Polynomial Bounds for Solutions to Boundary Value and Obstacle Problems with Applications to Financial Derivative Pricing

Abstract

This thesis introduces a new approach for obtaining smooth deterministic upper bounds for the solutions to bounded domain obstacle problems. These bounding functions are characterized by sufficient bounding conditions, under which the bounds may be optimized. These bounds are obtained by expressing the solution function as the solution to an optimization problem that is then formulated as computationally tractable semidefinite programming problem. In a single implementation, the proposed approach obtains explicit bounds in the form of piecewise polynomial functions, which bound the solution function from above over the whole problem domain both in time and state. The proposed approach achieves these bounds without discretizing the spatial or temporal variables, which is typical of current methodologies. We derive our bounds for a general problem setting; considering both elliptic and parabolic obstacle problems. Throughout this thesis we demonstrate, through numerical examples, the effectiveness of the proposed method in obtaining tight upper bounds for the solution to these types of problems. We then go on to discuss extensions of the proposed methodology to problems in financial derivative pricing. In particular, we examine American style options in several market models and with various payoff structures as well as derivative pricing problems in regime-switching models.