Physics and Derivatives: On Three Important Problems in Mathematical Finance

Abstract

In this article, we use recently developed extension of the classical heat potential method in order to solve three important but seemingly unrelated problems of financial engineering: (A) American put pricing; (B) default boundary determination for the structural default problem; and (C) evaluation of the hitting time probability distribution for the general time-dependent Ornstein–Uhlenbeck process. We show that all three problems boil down to analyzing behavior of a standard Wiener process in a semi-infinite domain with a quasi-square-root boundary.

TOPICS: Derivatives, options, credit default swaps

Key Findings

• • We introduce a powerful extension of the classical method of heat potentials designed for solving initial boundary value problems for the heat equation with moving boundaries.

• • We demonstrate the versatility of our method by solving several classical problems of financial engineering in a unified fashion.

• • In particular, we find the boundary corresponding to the constant default intensity in the structural default model, thus solving in the affirmative a long outstanding problem.