Analysis of non-linear approximated value equation under multiple risk factors and stochastic intensities

We develop a numerical method to approximate the adjusted value of a European contingent claim subject to multiple credit risks in a market model where the underlying's price is correlated with the stochastic default intensities of both 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 variables verifying a non-linear PDE. Thus, we build a numerical method to approximate the solution of this non-linear PDE, as an alternative to the commonly used Monte Carlo simulations, which require large computational times, especially when the number of the state variables grows. We construct this approximated solution by the simple method of finite differences and we show the method to be accurate and efficient.