On the distributional distance between the lognormal LIBOR and swap market models

Abstract

We consider the distributional difference in forward swap rates from the LIBOR market model (LFM) and the swap market model (LSM), the two fundamental market models for interest-rate derivatives. We explain how the Kullback–Leibler information (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 the projection of the LFM swap-rate distribution onto the lognormal family is compared to an industry synthetic swap volatility approximation in the LFM. Finally, we analyse how the above distance changes, in some cases, according to the parameter values and to the parameterizations themselves. We find a small distance in all cases.