The LIBOR Market Model

Short-rate models (Vasicek, Hull-White, G2++) price caps by computing a complicated non-linear function of their parameters — a function that almost never equals Black's cap formula, the market standard. The LIBOR Market Model (LFM, also called the BGM model after Brace, Gatarek and Musiela 1997) resolves this by modelling the rates that actually trade: discrete compounded forward LIBOR rates $L_i(t) = L(t;T_i,T_{i+1})$ on a fixed tenor grid $T_0 < T_1 < \cdots < T_n$. Under the $(T_{i+1})$-forward measure, each $L_i$ is a martingale and follows lognormal dynamics, making Black's caplet formula an exact result — not an approximation. The price paid is that the model lives on a discrete tenor grid and has no short-rate analogue: pricing a swaption requires Monte Carlo simulation (because the joint distribution of multiple LIBOR rates is not lognormal simultaneously) or one of several analytical approximations. This topic derives the BGM dynamics, works out the drift corrections under all relevant measures, calibrates the vol and correlation structure, and handles the swaption pricing challenge.