Volatility and Correlation

Riccardo Rebonato is Head of Group Market Risk for the Royal Bank of Scotland Group, and Head of The Royal Bank of Scotland Group Quantitative Research Centre. He is also a Visiting Lecturer at Oxford University for the Mathematical Finance Diploma and MSc. He holds Doctorates in Nuclear Engineering and Science of Materials/Solid State Physics. He sits on the Board of Directors of ISDA and on the Board of Trustees of GARP.Prior to joining the Royal Bank of Scotland, he was Head of Complex Derivatives Trading Europe and Head of Derivatives Research at Barclays Capital (BZW), where he worked for nine years.Before that he was a Research Fellow in Physics at Corpus Christi College, Oxford, UK. He is the author of three books, Modern Pricing of Interest-Rate Derivatives, Volatility and Correlation in Option Pricing and Interest-Rate Option Models. He has published several papers on finance in academic journals, and is on the editorial board of several journals. He is a regular speaker at conferences worldwide.

• Preface xxi0.1 Why a Second Edition? xxi0.2 What This Book Is Not About xxiii0.3 Structure of the Book xxiv0.4 The New Subtitle xxivAcknowledgements xxviiI Foundations 11 Theory and Practice of Option Modelling 31.1 The Role of Models in Derivatives Pricing 31.2 The Efficient Market Hypothesis and Why It Matters for Option Pricing 91.3 Market Practice 141.4 The Calibration Debate 171.5 Across-Markets Comparison of Pricing and Modelling Practices 271.6 Using Models 302 Option Replication 312.1 The Bedrock of Option Pricing 312.2 The Analytic (PDE) Approach 322.3 Binomial Replication 382.4 Justifying the Two-State Branching Procedure 652.5 The Nature of the Transformation between Measures: Girsanov’s Theorem 692.6 Switching Between the PDE, the Expectation and the Binomial Replication Approaches 733 The Building Blocks 753.1 Introduction and Plan of the Chapter 753.2 Definition of Market Terms 753.3 Hedging Forward Contracts Using Spot Quantities 773.4 Hedging Options: Volatility of Spot and Forward Processes 803.5 The Link Between Root-Mean-Squared Volatilities and the Time-Dependence of Volatility 843.6 Admissibility of a Series of Root-Mean-Squared Volatilities 853.7 Summary of the Definitions So Far 873.8 Hedging an Option with a Forward-Setting Strike 893.9 Quadratic Variation: First Approach 954 Variance and Mean Reversion in the Real and the Risk-Adjusted Worlds 1014.1 Introduction and Plan of the Chapter 1014.2 Hedging a Plain-Vanilla Option: General Framework 1024.3 Hedging Plain-Vanilla Options: Constant Volatility 1064.4 Hedging Plain-Vanilla Options: Time-Dependent Volatility 1164.5 Hedging Behaviour In Practice 1214.6 Robustness of the Black-and-Scholes Model 1274.7 Is the Total Variance All That Matters? 1304.8 Hedging Plain-Vanilla Options: Mean-Reverting Real-World Drift 1314.9 Hedging Plain-Vanilla Options: Finite Re-Hedging Intervals Again 1355 Instantaneous and Terminal Correlation 1415.1 Correlation, Co-Integration and Multi-Factor Models 1415.2 The Stochastic Evolution of Imperfectly Correlated Variables 1465.3 The Role of Terminal Correlation in the Joint Evolution of Stochastic Variables 1515.4 Generalizing the Results 1625.5 Moving Ahead 164II Smiles – Equity and FX 1656 Pricing Options in the Presence of Smiles 1676.1 Plan of the Chapter 1676.2 Background and Definition of the Smile 1686.3 Hedging with a Compensated Process: Plain-Vanilla and Binary Options 1696.4 Hedge Ratios for Plain-Vanilla Options in the Presence of Smiles 1736.5 Smile Tale 1: ‘Sticky’ Smiles 1806.6 Smile Tale 2: ‘Floating’ Smiles 1826.7 When Does Risk Aversion Make a Difference? 1847 Empirical Facts About Smiles 2017.1 What is this Chapter About? 2017.2 Market Information About Smiles 2037.3 Equities 2067.4 Interest Rates 2227.5 FX Rates 2277.6 Conclusions 2358 General Features of Smile-Modelling Approaches 2378.1 Fully-Stochastic-Volatility Models 2378.2 Local-Volatility (Restricted-Stochastic-Volatility) Models 2398.3 Jump–Diffusion Models 2418.4 Variance–Gamma Models 2438.5 Mixing Processes 2438.6 Other Approaches 2458.7 The Importance of the Quadratic Variation (Take 2) 2469 The Input Data: Fitting an Exogenous Smile Surface 2499.1 What is This Chapter About? 2499.2 Analytic Expressions for Calls vs Process Specification 2499.3 Direct Use of Market Prices: Pros and Cons 2509.4 Statement of the Problem 2519.5 Fitting Prices 2529.6 Fitting Transformed Prices 2549.7 Fitting the Implied Volatilities 2559.8 Fitting the Risk-Neutral Density Function – General 2569.9 Fitting the Risk-Neutral Density Function: Mixture of Normals 2599.10 Numerical Results 2659.11 Is the Term ∂C/∂S Really a Delta? 2759.12 Fitting the Risk-Neutral Density Function: The Generalized-Beta Approach 27710 Quadratic Variation and Smiles 29310.1 Why This Approach Is Interesting 29310.2 The BJN Framework for Bounding Option Prices 29310.3 The BJN Approach – Theoretical Development 29410.4 The BJN Approach: Numerical Implementation 30010.5 Discussion of the Results 31210.6 Conclusions (or, Limitations of Quadratic Variation) 31611 Local-Volatility Models: the Derman-and-Kani Approach 31911.1 General Considerations on Stochastic-Volatility Models 31911.2 Special Cases of Restricted-Stochastic-Volatility Models 32111.3 The Dupire, Rubinstein and Derman-and-Kani Approaches 32111.4 Green’s Functions (Arrow–Debreu Prices) in the DK Construction 32211.5 The Derman-and-Kani Tree Construction 32611.6 Numerical Aspects of the Implementation of the DK Construction 33111.7 Implementation Results 33411.8 Estimating Instantaneous Volatilities from Prices as an Inverse Problem 34312 Extracting the Local Volatility from Option Prices 34512.1 Introduction 34512.2 The Modelling Framework 34712.3 A Computational Method 34912.4 Computational Results 35512.5 The Link Between Implied and Local-Volatility Surfaces 35712.6 Gaining an Intuitive Understanding 36812.7 What Local-Volatility Models Imply about Sticky and Floating Smiles 37312.8 No-Arbitrage Conditions on the Current Implied Volatility Smile Surface 37512.9 Empirical Performance 38512.10 Appendix I: Proof that ∂2Call(St, K, T, t)/∂k2 = φ(ST)|K 38613 Stochastic-Volatility Processes 38913.1 Plan of the Chapter 38913.2 Portfolio Replication in the Presence of Stochastic Volatility 38913.3 Mean-Reverting Stochastic Volatility 40113.4 Qualitative Features of Stochastic-Volatility Smiles 40513.5 The Relation Between Future Smiles and Future Stock Price Levels 41613.6 Portfolio Replication in Practice: The Stochastic-Volatility Case 41813.7 Actual Fitting to Market Data 42713.8 Conclusions 43614 Jump–Diffusion Processes 43914.1 Introduction 43914.2 The Financial Model: Smile Tale 2 Revisited 44114.3 Hedging and Replicability in the Presence of Jumps: First Considerations 44414.4 Analytic Description of Jump–Diffusions 44914.5 Hedging with Jump–Diffusion Processes 45514.6 The Pricing Formula for Log-Normal Amplitude Ratios 47014.7 The Pricing Formula in the Finite-Amplitude-Ratio Case 47214.8 The Link Between the Price Density and the Smile Shape 48514.9 Qualitative Features of Jump–Diffusion Smiles 49414.10 Jump–Diffusion Processes and Market Completeness Revisited 50014.11 Portfolio Replication in Practice: The Jump–Diffusion Case 50215 Variance–Gamma 51115.1 Who Can Make Best Use of the Variance–Gamma Approach? 51115.2 The Variance–Gamma Process 51315.3 Statistical Properties of the Price Distribution 52215.4 Features of the Smile 52315.5 Conclusions 52716 Displaced Diffusions and Generalizations 52916.1 Introduction 52916.2 Gaining Intuition 53016.3 Evolving the Underlying with Displaced Diffusions 53116.4 Option Prices with Displaced Diffusions 53216.5 Matching At-The-Money Prices with Displaced Diffusions 53316.6 The Smile Produced by Displaced Diffusions 55316.7 Extension to Other Processes 56017 No-Arbitrage Restrictions on the Dynamics of Smile Surfaces 56317.1 A Worked-Out Example: Pricing Continuous Double Barriers 56417.2 Analysis of the Cost of Unwinding 57117.3 The Trader’s Dream 57517.4 Plan of the Remainder of the Chapter 58117.5 Conditions of No-Arbitrage for the Stochastic Evolution of Future Smile Surfaces 58217.6 Deterministic Smile Surfaces 58517.7 Stochastic Smiles 59317.8 The Strength of the Assumptions 59717.9 Limitations and Conclusions 598III Interest Rates – Deterministic Volatilities 60118 Mean Reversion in Interest-Rate Models 60318.1 Introduction and Plan of the Chapter 60318.2 Why Mean Reversion Matters in the Case of Interest-Rate Models 60418.3 A Common Fallacy Regarding Mean Reversion 60818.4 The BDT Mean-Reversion Paradox 61018.5 The Unconditional Variance of the Short Rate in BDT – the Discrete Case 61218.6 The Unconditional Variance of the Short Rate in BDT–the Continuous-Time Equivalent 61618.7 Mean Reversion in Short-Rate Lattices: Recombining vs Bushy Trees 61718.8 Extension to More General Interest-Rate Models 62018.9 Appendix I: Evaluation of the Variance of the Logarithm of the Instantaneous Short Rate 62219 Volatility and Correlation in the LIBOR Market Model 62519.1 Introduction 62519.2 Specifying the Forward-Rate Dynamics in the LIBOR Market Model 62619.3 Link with the Principal Component Analysis 63119.4 Worked-Out Example 1: Caplets and a Two-Period Swaption 63219.5 Worked-Out Example 2: Serial Options 63519.6 Plan of the Work Ahead 63620 Calibration Strategies for the LIBOR Market Model 63920.1 Plan of the Chapter 63920.2 The Setting 63920.3 Fitting an Exogenous Correlation Function 64320.4 Numerical Results 64620.5 Analytic Expressions to Link Swaption and Caplet Volatilities 65920.6 Optimal Calibration to Co-Terminal Swaptions 66221 Specifying the Instantaneous Volatility of Forward Rates 66721.1 Introduction and Motivation 66721.2 The Link between Instantaneous Volatilities and the Future Term Structure of Volatilities 66821.3 A Functional Form for the Instantaneous Volatility Function 67121.4 Ensuring Correct Caplet Pricing 67321.5 Fitting the Instantaneous Volatility Function: Imposing Time Homogeneity of the Term Structure of Volatilities 67721.6 Is a Time-Homogeneous Solution Always Possible? 67921.7 Fitting the Instantaneous Volatility Function: The Information from the Swaption Market 68021.8 Conclusions 68622 Specifying the Instantaneous Correlation Among Forward Rates 68722.1 Why Is Estimating Correlation So Difficult? 68722.2 What Shape Should We Expect for the Correlation Surface? 68822.3 Features of the Simple Exponential Correlation Function 68922.4 Features of the Modified Exponential Correlation Function 69122.5 Features of the Square-Root Exponential Correlation Function 69422.6 Further Comparisons of Correlation Models 69722.7 Features of the Schonmakers–Coffey Approach 69722.8 Does It Make a Difference (and When)? 698IV Interest Rates – Smiles 70123 How to Model Interest-Rate Smiles 70323.1 What Do We Want to Capture? A Hierarchy of Smile-Producing Mechanisms 70323.2 Are Log-Normal Co-Ordinates the Most Appropriate? 70423.3 Description of the Market Data 70623.4 Empirical Study I: Transforming the Log-Normal Co-ordinates 71523.5 The Computational Experiments 71823.6 The Computational Results 71923.7 Empirical Study II: The Log-Linear Exponent 72123.8 Combining the Theoretical and Experimental Results 72523.9 Where Do We Go From Here? 72524 (CEV) Processes in the Context of the LMM 72924.1 Introduction and Financial Motivation 72924.2 Analytical Characterization of CEV Processes 73024.3 Financial Desirability of CEV Processes 73224.4 Numerical Problems with CEV Processes 73424.5 Approximate Numerical Solutions 73524.6 Problems with the Predictor–Corrector Approximation for the LMM 74725 Stochastic-Volatility Extensions of the LMM 75125.1 Plan of the Chapter 75125.2 What is the Dog and What is the Tail? 75325.3 Displaced Diffusion vs CEV 75425.4 The Approach 75425.5 Implementing and Calibrating the Stochastic-Volatility LMM 75625.6 Suggestions and Plan of the Work Ahead 76426 The Dynamics of the Swaption Matrix 76526.1 Plan of the Chapter 76526.2 Assessing the Quality of a Model 76626.3 The Empirical Analysis 76726.4 Extracting the Model-Implied Principal Components 77626.5 Discussion, Conclusions and Suggestions for Future Work 78127 Stochastic-Volatility Extension of the LMM: Two-Regime Instantaneous Volatility 78327.1 The Relevance of the Proposed Approach 78327.2 The Proposed Extension 78327.3 An Aside: Some Simple Properties of Markov Chains 78527.4 Empirical Tests 78827.5 How Important Is the Two-Regime Feature? 79827.6 Conclusions 801Bibliography 805Index 813