Volatility and correlation : in the pricing of equity, FX, and interest-rate options

목차 (Table of Contents)

• CONTENTS
• Foreword = xiii
• Acknowledgements = xix
• Case Studies = xxi
• PART ONE FOUNDATIONS = 1
• 1 Volatility : Fundamental Concepts and Definitions = 3
• 1.1 Introduction and Plan of the Chapter = 3
• 1.2 Fundamental Concepts and Definitions = 4
• 1.3 Hedging Forward Contracts Using Spot Quantities = 6
• 1.4 Hedging Options : Volatilities of Spot and Forward Processes = 8
• 1.5 Definitions = 14
• 1.6 A Series of Options on Futures Contracts = 18
• 1.7 Hedging an Option with a Forward-Setting Strike = 18
• 1.8 Switching from the Real World to the Pricing Measure = 22
• 2 Variance and Mean Reversion in the Real and the Risk-Adjusted Worlds = 29
• 2.1 Introduction and Plan of the Chapter = 29
• 2.2 Hedging a Plain-Vanilla Option in the Presence of Constant Volatility = 30
• 2.3 Hedging a Plain-Vanilla Option in the Presence of Time-Dependent Volatility = 34
• 2.3.1 First View = 35
• 2.3.2 Second View = 36
• 2.3.3 Third View = 36
• 2.4 Hedging a Plain-Vanilla Option When the Real-World Process is Mean Reverting = 41
• 2.5 Hedging a Plain-Vanilla Option With Finite Re-Hedging Intervals = 44
• 3 Instantaneous and Terminal Correlations = 51
• 3.1 Introduction = 51
• 3.2 The Stochastic Evolution of Imperfectly, Correlated Variables = 52
• 3.3 The Role of Terminal Correlation in the Joint Evolution of Stochastic Variables = 57
• 3.3.1 Case 1 : European Option, One Underlying Asset = 58
• 3.3.2 Case 2 : Path-Dependent Option, One Asset = 61
• 3.3.3 Case 3 : Path-Dependent Option, Two Assets = 65
• 3.4 Generalising the Results = 68
• PART TWO DEALING WITH SMILES = 71
• 4 Pricing Options in the Presence of Smiles = 73
• 4.1 Introduction = 73
• 4.2 Hedging With a Compensated Process : Plain-Vanilla and Binary Options = 74
• 4.3 Smile Tale 1 : 'Sticky' Smiles = 78
• 4.4 Smile Tale 2 : 'Floating' Smiles = 80
• 4.5 Stylised Empirical Facts About Smiles = 83
• 4.5.1 Equities = 83
• 4.5.2 Interest Rates = 85
• 4.5.3 Foreign Exchange Rates = 87
• 4.6 General Features of the Smile-Modelling Approaches = 87
• 4.6.1 Fully Stochastic Volatility Models = 88
• 4.6.2 Complete-Markets Jump - Diffusion Models = 89
• 4.6.3 Random-Amplitude Jump - Diffusion Models = 90
• 4.6.4 Stochastic Volatility Functionally-Dependent on the Under-lying(Restricted-Stochastic-Volatility) Models = 91
• 4.7 Risk Derivatives for Plain-Vanilla Options in the Presence of Smiles = 93
• 5 Tree Methodologies for Smiley Option Prices = 97
• 5.1 Introduction = 97
• 5.2 General Considerations on Stochastic-Volatility Models = 97
• 5.3 The Dupire, Rubinstein and Derman and Kani Approaches = 100
• 5.4 Green's Function(Arrow-Debreu Prices) in the DK Construction = 101
• 5.5 The Derman and Kani Tree Construction = 104
• 5.6 Numerical Aspects of the Implementation of the DK Construction = 109
• 5.7 Implementation Results = 113
• 6 Efficient Extraction of the Future Local Volatility from Plain-Vanilla Option Prices = 129
• 6.1 Introduction = 129
• 6.2 The Computational Framework = 130
• 6.3 Computational Results = 135
• 6.4 The Link Between Implied and Local Volatility Surfaces = 139
• 6.4.1 Symmetric ('FX') Smiles = 140
• 6.4.2 Asymmetric ('Equity') Smile Surface = 144
• 6.4.3 Monotonic ('Interest-Rate') Smile Surface = 150
• 6.5 Gaining an Intuitive Understanding = 153
• 6.6 No-Arbitrage Conditions on the Implied Volatility Smile Surface = 161
• 6.7 A Worked-Out Example : Pricing Continuous Double Barriers in the Presence of Smiles = 174
• 6.8 Analysis of the Cost of Unwinding and Related Considerations About Option Pricing in the Presence of Smiles = 182
• Appendix 6.1 : Proof that ∂² Call($$S_t$$, K, T, t)／∂K² ＝ Φ($$S_T$$)$$
• _K$$ = 186
• 7 Closed-Form Solutions for Smiley Option Prices via Direct Modelling of the Density = 189
• 7.1 Introduction = 189
• 7.2 Estimating the Risk-Neutral Density Function = 195
• 7.3 Derivation of Analytic Formulae = 199
• 7.4 Results and Applications = 206
• 7.5 Conclusions and Range of Possible Applications = 213
• Appendix 7.1 Obtaining the Density of the Underlying from Quoted Option Prices = 214
• 8 Explaining Smiles by Means of Mixed Jump - Diffusion Processes = 215
• 8.1 Introduction = 215
• 8.2 The Financial Model : Smile Tale 2 Revisited = 216
• 8.3 Analytic Description of Mixed Jump - Diffusion Processes = 220
• 8.4 A General Framework for Option Pricing in Complete or Incomplete Markets = 229
• 8.5 Finding the Optimal Hedge = 235
• 8.6 Numerical Implernentation of the Britten-Jones - Neuberger Methodology = 236
• 8.7 Computational Results = 243
• 8.8 Discussion of the Results and Possible Developments = 249
• PART THREE INTEREST RATES = 251
• 9 The Role of Mean Reversion in Interest-Rate Models = 253
• 9.1 Introduction : Why Mean Reversion Matters in the Case of Interest-Rate Models = 253
• 9.2 The BDT Mean-Reversion Paradox = 256
• 9.3 The Unconditional Variance of the Short Rate in BDT - The Discrete Case = 259
• 9.4 The Unconditional Variance of the Short Rate in BDT - The Continuous-Time Equivalent = 261
• 9.5 Mean Reversion in Short-Rate Lattices : The Equi-Probable Binomial Versus the Bushy-Tree Approach = 263
• 9.6 Extension to More General Interest-Rate Models : The 'True' Role of Mean Reversion = 267
• Appendix 9.1 : Evaluation of the Variance of the Logarithm of the Instantaneous Short Rate = 269
• 10 Optimal Calibration of the Brace-Gatarek-Musiela Model = 271
• 10.1 Introduction and Statement of the Problem = 271
• 10.2 Constructing, the Most General BGM(Market) Model = 273
• 10.3 A Worked-Out Example : Caplets and a Two-Period Swaption = 278
• 10.4 A Worked-Out Example : Serial Options = 280
• 10.5 Reducing the Dimensionality of the BGM Model = 281
• 10.6 Numerical Results = 286
• 10.6.1 Fitting the Correlation Surface with a Three-Factor Model = 286
• 10.6.2 Fitting the Correlation Surface with a Four-Factor Model = 287
• 10.7 Conclusions = 298
• 11 Specifying the Instantaneous Volatility of Forward Rates = 303
• 11.1 The Link Between Instantaneous Volatility and the Future Term Structure of Volatilities = 303
• 11.2 A Functional Form for the Instantaneous Volatility Function = 306
• 11.3 Fitting the Instantaneous Volatility Function : Imposing Time-Homogeneity of the Term Structure of Volatilities = 311
• 11.4 Fitting the Instantaneous Volatility Function : Information from the Swaption Market = 318
• 11.5 Conclusions = 327
• References = 329
• Index = 333