Modelling, mathematical analysis and numerical simulation of problems related to counterparty risk and CVA

Abstract

This thesis deals with the modelling, mathematical analysis and numerical solution of partial di erential equation (PDE) problems for pricing European and American options when considering counterparty risk. Several valuation adjustments are considered, the most important one being the credit value adjustment (CVA). In the modelling, the intensity of default from each risky counterparty plays a relevant role. In the present work we analyze two situations. In the rst one constant intensities of default are assumed, leading to PDE models with one spatial dimension. In the second setting stochastic intensities are assumed, although only one counterparty can default so that PDE models with two spatial variables are deduced. Thus, Cauchy-boundary value PDE problems are posed for European options, while complementarity problems govern the pricing of American options. The two more usual choices for the mark-to-market value, risk-free and risky derivative values, lead to linear and nonlinear PDE problems, respectively. The mathematical analysis of the nonlinear models is one of the main achievements of this work, thus obtaining the existence and uniqueness of solution for the di erent problems. For the numerical solution, a method of characteristics jointly with a xed point iteration and nite elements are used. In the case of American options, an augmented Lagrangian active set method is additionally applied. Also, the equivalent formulations in terms of expectations have been posed and numerically solved by means of appropiate Monte Carlo techniques. Finally, we show illustrative results of the performance of the models and numerical methods that have been implemented.