Modern Pricing of Interest-Rate Derivatives

by Rebonato, Riccardo

• ISBN13: 9780691089737
• ISBN10: 0691089736
• Format: Hardcover
• Copyright: 2002-11-04
• Publisher: Princeton Univ Pr

Summary

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.

Table of Contents

Introduction	p. xi
Acknowledgements	p. xvii
The Structure of the LIBOR Market Model	p. 1
Putting the Modern Pricing Approach in Perspective	p. 3
Historical Developments	p. 3
Some Important Remarks	p. 21
The Mathematical and Financial Set-up	p. 25
The Modelling Framework	p. 25
Definition and Valuation of the Underlying Plain-Vanilla Instruments	p. 28
The Mathematical and Financial Description of the Securities Market	p. 40
Describing the Dynamics of Forward Rates	p. 57
A Working Framework for the Modern Pricing Approach	p. 57
Equivalent Descriptions of the Dynamics of Forward Rates	p. 65
Generalization of the Approach	p. 79
The Swap-Rate-Based LIBOR Market Model	p. 83
Characterizing and Valuing Complex LIBOR Products	p. 85
The Types of Product That Can be Handled Using the LIBOR Market Model	p. 85
Case Study: Pricing in a Three-Forward-Rate, Two-Factor World	p. 96
Overview of the Results So Far	p. 107
Determining the No-Arbitrage Drifts of Forward Rates	p. 111
General Derivation of the Drift Terms	p. 112
Expressing the No-Arbitrage Conditions in Terms of Market-Related Quantities	p. 118
Approximations of the Drift Terms	p. 123
Conclusions 131
The Inputs to the General Framework	p. 133
Instantaneous Volatilities	p. 135
Introduction and Motivation	p. 135
Instantaneous Volatility Functions: General Results	p. 141
Functional Forms for the Instantaneous Volatility Function - Financial Implications	p. 153
Analysis of Specific Functional Forms for the Instantaneous Volatility Functions	p. 167
Appendix I - Why Specification (6.11c) Fails to Satisfy Joint Conditions	p. 171
Appendix II - Indefinite Integral of the Instantaneous Covariance	p. 171
Specifying the Instantaneous Correlation Function	p. 173
General Considerations	p. 173
Empirical Data and Financial Plausibility	p. 180
Intrinsic Limitations of Low-Dimensionality Approaches	p. 185
Proposed Functional Forms for the Instantaneous Correlation Function	p. 189
Conditions for the Occurrence of Exponential Correlation Surfaces	p. 196
A Semi-Parametric Specification of the Correlation Surface 204
Calibration of the LIBOR Market Model	p. 209
Fitting the Instantaneous Volatility Functions	p. 211
General Calibration Philosophy and Plan of Part III	p. 211
A First Approach to Fitting the Caplet Market: Imposing Time-Homogeneity	p. 214
A Second Approach to Fitting the Caplet Market: Using Information from the Swaption Matrix	p. 218
A Third Approach to Fitting the Caplet Market: Assigning a Future Term Structure of Volatilities	p. 226
Results	p. 231
Conclusions	p. 248
Simultaneous Calibration to Market Caplet Prices and to an Exogenous Correlation Matrix	p. 249
Introduction and Motivation	p. 249
An Optimal Procedure to Recover an Exogenous Target Correlation Matrix	p. 254
Results and Discussion	p. 260
Conclusions	p. 274
Calibrating a Forward-Rate-Based LIBOR Market Model to Swaption Prices	p. 276
The General Context	p. 276
The Need for a Joint Description of the Forward-and Swap-Rate Dynamics	p. 280
Approximating the Swap-Rate Instantaneous Volatility	p. 294
Computational Results on European Swaptions	p. 306
Calibration to Co-Terminal European Swaption Prices	p. 312
An Application: Using an FRA-Based LIBOR Market Model for Bermudan Swaptions	p. 318
Quality of the Numerical Approximation in Realistic Market Cases	p. 326
Beyond the Standard Approach: Accounting for Smiles	p. 331
Extending the Standard Approach - I: CEV and Displaced Diffusion	p. 333
Practical and Conceptual Implications of Non-Flat Volatility Smiles	p. 333
Calculating Deltas and Other Risk Derivatives in the Presence of Smiles	p. 342
Accounting for Monotonically Decreasing Smiles	p. 349
Time-Homogeneity in the Context of Di
Table of Contents provided by Publisher. All Rights Reserved.