HJM framework (Heath-Jarrow-Morton)
Quantitative Finance glossary · Stochastic Calculus & Continuous Time Finance
Definition
Modelling the entire instantaneous forward-rate curve f(t,T) directly under no-arbitrage. Under mathbbQ, df(t,T) = α(t,T),dt + σ(t,T),dW_t, and the HJM drift condition forces α(t,T) = σ(t,T)int_t^T σ(t,u),du — the drift is uniquely determined by the volatility, so calibration is purely about specifying σ(·,·). The flip side is that f(t,T) is generally non-Markovian (infinite-dimensional state), making numerical pricing of path-dependent products expensive; LMM is the discrete-tenor cousin that restores tractability.
HJM framework (Heath-Jarrow-Morton) in plain English
Interest rates aren't a single number — there's a whole curve of them, one for every future date you might borrow or lend at, called the forward-rate curve. Instead of modeling one rate, the Heath-Jarrow-Morton (HJM) framework models how that entire curve wiggles over time, while forbidding any 'free money' (arbitrage) opportunities. Its big insight is the drift condition: once you decide how volatile each part of the curve is, the no-free-lunch rule automatically pins down which way the curve must tend to drift — you don't get to choose it separately. The catch is that tracking the whole curve means an enormous amount of state to keep track of, so pricing complicated products gets computationally heavy; the LIBOR Market Model (LMM) is the more practical version that tracks only a handful of real-world rates.
Also called: Heath-Jarrow-Morton
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