Algorithms for American XVA and free boundary calculations with stochastic counterparty default intensity

Credit and total valuation adjustments (CVA and XVA) are significant in equity markets, as parts of the risk management under Basel III framework. In addition, path-dependent derivatives, such as American-type ones, are heavily traded in markets. Therefore, it is important to accurately and efficiently compute valuation adjustments for American-type derivatives. In this paper, we derive a two-dimensional (2D) in space partial differential equation (PDE) for pricing American-type derivatives including the XVA, assuming the counterparty default risk follows a mean reversion stochastic process, while the self-party has constant default risk. We reformulate the time-dependent, 2D nonlinear PDE into penalty form, which includes two nonlinear source terms. We employ the double-penalty iteration for the 2D PDE to resolve the two nonlinear terms, while we use a finite difference scheme for the spatial discretization, and Crank–Nicolson-Rannacher timestepping. We introduce algorithms for the accurate calculation of the free boundary. We also formulate an asymptotic approximation technique, similar to the one developed for the European case problem, but adjusted for the American put option problem. A key step is to derive the asymptotic approximation to the free boundary for the American put option. We present numerical experiments in order to study the accuracy and effectiveness of the 2D PDE and asymptotic approximations.