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Discounted Optimal Stopping Problems for Maxima of Geometric Brownian Motions With Switching Payoffs

Part of: Miscellaneous topics - Partial differential equations Markov processes Stochastic processes Mathematical economics

Published online by Cambridge University Press:  17 March 2021

Pavel V. Gapeev ,
Peter M. Kort  and
Maria N. Lavrutich
Pavel V. Gapeev*
Affiliation:
London School of Economics
Peter M. Kort*
Affiliation:
Tilburg University and University of Antwerp
Maria N. Lavrutich*
Affiliation:
Norwegian University of Science and Technology
*
*Postal address: London School of Economics, Department of Mathematics, Houghton Street, LondonWC2A 2AE, United Kingdom.
**Postal address: Tilburg University, CentER, Department of Econometrics and Operations Research, PO Box 90153, 5000 LE Tilburg, The Netherlands; University of Antwerp, Department of Economics, Prinsstraat 13, 2000 Antwerp 1, Belgium.
***Postal address: Norwegian University of Science and Technology, Department of Industrial Economics and Technology Management, 7491Trondheim, Norway.
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Abstract

We present closed-form solutions to some discounted optimal stopping problems for the running maximum of a geometric Brownian motion with payoffs switching according to the dynamics of a continuous-time Markov chain with two states. The proof is based on the reduction of the original problems to the equivalent 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 maximal solutions of the associated two-dimensional systems of first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of real switching lookback options with fixed and floating sunk costs in the Black–Merton–Scholes model.

Keywords

Discounted optimal stopping problemgeometric Brownian motionrunning maximum processcontinuous-time Markov chainfree-boundary probleminstantaneous stopping and smooth fitnormal reflectionperpetual American and real optionschange-of-variable formula with local time on surfaces

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G44: Martingales with continuous parameter 60J65: Brownian motion
Secondary: 91B25: Asset pricing models 60J27: Continuous-time Markov processes on discrete state spaces 35R35: Free boundary problems
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 1 , March 2021 , pp. 189 - 219
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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