The Heath-Jarrow-Morton Framework

Every short-rate model so far — Vasicek, Hull-White, CIR, G2++ — specifies a stochastic process for the instantaneous short rate and then derives bond prices as conditional expectations. This works, but it conceals the geometry of the yield curve: different short-rate models imply different volatility shapes for forward rates at different maturities, and it is not obvious which model is capturing which empirical feature. Heath, Jarrow, and Morton (1992) reversed this: instead of modelling the short rate and deriving the curve, they directly model the entire instantaneous forward rate curve $f(t,T)$ for all maturities $T$ simultaneously. This one idea unifies all no-arbitrage term structure models. The short rate is a derived object — $r(t) = f(t,t)$ — and the bond price is $P(t,T) = e^{-\int_t^T f(t,u)\,du}$. The key insight, the HJM drift condition, is that no-arbitrage uniquely pins the drift of the forward curve as a function of its volatility: once the volatility surface $\sigma(t,T)$ is chosen, all derivative prices are determined.