Total value adjustment for European options with two stochastic factors. Mathematical model, analysis and numerical simulation

In the present paper we derive novel (non)linear PDE models for pricing European options and the associated total value adjustment (XVA), when incorporating the counterparty risk. The main innovative aspect is the consideration of stochastic spreads instead of less realistic constant spreads previously used in the literature. For the nonlinear model, a rigorous mathematical analysis based on sectorial differential operators allows to state the existence and uniqueness of a solution. Moreover, for the numerical solution we propose an appropriate set of techniques based on the method of characteristics for time discretization, finite element for spatial discretization and fixed point iteration for the nonlinear terms. Finally, numerical examples illustrate the expected behaviour of the option prices and the total value adjustment.