An unconditionally monotone numerical scheme for the two factor uncertain volatility model

1 Under the assumption that two asset prices follow an uncertain volatility model, the maximal and 2 minimal solution values of an option contract are given by a two dimensional Hamilton-Jacobi-Bellman 3 (HJB) PDE. A fully implicit, unconditionally monotone finite difference numerical scheme is developed 4 in this paper. Consequently, there are no time step restrictions due to stability considerations. The 5 discretized algebraic equations are solved using policy iteration. Our discretization method results in a 6 local objective function which is a discontinuous function of the control. Hence some care must be taken 7 when applying policy iteration. The main difficulty in designing a discretization scheme is development 8 of a monotonicity preserving approximation of the cross derivative term in the PDE. We derive a hybrid 9 numerical scheme which combines use of a fixed point stencil and a wide stencil based on a local coordinate 10 rotation. The algorithm uses the fixed point stencil as much as possible to take advantage of its accuracy 11 and computational efficiency. The analysis shows that our numerical scheme is l∞ stable, consistent in 12 the viscosity sense, and monotone. Thus, our numerical scheme guarantees convergence to the viscosity 13 solution. 14