Bilateral XVA pricing under stochastic default intensity: PDE modelling and computation

Abstract

Adjusting derivative prices to take into account default risk has attracted the attention of several researchers and practitioners, especially after the 2007-2008 financial crisis. We derive a novel partial differential equation (PDE) model for derivative pricing including the adjustment for default risk, assuming that the default risk of one of the counterparties (the buyer) follows a Cox-Ingersoll-Ross (CIR) process, while the other party has constant default risk. The time-dependent PDE derived is of Black-Scholes type and involves two “space” variables, namely the asset price and the buyer default intensity, as well as a nonlinear source term. We formulate boundary conditions appropriate for the default intensity variable. The numerical solution of the PDE is based on standard finite differences, and a penalty-like iteration for handling the nonlinearity. We also develop and analyze a novel asymptotic approximation formula for the adjusted price of derivatives, resulting in a very efficient, accurate, and convenient for practitioners formula. We present numerical results that indicate stable second order convergence for the 2D PDE solution in terms of the discretization size. We compare the effectiveness of the 2D PDE and asymptotic approximations. We study the effect of various numerical and market parameters to the values of the adjusted prices and to the accuracy of the computed solutions.