Numerical Solutions of Two-factor Hamilton-Jacobi-Bellman Equations in Finance

Abstract

In this thesis, we focus on solving multidimensional HJB equations which are derived from optimal stochastic control problems in the financial market. We develop a fully implicit, unconditionally monotone finite difference numerical scheme. Consequently, there are no time step restrictions due to stability considerations, and the fully implicit method has essentially the same complexity per step as the explicit method. The main difficulty in designing a discretization scheme is development of a monotonicity preserving approximation of cross derivative terms in the PDE. We primarily use a wide stencil based on a local coordination rotation. The analysis rigorously show that our numerical scheme is $l_\infty$ stable, consistent in the viscosity sense, and monotone. Therefore, our numerical scheme guarantees convergence to the viscosity solution.

Firstly, our numerical schemes are applied to pricing two factor options under an uncertain volatility model. For this application, a hybrid scheme which uses the fixed point stencil as much as possible is developed to take advantage of its accuracy and computational efficiency. Secondly, using our numerical method, we study the problem of optimal asset allocation where the risky asset follows stochastic volatility. Finally, we utilize our numerical scheme to carry out an optimal static hedge, in the case of an uncertain local volatility model.