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

Abel Azze, Bernardo D'Auria and Eduardo Garc\'ia-Portugu\'es

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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.

Date: 2022-11, Revised 2024-06
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Published in Stochastics, 96(1):921-946, 2024

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