PDE models and numerical methods for total value adjustment in European and American options with counterparty risk

Since the last financial crisis, a relevant effort in quantitative finance research concerns the consideration of counterparty risk in financial contracts, specially in the pricing of derivatives. As a consequence of this new ingredient, new models, mathematical tools and numerical methods are required. In the present paper, we mainly consider the problem formulation in terms of partial differential equations (PDEs) models to price the total credit value adjustment (XVA) to be added to the price of the derivative without counterparty risk. Thus, in the case of European options and forward contracts different linear and nonlinear PDEs arise. In the present paper we propose suitable boundary conditions and original numerical methods to solve these PDEs problems. Moreover, for the first time in the literature, we consider XVA associated to American options by the introduction of complementarity problems associated to PDEs, as well as numerical methods to be added in order to solve them. Finally, numerical examples are presented to illustrate the behavior of the models and numerical method to recover the expected qualitative and quantitative properties of the XVA adjustments in different cases. Also, the first order convergence of the numerical method is illustrated when applied to particular cases in which the analytical expression for the XVA is available.