Interest Rate Volatility and Derivatives (Libor Market Model)

Heath, Jarrow, & Morton (1989, HJM) develop a framework in which all interest rate models can be expressed. The starting point is a continuum of zero coupon bonds, or, equivalently, of instantaneous forward interest rates. In this way, the whole of the yield curve is modelled, according to some covariance model (between the bond prices or forward rates), in as many factors as required. It is not possible, however, to specify a strictly log-normal HJM model due to the well-known “blow up” effect; it is necessary to truncate the volatility function at some suitably high level of rates. Much greater problems arise with numerical implementation, since a computer will be unable to store the infinite number of forward rates in the continuous yield curve. The HJM approach can be modified, however, to deal with discretely compounded forward rates. Brace, Gatarek, and Musiela (1997, BGM) formally present a framework in which forward Libor rates are modelled, and Jamshidian (1997) describes a similar model in which forward-starting swap rates are modelled. It seems that generally, models of this class are referred to as BGM models. The advantages of this type of modelling come from the fact that it is market observables that are modelled, in a fashion which can be shown to be consistent with the Black pricing model used in the market. The model can be easily extended to more than one factor, and is very closely related to the principal component analysis of yield curve movements. Furthermore, since the volatilities of market observables are included directly in the model, much greater transparency is achieved, and calibration is not necessary. For these reasons, this model is often referred to as the market model.