PV and XVA Greeks for Callable Exotics by Algorithmic Differentiation

Abstract

We generalize the algorithmic differentiation method proposed by Antonov (2016) from price Greeks to XVA Greeks. This method, named Backward Differentiation (BD), was developed in the context of computing price or PV Greeks for individual callable exotic trades.We start by treating cases where cashflow derivatives are sufficient for computing PV/XVA Greeks, i.e., where the differentiation of conditional expectations (or regression functions) is not necessary. For example, PV Greeks for Bermudan swaptions can be computed without having to perform the complicated step of regression function differentiation. We modify the Backward Differentiation algorithm to calculate Greeks for such instruments: the method is applied during the backward pricing procedure and has almost no overhead with respect to a pure backward pricing (without the Greeks).A general XVA calculation cannot be done using only the cashflow derivatives - some exceptions are listed in this article - instead, the differentiation of future instrument values that are results of the regression may be required. We leverage the algorithmic calculation of future values (Algorithmic Exposures) and describe the Adjoint Differentiation (AD) and the new BD for XVA Greeks. The latter algorithm is much simpler than the former, in particular, it does require the use of the instrument tape, i.e., it does not require the storage of certain payoff derivatives during the pricing procedure as is the case for AD. At the same time, both AD and BD enjoy a similar level of performance.