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Optimal Double Stopping Problems for Maxima and Minima of Geometric Brownian Motions

Pavel V. Gapeev (), Peter Kort, Maria N. Lavrutich () and Jacco J. J. Thijssen ()
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Pavel V. Gapeev: London School of Economics
Maria N. Lavrutich: Norwegian University of Science and Technology
Jacco J. J. Thijssen: University of York

Methodology and Computing in Applied Probability, 2022, vol. 24, issue 2, 789-813

Abstract: Abstract We present closed-form solutions to some double optimal stopping problems with payoffs representing linear functions of the running maxima and minima of a geometric Brownian motion. It is shown that the optimal stopping times are th first times at which the underlying process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum. The proof is based on the reduction of the original double optimal stopping problems to sequences of single optimal stopping problems for the resulting three-dimensional continuous Markov process. The latter problems are solved as the equivalent free-boundary problems by means of the smooth-fit and normal-reflection conditions for the value functions at the optimal stopping boundaries and the edges of the three-dimensional state space. We show that the optimal stopping boundaries are determined as the extremal solutions of the associated first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of perpetual real double lookback options with floating sunk costs in the Black-Merton-Scholes model.

Keywords: Perpetual real double lookback options; the Black-Merton-Scholes model; Geometric Brownian motion; Double optimal stopping problem; First hitting time; Free-boundary problem; Instantaneous stopping and smooth fit; Normal reflection; A change-of-variable formula with local time on surfaces; Primary 60G40; 34B40; 91G20; Secondary 60J60; 60J65; 91B70 (search for similar items in EconPapers)
JEL-codes: G13 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s11009-022-09959-w

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