Swaptions, the Swap Market Model, and LFM-LSM Incompatibility

The LIBOR Market Model prices caplets exactly — Black's formula is the literal solution. For swaptions, however, the situation is fundamentally different. The swap rate is a weighted average of spanning forward LIBOR rates, and its distribution under any single measure is not lognormal. The Lognormal Forward-Swap Model (LSM) prices swaptions exactly by modelling the swap rate as lognormal directly, but this is an entirely separate model from the LFM. The two models are theoretically incompatible: they cannot both hold simultaneously for the same set of instruments. This lesson derives Black's swaption formula from the LSM, establishes the incompatibility formally, and then shows how the LFM prices swaptions in practice — via Monte Carlo simulation and an approximate analytical formula that converts the joint forward-rate distribution into a swaption price.