LMM-SABR: the LIBOR Market Model with SABR vol for rates
Model the quoted forward rates directly as lognormal under their own forward measures (Black caplet pricing). Add SABR vol on each forward and you have the standard swaption-desk model.
Pricing a Bermudan swaption requires a joint distribution over the whole path of forward rates — the LIBOR Market Model (LMM, BGM 1997) gives you exactly that: model each quoted LIBOR forward as lognormal under its own forward measure, so Black’s caplet formula falls out as an exact identity. Adding SABR stochastic vol to each forward (LMM-SABR) then gives a smiling caplet surface, so the desk can consistently price the full cap/floor/swaption complex from one model instead of patching short-rate and smile models together.
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Worked example
Compute the price of a caplet on the forward rate $L_i(0) = 4\%$ over the period $[T_i, T_{i+1}] = [1\text{y}, 1.5\text{y}]$ with strike $K = 4.5\%$, lognormal LIBOR volatility $\sigma_i = 20\%$, and discount factor $P(0, T_{i+1}) = e^{-0.04 \cdot 1.5} \approx 0.9418$. Use the Black formula directly. Express the answer as a fraction of notional (no scaling).
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Reflection
LMM is convenient because Black's caplet formula is exact in it. But the model loses the elegance of having one short-rate state. What is the trade-off: where does LMM win, and where do short-rate models still dominate?