Solving Semi-Linear Risky-Closeout PDE by Discount Boundary Optimization

Abstract

We consider a non-linear PDE that appears in a number of contexts including a one-way CSA, CVA with risky closeout, option pricing with differential borrowing and lending rates, accounting-consistent valuation, and constrained cash supply. We show that its solution is given as the minimum of solutions of certain related linear PDEs. An efficient numerical method in any number of dimensions that is based on our representation is presented.

Keywords: semi-linear PDE, non-linear PDE, one-way CSA, CVA with risky closeout, option pricing with differential rates, accounting-consistent valuation, FVA, CVA, XVA, DVA, obstacle problem