An obstacle problem arising from American options pricing: regularity of solutions

TL;DR

This paper investigates the regularity and existence of solutions to a nonlocal obstacle problem related to American options pricing, where the underlying process involves jump diffusion, providing new insights into the mathematical properties of such models.

Contribution

It establishes existence, uniqueness, and optimal regularity results for solutions to a nonlocal obstacle problem in the context of jump process-based American options pricing.

Findings

Proved existence and uniqueness of solutions.

Established optimal spatial regularity.

Achieved near-optimal temporal regularity.

Abstract

We analyse the obstacle problem for the nonlocal parabolic operator \[\partial_t u + (-\Delta)^{s} u - b \cdot \nabla u - \mathcal{I}u - ru,\] where $b\in\mathbb{R}^n$, $r\in\mathbb{R}$, and $\mathcal{I}$ is a nonlocal lower order diffusion operator with respect to the fractional Laplace operator $(-\Delta)^{s}$. This model appears in the study of American options pricing when the stochastic process governing the stock price is assumed to be a purely jump process. We study the existence and the uniqueness of solutions to the obstacle problem, and we prove optimal regularity of solutions in space, and almost optimal regularity in time.