Numerical convergence properties of option pricing PDEs with uncertain volatility

Abstract

The pricing equations derived from uncertain volatility models in finance are often cast in the form of nonlinear partial differential equations. Implicit timestepping leads to a set of nonlinear algebraic equations which must be solved at each timestep. To solve these equations, an iterative approach is employed. In this paper, we prove the convergence of a particular iterative scheme for one factor uncertain volatility models. We also demonstrate how non-monotone discretization schemes (such as standard Crank-Nicolson timestepping) can converge to incorrect solutions, or lead to instability. Numerical examples are provided.