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

Riccardo Rebonato
Pages: 480
https://www.jstor.org/stable/j.ctt7rpkk

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. Introduction
(pp. xi-xvi)

The aim of this book is to present my views as to the most satisfactory approach to pricing a wide class of interest-rate derivatives. This approach falls squarely within the framework of the LIBOR market model. However, many competing versions, and even more modes of implementation, exist. I have not attempted to present a comprehensive, unbiassed review of all these possible approaches. Rather, I have chosen the particular version and the overall calibration philosophy that I have found most conceptually satisfying and practically useful. I have not been shy to express my opinions, but having strong views on a subject...

4. Acknowledgements
(pp. xvii-xviii)
5. I The Structure of the LIBOR Market Model
• 1 Putting the Modern Pricing Approach in Perspective
(pp. 3-24)

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-marketmodel’ 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...

• 2 The Mathematical and Financial Set-up
(pp. 25-56)

The branch of finance called ‘asset pricing’ deals, not surprisingly, with the problem of assigning prices to assets. There are two main strands, traditionally referred to as ‘absolute’ and ‘relative’ pricing. With the first line of inquiry one seeks to explain prices in terms of fundamental macroeconomic variables; investors’ preferences and utility functions feature prominently in this type of approach. General-equilibrium models (such as the Cox, Ingersoll and Ross (Cox et al. 1985) or the Longstaff and Schwartz (1992) interest-rate models mentioned in Chapter 1) are typical examples of this approach. With relative pricing, on the other hand, one takes...

• 3 Describing the Dynamics of Forward Rates
(pp. 57-84)

The LIBOR market model allows the pricing of LIBOR products with discrete price-sensitive events by prescribing the continuous-time evolution of those forward rates that, together with one spot rate, define the relevant points of a discount curve. In this chapter I show that this is accomplished by requiring that the evolution of the forward rates should be a diffusion with deterministic volatility plus a drift term. I then show in Chapter 4 that the nature of the payoffs of the typical LIBOR products is such that they can be priced by moving the yield curve over ‘long’ time intervals. (‘Long’...

• 4 Characterizing and Valuing Complex LIBOR Products
(pp. 85-110)

From the discussion so far, the marginal and total covariance matrices, C and TOTC, can be expected to play a central role in the evolution of the forward rates that describe a yield curve. The next natural step is establishing the link between these matrices and the types of complex derivatives product the LIBOR market model has been specifically constructed to price. This analysis is important both from a conceptual and from a practical point of view, because it will inform the numerical techniques necessary to tackle the problem in practice. In fact, I will show that, for a very...

• 5 Determining the No-Arbitrage Drifts of Forward Rates
(pp. 111-132)

I have shown in the previous chapters that the LIBOR market model is ideally suited to the evaluation of derivative products whose payoffs satisfy certain mild measurability conditions. For these products, the yield curve need only be described on a finite set of points, which correspond to the price-sensitive events (and to the payoff times). In particular, I pointed out that the evolution of a finite set of spanning forward rates, and today’s value of a chosen numeraire, are all that is needed to price the payoffs above.

I have also shown (Section 3.1) that, if the set of these...

6. II The Inputs to the General Framework
• 6 Instantaneous Volatilities
(pp. 135-172)

The treatment of the no-arbitrage evolution of forward rates in the standard LIBOR-market-model framework presented in Part I has led to the following results:

1. The coefficients that multiply the Brownian increments, and that therefore affect the stochastic part of the evolution of the logarithms of the forward rates, are purely deterministic functions of time (the forward rates, in particular, do not appear). They can be expressed in terms of deterministic instantaneous volatilities and correlation functions. These two deterministic functions have been shown to appear together in the form of the total or marginal covariance elements defined in Chapter 3...

• 7 Specifying the Instantaneous Correlation Function
(pp. 173-208)

A great deal of attention has been devoted in the previous chapter to forward-rate instantaneous volatilities. Nothing, however, has so far been said about the associated instantaneous correlation. In general, this correlation function can be assigned a functional dependence on calendar time and on the maturities of the two forward rates:¹

${p_{ij}} = p\left( {t,{T_i},{T_j}} \right).$

In order for the covariance element$\int_{{T_k}}^{{T_{k + 1}}} {\sigma i\left( u \right)} \sigma j\left( u \right)pij\left( u \right)du$to be well defined, once suitably square-integrable volatility functions have been chosen, it is enough to assign that the correlation function should be (Riemann or Lebesgue) integrable over any interval$[{T_k},{T_{k + 1}}]$(by Schwartz inequality). For the interpretation of the function p as...

7. III Calibration of the LIBOR Market Model
• 8 Fitting the Instantaneous Volatility Functions
(pp. 211-248)

In Parts I and II I have gathered the tools required to undertake a robust and financially appealing calibration strategy of the LIBOR market model. More precisely, after laying out the conceptual foundations of the approach in Part I, I have highlighted in Part II the financial criteria that can guide the user in her choice of instantaneous volatility and correlation functions (see Chapters 6 and 7, respectively). Once this choice has been made, the task remains to pin down the free parameters that appear in the specification of these functions. This task is often referred to as the calibration...

• 9 Simultaneous Calibration to Market Caplet Prices and to an Exogenous Correlation Matrix
(pp. 249-275)

One of the most important messages conveyed in Part I has been that the marginal coveriance matrices, C, of elements

$C\left( {i,j,k + 1} \right) = \int_{{T_k}}^{{T_{k + 1}}} {\sigma i\left( u \right)} \sigma j\left( u \right)pij\left( u \right)du$

(and sometimes just the total covariance matrix, TOTC), are all that matters for option pricing using the standard LIBOR market model. These quantities are in turn fully determined once the time-dependent (but deterministic) instantaneous volatilities and correlations are specified. A great deal of attention has therefore been devoted to the specification of desirable general functional forms for these two sets of functions (see Chapters 6 and 7). Given the parametric approach that I have chosen to embrace, I have...

• 10 Calibrating a Forward-Rate-Based LIBOR Market Model to Swaption Prices
(pp. 276-330)

The no-arbitrage restrictions on the drifts of the forward rates do not fully characterize, by themselves, the LIBOR market model, but only do so when specific functional forms for the volatility and correlation functions are chosen. I have shown that these functions are under-determined by the market prices of the plain-vanilla instruments (caplets and European swaptions). This is of great practical relevance, because a remarkable richness of behaviors (and exotic option prices) can be generated by relatively simple choices of market-consistent instantaneous volatility or correlation functions (see, e.g., Sidenius 1998, 2000). The trader is therefore left with a variety of...

8. IV Beyond the Standard Approach:: Accounting for Smiles
• 11 Extending the Standard Approach - I: CEV and Displaced Diffusion
(pp. 333-366)

In Parts I—III of this book I have presented a modelling approach for a set of spanning forward rates that is underpinned by the assumption of lognormality. More precisely, the crucial assumption was made in Chapter 2 that it was possible to find a measure (the ‘terminal’ measure) under which one forward rate at a time could be described in terms of a (strictly positive) exponential martingale with deterministic volatility (see Condition 2.3 and Equations (2.28) and (2.28’) in Section 2.3). Despite the fact that other forward rates were shown not to be log-normal under the same terminal measure...

• 12 Extending the Standard Approach - II: Stochastic Instantaneous Volatilities
(pp. 367-414)

In the previous chapter, I have presented the possible financial reasons for describing the forward-rate dynamics in terms of a displaced-diffusion or CEV model. I would like to stress that I find the CEV approach theoretically more appealing, since it is guaranteed to preserve non-negative forward rates (for β ≠ 0). Unfortunately, for arbitrary values of the exponent β (see Equation (11.8)), it does not readily lend itself to simple closed-form solutions. As pointed out in Section 11.3, however, there exists a simple relationship (see Marris 1999) between the CEV exponent and the displacement factor of the displaced diffusion model...

• 13 A Joint Empirical and Theoretical Analysis of the Stochastic-Volatility LIBOR Market Model
(pp. 415-444)

One of the underlying themes of this book has been that caplets and European swaptions enjoy a privileged status in the interest-rate derivatives world, in that they are the liquid plain-vanilla reference instruments used by the complex derivatives traders to hedge the ‘convexity’ of their positions (i.e., for hedging beyond the delta level). In this respect, the exotic interest-rate option trader tends to regard cap (let)s and European swaptions almost as their most natural set of ‘underlying’ instruments, and one of the reasons for the appeal of the LIBOR market model is its ability to recover almost by construction the...

9. Bibliography
(pp. 445-452)
10. Index
(pp. 453-467)