Modern Pricing of Interest-Rate Derivatives: The LIBOR Market Model and Beyond by Riccardo RebonatoModern Pricing of Interest-Rate Derivatives: The LIBOR Market Model and Beyond by Riccardo Rebonato

Modern Pricing of Interest-Rate Derivatives: The LIBOR Market Model and Beyond

byRiccardo Rebonato

Hardcover | November 24, 2002

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In recent years, interest-rate modeling has developed rapidly in terms of both practice and theory. The academic and practitioners' communities, however, have not always communicated as productively as would have been desirable. As a result, their research programs have often developed with little constructive interference. In this book, Riccardo Rebonato draws on his academic and professional experience, straddling both sides of the divide to bring together and build on what theory and trading have to offer.

Rebonato begins by presenting the conceptual foundations for the application of the LIBOR market model to the pricing of interest-rate derivatives. Next he treats in great detail the calibration of this model to market prices, asking how possible and advisable it is to enforce a simultaneous fitting to several market observables. He does so with an eye not only to mathematical feasibility but also to financial justification, while devoting special scrutiny to the implications of market incompleteness.

Much of the book concerns an original extension of the LIBOR market model, devised to account for implied volatility smiles. This is done by introducing a stochastic-volatility, displaced-diffusion version of the model. The emphasis again is on the financial justification and on the computational feasibility of the proposed solution to the smile problem. This book is must reading for quantitative researchers in financial houses, sophisticated practitioners in the derivatives area, and students of finance.

Riccardo Rebonato is Head of Group Market Risk and Head of the Quantitative Research Centre (QUARC) for the Royal Bank of Scotland Group. He is also a Visiting Lecturer at Oxford University's Mathematical Institute, where he teaches for the MSC/Diploma in Mathematical Finance. His books include Interest-Rate Option Models and Volatili...
Title:Modern Pricing of Interest-Rate Derivatives: The LIBOR Market Model and BeyondFormat:HardcoverDimensions:488 pagesPublished:November 24, 2002Publisher:Princeton University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0691089736

ISBN - 13:9780691089737


Table of Contents

Introduction xi

Acknowledgements xvii

I. The Structure of the LIBOR Market Model 1

1. Putting the Modern Pricing Approach in Perspective 3

1.1. Historical Developments 3

1.2. Some Important Remarks 21

2. The Mathematical and Financial Set-up 25

2.1. The Modelling Framework 25

2.2. Definition and Valuation of the Underlying Plain-Vanilla Instruments 28

2.3. The Mathematical and Financial Description of the Securities Market 40

3. Describing the Dynamics of Forward Rates 57

3.1. A Working Framework for the Modern Pricing Approach 57

3.2. Equivalent Descriptions of the Dynamics of Forward Rates 65

3.3. Generalization of the Approach 79

3.4. The Swap-Rate-Based LIBOR Market Model 83

4. Characterizing and Valuing Complex LIBOR Products 85

4.1. The Types of Product That Can be Handled Using the LIBOR Market Model 85

4.2. Case Study: Pricing in a Three-Forward-Rate, Two-Factor World 96

4.3. Overview of the Results So Far 107

5. Determining the No-Arbitrage Drifts of Forward Rates 111

5.1. General Derivation of the Drift Terms 112

5.2. Expressing the No-Arbitrage Conditions in Terms of Market-Related Quantities 118

5.3. Approximations of the Drift Terms 123

5.4. Conclusions 131

II. The Inputs to the General Framework 133

6. Instantaneous Volatilities 135

6.1. Introduction and Motivation 135

6.2. Instantaneous Volatility Functions: General Results 141

6.3. Functional Forms for the Instantaneous Volatility Function - Financial Implications 153

6.4. Analysis of Specific Functional Forms for the Instantaneous Volatility Functions 167

6.5. Appendix I - Why Specification (6.11c) Fails to Satisfy Joint Conditions 171

6.6. Appendix II - Indefinite Integral of the Instantaneous Covariance 171

7. Specifying the Instantaneous Correlation Function 173

7.1. General Considerations 173

7.2. Empirical Data and Financial Plausibility 180

7.3. Intrinsic Limitations of Low-Dimensionality Approaches 185

7.4. Proposed Functional Forms for the Instantaneous Correlation Function 189

7.5. Conditions for the Occurrence of Exponential Correlation Surfaces 196

7.6. A Semi-Parametric Specification of the Correlation Surface 204

III Calibration of the LIBOR Market Model 209

8. Fitting the Instantaneous Volatility Functions 211

8.1. General Calibration Philosophy and Plan of Part III 211

8.2. A First Approach to Fitting the Caplet Market: Imposing Time-Homogeneity 214

8.3. A Second Approach to Fitting the Caplet Market: Using Information from the Swaption Matrix 218

8.4. A Third Approach to Fitting the Caplet Market: Assigning a Future Term Structure of Volatilities 226

8.5. Results 231

8.6. Conclusions 248

9. Simultaneous Calibration to Market Caplet Prices and to an Exogenous Correlation Matrix 249

9.1. Introduction and Motivation 249

9.2. An Optimal Procedure to Recover an Exogenous Target Correlation Matrix 254

9.3. Results and Discussion 260

9.4. Conclusions 274

10 Calibrating a Forward-Rate-Based LIBOR Market Model to Swaption Prices 276

10.1. The General Context 276

10.2. The Need for a Joint Description of the Forward-and Swap-Rate Dynamics 280

10.3. Approximating the Swap-Rate Instantaneous Volatility 294

10.4. Computational Results on European Swaptions 306

10.5. Calibration to Co-Terminal European Swaption Prices 312

10.6. An Application: Using an FRA-Based LIBOR Market Model for Bermudan Swaptions 318

10.7. Quality of the Numerical Approximation in Realistic Market Cases 326

IV. Beyond the Standard Approach: Accounting for Smiles 331

11. Extending the Standard Approach - I: CEV and Displaced Diffusion 333

11.1. Practical and Conceptual Implications of Non-Flat Volatility Smiles 333

11.2. Calculating Deltas and Other Risk Derivatives in the Presence of Smiles 342

11.3. Accounting for Monotonically Decreasing Smiles 349

11.4. Time-Homogeneity in the Context of Displaced Diffusions 363

12. Extending the Standard Approach - II: Stochastic Instantaneous Volatilities 367

12.1. Introduction and Motivation 367

12.2. The Modelling Framework 372

12.3. Numerical Techniques 382

12.4. Numerical Results 397

12.5. Conclusions and Suggestions for Future Work 413

13. A Joint Empirical and Theoretical Analysis of the Stochastic-Volatility LIBOR Market Model 415

13.1. Motivation and Plan of the Chapter 415

13.2. The Empirical Analysis 420

13.3. The Computer Experiments 437

13.4. Conclusions and Suggestions for Future Work 442

Bibliography 445

Index 453

Editorial Reviews

"There are many books that get bogged down in mathematical technicalities before they get to the point and are therefore of little use to practitioners. Rebonato takes the opposite approach: he gets to the point. People working in the mathematical finance industry will love this book."-Jeff Dewynne, Oxford University