  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.
