Approximated moment-matching dynamics for basket-options pricing

The aim of this paper is to present two different approximated dynamics for basket-options pricing. The approximations are based on a moment-matching procedure. Single names in the basket are modelled according to geometric Brownian motions (GBMs), as in the classical Black and Scholes framework, driven by instantaneously correlated Brownian motions. The basket distribution resulting from such GBMs for the single assets is unknown. By resorting to Monte Carlo simulation, we compare this distribution to the known approximated distribution coming from the moment-matching dynamics. The comparison is carried out through distances on the space of probability densities, namely the Kullback-Leibler information (KLI) and the Hellinger distance (HD). We are interested in measuring the KLI and the HD between the 'real' simulated basket distribution at terminal time and the distributions used for the approximation, both in the log-normal and shifted log-normal families. We isolate influences of instantaneous volatilities and instantaneous correlations, in order to assess which configurations of these variables have a major impact on the KLI and HD and therefore on the quality of the approximation.