ABSTRACT
Abstract In this chapter, we focus on the last generation of interest-rate models, the Libor Market Models (LMMs). After a quick review of LMMs, we discuss the calibration of such models. In practice, this model is calibrated to a swaption matrix\cube. As a Monte-Carlo (MC) calibration routine is fairly time-consuming and noisy, the calibration requires an analytical approximation for the swaption implied volatility. For the BGM-LMM, some approximations have been found which are based on the Hull-White-Rebonato freezing arguments. Using our geometrical framework, we give a justification for the freezing arguments and derive a more accurate asymptotic swaption implied volatility at the first-order for general stochastic volatility LMMs. We apply this formula to a specific model where the forward rates are assumed to follow a multi-dimensional CEV process correlated to a unique SABR process. The geometry underlying this model is the hyperbolic manifold Hn+1 with n the number of Libor forward rates. Parts of this research were published by the author in Risk magazine [104].
