Heath–Jarrow–Morton (HJM) models are driven by the evolution in time t of the instantaneous forward-rate curve f(t, T) parameterised by the maturity date T. The entire curve serves as the state variable. This is in contrast to short-rate models, which are driven by the evolution of a single point on the curve, the short rate r(t).

Just like in the case of short-rate models, we adopt Assumption 2.1, i.e. we assume the existence of a risk-neutral measure Q, which transforms all security prices discounted by the money market account into martingales.

The key result in this framework is that the drift of the forward rate f(t, T) under the risk-neutral measure Q is determined by the volatility. This is different to short-rate models, where we are free to specify the drift for the short rate. Compare this to the classical Black–Scholes model, where the drift of the underlying stock price process under the risk-neutral measure is equal to the spot interest rate; see [BSM]. In the HJM framework, just like in the Black–Scholes model, the drift of the underlying process (the instantaneous forward rate and the stock price, respectively) is fixed.

The main benefit of HJM models is that they allow for a perfect fit to the initial interest rate term structure and offer more fiexibility than short-rate models. However, they can be difficult to apply in practice.

One-factor HJM models

Heath, Jarrow and Morton proposed a framework for modelling stochastic interest rates based on the dynamics of the instantaneous forward rate f(t, T).