Preface xxi 0.1 Why a Second Edition? xxi 0.2 What This Book Is Not About xxiii 0.3 Structure of the Book xxiv 0.4 The New Subtitle xxiv Acknowledgements xxvii I Foundations 1 1 Theory and Practice of Option Modelling 3 1.1 The Role of Models in Derivatives Pricing 3 1.2 The Efficient Market Hypothesis and Why It Matters for Option Pricing 9 1.3 Market Practice 14 1.

4 The Calibration Debate 17 1.5 Across-Markets Comparison of Pricing and Modelling Practices 27 1.6 Using Models 30 2 Option Replication 31 2.1 The Bedrock of Option Pricing 31 2.2 The Analytic (PDE) Approach 32 2.3 Binomial Replication 38 2.4 Justifying the Two-State Branching Procedure 65 2.5 The Nature of the Transformation between Measures: Girsanov''s Theorem 69 2.

6 Switching Between the PDE, the Expectation and the Binomial Replication Approaches 73 3 The Building Blocks 75 3.1 Introduction and Plan of the Chapter 75 3.2 Definition of Market Terms 75 3.3 Hedging Forward Contracts Using Spot Quantities 77 3.4 Hedging Options: Volatility of Spot and Forward Processes 80 3.5 The Link Between Root-Mean-Squared Volatilities and the Time-Dependence of Volatility 84 3.6 Admissibility of a Series of Root-Mean-Squared Volatilities 85 3.7 Summary of the Definitions So Far 87 3.

8 Hedging an Option with a Forward-Setting Strike 89 3.9 Quadratic Variation: First Approach 95 4 Variance and Mean Reversion in the Real and the Risk-Adjusted Worlds 101 4.1 Introduction and Plan of the Chapter 101 4.2 Hedging a Plain-Vanilla Option: General Framework 102 4.3 Hedging Plain-Vanilla Options: Constant Volatility 106 4.4 Hedging Plain-Vanilla Options: Time-Dependent Volatility 116 4.5 Hedging Behaviour In Practice 121 4.6 Robustness of the Black-and-Scholes Model 127 4.

7 Is the Total Variance All That Matters? 130 4.8 Hedging Plain-Vanilla Options: Mean-Reverting Real-World Drift 131 4.9 Hedging Plain-Vanilla Options: Finite Re-Hedging Intervals Again 135 5 Instantaneous and Terminal Correlation 141 5.1 Correlation, Co-Integration and Multi-Factor Models 141 5.2 The Stochastic Evolution of Imperfectly Correlated Variables 146 5.3 The Role of Terminal Correlation in the Joint Evolution of Stochastic Variables 151 5.4 Generalizing the Results 162 5.5 Moving Ahead 164 II Smiles - Equity and FX 165 6 Pricing Options in the Presence of Smiles 167 6.

1 Plan of the Chapter 167 6.2 Background and Definition of the Smile 168 6.3 Hedging with a Compensated Process: Plain-Vanilla and Binary Options 169 6.4 Hedge Ratios for Plain-Vanilla Options in the Presence of Smiles 173 6.5 Smile Tale 1: ''Sticky'' Smiles 180 6.6 Smile Tale 2: ''Floating'' Smiles 182 6.7 When Does Risk Aversion Make a Difference? 184 7 Empirical Facts About Smiles 201 7.1 What is this Chapter About? 201 7.

2 Market Information About Smiles 203 7.3 Equities 206 7.4 Interest Rates 222 7.5 FX Rates 227 7.6 Conclusions 235 8 General Features of Smile-Modelling Approaches 237 8.1 Fully-Stochastic-Volatility Models 237 8.2 Local-Volatility (Restricted-Stochastic-Volatility) Models 239 8.3 Jump-Diffusion Models 241 8.

4 Variance-Gamma Models 243 8.5 Mixing Processes 243 8.6 Other Approaches 245 8.7 The Importance of the Quadratic Variation (Take 2) 246 9 The Input Data: Fitting an Exogenous Smile Surface 249 9.1 What is This Chapter About? 249 9.2 Analytic Expressions for Calls vs Process Specification 249 9.3 Direct Use of Market Prices: Pros and Cons 250 9.4 Statement of the Problem 251 9.

5 Fitting Prices 252 9.6 Fitting Transformed Prices 254 9.7 Fitting the Implied Volatilities 255 9.8 Fitting the Risk-Neutral Density Function - General 256 9.9 Fitting the Risk-Neutral Density Function: Mixture of Normals 259 9.10 Numerical Results 265 9.11 Is the Term ∂C/∂ S Really a Delta? 275 9.12 Fitting the Risk-Neutral Density Function: The Generalized-Beta Approach 277 10 Quadratic Variation and Smiles 293 10.

1 Why This Approach Is Interesting 293 10.2 The BJN Framework for Bounding Option Prices 293 10.3 The BJN Approach - Theoretical Development 294 10.4 The BJN Approach: Numerical Implementation 300 10.5 Discussion of the Results 312 10.6 Conclusions (or, Limitations of Quadratic Variation) 316 11 Local-Volatility Models: the Derman-and-Kani Approach 319 11.1 General Considerations on Stochastic-Volatility Models 319 11.2 Special Cases of Restricted-Stochastic-Volatility Models 321 11.

3 The Dupire, Rubinstein and Derman-and-Kani Approaches 321 11.4 Green''s Functions (Arrow-Debreu Prices) in the DK Construction 322 11.5 The Derman-and-Kani Tree Construction 326 11.6 Numerical Aspects of the Implementation of the DK Construction 331 11.7 Implementation Results 334 11.8 Estimating Instantaneous Volatilities from Prices as an Inverse Problem 343 12 Extracting the Local Volatility from Option Prices 345 12.1 Introduction 345 12.2 The Modelling Framework 347 12.

3 A Computational Method 349 12.4 Computational Results 355 12.5 The Link Between Implied and Local-Volatility Surfaces 357 12.6 Gaining an Intuitive Understanding 368 12.7 What Local-Volatility Models Imply about Sticky and Floating Smiles 373 12.8 No-Arbitrage Conditions on the Current Implied Volatility Smile Surface 375 12.9 Empirical Performance 385 12.10 Appendix I: Proof that ∂ 2 Call ( St, K, T, t )/ ∂k 2 = φ ( ST ) K 386 13 Stochastic-Volatility Processes 389 13.

1 Plan of the Chapter 389 13.2 Portfolio Replication in the Presence of Stochastic Volatility 389 13.3 Mean-Reverting Stochastic Volatility 401 13.4 Qualitative Features of Stochastic-Volatility Smiles 405 13.5 The Relation Between Future Smiles and Future Stock Price Levels 416 13.6 Portfolio Replication in Practice: The Stochastic-Volatility Case 418 13.7 Actual Fitting to Market Data 427 13.8 Conclusions 436 14 Jump-Diffusion Processes 439 14.

1 Introduction 439 14.2 The Financial Model: Smile Tale 2 Revisited 441 14.3 Hedging and Replicability in the Presence of Jumps: First Considerations 444 14.4 Analytic Description of Jump-Diffusions 449 14.5 Hedging with Jump-Diffusion Processes 455 14.6 The Pricing Formula for Log-Normal Amplitude Ratios 470 14.7 The Pricing Formula in the Finite-Amplitude-Ratio Case 472 14.8 The Link Between the Price Density and the Smile Shape 485 14.

9 Qualitative Features of Jump-Diffusion Smiles 494 14.10 Jump-Diffusion Processes and Market Completeness Revisited 500 14.11 Portfolio Replication in Practice: The Jump-Diffusion Case 502 15 Variance-Gamma 511 15.1 Who Can Make Best Use of the Variance-Gamma Approach? 511 15.2 The Variance-Gamma Process 513 15.3 Statistical Properties of the Price Distribution 522 15.4 Features of the Smile 523 15.5 Conclusions 527 16 Displaced Diffusions and Generalizations 529 16.

1 Introduction 529 16.2 Gaining Intuition 530 16.3 Evolving the Underlying with Displaced Diffusions 531 16.4 Option Prices with Displaced Diffusions 532 16.5 Matching At-The-Money Prices with Displaced Diffusions 533 16.6 The Smile Produced by Displaced Diffusions 553 16.7 Extension to Other Processes 560 17 No-Arbitrage Restrictions on the Dynamics of Smile Surfaces 563 17.1 A Worked-Out Example: Pricing Continuous Double Barriers 564 17.

2 Analysis of the Cost of Unwinding 571 17.3 The Trader''s Dream 575 17.4 Plan of the Remainder of the Chapter 581 17.5 Conditions of No-Arbitrage for the Stochastic Evolution of Future Smile Surfaces 582 17.6 Deterministic Smile Surfaces 585 17.7 Stochastic Smiles 593 17.8 The Strength of the Assumptions 597 17.9 Limitations and Conclusions 598 III Interest Rates - Deterministic Volatilities 601 18 Mean Reversion in Interest-Rate Models 603 18.

1 Introduction and Plan of the Chapter 603 18.2 Why Mean Reversion Matters in the Case of Interest-Rate Models 604 18.3 A Common Fallacy Regarding Mean Reversion 608 18.4 The BDT Mean-Reversion Paradox 610 18.5 The Unconditional Variance of the Short Rate in BDT - the Discrete Case 612 18.6 The Unconditional Variance of the Short Rate in BDT-the Continuous-Time Equivalent 616 18.7 Mean Reversion in Short-Rate Lattices: Recombining vs Bushy Trees 617 18.8 Extension to More General Interest-Rate Models 620 18.

9 Appendix I: Evaluation of the Variance of the Logarithm of the Instantaneous Short Rate 622 19 Volatility and Correlation in the LIBOR Market Model 625 19.1 Introduction 625 19.2 Specifying the Forward-Rate Dynamics in the LIBOR Market Model 626 19.3 Link with the Principal Component Analysis 631 19.4 Worked-Out Example 1: Caplets and a Two-Period Swaption 632 19.5 Worked-Out Example 2: Serial Options 635 19.6 Plan of the Work Ahead 636 20 Calibration Str.