Evaluation of Non-linear Credit Value Adjustments under multiple credit risks

In this work, we consider the problem of correctly pricing financial derivatives that might be subject to multiple credit risks, such as default or liquidity risk, that gained importance after the financial crisis of 2008-09. An important achievement in the mathematical modelling of the problem was the representation of the value of such derivatives as solutions of appropriate Backward Stochastic Differential Equations (BSDE), which might be solvable in the samples cases. When various risks are taken into account simultaneously, and correlation is admitted between the processes underlying the price formation, the picture becomes much more complex, and although the BSDE representation still applies, explicit solvability becomes impossible. Monte Carlo simulations, usually requiring long computational times, are often the only way to get an approximation of the solution, so it might be important to develop alternative approximation techniques that require shorter computational times yet preserving accuracy. Once the BSDE's representation is developed, in a Markovian setting, the derivative's can be rewritten as a deterministic function of the state variables, which verifies a non-linear PDE. Thus, we decided to employ a PDE discretization approach to approximate the PDE solution. By employing an adaptation of the simple method of lines, we were able to construct an approximation method that turned out to be accurate and efficient, thus producing a valid alternative to Monte Carlo simulations.