Discounted optimal stopping for maxima in diffusion models with finite horizon
Pavel V. Gapeev, WIAS Berlin
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 surfaces 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.
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