Non-linear Approximated Value Adjustments for Derivatives Under Multiple Risk Factors

We develop a numerical method to approximate the adjusted value of a European contingent claim in a market model where the underlying's price is correlated with the stochastic default intensities of the two parties of the contract. When the close-out value of the contract is chosen as a fraction of the adjusted value, the latter verifies a non linear, not explicitly solvable BSDE. In a Markovian setting, this adjusted value is a deterministic function of the state variable verifying a non-linear PDE. We develop here a numerical method to approximate the PDE solution, as an alternative choice to the commonly used Monte Carlo simulations, which require large computational times, especially when the number of the state variables grows. We construct the approximated solution by the simple method of lines and we show the method to be accurate and efficient in a simplified cases. We show numerical results in the case of both constant intensities and the situation where only one is diffusive.