The pricing of average options with jump diffusion processes in the uncertain volatility model

Abstract

The pricing equations of the average options with jump diffusion processes can be formulated as two-dimensional partial integro-differential equations (PIDEs). In the uncertain volatility model, for options with non-convex and non-concave payoffs, such as the butterfly spread, the PIDEs are nonlinear. We use the semi-Lagrangian method to reduce the two-dimensional nonlinear PIDE to a one-dimensional nonlinear PIDE along the trajectory of the average price, and use a Newton-type iteration to guarantee the convergence of the discrete solution to the viscosity solution. Monotonicity and stability as well as the convergence results are derived. Numerical tests of convergence for a variety of cases, including average butterfly spread and ordinary butterfly spread, are presented.