Optimal stopping problems for running minima with positive discounting rates

Pavel V. Gapeev

LSE Research Online Documents on Economics from London School of Economics and Political Science, LSE Library

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: a change-of-variable formula with local time on surfaces; Brownian motion; exponential positive discounting rate; free-boundary problem; optimal stopping problem; running minimum process (search for similar items in EconPapers)
JEL-codes: C1 (search for similar items in EconPapers)
Pages: 12 pages
Date: 2020-12-01
New Economics Papers: this item is included in nep-isf and nep-ore
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Published in Statistics and Probability Letters, 1, December, 2020, 167. ISSN: 0167-7152

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