Description
Book SynopsisIn Volatility and Correlation 2nd edition: The Perfect Hedger and the Fox, Rebonato looks at derivatives pricing from the angle of volatility and correlation.
Table of ContentsPreface 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
