The Volatility Surface: A Practitioner's Guide (Wiley Finance)

A good book for its wide view of all the topics linked to the volatility trading : stochastic volatility and jumps with a quite rare study of the impact of stochastic vol over the pricing of most commons exotic products like cliquets, lookback etc... The main problem of this book are : 1/ the incredible numbers of errors and typos in the proofs (typos for the two first chapters has been published by the author on the website of the Imperial College where Mr Gatheral has given lessons) 2/ the fact it's not so useful as the link between dividends and vol stochastic isn't treated at all. 3/ Most of the interesting theoretical exposee are clearly uncomplete and needs further investigation. But as the author try to do a wide tour of the subject in a rather limited number of pages, this is not surprising. 4/ This isn't explicit but this is of a weaker interest for stochastic volatility of underlyings other than Equities, index and funds... So finally, it's a good tour on the subject but for a lot of subjects it look likes a summary. Others books are needed. mainly the Alan Lewis book for Fourier transform's methods of pricing, the Cont/tankov on the Jumps process (see my review of this one) and the Alireza Javaheri's one for a more deeper work on THE calibration problem of stochastic volatility. 1. Stochastic Volatility And Local Volatility -- Stochastic Volatility -- Derivation Of The Valuation Equation -- Local Volatility -- History -- A Brief Review Of Dupire's Work -- Derivation Of The Dupire Equation -- Local Volatility In Terms Of Implied Volatility -- Special Case: No Skew -- Local Variance As A Conditional Expectation Of Instantaneous Variance -- 2. The Heston Model -- The Process -- The Heston Solution For European Options -- A Digression: The Complex Logarithm In The Integration (2.13) -- Derivation Of The Heston Characteristic Function -- Simulation Of The Heston Process -- Milstein Discretization -- Sampling From The Exact Transition Law -- Why The Heston Model Is So Popular -- 3. The Implied Volatility Surface -- Getting Implied Volatility From Local Volatilities -- Model Calibration -- Understanding Implied Volatility -- Local Volatility In The Heston Model -- Ansatz -- Implied Volatility In The Heston Model --^ The Term Structure Of Black-scholes Implied Volatility In The Heston Model -- The Black-scholes Implied Volatility Skew In The Heston Model -- The Spx Implied Volatility Surface -- Another Digression: The Svi Paramaterization -- A Heston Fit To The Data -- Final Remarks On Sv Models And Fitting The Volatility Surface -- 4. The Heston-nandi Model -- Local Variance In The Heston-nandi Model -- A Numerical Example -- The Heston-nandi Density -- Computation Of Local Volatilities -- Computation Of Implied Volatilities -- Discussion Of Results -- 5. Adding Jumps -- Why Jumps Are Needed -- Jump Diffusion -- Derivation Of The Valuation Equation -- Uncertain Jump Size -- Characteristic Function Methods -- Lévy Processes -- Examples Of Characteristic Functions For Specific Processes -- Computing Option Prices From The Characteristic Function -- Proof Of (5.6) -- Computing Implied Volatility -- Computing The At-the-money Volatility Skew -- How Jumps Impact The Volatility Skew --^ Stochastic Volatility Plus Jumps -- Stochastic Volatility Plus Jumps In The Underlying Only (svj) -- Some Empirical Fits To The Spx Volatility Surface -- Stochastic Volatility With Simultaneous Jumps In Stock Price And Volatility (svjj) -- Svj Fit To The September 15, 2005, Spx Option Data -- Why The Svj Model Wins -- 6. Modeling Default Risk -- Merton's Model Of Default -- Intuition -- Implications For The Volatility Skew -- Capital Structure Arbitrage -- Put-call Parity -- The Arbitrage -- Local And Implied Volatility In The Jump-to-ruin Model -- The Effect Of Default Risk On Option Prices -- The Creditgrades Model -- Model Setup -- Survival Probability -- Equity Volatility -- Model Calibration -- 7. Volatility Surface Asymptotics -- Short Expirations -- The Medvedev-scaillet Result -- The Sabr Model -- Including Jumps -- Corollaries -- Long Expirations: Fouque, Papanicolaou, And Sircar -- Small Volatility Of Volatility: Lewis -- Extreme Strikes: Roger Lee --^ Example: Black-scholes -- Stochastic Volatility Models -- Asymptotics In Summary -- 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 -- Valuing Digital Options -- Digital Cliquets -- 9. Barrier Options -- Definitions -- Limiting Cases -- Limit Orders -- European Capped Calls -- The Reflection Principle -- The Lookback Hedging Argument -- One-touch Options Again -- Put-call Symmetry -- Quasi-static Hedging And Qualitative Valuation -- Out-of The-money Barrier Options -- One-touch Options -- Live-out Options -- Lookback Options -- Adjusting For Discrete Monitoring -- Discretely Monitored Lookback Options -- Parisian Options -- Some Applications Of Barrier Options -- Ladders -- Ranges -- Conclusion -- 10. Exotic Cliquets -- Locally Capped Globally Floored Cliquet --^ Valuation Under Heston And Local Volatility Assumptions -- Performance -- Reverse Cliquet -- Valuation Under Heston And Local Volatility Assumptions -- Performance -- Napoleon -- Valuation Under Heston And Local Volatility Assumptions -- Performance -- Investor Motivation -- More On Napoleons -- 11. Volatility Derivatives -- Spanning Generalized European Payoffs -- Example: European Options -- Example: Amortizing Options -- The Log Contract -- Variance And Volatility Swaps -- Variance Swaps -- Variance Swaps In The Heston Model -- Dependence On Skew And Curvature -- The Effect Of Jumps -- Volatility Swaps -- Convexity Adjustment In The Heston Model -- Valuing Volatility Derivatives -- Fair Value Of The Power Payoff -- The Laplace Transform Of Quadratic Variation Under Zero Correlation -- The Fair Value Of Volatility Under Zero Correlation -- A Simple Lognormal Model -- Options On Volatility: More On Model-independence -- Listed Quadratic-variation Based Securities -- The Vix Index --^ Vxb Futures -- Knock-on Benefits -- Summary. Jim Gatheral ; Foreword By Nassim Nicholas Taleb. Includes Bibliographical References (pages 163-167) And Index. The Volatility Surface: A Practitioner's Guide......Page 6 Contents......Page 10 Figures......Page 16 Tables......Page 22 Foreword......Page 24 Preface......Page 26 HOW THIS BOOK IS ORGANIZED......Page 28 Acknowledgments......Page 30 STOCHASTIC VOLATILITY......Page 32 LOCAL VOLATILITY......Page 38 THE PROCESS......Page 46 THE HESTON SOLUTION FOR EUROPEAN OPTIONS......Page 47 DERIVATION OF THE HESTON CHARACTERISTIC FUNCTION......Page 51 SIMULATION OF THE HESTON PROCESS......Page 52 GETTING IMPLIED VOLATILITY FROM LOCAL VOLATILITIES......Page 56 LOCAL VOLATILITY IN THE HESTON MODEL......Page 62 IMPLIED VOLATILITY IN THE HESTON MODEL......Page 64 THE SPX IMPLIED VOLATILITY SURFACE......Page 67 LOCAL VARIANCE IN THE HESTON-NANDI MODEL......Page 74 A NUMERICAL EXAMPLE......Page 75 DISCUSSION OF RESULTS......Page 80 WHY JUMPS ARE NEEDED......Page 81 JUMP DIFFUSION......Page 83 CHARACTERISTIC FUNCTION METHODS......Page 87 STOCHASTIC VOLATILITY PLUS JUMPS......Page 96 MERTON’S MODEL OF DEFAULT......Page 105 CAPITAL STRUCTURE ARBITRAGE......Page 108 LOCAL AND IMPLIED VOLATILITY IN THE JUMP-TO-RUIN MODEL......Page 110 THE EFFECT OF DEFAULT RISK ON OPTION PRICES......Page 113 THE CREDITGRADES MODEL......Page 115 SHORT EXPIRATIONS......Page 118 THE MEDVEDEV-SCAILLET RESULT......Page 120 INCLUDING JUMPS......Page 124 LONG EXPIRATIONS: FOUQUE, PAPANICOLAOU, AND SIRCAR......Page 126 SMALL VOLATILITY OF VOLATILITY: LEWIS......Page 127 EXTREME STRIKES: ROGER LEE......Page 128 ASYMPTOTICS IN SUMMARY......Page 131 DYNAMICS OF THE VOLATILITY SKEW UNDER STOCHASTIC VOLATILITY......Page 132 DYNAMICS OF THE VOLATILITY SKEW UNDER LOCAL VOLATILITY......Page 133 DIGITAL OPTIONS AND DIGITAL CLIQUETS......Page 134 DEFINITIONS......Page 138 LIMITING CASES......Page 139 THE REFLECTION PRINCIPLE......Page 140 THE LOOKBACK HEDGING ARGUMENT......Page 143 PUT-CALL SYMMETRY......Page 144 QUASISTATIC HEDGING AND QUALITATIVE VALUATION......Page 145 ADJUSTING FOR DISCRETE MONITORING......Page 148 SOME APPLICATIONS OF BARRIER OPTIONS......Page 151 CONCLUSION......Page 152 LOCALLY CAPPED GLOBALLY FLOORED CLIQUET......Page 153 REVERSE CLIQUET......Page 156 NAPOLEON......Page 158 SPANNING GENERALIZED EUROPEAN PAYOFFS......Page 164 VARIANCE AND VOLATILITY SWAPS......Page 167 VALUING VOLATILITY DERIVATIVES......Page 177 LISTED QUADRATIC-VARIATION BASED SECURITIES......Page 187 SUMMARY......Page 192 Postscript......Page 193 Bibliography......Page 194 Index......Page 200 Praise for The Volatility Surface "I'm thrilled by the appearance of Jim Gatheral's new book The Volatility Surface. The literature on stochastic volatility is vast, but difficult to penetrate and use. Gatheral's book, by contrast, is accessible and practical. It successfully charts a middle ground between specific examples and general models--achieving remarkable clarity without giving up sophistication, depth, or breadth." --Robert V. Kohn, Professor of Mathematics and Chair, Mathematical Finance Committee, Courant Institute of Mathematical Sciences, New York University "Concise yet comprehensive, equally attentive to both theory and phenomena, this book provides an unsurpassed account of the peculiarities of the implied volatility surface, its consequences for pricing and hedging, and the theories that struggle to explain it." --Emanuel Derman, author of My Life as a Quant "Jim Gatheral is the wiliest practitioner in the business. This very fine book is an outgrowth of the lecture notes prepared for one of the most popular classes at NYU's esteemed Courant Institute. The topics covered are at the forefront of research in mathematical finance and the author's treatment of them is simply the best available in this form." --Peter Carr, PhD, head of Quantitative Financial Research, Bloomberg LP Director of the Masters Program in Mathematical Finance, New York University "Jim Gatheral is an acknowledged master of advanced modeling for derivatives. In The Volatility Surface he reveals the secrets of dealing with the most important but most elusive of financial quantities, volatility." --Paul Wilmott, author and mathematician "As a teacher in the field of mathematical finance, I welcome Jim Gatheral's book as a significant development. Written by a Wall Street practitioner with extensive market and teaching experience, The Volatility Surface gives students access to a level of knowledge on derivatives which was not previously available. I strongly recommend it." --Marco Avellaneda, Director, Division of Mathematical Finance Courant Institute, New York University "Jim Gatheral could not have written a better book." --Bruno Dupire, winner of the 2006 Wilmott Cutting Edge Research Award Quantitative Research, Bloomberg LP The volatility surface, formed from implied volatilities of all strikes and expirations, moves around. This randomness needs to be explicitly modeled for the effective pricing, trading, and risk management of equity derivatives. Focusing on equity derivatives, author Jim Gatheral examines why options are priced as they are and, starting from a powerful representation of implied volatility in terms of a weighted average of realized volatilities, explores the implications of various popular models for pricing. Along the way he also discusses default risk models, capital structure arbitrage, quadratic variation-based payoffs, VIX futures contracts, and much more. Throughout The Volatility Surface, specific examples are considered to make theory come to life for practitioners