Mathematical models and numerical methods for XVA in multicurrency setting

Abstract

This thesis is devoted to the mathematical modelling and numerical solution of problems related to the valuation of financial options including total value adjustment (XVA) in a multicurrency setting. In order to build the models, we assume European options with underlying as- sets written in di↵erent currencies, stochastic credit spread of the counterparty and, eventually, stochastic foreign exchange rates. Depending on the choice of the mark- to-market value, nonlinear or linear partial di↵erential equations (PDEs) are derived. We also make use of the nonlinear and linear Feynman-Kac theorems to deduce the equivalent models in terms of expectations. For each derived model, we propose numerical methods. When the number of stochastic factors is no greater than two, we propose a Lagrange-Galerkin scheme (based on the method of characteristics and the finite element method) for solving the PDEs, eventually combined with fixed point techniques for the nonlinear problems. For problems that include more than two underlying assets and/or stochastic FX rates, we propose the use of Monte Carlo simulations applied to the formulations based on expectations, combined with a Picard method and the more effi cient multilevel Picard iteration (MPI) scheme for the nonlinear cases. We apply these techniques to di↵erent options of European type that validate the performance of the models as well as the proposed numerical methods.