Obstacle problems for nonlocal operators: A brief overview

TL;DR

This paper provides an overview of obstacle problems for nonlocal operators, emphasizing their applications in financial mathematics, particularly in modeling option prices with non-Gaussian asset dynamics.

Contribution

It summarizes recent results on existence, uniqueness, and regularity of viscosity solutions for obstacle problems related to nonlocal operators in finance.

Findings

Existence and uniqueness of solutions established

Regularity results for viscosity solutions

Applications to American option pricing models

Abstract

In this note, we give a brief overview of obstacle problems for nonlocal operators, focusing on the applications to financial mathematics. The class of nonlocal operators that we consider can be viewed as infinitesimal generators of non-Gaussian asset price models, such as Variance Gamma Processes and Regular L\'evy Processes of Exponential type. In this context, we analyze the existence, uniqueness and regularity of viscosity solutions to obstacle problems which correspond to prices of perpetual and finite expiry American options. Complete proofs can be found in arXiv:1709.10384, where these results have originally appeared.