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Discounted optimal stopping problems for the maximum process

Part of: Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

Jesper Lund Pedersen
Jesper Lund Pedersen*
Affiliation:
University of Aarhus
*
∗ Postal address: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, 8000 Aarhus C, Denmark. Email address: jesper1@imf.au.dk
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Abstract

The maximality principle [6] is shown to be valid in some examples of discounted optimal stopping problems for the maximum process. In each of these examples explicit formulas for the value functions are derived and the optimal stopping times are displayed. In particular, in the framework of the Black-Scholes model, the fair prices of two lookback options with infinite horizon are calculated. The main aim of the paper is to show that in each considered example the optimal stopping boundary satisfies the maximality principle and that the value function can be determined explicitly.

Keywords

Optimal stoppingdiscountingvalue functionmaximum processdiffusionmaximality principlefree boundary problemprinciple of smooth fitinfinitesimal operator(reflected) Brownian motionBessel processBlack-Scholes modelgeometric Brownian motionlookback option

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60G44: Martingales with continuous parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 4 , December 2000 , pp. 972 - 983
Copyright
Copyright © by the Applied Probability Trust 2000 

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References

Beibel, M. and Lerche, H. R. (1997). A new look at warrant pricing and related optimal stopping problems. Empirical Bayes, sequential analysis and related topics in statistics and probability. Statist. Sinica 7, 93–108.Google Scholar
Conze, A. and Viswanathan, (1991). Path dependent options: the case of lookback options. J. Finance 46, 1893–1907.CrossRefGoogle Scholar
Graversen, S. E. and Peskir, G. (1997). On the Russian option: the expected waiting time. Theory Prob. Appl. 42, 416–425.Google Scholar
Graversen, S. E. and Peskir, G. (1998). Optimal stopping and maximal inequalities for geometric Brownian motion. J. Appl. Prob. 35, 856–872.Google Scholar
Pedersen, J. L. (2000). Best bounds in Doob's maximal inequality for Bessel processes. J. Multivar. Anal. 75, 36–46.CrossRefGoogle Scholar
Peskir, G. (1998). Optimal stopping of the maximum process: the maximality principle. Ann. Prob. 26, 1614–1640.CrossRefGoogle Scholar
Revuz, D., and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, New York.Google Scholar
Rogers, L. C. G., and Williams, D. (1994). Diffusions, Markov Processes, and Martingales; Vol. 1: Foundations, 2nd edn. John Wiley, New York.Google Scholar
Shepp, L. A., and Shiryaev, A. N. (1993). The Russian option: reduced regret. Ann. Appl. Prob. 3, 631–640.Google Scholar
Shepp, L. A., and Shiryaev, A. N. (1994). A new look at the Russian option. Theory Prob. Appl. 39, 103–119.Google Scholar