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…mehr
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.
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
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
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