t) as the expectation of 2t withrespect to the probability measure Gt dened, relative to the pricing measureP, by the Radon-Nikodym derivativedGtdP := S2t BS(St, (t))E_S2t BS(St, (t)) |F0_Note in passing that equations (3.1) and (3.5) are implicit because thegamma BS(St) of the option depends on all the forward implied variancesvK,T (t).Speci al Case (Bl ack-Schol es) Suppose t = (t), a function of t only. ThenvK,T (t) = E_(t)2S2t BS(St) |F0_E_S2t BS(St) |F0_ = (t)2The forward implied variance vK,T(t) and the forward variance (t)2coin-cide. As expected, vK,T(t) has no dependence on the strike K or the optionexpiration T.Interpretation In order to get better intuition for equation (3.1), rst recallhow to compute a risk-neutral expectation:EP_f (St)_=_ dSt p(St, t; S0) f (St)We get the risk-neutral pdf of the stock price at time t by taking thesecond derivative of the market price of European options with respect tostrike price.p(St, t; S0) = 2C(S0, K, t)K2K=StThen from equation (3.6) we havevK,T (t) = EGt_2t_= EP_2tdGtdP_30 THE VOLATILITY SURFACE=_ dSt q(St; S0, K, T) EP_2t |St_=_ dSt q(St; S0, K, T) vL(St, t) (3.7)where we further deneq(St, t; S0, K, T) := p(St, t; S0) S2t BS(St)E_S2t BS(St) |F0_and vL(St, t) = EP_2t |St_ is the local variance.We see that q(St, t; S0, K, T) looks like a Brownian Bridge density forthe stock price: p(St, t; S0) has a delta function peak at S0 at time 0 and

BS(St) has a delta function peak at K at expiration T.For convenience in what follows, we now rewrite equation (3.7) interms of xt := log (St/S0):vK,T (t) =_ dxt q(xt, t; xT, T) vL(xt, t) (3.8)Figure 3.1 shows how q(xt, t; xT, T) looks in the case of a 1-yearEuropean option struck at 1.3 with a at 20% volatility. We see thatq(xt, t; xT, T) peaks on a line, which we will denote by xt, joining the stockprice today with the strike price at expiration. Moreover, the density looksroughly symmetric around the peak. This suggests an expansion around thepeak xt, at which the derivative of q(xt, t; xt, T) with respect to xt is zero.Then we writeq(xt, t; xT, T) q( xt, t; xT, T) + 12 (xt xt)2 2qx2txt= xt(3.9)In practice, the local variance vL(xt, t) is typically not so far from linear inxt in the region where q(xt, t; xT, T) is signicant, so we may further writevL(xt, t) vL( xt, t) +(xt xt)vLxtxt= xt(3.10)Substituting (3.9) and (3.10) into the integrand in equation (3.8) givesvK,T (t) vL( xt, t)and we may rewrite equation (3.5) asBS(K, T)2 1T_ T0vL( xt) dt (3.11)The Implied Volatility Surface 310.40.200.20.4x00.250.50.751t00.2xFIGURE 3.1 Graph of the pdf of xt conditional on xT = log(K) for a 1-yearEuropean option, strike 1.3 with current stock price = 1 and 20% volatility.In words, equation (3.11) says that the Black-Scholes implied variance of anoption with strike K is given approximately by the integral from valuationdate (t = 0) to the expiration date (t = T) of the local variances along thepath xt that maximizes the Brownian Bridge density q(xt, t; xT, T).Of course in practice, its not easy to compute the path xt. Nevertheless,we now have a very simple and intuitive picture for the meaning of Black-Scholes implied variance of a European option with a given strike andexpiration: It is approximately the integral from today to expiration of localvariances along the most probable path for the stock price conditional onthe stock price at expiration being the strike price of the option.LOCAL VOLATILITY I N THE HESTON MODELFrom equations (2.1) and (2.2) with xt := log (St/K) and = 0, we havedxt = vt2 dt +vt dZt32 THE VOLATILITY SURFACEdvt = (vtv)dt +vtdZt+_1 2vt dWt (3.12)where dWt and dZt are orthogonal. EliminatingvtdZt, we getdvt = (vtv) dt + _dxt+ 12 vt dt_+_1 2vt dWt (3.13)Our strategy will be to compute local variances in the Heston model andthen integrate local variance from valuation date to expiration date toapproximate the BS implied variance using equation (3.11).First, consider the unconditional expectation vs of the instantaneousvariance at time s. Solving equation (3.13) gives vs = (v0v) e s+vThen dene the expected total variance to time t through the relation wt :=_ t0 vsds = (v0v)_1 et_+v tFinally, let ut := E[vt|xT] be the expectation of the instantaneous varianceat time t conditional on the nal value xT of x.AnsatzAnsatz means here, Lets just suppose this were true so that we canproceed. Without loss of generality, assume x0 = 0. ThenE[xs|xT] = xT ws wTwhere wt :=_t0 ds vs is the expected total variance to time t. To see that thisansatz is at least a plausible approximation, note thatE(xs) = E(xT) ws wT= wT2 ws wT= ws2In fact, if the process for xt were a conventional Brownian Bridge process,the result would be true but in this case, the ansatz is only an approximationwhich is reasonable when |xT| is small (i.e. not too far from at-the-money).The Implied Volatility Surface 33Building on the ansatz, we may take the conditional expectation of(3.13) to get:dut = (utv)dt + 2 utdt +xT wTd wt+_1 2vtE[dWt|xT] (3.14)If the dependence of dWt on xT is weak or if _1 2is very small, we maydrop the last term to getdut

(utv

)dt +xT wT vtdtwith

= /2, v

= v/

. The solution to this equation isuT v

T+xT wT_ T0 vse

(Ts)ds (3.15)with v

s :=_v v

_e

s+v

.From equation (1.12), we know that the local variance 2(K, T, S0) =E[vT|ST = K]. Then, equation (3.15) gives us an approximate but sur-prisingly accurate formula for local variance within the Heston model (anextremely accurate approximation when = 1). We see that in the Hestonmodel, local variance is approximately linear in x = log_FK_.In summary, we have made two approximations: the ansatz and drop-ping the last term in equation (3.14). For reasonable parameters, equation(3.15) gives good intuition for the functional form of local variance andwhen = 1, as we will see in Chapter 4, it is almost exact. Peter Friz hasshown that equation (3.15) is in fact exact to rst order in whether ornot the ansatz holds or _1 2is small. Given this, equation (3.15) canalso be shown to agree with the general perturbative expansions of Lewis(2000).IMPLIED VOLATILITY I N THE HESTON MODELNow, to get implied variance in the Heston model, following our earlierexplanation as summarized in equation (3.11), we need to integrate theHeston local variance along the most probable stock price path joining theinitial stock price to the strike price at expiration (the one which maximizesthe Brownian Bridge probability density).34 THE VOLATILITY SURFACEUsing our earlier notation, the Black-Scholes implied variance is given byBS(K, T)2 1T_ T02 xt,tdt = 1T_ T0ut( xt)dt (3.16)where { xt} is the most probable path (as dened earlier).Recall that the Brownian Bridge density q(xt, t; xT, T) is roughly sym-metric and peaked around xt, so E[xt xt|xT] 0. Applying the ansatzonce again, we obtain xt = E[ xt|xT] = E[ xtxt|xT] +E[xt|xT] wt wTxTWe substitute this expression back into equations (3.15) and (3.16)to getBS(K, T)2 1T_ T0ut( xt)dt 1T_ T0 v

tdt +xT wT1T_ T0dt_ t0 vse

(ts)ds (3.17)The Term Structure of Black-Scholes Implied Volatilityin the Heston ModelThe at-the-money termstructure of BS implied variance in the Heston modelis obtained by setting xT = 0 in equation (3.17). Performing the integrationexplicitly givesBS(K, T)2K=FT 1T_ T0 v

tdt = 1T_ T0__v v

_e

t+v

_dt=_v v

_ 1 e

T

T +v

(3.18)We see that in the Heston model, the at-the-money Black-Scholes impliedvarianceBS(K, T)2K=FTv(the instantaneous variance) as the time to expiration T 0 and as T ,the at-the-money Black-Scholes implied variance reverts to v

.The Implied Volatility Surface 35The Black-Schol es Implied Volatility Skew in the Heston ModelIt is possible (but not very illuminating) to integrate the second term ofequation (3.17) explicitly. Even without doing that, we can see that theimplied variance skew in the Heston model is approximately linear in thecorrelation and the volatility of volatility .In the special case where v0 = v, the implied variance skew has aparticularly simple form. Then vs = v and wt = v t. The most probable path xt tTxT is exactly a straight line in log-space between the initial stockprice on valuation date and the strike price at expiration. Performing theintegrations in equation (3.17) explicitly, we getBS(K, T)2 w

TT +xTT2_ T0dt 1T_ t0e

(ts)ds= w

TT +xT

T___1 _1 e

T_

T___ (3.19)From equation (3.19), we see that the implied variance skew xtBS(K, T)2is independent of the level of instantaneous variance v or long-termmean variance v. In fact, this remains approximately true even when v = v.It follows that we now have a fast way of calibrating the Heston model toobserved implied volatility skews. Just two expirations would in principleallow us to determine

and the product . We can then t the termstructure of volatility to determine the long-term mean variance v and theinstantaneous variance v0. The curvature of the skew (not discussed here)would allow us to determine and separately.We note that as we increase either the correlation or the volatility ofvolatility , the skew increases.Also, the very short-dated skew is independent of and T:xtBS(K, T)2= 1

T___1 _1 e

T_

T___ 2 as T 0and the long-dated skew is inversely proportional to T:xtBS(K, T)2= 1

T___1 _1 e

T_

T___

T as T 36 THE VOLATILITY SURFACE00