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The Volatility Surface


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Table of Contents

List of Figures xiii List of Tables xix Foreword xxi Preface xxiii Acknowledgments xxvii CHAPTER 1 Stochastic Volatility and Local Volatility 1 Stochastic Volatility 1 Derivation of the Valuation Equation 4 Local Volatility 7 History 7 A Brief Review of Dupire's Work 8 Derivation of the Dupire Equation 9 Local Volatility in Terms of Implied Volatility 11 Special Case: No Skew 13 Local Variance as a Conditional Expectation of Instantaneous Variance 13 CHAPTER 2 The Heston Model 15 The Process 15 The Heston Solution for European Options 16 A Digression: The Complex Logarithm in the Integration (2.13) 19 Derivation of the Heston Characteristic Function 20 Simulation of the Heston Process 21 Milstein Discretization 22 Sampling from the Exact Transition Law 23 Why the Heston Model Is so Popular 24 CHAPTER 3 The Implied Volatility Surface 25 Getting Implied Volatility from Local Volatilities 25 Model Calibration 25 Understanding Implied Volatility 26 Local Volatility in the Heston Model 31 Ansatz 32 Implied Volatility in the Heston Model 33 The Term Structure of Black-Scholes Implied Volatility in the Heston Model 34 The Black-Scholes Implied Volatility Skew in the Heston Model 35 The SPX Implied Volatility Surface 36 Another Digression: The SVI Parameterization 37 A Heston Fit to the Data 40 Final Remarks on SV Models and Fitting the Volatility Surface 42 CHAPTER 4 The Heston-Nandi Model 43 Local Variance in the Heston-Nandi Model 43 A Numerical Example 44 The Heston-Nandi Density 45 Computation of Local Volatilities 45 Computation of Implied Volatilities 46 Discussion of Results 49 CHAPTER 5 Adding Jumps 50 Why Jumps are Needed 50 Jump Diffusion 52 Derivation of the Valuation Equation 52 Uncertain Jump Size 54 Characteristic Function Methods 56 Levy Processes 56 Examples of Characteristic Functions for Specific Processes 57 Computing Option Prices from the Characteristic Function 58 Proof of (5.6) 58 Computing Implied Volatility 60 Computing the At-the-Money Volatility Skew 60 How Jumps Impact the Volatility Skew 61 Stochastic Volatility Plus Jumps 65 Stochastic Volatility Plus Jumps in the Underlying Only (SVJ) 65 Some Empirical Fits to the SPX Volatility Surface 66 Stochastic Volatility with Simultaneous Jumps in Stock Price and Volatility (SVJJ) 68 SVJ Fit to the September 15, 2005, SPX Option Data 71 Why the SVJ Model Wins 73 CHAPTER 6 Modeling Default Risk 74 Merton's Model of Default 74 Intuition 75 Implications for the Volatility Skew 76 Capital Structure Arbitrage 77 Put-Call Parity 77 The Arbitrage 78 Local and Implied Volatility in the Jump-to-Ruin Model 79 The Effect of Default Risk on Option Prices 82 The CreditGrades Model 84 Model Setup 84 Survival Probability 85 Equity Volatility 86 Model Calibration 86 CHAPTER 7 Volatility Surface Asymptotics 87 Short Expirations 87 The Medvedev-Scaillet Result 89 The SABR Model 91 Including Jumps 93 Corollaries 94 Long Expirations: Fouque, Papanicolaou, and Sircar 95 Small Volatility of Volatility: Lewis 96 Extreme Strikes: Roger Lee 97 Example: Black-Scholes 99 Stochastic Volatility Models 99 Asymptotics in Summary 100 CHAPTER 8 Dynamics of the Volatility Surface 101 Dynamics of the Volatility Skew under Stochastic Volatility 101 Dynamics of the Volatility Skew under Local Volatility 102 Stochastic Implied Volatility Models 103 Digital Options and Digital Cliquets 103 Valuing Digital Options 104 Digital Cliquets 104 CHAPTER 9 Barrier Options 107 Definitions 107 Limiting Cases 108 Limit Orders 108 European Capped Calls 109 The Reflection Principle 109 The Lookback Hedging Argument 112 One-Touch Options Again 113 Put-Call Symmetry 113 QuasiStatic Hedging and Qualitative Valuation 114 Out-of-the-Money Barrier Options 114 One-Touch Options 115 Live-Out Options 116 Lookback Options 117 Adjusting for Discrete Monitoring 117 Discretely Monitored Lookback Options 119 Parisian Options 120 Some Applications of Barrier Options 120 Ladders 120 Ranges 120 Conclusion 121 CHAPTER 10 Exotic Cliquets 122 Locally Capped Globally Floored Cliquet 122 Valuation under Heston and Local Volatility Assumptions 123 Performance 124 Reverse Cliquet 125 Valuation under Heston and Local Volatility Assumptions 126 Performance 127 Napoleon 127 Valuation under Heston and Local Volatility Assumptions 128 Performance 130 Investor Motivation 130 More on Napoleons 131 CHAPTER 11 Volatility Derivatives 133 Spanning Generalized European Payoffs 133 Example: European Options 134 Example: Amortizing Options 135 The Log Contract 135 Variance and Volatility Swaps 136 Variance Swaps 137 Variance Swaps in the Heston Model 138 Dependence on Skew and Curvature 138 The Effect of Jumps 140 Volatility Swaps 143 Convexity Adjustment in the Heston Model 144 Valuing Volatility Derivatives 146 Fair Value of the Power Payoff 146 The Laplace Transform of Quadratic Variation under Zero Correlation 147 The Fair Value of Volatility under Zero Correlation 149 A Simple Lognormal Model 151 Options on Volatility: More on Model Independence 154 Listed Quadratic-Variation Based Securities 156 The VIX Index 156 VXB Futures 158 Knock-on Benefits 160 Summary 161 Postscript 162 Bibliography 163 Index 169

About the Author

JIM GATHERAL is a Managing Director at Merrill Lynch and also an Adjunct Professor at the Courant Institute of Mathematical Sciences, New York University.Dr. Gatheral obtained a PhD in theoretical physics from Cambridge Universityin 1983. Since then, he has been involved in all of the major derivative product areasas a bookrunner, risk manager, and quantitative analyst in London, Tokyo, and New York. From 1997 to 2005, Dr. Gatheral headed the Equity Quantitative Analytics group at Merrill Lynch. His current research focus is equity market microstructure and algorithmic trading. With a foreword by Nassim Nicholas Taleb Taleb is the Dean's Professor in the Sciences of Uncertainty at the University of Massachusetts at Amherst. He is also author of Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets (Random House, 2005).


??I do recommend this book?? (Zentralblatt MATH , Vol. 1118 2007/20)

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