Models and numerical methods for XVA pricing under mean reversion spreads in a multicurrency framework

In this article we make some new relevant contributions to the computation of total valuation adjustments (XVA) for financial derivatives involving several currencies. From the modelling point of view, for the credit spreads we consider the more realistic exponential Vasicek and CIR positive mean reversion processes. Moreover, the derivative is partially collateralized in cash in a foreign currency and the collateral value is a percentage of the derivative prices. Under this modelling assumptions and using appropriate dynamic hedging methodologies, we obtain formulations in terms of linear and nonlinear partial differential equations, which are solved with Lagrange-Galerkin methods in low dimension. For higher dimensions, we use the Monte Carlo techniques for the equivalent formulations in terms of expectations. These techniques include a multilevel Picard iteration method for the nonlinear case. Finally, the methodologies are applied to several European options with different payoffs and the numerical results are discussed.