Forward Rate Curve Dynamics and the HJM Specification

The Heath-Jarrow-Morton (HJM) framework is the answer to a question that every short-rate model sidesteps: what drives the shape of the yield curve? A one-factor short-rate model like Hull-White collapses the curve to a single random variable — the short rate — and recovers the curve mechanically as a conditional expectation. The curve's shape is an output, not an input, and the only "free" inputs are the mean-reversion speed and volatility of the short rate. HJM inverts this: it takes the forward rate curve $T \mapsto f(t,T)$ as the fundamental state variable and specifies directly how each point on the curve evolves stochastically. Short-rate models become special cases — particular volatility specifications within the HJM class. This first lesson sets up the general specification, derives the bond price as an exponential functional of the forward curve, and introduces the Musiela parametrisation.