Volatility and Correlation The Perfect Hedger and the Fox

by Rebonato, Riccardo

• ISBN13: 9780470091395
• ISBN10: 0470091398
• Edition: 2nd
• Format: Hardcover
• Copyright: 2004-09-03
• Publisher: WILEY

Summary

The new edition of Volatility and Correlation has been thoroughly updated and expanded with over 80% new or reworked material, reflecting the changes and developments that have taken place in the field. The new and updated material includes: empirical and theoretical analysis of the smile dynamics; examination of the perfect-replication model in relation to exotic options; treatment of additional important models, namely, Variance Gamma, displaced diffusion, CEV, stochastic volatility for interest-rate smiles and equity/FX options; questioning of the informational efficiency of markets in commonly-used calibration and hedging practices.

Author Biography

<b>Riccardo Rebonato</b> 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.<br> 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.<br> Before that he was a Research Fellow in Physics at Corpus Christi College, Oxford, UK. He is the author of three books, <i>Modern Pricing of Interest-Rate Derivatives</i>, <i>Volatility and Correlation in Option Pricing</i> and <i>Interest-Rate Option Models</i>. 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.

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	(164)
1 Theory and Practice of Option Modelling	3	(28)
1.1 The Role of Models in Derivatives Pricing	3	(6)
1.1.1 What Are Models For?	3	(2)
1.1.2 The Fundamental Approach	5	(2)
1.1.3 The Instrumental Approach	7	(1)
1.1.4 A Conundrum (or, 'What is Vega Hedging For?')	8	(1)
1.2 The Efficient Market Hypothesis and Why It Matters for Option Pricing	9	(5)
1.2.1 The Three Forms of the EMH	9	(1)
1.2.2 Pseudo-Arbitrageurs in Crisis	10	(1)
1.2.3 Model Risk for Traders and Risk Managers	11	(1)
1.2.4 The Parable of the Two Volatility Traders	12	(2)
1.3 Market Practice	14	(3)
1.3.1 Different Users of Derivatives Models	14	(1)
1.3.2 In-Model and Out-of-Model Hedging	15	(2)
1.4 The Calibration Debate	17	(10)
1.4.1 Historical vs Implied Calibration	18	(1)
1.4.2 The Logical Underpinning of the Implied Approach	19	(2)
1.4.3 Are Derivatives Markets Informationally Efficient?	21	(5)
1.4.4 Back to Calibration	26	(1)
1.4.5 A Practical Recommendation	27	(1)
1.5 Across-Markets Comparison of Pricing and Modelling Practices	27	(3)
1.6 Using Models	30	(1)
2 Option Replication	31	(44)
2.1 The Bedrock of Option Pricing	31	(1)
2.2 The Analytic (PDE) Approach	32	(6)
2.2.1 The Assumptions	32	(1)
2.2.2 The Portfolio-Replication Argument (Deterministic Volatility)	32	(2)
2.2.3 The Market Price of Risk with Deterministic Volatility	34	(2)
2.2.4 Link with Expectations - the Feynman-Kac Theorem	36	(2)
2.3 Binomial Replication	38	(1)
2.3.1 First Approach - Replication Strategy	39	(2)
2.3.2 Second Approach - 'Naive Expectation'	41	(1)
2.3.3 Third Approach - 'Market Price of Risk'	42	(3)
2.3.4 A Worked-Out Example	45	(1)
2.3.5 Fourth Approach Risk-Neutral Valuation	46	(2)
2.3.6 Pseudo-Probabilities	48	(1)
2.3.7 Are the Quantities π1 andπ2 Really Probabilities?	49	(2)
2.3.8 Introducing Relative Prices	51	(2)
2.3.9 Moving to a Multi-Period Setting	53	(3)
2.3.10 Fair Prices as Expectations	56	(2)
2.3.11 Switching Numeraires and Relating Expectations Under Different Measures	58	(3)
2.3.12 Another Worked-Out Example	61	(3)
2.3.13 Relevance of the Results	64	(1)
2.4 Justifying the Two-State Branching Procedure	65	(4)
2.4.1 How To Recognize a Jump When You See One	65	(4)
2.5 The Nature of the Transformation between Measures: Girsanov's Theorem	69	(4)
2.5.1 An Intuitive Argument	69	(1)
2.5.2 A Worked-Out Example	70	(3)
2.6 Switching Between the PDE, the Expectation and the Binomial Replication Approaches	73	(2)
3 The Building Blocks	75	(26)
3.1 Introduction and Plan of the Chapter	75	(1)
3.2 Definition of Market Terms	75	(2)
3.3 Hedging Forward Contracts Using Spot Quantities	77	(3)
3.3.1 Hedging Equity Forward Contracts	78	(1)
3.3.2 Hedging Interest-Rate Forward Contracts	79	(1)
3.4 Hedging Options: Volatility of Spot and Forward Processes	80	(4)
3.5 The Link Between Root-Mean-Squared Volatilities and the Time-Dependence of Volatility	84	(1)
3.6 Admissibility of a Series of Root-Mean-Squared Volatilities	85	(2)
3.6.1 The Equity/FX Case	85	(1)
3.6.2 The Interest-Rate Case	86	(1)
3.7 Summary of the Definitions So Far	87	(2)
3.8 Hedging an Option with a Forward-Setting Strike	89	(6)
3.8.1 Why Is This Option Important? (And Why Is it Difficult to Hedge?)	90	(1)
3.8.2 Valuing a Forward-Setting Option	91	(4)
3.9 Quadratic Variation: First Approach	95	(6)
3.9.1 Definition	95	(1)
3.9.2 Properties of Variations	96	(1)
3.9.3 First and Second Variation of a Brownian Process	97	(1)
3.9.4 Links between Quadratic Variation and ƒT/t σ (u)²du	97	(1)
3.9.5 Why Quadratic Variation Is So Important (Take 1)	98	(3)
4 Variance and Mean Reversion in the Real and the Risk-Adjusted Worlds	101	(40)
4.1 Introduction and Plan of the Chapter	101	(1)
4.2 Hedging a Plain-Vanilla Option: General Framework	102	(4)
4.2.1 Trading Restrictions and Model Uncertainty: Theoretical Results	103	(1)
4.2.2 The Setting	104	(1)
4.2.3 The Methodology	104	(2)
4.2.4 Criterion for Success	106	(1)
4.3 Hedging Plain-Vanilla Options: Constant Volatility	106	(10)
4.3.1 Trading the Gamma: One Step and Constant Volatility	108	(6)
4.3.2 Trading the Gamma: Several Steps and Constant Volatility	114	(2)
4.4 Hedging Plain-Vanilla Options: Time-Dependent Volatility	116	(5)
4.4.1 Views on Gamma Trading When the Volatility is Time Dependent	116	(3)
4.4.2 Which View Is the Correct One? (and the Feynman-Kac Theorem Again)	119	(2)
4.5 Hedging Behaviour In Practice	121	(6)
4.5.1 Analysing the Replicating Portfolio	121	(1)
4.5.2 Hedging Results: the Time-Dependent Volatility Case	122	(3)
4.5.3 Hedging with the Wrong Volatility	125	(2)
4.6 Robustness of the Black-and-Scholes Model	127	(3)
4.7 Is the Total Variance All That Matters?	130	(1)
4.8 Hedging Plain-Vanilla Options: Mean-Reverting Real-World Drift	131	(4)
4.9 Hedging Plain-Vanilla Options: Finite Re-Hedging Intervals Again	135	(6)
4.9.1 The Crouhy-Galai Set-Up	135	(6)
5 Instantaneous and Terminal Correlation	141	(26)
5.1 Correlation, Co-Integration and Multi-Factor Models	141	(5)
5.1.1 The Multi-Factor Debate	144	(2)
5.2 The Stochastic Evolution of Imperfectly Correlated Variables	146	(5)
5.3 The Role of Terminal Correlation in the Joint Evolution of Stochastic Variables	151	(11)
5.3.1 Defining Stochastic Integrals	151	(2)
5.3.2 Case 1: European Option, One Underlying Asset	153	(2)
5.3.3 Case 2: Path-Dependent Option, One Asset	155	(1)
5.3.4 Case 3: Path-Dependent Option, Two Assets	156	(6)
5.4 Generalizing the Results	162	(2)
5.5 Moving Ahead	164	(1)
II Smiles - Equity and FX	165	(436)
6 Pricing Options in the Presence of Smiles	167	(34)
6.1 Plan of the Chapter	167	(1)
6.2 Background and Definition of the Smile	168	(1)
6.3 Hedging with a Compensated Process: Plain-Vanilla and Binary Options	169	(4)
6.3.1 Delta- and Vega-Hedging a Plain-Vanilla Option	169	(3)
6.3.2 Pricing a European Digital Option	172	(1)
6.4 Hedge Ratios for Plain-Vanilla Options in the Presence of Smiles	173	(7)
6.4.1 The Relationship Between the True Call Price Functional and the Black Formula	174	(1)
6.4.2 Calculating the Delta Using the Black Formula and the Implied Volatility	175	(1)
6.4.3 Dependence of Implied Volatilities on the Strike and the Underlying	176	(2)
6.4.4 Floating and Sticky Smiles and What They Imply about Changes in Option Prices	178	(2)
6.5 Smile Tale 1: 'Sticky' Smiles	180	(2)
6.6 Smile Tale 2: 'Floating' Smiles	182	(2)
6.6.1 Relevance of the Smile Story for Floating Smiles	183	(1)
6.7 When Does Risk Aversion Make a Difference?	184	(17)
6.7.1 Motivation	184	(1)
6.7.2 The Importance of an Assessment of Risk Aversion for Model Building	185	(1)
6.7.3 The Principle of Absolute Continuity	186	(1)
6.7.4 The Effect of Supply and Demand	187	(1)
6.7.5 A Stylized Example: First Version	187	(7)
6.7.6 A Stylized Example: Second Version	194	(2)
6.7.7 A Stylized Example: Third Version	196	(1)
6.7.8 Overall Conclusions	196	(3)
6.7.9 The EMH Again	199	(2)
7 Empirical Facts About Smiles	201	(36)
7.1 What is this Chapter About?	201	(2)
7.1.1 'Fundamental' and 'Derived' Analyses	201	(1)
7.1.2 A Methodological Caveat	202	(1)
7.2 Market Information About Smiles	203	(3)
7.2.1 Direct Static Information	203	(1)
7.2.2 Semi-Static Information	204	(1)
7.2.3 Direct Dynamic Information	204	(1)
7.2.4 Indirect Information	205	(1)
7.3 Equities	206	(16)
7.3.1 Basic Facts	206	(1)
7.3.2 Subtler Effects	206	(16)
7.4 Interest Rates	222	(5)
7.4.1 Basic Facts	222	(2)
7.4.2 Subtler Effects	224	(3)
7.5 FX Rates	227	(8)
7.5.1 Basic Facts	227	(1)
7.5.2 Subtler Effects	227	(8)
7.6 Conclusions	235	(2)
8 General Features of Smile-Modelling Approaches	237	(12)
8.1 Fully-Stochastic-Volatility Models	237	(2)
8.2 Local-Volatility (Restricted-Stochastic-Volatility) Models	239	(2)
8.3 Jump-Diffusion Models	241	(2)
8.3.1 Discrete Amplitude	241	(1)
8.3.2 Continuum of Jump Amplitudes	242	(1)
8.4 Variance-Gamma Models	243	(1)
8.5 Mixing Processes	243	(2)
8.5.1 A Pragmatic Approach to Mixing Models	244	(1)
8.6 Other Approaches	245	(1)
8.6.1 Tight Bounds with Known Quadratic Variation	245	(1)
8.6.2 Assigning Directly the Evolution of the Smile Surface	246	(1)
8.7 The Importance of the Quadratic Variation (Take 2)	246	(3)
9 The Input Data: Fitting an Exogenous Smile Surface	249	(44)
9.1 What is This Chapter About?	249	(1)
9.2 Analytic Expressions for Calls vs Process Specification	249	(1)
9.3 Direct Use of Market Prices: Pros and Cons	250	(1)
9.4 Statement of the Problem	251	(1)
9.5 Fitting Prices	252	(2)
9.6 Fitting Transformed Prices	254	(1)
9.7 Fitting the Implied Volatilities	255	(1)
9.7.1 The Problem with Fitting the Implied Volatilities	255	(1)
9.8 Fitting the Risk-Neutral Density Function - General	256	(3)
9.8.1 Does It Matter if the Price Density Is Not Smooth?	257	(1)
9.8.2 Using Prior Information (Minimum Entropy)	258	(1)
9.9 Fitting the Risk-Neutral Density Function: Mixture of Normals	259	(6)
9.9.1 Ensuring the Normalization and Forward Constraints	261	(3)
9.9.2 The Fitting Procedure	264	(1)
9.10 Numerical Results	265	(10)
9.10.1 Description of the Numerical Tests	265	(1)
9.10.2 Fitting to Theoretical Prices: Stochastic-Volatility Density	265	(3)
9.10.3 Fitting to Theoretical Prices: Variance-Gamma Density	268	(2)
9.10.4 Fitting to Theoretical Prices: Jump-Diffusion Density	270	(2)
9.10.5 Fitting to Market Prices	272	(3)
9.11 Is the Term ac/as Really a Delta?	275	(2)
9.12 Fitting the Risk-Neutral Density Function: The Generalized-Beta Approach	277	(16)
9.12.1 Derivation of Analytic Formulae	280	(7)
9.12.2 Results and Applications	287	(4)
9.12.3 What Does This Approach Offer?	291	(2)
10 Quadratic Variation and Smiles	293	(26)
10.1 Why This Approach Is Interesting	293	(1)
10.2 The BJN Framework for Bounding Option Prices	293	(1)
10.3 The BJN Approach - Theoretical Development	294	(6)
10.3.1 Assumptions and Definitions	294	(3)
10.3.2 Establishing Bounds	297	(1)
10.3.3 Recasting the Problem	298	(1)
10.3.4 Finding the Optimal Hedge	299	(1)
10.4 The BJN Approach: Numerical Implementation	300	(12)
10.4.1 Building a 'Traditional' Tree	301	(1)
10.4.2 Building a BJN Tree for a Deterministic Diffusion	301	(3)
10.4.3 Building a BJN Tree for a General Process	304	(3)
10.4.4 Computational Results	307	(2)
10.4.5 Creating Asymmetric Smiles	309	(2)
10.4.6 Summary of the Results	311	(1)
10.5 Discussion of the Results	312	(4)
10.5.1 Resolution of the Crouhy-Galai Paradox	312	(1)
10.5.2 The Difference Between Diffusions and Jump-Diffusion Processes: the Sample Quadratic Variation	312	(2)
10.5.3 How Can One Make the Approach More Realistic?	314	(1)
10.5.4 The Link with Stochastic-Volatility Models	314	(1)
10.5.5 The Link with Local-Volatility Models	315	(1)
10.5.6 The Link with Jump-Diffusion Models	315	(1)
10.6 Conclusions (or, Limitations of Quadratic Variation)	316	(3)
11 Local-Volatility Models: the Derman-and-Kani Approach	319	(26)
11.1 General Considerations on Stochastic-Volatility Models	319	(2)
11.2 Special Cases of Restricted-Stochastic-Volatility Models	321	(1)
11.3 The Dupire, Rubinstein and Derman-and-Kani Approaches	321	(1)
11.4 Green's Functions (Arrow-Debreu Prices) in the DK Construction	322	(4)
11.4.1 Definition and Main Properties of Arrow-Debreu Prices	322	(2)
11.4.2 Efficient Computation of Arrow-Debreu Prices	324	(2)
11.5 The Derman-and-Kani Tree Construction	326	(5)
11.5.1 Building the First Step	327	(2)
11.5.2 Adding Further Steps	329	(2)
11.6 Numerical Aspects of the Implementation of the DK Construction	331	(3)
11.6.1 Problem 1: Forward Price Greater Than 1/2 S(up) or Smaller Than S(down)	331	(1)
11.6.2 Problem 2: Local Volatility Greater Than 1/2 S(up) - S(down)	332	(1)
11.6.3 Problem 3: Arbitrariness of the Choice of the Strike	332	(2)
11.7 Implementation Results	334	(9)
11.7.1 Benchmarking 1: The No-Smile Case	334	(1)
11.7.2 Benchmarking 2: The Time-Dependent-Volatility Case	335	(1)
11.7.3 Benchmarking 3: Purely Strike-Dependent Implied Volatility	336	(1)
11.7.4 Benchmarking 4: Strike-and-Maturity-Dependent Implied Volatility	337	(1)
11.7.5 Conclusions	338	(5)
11.8 Estimating Instantaneous Volatilities from Prices as an Inverse Problem	343	(2)
12 Extracting the Local Volatility from Option Prices	345	(44)
12.1 Introduction	345	(2)
12.1.1 A Possible Regularization Strategy	346	(1)
12.1.2 Shortcomings	346	(1)
12.2 The Modelling Framework	347	(2)
12.3 A Computational Method	349	(6)
12.3.1 Backward Induction	349	(1)
12.3.2 Forward Equations	350	(2)
12.3.3 Why Are We Doing Things This Way?	352	(2)
12.3.4 Related Approaches	354	(1)
12.4 Computational Results	355	(2)
12.4.1 Are We Looking at the Same Problem?	356	(1)
12.5 The Link Between Implied and Local-Volatility Surfaces	357	(11)
12.5.1 Symmetric ('FX') Smiles	358	(3)
12.5.2 Asymmetric ('Equity') Smiles	361	(7)
12.5.3 Monotonic (Interest-Rate') Smile Surface	368	(1)
12.6 Gaining an Intuitive Understanding	368	(5)
12.6.1 Symmetric Smiles	369	(1)
12.6.2 Asymmetric Smiles: One-Sided Parabola	370	(2)
12.6.3 Asymmetric Smiles: Monotonically Decaying	372	(1)
12.7 What Local-Volatility Models Imply about Sticky and Floating Smiles	373	(2)
12.8 No-Arbitrage Conditions on the Current Implied Volatility Smile Surface	375	(10)
12.8.1 Constraints on the Implied Volatility Surface	375	(6)
12.8.2 Consequences for Local Volatilities	381	(4)
12.9 Empirical Performance	385	(1)
12.10 Appendix 1: Proof that a²call (St,K,T,t)/Ak² = φ(ST)|K	386	(3)
13 Stochastic-Volatility Processes	389	(50)
13.1 Plan of the Chapter	389	(1)
13.2 Portfolio Replication in the Presence of Stochastic Volatility	389	(12)
13.2.1 Attempting to Extend the Portfolio Replication Argument	389	(7)
13.2.2 The Market Price of Volatility Risk	396	(2)
13.2.3 Assessing the Financial Plausibility of λσ	398	(3)
13.3 Mean-Reverting Stochastic Volatility	401	(4)
13.3.1 The Ornstein-Uhlenbeck Process	402	(1)
13.3.2 The Functional Form Chosen in This Chapter	403	(1)
13.3.3 The High-Reversion-Speed, High-Volatility Regime	404	(1)
13.4 Qualitative Features of Stochastic-Volatility Smiles	405	(11)
13.4.1 The Smile as a Function of the Risk-Neutral Parameters	406	(10)
13.5 The Relation Between Future Smiles and Future Stock Price Levels	416	(2)
13.5.1 An Intuitive Explanation	417	(1)
13.6 Portfolio Replication in Practice: The Stochastic-Volatility Case	418	(9)
13.6.1 The Hedging Methodology	418	(2)
13.6.2 A Numerical Example	420	(7)
13.7 Actual Fitting to Market Data	427	(9)
13.8 Conclusions	436	(3)
14 Jump-Diffusion Processes	439	(72)
14.1 Introduction	439	(2)
14.2 The Financial Model: Smile Tale 2 Revisited	441	(3)
14.3 Hedging and Replicability in the Presence of Jumps: First Considerations	444	(5)
14.3.1 What Is Really Required To Complete the Market?	445	(4)
14.4 Analytic Description of Jump-Diffusions	449	(6)
14.4.1 The Stock Price Dynamics	449	(6)
14.5 Hedging with Jump-Diffusion Processes	455	(15)
14.5.1 Hedging with a Bond and the Underlying Only	455	(2)
14.5.2 Hedging with a Bond, a Second Option and the Underlying	457	(3)
14.5.3 The Case of a Single Possible Jump Amplitude	460	(5)
14.5.4 Moving to a Continuum of Jump Amplitudes	465	(1)
14.5.5 Determining the Function g Using the Implied Approach	465	(5)
14.5.6 Comparison with the Stochastic-Volatility Case (Again)	470	(1)
14.6 The Pricing Formula for Log-Normal Amplitude Ratios	470	(2)
14.7 The Pricing Formula in the Finite-Amplitude-Ratio Case	472	(13)
14.7.1 The Structure of the Pricing Formula for Discrete Jump Amplitude Ratios	474	(1)
14.7.2 Matching the Moments	475	(1)
14.7.3 Numerical Results	476	(9)
14.8 The Link Between the Price Density and the Smile Shape	485	(9)
14.8.1 A Qualitative Explanation	491	(3)
14.9 Qualitative Features of Jump-Diffusion Smiles	494	(6)
14.9.1 The Smile as a Function of the Risk-Neutral Parameters	494	(5)
14.9.2 Comparison with Stochastic-Volatility Smiles	499	(1)
14.10 Jump-Diffusion Processes and Market Completeness Revisited	500	(2)
14.11 Portfolio Replication in Practice: The Jump-Diffusion Case	502	(9)
14.11.1 A Numerical Example	503	(1)
14.11.2 Results	504	(5)
14.11.3 Conclusions	509	(2)
15 Variance-Gamma	511	(18)
15.1 Who Can Make Best Use of the Variance-Gamma Approach?	511	(2)
15.2 The Variance-Gamma Process	513	(9)
15.2.1 Definition	513	(1)
15.2.2 Properties of the Gamma Process	514	(1)
15.2.3 Properties of the Variance-Gamma Process	514	(3)
15.2.4 Motivation for Variance-Gamma Modelling	517	(1)
15.2.5 Properties of the Stock Process	518	(1)
15.2.6 Option Pricing	519	(3)
15.3 Statistical Properties of the Price Distribution	522	(1)
15.3.1 The Real-World (Statistical) Distribution	522	(1)
15.3.2 The Risk-Neutral Distribution	522	(1)
15.4 Features of the Smile	523	(4)
15.5 Conclusions	527	(2)
16 Displaced Diffusions and Generalizations	529	(34)
16.1 Introduction	529	(1)
16.2 Gaining Intuition	530	(1)
16.2.1 First Formulation	530	(1)
16.2.2 Second Formulation	531	(1)
16.3 Evolving the Underlying with Displaced Diffusions	531	(1)
16.4 Option Prices with Displaced Diffusions	532	(1)
16.5 Matching At-The-Money Prices with Displaced Diffusions	533	(20)
16.5.1 A First Approximation	533	(1)
16.5.2 Numerical Results with the Simple Approximation	534	(1)
16.5.3 Refining the Approximation	534	(10)
16.5.4 Numerical Results with the Refined Approximation	544	(9)
16.6 The Smile Produced by Displaced Diffusions	553	(7)
16.6.1 How Quickly is the Normal-Diffusion Limit Approached?	553	(7)
16.7 Extension to Other Processes	560	(3)
17 No-Arbitrage Restrictions on the Dynamics of Smile Surfaces	563	(38)
17.1 A Worked-Out Example: Pricing Continuous Double Barriers	564	(7)
17.1.1 Money For Nothing: A Degenerate Hedging Strategy for a Call Option	564	(2)
17.1.2 Static Replication of a Continuous Double Barrier	566	(5)
17.2 Analysis of the Cost of Unwinding	571	(4)
17.3 The Trader's Dream	575	(6)
17.4 Plan of the Remainder of the Chapter	581	(1)
17.5 Conditions of No-Arbitrage for the Stochastic Evolution of Future Smile Surfaces	582	(3)
17.5.1 Description of the Market	582	(2)
17.5.2 The Building Blocks	584	(1)
17.6 Deterministic Smile Surfaces	585	(8)
17.6.1 Equivalent Descriptions of a State of the World	585	(2)
17.6.2 Consequences of Deterministic Smile Surfaces	587	(1)
17.6.3 Kolmogorov-Compatible Deterministic Smile Surfaces	588	(1)
17.6.4 Conditions for the Uniqueness of Kolmogorov-Compatible Densities	589	(2)
17.6.5 Floating Smiles	591	(2)
17.7 Stochastic Smiles	593	(4)
17.7.1 Stochastic Floating Smiles	594	(1)
17.7.2 Introducing Equivalent Deterministic Smile Surfaces	595	(1)
17.7.3 Implications of the Existence of an Equivalent Deterministic Smile Surface	596	(1)
17.7.4 Extension to Displaced Diffusions	597	(1)
17.8 The Strength of the Assumptions	597	(1)
17.9 Limitations and Conclusions	598	(3)
III Interest Rates - Deterministic Volatilities	601	(100)
18 Mean Reversion in Interest-Rate Models	603	(22)
18.1 Introduction and Plan of the Chapter	603	(1)
18.2 Why Mean Reversion Matters in the Case of Interest-Rate Models	604	(4)
18.2.1 What Does This Mean for Forward-Rate Volatilities?	606	(2)
18.3 A Common Fallacy Regarding Mean Reversion	608	(2)
18.3.1 The Grain of Truth in the Fallacy	609	(1)
18.4 The BDT Mean-Reversion Paradox	610	(2)
18.5 The Unconditional Variance of the Short Rate in BDT - the Discrete Case	612	(4)
18.6 The Unconditional Variance of the Short Rate in BDT-the Continuous-Time Equivalent	616	(1)
18.7 Mean Reversion in Short-Rate Lattices: Recombining vs Bushy Trees	617	(3)
18.8 Extension to More General Interest-Rate Models	620	(2)
18.9 Appendix I: Evaluation of the Variance of the Logarithm of the Instantaneous Short Rate	622	(3)
19 Volatility and Correlation in the LIBOR Market Model	625	(14)
19.1 Introduction	625	(1)
19.2 Specifying the Forward-Rate Dynamics in the LIBOR Market Model	626	(5)
19.2.1 First Formulation: Each Forward Rate in Isolation	626	(2)
19.2.2 Second Formulation: The Covariance Matrix	628	(2)
19.2.3 Third Formulation: Separating the Correlation from the Volatility Term	630	(1)
19.3 Link with the Principal Component Analysis	631	(1)
19.4 Worked-Out Example 1: Caplets and a Two-Period Swaption	632	(3)
19.5 Worked-Out Example 2: Serial Options	635	(1)
19.6 Plan of the Work Ahead	636	(3)
20 Calibration Strategies for the LIBOR Market Model	639	(28)
20.1 Plan of the Chapter	639	(1)
20.2 The Setting	639	(4)
20.2.1 A Geometric Construction: The Two-Factor Case	640	(2)
20.2.2 Generalization to Many Factors	642	(1)
20.2.3 Re-Introducing the Covariance Matrix	642	(1)
20.3 Fitting an Exogenous Correlation Function	643	(3)
20.4 Numerical Results	646	(13)
20.4.1 Fitting the Correlation Surface with a Three-Factor Model	646	(4)
20.4.2 Fitting the Correlation Surface with a Four-Factor Model	650	(4)
20.4.3 Fitting Portions of the Target Correlation Matrix	654	(5)
20.5 Analytic Expressions to Link Swaption and Caplet Volatilities	659	(3)
20.5.1 What Are We Trying to Achieve?	659	(1)
20.5.2 The Set-Up	659	(3)
20.6 Optimal Calibration to Co-Terminal Swaptions	662	(5)
20.6.1 The Strategy	662	(5)
21 Specifying the Instantaneous Volatility of Forward Rates	667	(20)
21.1 Introduction and Motivation	667	(1)
21.2 The Link between Instantaneous Volatilities and the Future Term Structure of Volatilities	668	(3)
21.3 A Functional Form for the Instantaneous Volatility Function	671	(2)
21.3.1 Financial Justification for a Humped Volatility	672	(1)
21.4 Ensuring Correct Caplet Pricing	673	(4)
21.5 Fitting the Instantaneous Volatility Function: Imposing Time Homogeneity of the Term Structure of Volatilities	677	(2)
21.6 Is a Time-Homogeneous Solution Always Possible?	679	(1)
21.7 Fitting the Instantaneous Volatility Function: The Information from the Swaption Market	680	(6)
21.8 Conclusions	686	(1)
22 Specifying the Instantaneous Correlation Among Forward Rates	687	(14)
22.1 Why Is Estimating Correlation So Difficult?	687	(1)
22.2 What Shape Should We Expect for the Correlation Surface'?	688	(1)
22.3 Features of the Simple Exponential Correlation Function	689	(2)
22.4 Features of the Modified Exponential Correlation Function	691	(3)
22.5 Features of the Square-Root Exponential Correlation Function	694	(3)
22.6 Further Comparisons of Correlation Models	697	(1)
22.7 Features of the Schonmakers-Coffey Approach	697	(1)
22.8 Does It Make a Difference (and When)?	698	(3)
IV Interest Rates - Smiles	701	(104)
23 How to Model Interest-Rate Smiles	703	(26)
23.1 What Do We Want to Capture? A Hierarchy of Smile-Producing Mechanisms	703	(1)
23.2 Are Log-Normal Co-Ordinates the Most Appropriate?	704	(2)
23.2.1 Defining Appropriate Co-ordinates	705	(1)
23.3 Description of the Market Data	706	(9)
23.4 Empirical Study I: Transforming the Log-Normal Co-ordinates	715	(3)
23.5 The Computational Experiments	718	(1)
23.6 The Computational Results	719	(2)
23.7 Empirical Study II: The Log-Linear Exponent	721	(4)
23.8 Combining the Theoretical and Experimental Results	725	(1)
23.9 Where Do We Go From Here?	725	(4)
24 (CEV) Processes in the Context of the LMM	729	(22)
24.1 Introduction and Financial Motivation	729	(1)
24.2 Analytical Characterization of CEV Processes	730	(2)
24.3 Financial Desirability of CEV Processes	732	(2)
24.4 Numerical Problems with CEV Processes	734	(1)
24.5 Approximate Numerical Solutions	735	(12)
24.5.1 Approximate Solutions: Mapping to Displaced Diffusions	735	(1)
24.5.2 Approximate Solutions: Transformation of Variables	735	(1)
24.5.3 Approximate Solutions: the Predictor-Corrector Method	736	(11)
24.6 Problems with the Predictor-Corrector Approximation for the LMM	747	(4)
25 Stochastic-Volatility Extensions of the LMM	751	(14)
25.1 Plan of the Chapter	751	(2)
25.2 What is the Dog and What is the Tail?	753	(1)
25.3 Displaced Diffusion vs CEV	754	(1)
25.4 The Approach	754	(2)
25.5 Implementing and Calibrating the Stochastic-Volatility LMM	756	(8)
25.5.1 Evolving the Forward Rates	759	(1)
25.5.2 Calibrating to Caplet Prices	759	(5)
25.6 Suggestions and Plan of the Work Ahead	764	(1)
26 The Dynamics of the Swaption Matrix	765	(18)
26.1 Plan of the Chapter	765	(1)
26.2 Assessing the Quality of a Model	766	(1)
26.3 The Empirical Analysis	767	(1)
26.3.1 Description of the Data	767	(1)
26.3.2 Results	768	(8)
26.4 Extracting the Model-Implied Principal Components	776	(5)
26.4.1 Results	778	(3)
26.5 Discussion, Conclusions and Suggestions for Future Work	781	(2)
27 Stochastic-Volatility Extension of the LMM: Two-Regime Instantaneous Volatility	783	(22)
27.1 The Relevance of the Proposed Approach	783	(1)
27.2 The Proposed Extension	783	(2)
27.3 An Aside: Some Simple Properties of Markov Chains	785	(3)
27.3.1 The Case of Two-State Markov Chains	787	(1)
27.4 Empirical Tests	788	(10)
27.4.1 Description of the Test Methodology	788	(2)
27.4.2 Results	790	(8)
27.5 How Important Is the Two-Regime Feature?	798	(3)
27.6 Conclusions	801	(4)
Bibliography	805	(8)
Index	813