Optimal exercise of American options under time-dependent Ornstein-Uhlenbeck processes

TL;DR

This paper models the optimal exercise boundary for American options on time-dependent Ornstein-Uhlenbeck processes, providing a new integral equation, bounds, and a numerical solution method.

Contribution

It introduces a novel integral equation for the exercise boundary, along with bounds and a Picard iteration algorithm for solutions.

Findings

Derived a non-linear Volterra integral equation for the exercise boundary.

Proved Lipschitz continuity and differentiability properties of the boundary.

Demonstrated the boundary's dependence on process parameters through examples.

Abstract

We study the barrier that gives the optimal time to exercise an American option written on a time-dependent Ornstein--Uhlenbeck process, a diffusion often adopted by practitioners to model commodity prices and interest rates. By framing the optimal exercise of the American option as a problem of optimal stopping and relying on probabilistic arguments, we provide a non-linear Volterra-type integral equation characterizing the exercise boundary, develop a novel comparison argument to derive upper and lower bounds for such a boundary, and prove its Lipschitz continuity in any closed interval that excludes the expiration date and, thus, its differentiability almost everywhere. We implement a Picard iteration algorithm to solve the Volterra integral equation and show illustrative examples that shed light on the boundary's dependence on the process's drift and volatility.