LIBOR and swap market models

Abstract

In the previous chapters we presented models based on the instantaneous short rate and the instantaneous forward rate. These models suffer from a number of drawbacks. Firstly, calibration to the prices of commonly traded vanilla instruments such as caps, floors or swaptions can be quite involved. Exotic derivatives depending on the volatilities of many different rates may need to be calibrated to a large set of market instruments, which is difficult when using a short-rate model. Secondly, although instantaneous rates are mathematically convenient, they are not directly observable in the market, nor are they related in a straightforward manner to the prices of any traded instruments. It can be difficult to relate the model parameters, such as mean-reversion in the Hull–White model, to a market-observable quantity. In the LIBOR market model (LMM) we are going to use market rates, namely the forward LIBOR rates, as state variables modelled by a set of stochastic differential equations. For a suitable choice of numeraire we will express the drifts in these SDEs as functions of the volatilities and correlations among the forward rates. A remarkable feature of the LMM is that the model prices are consistent with Black's formula. For a given forward LIBOR rate setting at time S and maturing at time T > S , the forward-rate dynamics is driftless under the forward measure P T . This is consistent with Black's formula for caplets, where the LIBOR rate underlying each caplet is a log-normal process. Using these facts, we shall see how Black's formula arises naturally in the LMM framework. This is a major advantage of the LMM. It means that we can calibrate to implied (at-the-money) cap volatilities automatically. The swap market model (SMM) makes it possible to derive Black's formula for swaptions. Moreover, it is possible to apply the LMM to obtain an analytic approximation (known as Rebonato's formula) for the volatility in Black's swaption formula. This facilitates efficient calibration of the LMM.