Modelling, mathematical analysis and computation of the total value adjustment including KVA and multi-currency

Abstract

[Abstract]: Since the introduction of Basel III, capital valuation adjustment (KVA) has been one of the most debated valuation adjustments used to price financial derivatives. Although financial institutions operate across several currencies, the study of the multi-currency setting has received limited attention in the literature on valuation adjustments. In the present article, we rigorously propose models for total valuation adjustments (XVA), including KVA, within a multi-currency framework. We develop mathematical analysis techniques to prove the classical well-posedness of the corresponding high-dimensional nonlinear partial differential equation (PDE). Finally, we adapt numerical methods based on a Monte Carlo technique and a deep BSDE algorithm to compute the value of the XVA for the linear and the nonlinear models. We also demonstrate the efficiency of these methods in high-dimensional settings by pricing two examples of average options on a basket with underlying assets in different currencies.