New developments on the analytical cascade swaption calibration of the Libor Market Model

In this work we focus on the swaptions “second-order automatic cascade calibration” methodology for the Libor Market Model (LMM), first appeared in Brigo and Mercurio (2001). Such analytical calibration induces a direct correspondence between swaption market volatilities and LMM volatility parameters. We present the basic algorithm and a possible extension, pointing out its main features and some typical problems. The procedure requires an exogenous forward rates instantaneous correlation matrix, so that a historically estimated matrix is considered. This is used to find the parameters of parsimonious parametric forms, in order to make them reflect some key characteristics of the historical matrix. A calibration case study is presented, using different correlations and analyzing their impact on the results. We also consider the consequences of techniques used for problems related to missing or illiquid input data, and analyze calibration results in terms of terminal correlations and evolution of the term structure of volatilities. We further assess, via Monte Carlo simulation of the LMM dynamics,the reliability of the underlying approximation leading to the cascade calibration in this specific context, and finally present some possibilities to include information coming from the cap market.