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

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.

Keywords:

• American option
• free-boundary problem
• optimal stopping
• Ornstein–Uhlenbeck
• time-inhomogeneity

Acknowledgments

The authors acknowledge the comments of two anonymous referees that led to significant improvements in the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

The authors acknowledge support from grants PID2020-116694GB-I00 (first and second authors), and PGC2018-097284-B-100 (third author), funded by MCIN/AEI/10.13039/501100011033 and by ‘ERDF A way of making Europe’. The research of the first and third was also supported by the Community of Madrid through the framework of the multi-year agreement with Universidad Carlos III de Madrid in its line of action ‘Excelencia para el Profesorado Universitario’ (EPUC3M13).