Discounted Optimal Stopping Problems for Maxima of Geometric Brownian Motions With Switching Payoffs. (March 2021)

• Record Type: Journal Article
• Authors: Gapeev, Pavel V.; Kort, Peter M.; Lavrutich, Maria N.
• Is Part Of: Advances in applied probability. Volume 53:Number 1(2021)
• Page Start: 189
• Page End: 219
• Publication Date: 2021-03
• Subjects: Discounted optimal stopping problem, -- geometric Brownian motion, -- running maximum process, -- continuous-time Markov chain, -- free-boundary problem, -- instantaneous stopping and smooth fit, -- normal reflection, -- perpetual American and real options, -- change-of-variable formula with local time on surfaces; 60G40, -- 60G44, -- 60J65, -- 91B25, -- 60J27, -- 35R35; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2020.57
• Languages: English
• ISSNs: 0001-8678

Abstract

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.