Discounted optimal stopping for maxima in diffusion models with finite horizon

Pavel V. Gapeev

No 2006-057, SFB 649 Discussion Papers from Humboldt University Berlin, Collaborative Research Center 649: Economic Risk

Abstract: 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.

Date: 2006
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