ABSTRACT
Abstract A standard method to price a multi-asset European option incorporating an implied volatility is to use a local volatility Monte-Carlo computation. Although straightforward, this method is quite time-consuming, particularly when the number of assets is large and we evaluate the Greeks. Applying our geometrical framework to this multi-dimensional problem, we explain how to obtain accurate approximations of multi-asset European options. We use the heat kernel expansion to obtain an asymptotic solution to the Kolmogorov equation for a n-dimensional local volatility model. The resulting manifold is the flat Euclidean space Rn. We present two applications. The first application we look at is the derivation of an asymptotic implied volatility for a basket option. In particular, we try to reconstruct the basket implied volatility from the implied volatility of each asset. In the second application, we obtain accurate approximation for Collateralized Commodity Obligations (CCO), which are recent commodity derivatives that mimic the Collateralized Debt Obligations (CDO).
