Discounted optimal stopping for maxima in diffusion models with finite horizon

We present a solution to some discounted optimal stopping problem for the maximum of a geometric Brownian motion on a finite time interval. The method of proof is based on reducing the initial optimal stopping problem with the continuation region determined by an increasing continuous boundary surface to a parabolic free-boundary problem. Using the change-of-variable formula with local time on surface we show that the optimal boundary can be characterized as a unique solution of a nonlinear integral equation. The result can be interpreted as pricing American fixed-strike lookback option in a diffusion model with finite time horizon. -- Discounted optimal stopping problem ; finite horizon ; geometric Brownian motion ; maximum process ; parabolic free-boundary problem ; smooth fit ; normal reflection ; a nonlinear Volterra integral equation of the second kind ; boundary surface ; a change-of-variable formula with local time on surfaces ; American lookback option problem