Inexact arithmetic considerations for direct control and penalty methods: American options under jump diffusion

Solutions of Hamilton–Jacobi–Bellman (HJB) Partial Integro-Differential Equations (PIDEs) arising in financial option problems are not necessarily unique. In order to ensure convergence of a numerical scheme to the viscosity solution, it is common to use a positive coefficient discretization for such PIDEs. However in finite precision arithmetic one often encounters difficulties in solving the discretized nonlinear algebraic equations. In this paper we focus on a specific HJB PIDE, arising from pricing American options under jump diffusion. We use two formulations of this problem, the first a penalty method and the second a direct control formulation. In each case we use a positive coefficient discretization which implies that a fixed point policy iteration will converge when used to solve the nonlinear discretized algebraic equations, under very mild restrictions on parameters. However, when using finite precision arithmetic, we observe that convergence may not occur for either formulation, even if the theoretical conditions are satisfied. We estimate bounds for the penalty parameter (penalty method) and the scaling parameter (direct control formulation) so that convergence of the fixed point policy iteration in inexact arithmetic can be expected. Numerical tests verify that these bounds are conservative. The lower bound is of more practical importance, and conveniently this has a very simple form. We remark that similar issues also arise in more complicated HJB PIDEs in finance, for example when pricing American options under regime switching or guaranteed minimum withdrawal benefits (GMWB) under jump diffusion.