The HJM Drift Restriction: No-Arbitrage Constrains the Forward Rate Drift

The HJM specification allows the forward rate drift $\alpha(t,T)$ to be almost anything — a remarkable degree of freedom. But the requirement of no arbitrage imposes a precise, unavoidable constraint: $\alpha(t,T)$ must equal $\sigma(t,T)\cdot\int_t^T\sigma(t,u)\,du$. This one equation is the HJM drift condition, and it is the central result of the Heath-Jarrow-Morton framework. Its consequence is striking: the entire forward rate dynamics under the risk-neutral measure are completely determined by the volatility surface $\sigma(t,T)$. There is no free choice of drift. A quant building a term structure model has exactly one degree of freedom per maturity: the volatility $\sigma(t,T)$. This lesson states, interprets, and illustrates the drift condition.