In Volatility and Correlation 2nd edition: The Perfect Hedger and the Fox, Rebonato looks at derivatives pricing from the angle of volatility and correlation. With both practical and theoretical applications, this is a thorough update of the highly successful Volatility & Correlation – with over 80% new or fully reworked material and is a must have both for practitioners and for students. The new and updated material includes a critical examination of the ‘perfect-replication’ approach to derivatives pricing, with special attention given to exotic options; a thorough analysis of the role of quadratic variation in derivatives pricing and hedging; a discussion of the informational efficiency of markets in commonly-used calibration and hedging practices. Treatment of new models including Variance Gamma, displaced diffusion, stochastic volatility for interest-rate smiles and equity/FX options. The book is split into four parts. Part I deals with a Black world without smiles, sets out the author’s ‘philosophical’ approach and covers deterministic volatility. Part II looks at smiles in equity and FX worlds. It begins with a review of relevant empirical information about smiles, and provides coverage of local-stochastic-volatility, general-stochastic-volatility, jump-diffusion and Variance-Gamma processes. Part II concludes with an important chapter that discusses if and to what extent one can dispense with an explicit specification of a model, and can directly prescribe the dynamics of the smile surface. Part III focusses on interest rates when the volatility is deterministic. Part IV extends this setting in order to account for smiles in a financially motivated and computationally tractable manner. In this final part the author deals with CEV processes, with diffusive stochastic volatility and with Markov-chain processes. Praise for the First Edition: “In this book, Dr Rebonato brings his penetrating eye to bear on option pricing and hedging.… The book is a must-read for those who already know the basics of options and are looking for an edge in applying the more sophisticated approaches that have recently been developed.” —Professor Ian Cooper, London Business School “Volatility and correlation are at the very core of all option pricing and hedging. In this book, Riccardo Rebonato presents the subject in his characteristically elegant and simple fashion…A rare combination of intellectual insight and practical common sense.” —Anthony Neuberger, London Business School

Table of Contents:
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 ∂2Call(St, K, T, t)/∂k2 = 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 Strategies for the LIBOR Market Model 639 20.1 Plan of the Chapter 639 20.2 The Setting 639 20.3 Fitting an Exogenous Correlation Function 643 20.4 Numerical Results 646 20.5 Analytic Expressions to Link Swaption and Caplet Volatilities 659 20.6 Optimal Calibration to Co-Terminal Swaptions 662 21 Specifying the Instantaneous Volatility of Forward Rates 667 21.1 Introduction and Motivation 667 21.2 The Link between Instantaneous Volatilities and the Future Term Structure of Volatilities 668 21.3 A Functional Form for the Instantaneous Volatility Function 671 21.4 Ensuring Correct Caplet Pricing 673 21.5 Fitting the Instantaneous Volatility Function: Imposing Time Homogeneity of the Term Structure of Volatilities 677 21.6 Is a Time-Homogeneous Solution Always Possible? 679 21.7 Fitting the Instantaneous Volatility Function: The Information from the Swaption Market 680 21.8 Conclusions 686 22 Specifying the Instantaneous Correlation Among Forward Rates 687 22.1 Why Is Estimating Correlation So Difficult? 687 22.2 What Shape Should We Expect for the Correlation Surface? 688 22.3 Features of the Simple Exponential Correlation Function 689 22.4 Features of the Modified Exponential Correlation Function 691 22.5 Features of the Square-Root Exponential Correlation Function 694 22.6 Further Comparisons of Correlation Models 697 22.7 Features of the Schonmakers–Coffey Approach 697 22.8 Does It Make a Difference (and When)? 698 IV Interest Rates – Smiles 701 23 How to Model Interest-Rate Smiles 703 23.1 What Do We Want to Capture? A Hierarchy of Smile-Producing Mechanisms 703 23.2 Are Log-Normal Co-Ordinates the Most Appropriate? 704 23.3 Description of the Market Data 706 23.4 Empirical Study I: Transforming the Log-Normal Co-ordinates 715 23.5 The Computational Experiments 718 23.6 The Computational Results 719 23.7 Empirical Study II: The Log-Linear Exponent 721 23.8 Combining the Theoretical and Experimental Results 725 23.9 Where Do We Go From Here? 725 24 (CEV) Processes in the Context of the LMM 729 24.1 Introduction and Financial Motivation 729 24.2 Analytical Characterization of CEV Processes 730 24.3 Financial Desirability of CEV Processes 732 24.4 Numerical Problems with CEV Processes 734 24.5 Approximate Numerical Solutions 735 24.6 Problems with the Predictor–Corrector Approximation for the LMM 747 25 Stochastic-Volatility Extensions of the LMM 751 25.1 Plan of the Chapter 751 25.2 What is the Dog and What is the Tail? 753 25.3 Displaced Diffusion vs CEV 754 25.4 The Approach 754 25.5 Implementing and Calibrating the Stochastic-Volatility LMM 756 25.6 Suggestions and Plan of the Work Ahead 764 26 The Dynamics of the Swaption Matrix 765 26.1 Plan of the Chapter 765 26.2 Assessing the Quality of a Model 766 26.3 The Empirical Analysis 767 26.4 Extracting the Model-Implied Principal Components 776 26.5 Discussion, Conclusions and Suggestions for Future Work 781 27 Stochastic-Volatility Extension of the LMM: Two-Regime Instantaneous Volatility 783 27.1 The Relevance of the Proposed Approach 783 27.2 The Proposed Extension 783 27.3 An Aside: Some Simple Properties of Markov Chains 785 27.4 Empirical Tests 788 27.5 How Important Is the Two-Regime Feature? 798 27.6 Conclusions 801 Bibliography 805 Index 813