Markovianity, HJM Model Classes, and the Bridge to LMM

The HJM framework is exact: every no-arbitrage term structure model is an HJM model for some volatility function $\sigma(t,T)$. But exactness is not the same as tractability. Once you write down $\sigma(t,T)$, you inherit the short rate $r(t) = f(t,t)$, which — unless $\sigma$ has a very specific structure — is not a Markov process. A non-Markovian short rate means that pricing an interest-rate derivative requires tracking the full history of the forward curve, not just its current value. The binomial tree grows exponentially, and simulation requires storing the entire path. This lesson characterises exactly when an HJM model generates a Markovian short rate (Carverhill's separability condition), when a two-dimensional enlargement restores Markovianity at reduced computational cost (the Ritchken–Sankarasubramanian framework), and why the natural log-normal specification $\sigma(t,T) = \sigma f(t,T)$ is mathematically forbidden — it produces a model that explodes in finite time. That explosion result is what forced the development of the LIBOR Market Model (LMM), which models discrete compounded LIBOR rates instead of the instantaneous forward curve.