How does the Libor Market Model (BGM) differ from HJM?

Question

AcadiFi · Accepted Answer

BGM (Brace-Gatarek-Musiela), also called the Libor Market Model (LMM), models discrete forward LIBOR rates L_i(t) — the rates underlying caps, floors, and swaptions — rather than instantaneous forwards.

Key innovation: each forward rate L_i is assumed lognormal under its OWN forward measure (the T_i+1 forward measure). This produces Black-like closed-form formulas for caplets, matching the way traders already quote cap vol.

dL_i(t) = sigma_i(t) * L_i(t) * dW_i under measure Q_i+1

Why it beat HJM in practice:
- Direct market quotes for cap and swaption vols can be fed in without transformation
- Prices caps using the Black-76 formula the market already uses
- Forward rates are observable and tradeable; instantaneous forwards are not

Complication: different forward rates L_i live under different measures. To simulate them jointly, you must change measure — each L_i picks up a drift term under a common measure (spot or terminal). These drifts depend on correlations between the forwards, which must be specified.

Calibration: fit caplet vols for each tenor, then fit forward-forward correlations to swaption vols. At LondonBridge Derivatives, the LMM is the workhorse for pricing exotic interest rate products like Bermudan swaptions and CMS spread options.

For CFA, remember BGM = market-observable rates, lognormal, Black-formula compatible, industry standard for caps and floors.

How does the Libor Market Model (BGM) differ from HJM?

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