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

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