Discounted optimal stopping problems for maxima of geometric Brownian motions with switching payoffs. (English) Zbl 1497.60059
The authors present closed-form solutions to some discounted optimal stopping problems for running the maximum of geometric Brownian motion with payoff switching based on the dynamics of a two-state continuous-time Markov chain [X. Guo, J. Appl. Probab. 38, No. 2, 464–481 (2001; Zbl 0988.60038); X. Guo and Q. Zhang, SIAM J. Appl. Math. 64, No. 6, 2034–2049 (2004; Zbl 1061.90082)]. The proof is based on reducing the original problems to equivalent free-boundary problems and solving the latter problems using smooth fit and normal refractive conditions. It is shown that the optimal termination boundary is determined as the maximum solution of all related two-dimensional systems associated with the first-order nonlinear ordinary differential equations of the two-dimensional solution of the maximum boundary. The obtained results relate to the valuation of real conversion look-back options with fixed and floating sunk costs with real switching look-back options in the Black-Merton-Scholes model (see [A. K. Dixit and R. S. Pindyck, Investment under uncertainty. Princeton: Princeton University Press. 303–309 (1994)] for examples of standard real options with switching payoffs).
Reviewer: Krzysztof J. Szajowski (Wrocław)
MSC:
| 60G40 | Stopping times; optimal stopping problems; gambling theory |
| 60G44 | Martingales with continuous parameter |
| 60J65 | Brownian motion |
| 60J27 | Continuous-time Markov processes on discrete state spaces |
| 35R35 | Free boundary problems for PDEs |
Keywords:
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 surfacesReferences:
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