A Multi-Level Monte Carlo Lagrange-Galerkin Method for Solving a XVA Hybrid Model in European Options

In this article we consider XVA pricing models for European options that incorporate three stochastic factors, namely, the price of the underlying asset and the intensities of default of the investor and the hedger, with the corresponding stochastic differential equations (SDEs) that govern their dynamics. By using a dynamic hedging methodology, linear and nonlinear partial differential equations (PDEs) have been posed in L.M. Garcia (2013). In order to avoid the numerical solution PDEs with three spatial variables, we propose a hybrid model based on PDEs in one spatial variable, keeping the two intensities of default as stochastic coefficients of the PDEs satisfying their respective associated SDEs. For the numerical solution, we propose a Multi-Level Monte Carlo Lagrange-Galerkin method, where the PDEs obtained for each sample path of the stochastic coefficients is solved by combining a semi-Lagrangian method for time discretization and a finite element technique for the spatial discretization, adding a fixed point iteration in the nonlinear case. Numerical results allow to compare the different models and the qualitative and quantitative behaviour of the XVA