LIBOR Market Model (LMM / BGM)
Quantitative Finance glossary · Quantitative Finance
Definition
Models each discrete forward rate L_i(t) = L(t; T_i-1, T_i) as log-normal under its own forward measure mathbbQ^T_i: dL_i(t) = σ_i(t) L_i(t),dW_t^T_i, recovering Black's caplet formula by construction. Under a single terminal measure the drifts become non-zero and tenor-dependent (LMM drift cascade). Calibrated jointly to ATM cap and swaption vols; instantaneous-correlation matrix among forwards drives swaption smile. Post-LIBOR the analogue prices SOFR-based caps and term-SOFR products.
LIBOR Market Model (LMM / BGM) in plain English
A "cap" or "caplet" is basically insurance on a floating interest rate: if the rate on a loan jumps above some ceiling, the caplet pays you the difference, so you're protected. To price that insurance you need a model of how future short-term interest rates (like the old LIBOR benchmark, now replaced by SOFR) wiggle over time. The LIBOR Market Model is the industry-standard approach: instead of modeling one abstract rate, it directly models each real, tradeable forward rate — the rate you can lock in today for a future period — and lets each one drift and jump randomly. Its big selling point is that it's built to exactly match the simple pricing formula traders already use, so it fits real market prices right out of the box; the tricky math ("drift cascade") just handles how all these rates move together in a consistent, no-free-lunch way.
Also called: LMM / BGM
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