Libor Market Model Based on Swaption Data

An empirically efficient analytical cascade calibration of the LIBOR Market Model based only on directly quoted swaptions data

Damiano Brigo Credit Models Banca IMI Corso Matteotti 6 20121 Milano, Italy

damiano.brigo@bancaimi.it

Massimo Morini City University London Cass Business School and Universit` di Milano a Bicocca

massimo.morini@unimib.it

First Version: December, 2002. This version: January, 2005

Abstract This work focuses on the swaptions automatic cascade calibration algorithm (CCA) for the LIBOR Market Model (LMM) first appeared in Brigo and Mercurio (2001). This method induces a direct analytical correspondence between market swaption volatilities and LMM parameters, and allows for a perfect recovery of market quoted swaption volatilities if a common industry swaptions approximation is used. We present explicitly an extension of the CCA to calibrate the entire swaption matrix rather than its upper triangular part. Then, while previous tests on earlier data showed the appearance of numerical problems, we present here different calibration cases leading to acceptable results. We analyze the characteristics of the configurations used and concentrate on the effects of different exogenous instantaneous historical or parametric correlation matrices. We also investigate the influence of manipulations in input swaptions data for missing quotes, and devise a new algorithm maintaining all the positive characteristics of the CCA while relying only on directly quoted market data. Empirical results on a larger range of market situations and instantaneous covariance assumptions show this algorithm to be more robust and efficient than the previous version. Calibrated σ parameters are in general regular and financially satisfactory, as confirmed by the analysis of various diagnostics implied structures. Finally we Monte Carlo investigate the reliability of the underlying LMM swaption analytical approximation in the new context, and present some possibilities to include...