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Discounted optimal stopping problems in first-passage time models with random thresholds

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

Published online by Cambridge University Press:  27 June 2022

Pavel V. Gapeev  and
Hessah Al Motairi
Pavel V. Gapeev*
Affiliation:
London School of Economics
Hessah Al Motairi*
Affiliation:
Kuwait University
*
*Postal address: London School of Economics, Department of Mathematics, Houghton Street, London WC2A 2AE, UK. Email address: p.v.gapeev@lse.ac.uk
**Postal address: Kuwait University, Faculty of Science, Department of Mathematics, PO Box 5969, Safat 13060, Kuwait
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Abstract

We derive closed-form solutions to some discounted optimal stopping problems related to the perpetual American cancellable dividend-paying put and call option pricing problems in an extension of the Black–Merton–Scholes model. The cancellation times are assumed to occur when the underlying risky asset price process hits some unobservable random thresholds. The optimal stopping times are shown to be the first times at which the asset price reaches stochastic boundaries depending on the current values of its running maximum and minimum processes. The proof is based on the reduction of the original optimal stopping problems to the associated free-boundary problems and the solution of the latter problems by means of the smooth-fit and modified normal-reflection conditions. We show that the optimal stopping boundaries are characterised as the maximal and minimal solutions of certain first-order nonlinear ordinary differential equations.

Keywords

Discounted optimal stopping problemgeometric Brownian motionfirst passage timesrunning maximum and minimum processesfree-boundary problemsmooth fit and normal reflectiona change-of-variable formula with local time on surfacesperpetual American optionsrandom dividends

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G44: Martingales with continuous parameter 60J65: Brownian motion
Secondary: 91B25: Asset pricing models 60J60: Diffusion processes 35R35: Free boundary problems
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 3 , September 2022 , pp. 714 - 733
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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