Some results on optimally exercising American put options for time-inhomogeneous processes

Abstract

We solve the finite-horizon, discounted, Mayer optimal stopping problem, with the gain function coming for exercising an American put option, and the underlying process modeled by adiffusion with constant volatility and a time-dependent drift satisfying certain regularity conditions. Both the corresponding value function and optimal stopping boundary are proved to be Lipschitz continuous away from the terminal time. The optimal stopping boundary is characterizedas the unique solution, up to mild regularity conditions, of the free-boundary equation. When the underlying process has Gaussian marginal distributions, more tractable expressions for the pricing formula and free-boundary equation are provided. Finally, we check that an Ornstein&-Uhlenbeck process with time-dependent parameters fulfills the required conditions assumed throughout the paper.