A Forward Solution for Computing Risk-Neutral Derivatives Exposure
Résumé
In this paper, we derive a forward analytical formula for computing the expected exposure of financial derivatives. Under general assumptions about the underlying diffusion process, we give a convenient decomposition of the exposure into two terms: The first term is an intrinsic value part which is directly deduced from the term structure of the forward mark-to-market. The second term expresses the variability of the future mark-to-market and represents the time value part.
Abstract In the spirit of Dupire's equation for local volatility, our representation establishes a differential equation for the evolution of the expected exposure with respect to the observation dates. Our results are twofold: First, we derive analytically an integral formula for the exposure's expectation and we highlight straightforward links with local times and the co-area formula. Second, we show that from a numerical perspective, our solution can be significantly efficient when compared to standard numerical methods. The accuracy and time-efficiency of the forward representation are of special interest in benchmarking XVA valuation adjustments at the trade level.