XVA in a multi-currency setting with stochastic foreign exchange rates

In the present article we address the modelling and the numerical computation of the total value adjustment for European options in a multi-currency setting when the foreign exchange rates between the different involved currencies are assumed to be stochastic. Thus, we extend to a more realistic approach a previous work where constant exchange rates have been considered. New models are formulated both in terms of linear and nonlinear PDEs and expectations, the hedging arguments requiring the additional consideration of the exposure to foreign exchange risk. For the nonlinear models, Picard iteration methods are applied to the formulation in terms of expectations and compared with multilevel Picard iteration methods. In this way, we avoid the curse of dimensionality associated to the use of deterministic numerical methods (such as finite differences or finite element methods) for solving high dimensional PDEs. Some examples of option pricing problems illustrate the performance of the proposed models and numerical methods.