Combining the SABR and LMM models

Pierre Henry-Labordere analyses a stochastic volatility Libor market model that combines the SABR and Brace-Gatarek-Musiela (BGM) models in a natural way. Using an innovative geometrical method, he explains how to obtain analytical formulas for swaption implied volatilities. For a caplet, the resulting asymptotic implied volatility formula degenerates to the classical SABR formula

The BGM model is a special version of the Libor market model (LMM), in which all forward rates are lognormal. While the LMM is a very flexible framework as it enables us to specify an individual dynamics for each Libor, the calibration of swaption smiles is difficult as, even in the simple case of lognormal forward rates, swaption prices are not known analytically and straight Monte Carlo calibration is too time-consuming.

In such a model, the instantaneous variance of a swap rate is a weighted

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