Numerical Solutions to PDE Representations of Derivatives with Bilateral Counterparty Risk and Funding Costs

The purpose of this paper is to present numerical solutions to PDE representations for derivatives pricing including bilateral credit valuation adjustments and funding costs valuation adjustment as presented in Burgard and Kjaer (2011). In particular, we use Crank-Nicolson finite-difference scheme to solve Black-Scholes risk-free PDE, for European and American options, and show how this numerical solution approach is extendable to solve the risky PDE for the value of the same derivative using the same finite-difference scheme and algorithm. Also, we present numerical solutions to valuation adjustments derived from PDE representations for European options through Monte Carlo simulation and numerical integration and we explore an empirical approach for American options through Monte Carlo simulation, least-squares and numerical integration