A FORWARD EQUATION FOR COMPUTING DERIVATIVES EXPOSURE

Abstract

We derive a forward equation for computing the expected exposure of financial derivatives. Under general assumptions about the underlying diffusion process, we give an explicit decomposition of the exposure into an intrinsic value which can be directly deduced from the term structure of the forward mark-to-market, and a time value which expresses the variability of the future mark-to-market. Our approach is inspired by Dupire’s equation for local volatility and leads to an ordinary differential equation qualifying the evolution of the expected exposure with respect to the observation dates. We show how this approach can be linked with local times theory in dimension one and to the co-area formula in a higher dimension. As for numerical considerations, we show how this approach leads to an efficient numerical method in the case of one or two risk factors. The accuracy and time-efficiency of this forward representation in small dimension are of special interest in benchmarking XVA valuation adjustments at trade level.