On the distributional distance between the Libor and the Swap market models

In this paper we are concerned with the distributional difference of forward swap rates between the lognormal forward–Libor model (LFM) or “Libor market model ” and the lognormal forward-swap model (LSM) or “swap market model”, the two fundamental models for interest-rate derivatives. To measure this distributional difference, we resort to a “metric ” in the space of distributions, the well known Kullback-Leibler information Preliminary version. We are grateful towards Lorenzo Finesso for introducing us to the literature on entropy estimators. We are also grateful towards Francesco Rapisarda for suggestions on improvements of Monte Carlo simulations and to Fabio Mercurio for general discussions. Jan Liinev is supported by the BOF-project 001104599, whose financial support is gratefully acknowledged. Damiano Brigo and Jan Liinev: Distributional distance between LIBOR and Swap market models 2 (KLI). We explain how the KLI can be used to measure the distance of a given distribution from the lognormal (exponential) family of densities, and then apply this to our models’ comparison. The volatility of (i.e. standard deviation associated with) the projection of the LFM swap-rate distribution onto the lognormal family is compared to a industry synthetic swap volatility approximation obtained via “freezing the drift ” techniques in the LFM. Finally, for some instantaneous covariance parameterizations of the LFM we analyze how the above distance changes according to the parameter values and to the parameterizations themselves, in an attempt to characterize the situations where LFM and LSM are really distributionally close, as is assumed by the market. 1