The Heath–Jarrow–Morton Framework

Interest rate modelling can also be performed by starting from the dynamics of the instantaneous forward rate. As we shall see the dynamics of all other quantities of interest can then be derived from it. This approach has its origin in Ho and Lee (J Finance XLI:1011–1029, 1986) but was most clearly articulated in Heath et al. (Econometrica 60(1):77–105, 1992a), to which we shall subsequently refer as Heath–Jarrow–Morton. In this framework, the condition of no riskless arbitrage results in the drift coefficient of the forward rate dynamics being expressed in terms of the forward rate volatility function. The major weakness in implementing the Heath–Jarrow–Morton approach is that the spot rate dynamics are usually path dependent (non-Markovian). We consider a class of functional forms of the forward rate volatility that allow the model to be reduced to a finite dimensional Markovian system of stochastic differential equations. This class contains some important models considered in the literature.