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

作者：Ma K; Forsyth P A^*

来源：IMA Journal of Numerical Analysis, 2017, 37(2): 905-944.

DOI：10.1093/imanum/drw025

• monotone scheme
• fully implicit
• uncertain volatility
• HJB equation
• policy iteration
• hybrid scheme

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

• 出版日期2017-4