Motivation and the Discrete Forward Rate Framework

Before the LIBOR Market Model (LFM), no interest-rate model could price caplets and recover Black's formula exactly. The models that were used — G2++, Hull-White, Black-Karasinski — approximated cap prices as complicated non-linear functions of model parameters, and practitioners had to calibrate these parameters empirically to match market cap quotes. The mismatch between the model's pricing formula and Black's market formula created persistent hedging errors. The LFM resolves this by starting from the rates that actually trade: discrete compounded LIBOR rates $F_k(t) = F(t;T_{k-1},T_k)$ on a fixed tenor grid, instead of the unobservable instantaneous forward rate $f(t,T)$. Under the $T_k$-forward measure, $F_k$ is a martingale; specifying lognormal dynamics gives Black's caplet formula as an exact result. This lesson sets up the tenor grid, derives the key martingale property that makes everything work, and explains why the LFM escapes the explosion that dooms the HJM lognormal model.