Optimal stopping problems for running minima with positive discounting rates

Pavel V. Gapeev

Statistics & Probability Letters, 2020, vol. 167, issue C

Abstract: We present analytic solutions to some optimal stopping problems for the running minimum of a geometric Brownian motion with exponential positive discounting rates. The proof is based on the reduction of the original problems to the associated free-boundary problems and the solution of the latter problems by means of the smooth-fit and normal-reflection conditions. We show that the optimal stopping boundaries are determined as the minimal solutions of certain first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of perpetual dual American lookback options with fixed and floating strikes in the Black–Merton–Scholes model from the point of view of short sellers.

Keywords: Optimal stopping problem; Exponential positive discounting rate; Brownian motion; Running minimum process; Free-boundary problem; A change-of-variable formula with local time on surfaces (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (2)

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DOI: 10.1016/j.spl.2020.108899

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