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 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