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Discounted Optimal Stopping for Maxima in Diffusion Models with Finite Horizon

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  • Pavel V. Gapeev
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    Abstract

    We present a solution to some discounted optimal stopping problem for the maximum of a geometric Brownian motion on a finite time interval. The method of proof is based on reducing the initial optimal stopping problem with the continuation region determined by an increasing continuous boundary surface to a parabolic free-boundary problem. Using the change-of-variable formula with local time on surfaces we show that the optimal boundary can be characterized as a unique solution of a nonlinear integral equation. The result can be interpreted as pricing American fixed-strike lookback option in a diffusion model with finite time horizon.

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    File URL: http://sfb649.wiwi.hu-berlin.de/papers/pdf/SFB649DP2006-057.pdf
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    Bibliographic Info

    Paper provided by Sonderforschungsbereich 649, Humboldt University, Berlin, Germany in its series SFB 649 Discussion Papers with number SFB649DP2006-057.

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    Length: 22 pages
    Date of creation: Sep 2006
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    Handle: RePEc:hum:wpaper:sfb649dp2006-057

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    Related research

    Keywords: Discounted optimal stopping problem; finite horizon; geometric Brownian motion; maximum process; parabolic free-boundary problem; smooth fit; normal reflection; a nonlinear Volterra integral equation of the second kind; boundary surface; a change-of-variable formula with local time on surfaces; American lookback option problem;

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    1. S. D. Jacka, 1991. "Optimal Stopping and the American Put," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 1(2), pages 1-14.
    2. Gapeev, P.V. & Peskir, G., 2006. "The Wiener disorder problem with finite horizon," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 116(12), pages 1770-1791, December.
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