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

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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.

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Editorial Reviews

Quantitative Finance
Rebonato's writing style is probably the most elegant I have ever seen in a quantitative finance book. His ideas are conveyed in a brief and clear manner. . . . I thoroughly enjoyed this book since it allowed me to discover a whole new world in a fast and painless fashion. I would therefore recommend it to everyone who has any interest in the fascinating universe of fixed-income derivatives.
— Alireza Javaheri
Quantitative Finance - Alireza Javaheri
Rebonato's writing style is probably the most elegant I have ever seen in a quantitative finance book. His ideas are conveyed in a brief and clear manner. . . . I thoroughly enjoyed this book since it allowed me to discover a whole new world in a fast and painless fashion. I would therefore recommend it to everyone who has any interest in the fascinating universe of fixed-income derivatives.
From the Publisher
"Rebonato's writing style is probably the most elegant I have ever seen in a quantitative finance book. His ideas are conveyed in a brief and clear manner. . . . I thoroughly enjoyed this book since it allowed me to discover a whole new world in a fast and painless fashion. I would therefore recommend it to everyone who has any interest in the fascinating universe of fixed-income derivatives."—Alireza Javaheri, Quantitative Finance
Quantitative Finance

Rebonato's writing style is probably the most elegant I have ever seen in a quantitative finance book. His ideas are conveyed in a brief and clear manner. . . . I thoroughly enjoyed this book since it allowed me to discover a whole new world in a fast and painless fashion. I would therefore recommend it to everyone who has any interest in the fascinating universe of fixed-income derivatives.
— Alireza Javaheri
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Product Details

  • ISBN-13: 9780691089737
  • Publisher: Princeton University Press
  • Publication date: 11/4/2002
  • Edition description: New Edition
  • Pages: 480
  • Product dimensions: 6.20 (w) x 9.40 (h) x 1.50 (d)

Meet the Author

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 "Volatility and Correlation in Option Pricing".
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Read an Excerpt

Modern Pricing of Interest-Rate Derivatives

The LIBOR Market Model and Beyond
By Riccardo Rebonato


Copyright © 2002
All right reserved.

ISBN: 0691089736

Chapter One


1.1 Historical Developments

1.1.1 Introduction

The set of techniques to price interest-rate derivatives that stemmed from the original work of Heath, Jarrow and Morton (HJM) in the late 1980s (HJM 1989) are referred to in this book as the 'modern' or the 'LIBOR-market-model' approach. At a superficial glance, the differences between the various 'incarnations' of the approach might appear greater than what they have in common. The state variables could be instantaneous or discretely compounded rates; they could be swap rates or forward rates; they might be normally or log-normally (or otherwise) distributed; the associated numeraire could be a zero-coupon bond, a swap annuity or the money-market account; and so on. Despite these non-trivial differences, these approaches share one essential common feature: the recognition that the no-arbitrage evolution of the state variables (however chosen) can be expressed purely as a function of the volatilities of, and of the correlations among, the state variables themselves. Different choices of numeraires will give rise to different combinations for these covariance elements, but this fundamental result, which goes back to the original insight of HJM, is shared by all the approaches that will be dealt with in this book. This result and its implications are sufficiently fundamental and far-reaching to justify a self-contained and unified treatment.

Given the various 'versions', 'implementations' and choices of numeraires, no general agreement exists in the financial community on how to call this set of approaches: the terms 'BGM (Brace, Gatarek and Musiela) model' and 'Jamshidian approach' are often used, but 'pricing in the forward measure', the 'LIBOR market model' and other terms are also frequently encountered. Some purists insist on calling the approach simply the 'HJM model'. The difficulty in establishing intellectual priority is compounded by the fact that many of the key results were first obtained (and used) by practitioners but, for obvious reasons, not published in the academic press. I have therefore avoided any identification of the approach with names of academics or market professionals, and used the more neutral terms 'LIBOR market model' or 'modern pricing approach' - much as I am sure the latter may read rather quaint in a few years' time.

This introductory chapter is meant to provide a brief map of the development of interest-rate derivative pricing from its earliest (modern) days to the present. I have chosen to present such an introduction not only for its intrinsic historical interest, but also because it illustrates rather clearly that an uneven combination of market events, 'right choices made for the wrong reasons', computational expediency and sound judgement have conspired to produce the market standard that the later, more sophisticated, models have striven to recover. In other words, the modern approach is, justifiably, so loved by practitioners because of its ability to price exotic products while at the same time recovering exactly the prices of the relevant plain-vanilla options (caplets or European swaptions). I shall explain below how the market consensus has crystallized around the Black framework, partly for sound financial reasons, but partly also by historical accident. If this analysis is correct, there is nothing 'inevitable' about the current market standard, and it is quite possible that the target the modern approach has been trying to hit might in the near future turn out to be a rather rapidly moving one.

Indeed, this phenomenon is already becoming apparent: as discussed in Part IV of this book, in the last few years the prices of plain-vanilla options have been able to be strait-jacketed into their log-normal-rate Black framework only by increasingly complex ad hoc adjustments.1 As a consequence, just when the pricing of exotic products had finally been successfully tuned onto the log-normal-rate wavelength, the prices of the underlying vanilla instruments have ceased to inhabit the same (Black) world. The brief account of the developments that brought about this state of affairs is presented below, and should give a clear indication of the fact that the 'modern' approach is virtually certain to be anything but the last step in interest-rate derivative pricing. The reader keen to delve into the quantitative aspects of the pricing can safely skip these pages. She would miss, however, not only a good story, but also some perspective useful in appreciating what aspects of today's market consensus are more likely to be challenged and superseded tomorrow.

1.1.2 The Early Days

The relatively brief history of the evolution of the pricing of interest-rate derivatives can be subdivided into four distinct periods. The very first one corresponds to the use of the Black and Scholes (1973) (BS), Black (1976) and Merton (1973) approaches. In all these cases, the same distributional assumption (namely, the log-normal) was made for the underlying variable, and the resulting expiry-time distribution was integrated over the terminal payoff of a European option. For all of the three above-mentioned models, the solutions have a very (and deceptively) similar appearance, with the integration over the appropriate log-normal probability densities giving rise to the familiar cumulative-normal-distribution terms. The quantities that were typically assumed to be log-normal were bond prices (spot or forward), forward rates, forward swap rates, or bond yields. As for the products priced using these modelling approaches, they belonged to two rather distinct markets and yield curves: the Treasury/repo world, on the one hand, was the relevant environment for the pricing of plain-vanilla bond options, either directly, or embedded in once-callable or extendable-maturity structures; and the LIBOR environment, on the other, which provided caps and European swaptions.

The most straightforward approach (i.e., the use of the Black and Scholes formula with the spot bond price as the underlying) was popular, but it also came under early criticism because of the so-called pull-to-par phenomenon: in its original form the Black and Scholes formula requires a constant percentage volatility of the underlying. For a coupon or a discount bond, however, the volatility is certainly not constant (since the price has to converge to par at maturity). This fact was considered to be cause for little worry if the expiry of the bond option was much shorter than the maturity of the underlying bond (e.g., for a few weeks' or months' option on a, say, 10-year bond); but it would create discomfort when the times to expiry and maturity were comparable.

The 'easy-fix' solution was to consider a non-traded quantity (the bond yield) as the underlying log-normal variable. The main advantage of this approach was that a yield does not exhibit a deterministic pull to par. The most obvious drawback, on the other hand, was that the yield is not a traded asset, and, therefore, the Black and Scholes reasoning behind the self-financing dynamic trading strategy that reproduces the final payoff of the option could not be easily adapted. Despite being theoretically not justifiable, the approach was for a period widely used because it allowed the trader to think in terms of a volatility (the volatility of the yield) that was more independent of the maturity of the underlying instrument than the volatility of the spot price. With hindsight, and much to the relief of the academic community, this route was to prove a blind alley, and very little more will be said in this book about it.2 Academics, of course and correctly, 'hated' the log-normal yield approach: not only did it make use as its driving variable of a quantity (the yield itself) of poor theoretical standing (see, e.g., Schaefer 1977), but it was close to impossible to produce a satisfactory financial justification in terms of risk-neutral valuation for its use in a Black-like formula. In all likelihood, however, the trading community abandoned the log-normal yield model for reasons other than its lack of sound theoretical standing. As discussed below, in fact, the cap and swaption pricing formulas, which used log-normal forward and swap rates, were at the very same time gaining general market acceptance. Prima facie, these formulas appear deceptively similar to the log-normal yield approach. It so happens that, unlike the log-normal yield approach, these latter formulas can be justified theoretically. In the early days, however, this was rarely recognized, and papers showing the 'correctness' of the Black swaption formula were still appearing as late as the mid-1990s (see Neuberger 1990, Gustavsson 1997). Traders, although with a somewhat guilty conscience, were nonetheless still using the Black formula for caplets and European swaptions well before its theoretical justification had become common knowledge.

The reasons for the abandonment of the log-normal yield model must therefore have been rooted in causes other than its lack of theoretical justifiability. This constitutes more than a piece of historical curiosity: the fact that traders carried on using the Black formula for caplets and swaptions despite its then-perceived lack of theoretical standing turned the approach into a market standard. The subsequent realization that this market standard could be put on solid theoretical ground then provided the impetus behind the development of more general approaches (such as the LIBOR market model) capable of pricing complex interest-rate derivatives consistently with the (Black) caplet market. Therefore, without the somewhat fortuitous choices made during the establishment of a market standard, the current 'modern' pricing approach would probably not exist in its present form.

Going back to the pull-to-par problem, its 'correct' solution was, of course, to use the Black - rather than the Black and Scholes - formula, with the forward (as opposed to spot) price and the volatility of the forward as inputs. This route was often overlooked, and for different reasons, both by naive and by sophisticated market players. The naive traders simply did not appreciate the subtle, but fundamental, difference between the Black and the Black and Scholes formulas and the volatilities used as input for both, and believed the pull-to-par phenomenon to be relevant to the Black formula as well. The sophisticated traders understood that the Black formula, with the appropriate inputs, is perfectly correct and justifiable, and that no pull-to-par effect applies to a forward bond price (which remains of the same residual maturity throughout its life). They also realized, however, that something important was still missing. Much as the Black formula can give a perfectly correct answer for a series of options considered independently of each other, there is no way of telling whether these options inhabit a plausible, or even logically consistent, universe. In reality, despite the fact that different forward bond prices correspond to different assets, we also know that these prices are significantly correlated, but there is no mechanism within the Black formula to incorporate views about this joint dynamics. To make this important point clearer, let us assume that a trader were to give two different forward bond price volatilities as inputs for the Black formula for two options with different expiries on forward bonds with the same residual maturities. By so doing the trader is implicitly saying 'something' about the volatilities of the forward bonds spanning the period between the first and the second expiry, and the first and second maturity. Notice, however, the vagueness of the term 'something'. The relationship between the various volatilities is not strong enough to be of deterministic nature: given the volatilities for two same-residual-maturity, different-expiry forward bonds, the volatility of the forward bond spanning, say, the two expiries is by no means uniquely determined. At the same time, the relationship between the various volatilities is not weak enough for the various underlying 'assets' to be considered completely independent.

Sure enough, if an exogenous model existed, capable of providing self-consistent inputs (i.e., volatilities) to the Black formula, then all problems of consistency would be solved. Indeed, I have shown elsewhere (Rebonato 1998) that the Black formula can be rigorously extended to a surprising variety of path-dependent and compound-option cases (at least, as long as one is prepared to work with higher-and-higher-dimension cumulative normal distributions). However, paraphrasing what I wrote in a slightly different context (Rebonato 1998):

. . . what is needed is some strong structure to be imposed on the co-movements of the financial quantities of interest; . . . this structure can be provided by specifying the dynamics of a small number of variables Once the process for all these driving factors has been chosen, the variances of and correlations among all the financial observables can be obtained, analytically or numerically as the case might be, as a by-product of the model itself. The implied co-dynamics of these quantities might turn out to be simplified to the point of becoming simplistic, but, at least, the pricing of different options can be undertaken on a consistent basis . . .

These reasons for the early dissatisfaction with the Black approach are of more than historical interest because, paradoxically, the modern approach to pricing derivatives can suffer from the same over-abundance of degrees of freedom. I will argue that the need to impose the strong structure on the co-movements of the yield curve mentioned above is absolutely central to the modern implementation of the pricing approaches derived from the HJM family. But more about this later.

In parallel with the Treasury/repo market, the LIBOR-related market was also providing interesting pricing challenges. The demand for LIBOR-based derivative products was coming on the one hand from liability managers (borrowers) seeking interest-rate insurance via the purchase of caps, and on the other from issuers and investors seeking better returns or funding rates, respectively, via callable or puttable bonds. The coupons from the latter would then be swapped with an investment firm, who would also receive the optionality (typically a swaption) embedded in the structure. The mechanics of, and financial rationale for, these trades is an interesting topic in itself: the story of the search for funding 'advantage' that drove the evolution of financial instruments from the early once-callable coupon bonds to the 30-year multi-callable zero-coupon swaptions of late 1997/early 1998 deserves to be told, but elsewhere. For the present discussion it is sufficient to point out two salient features.


Excerpted from Modern Pricing of Interest-Rate Derivatives by Riccardo Rebonato Copyright © 2002 by
Excerpted by permission. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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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

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"Dr. Rebonato has blended technical mastery with many years of practical experience to produce what should become the standard handbook for anyone wanting to value, hedge or control the risks of interest rate derivatives."—Ian Cooper, Professor of Finance, London Business School

"This eagerly awaited book fills an important need. It covers the pressing but technically difficult issues of how to implement 'market' models of the term structure for the purposes of valuing and hedging interest-rate-sensitive derivatives. Dr. Rebonato is a leading expert in the field. His treatment is exceptionally lucid as well as authoritative."—Stewart Hodges, University of Warwick

"Riccardo Rebonato succeeds admirably in combining an accessible exposition of the foundations of the LIBOR market model framework with extensive guidance on the calibration and implementation of the models in practice. The book's many insights into the dynamics of fixed income markets and models should provide industry professionals with valuable tools and offer academics a rare glimpse of the market as viewed by a practitioner-theorist, all presented in the author's elegant and lively style."—Paul Glasserman, Columbia University

"This book is a significant contribution to the field. It offers plenty of empirical work and case studies illustrating the application of the models each step of the way. Unlike other treatments, it emphasizes the market rationale for modeling choices, and is not driven by purely mathematical considerations. Reference is continually made to market features, the behaviour of instruments, and empirical features, with all of this backed up by the author'sconsiderable experience."—Nick Webber, University of Warwick

"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

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