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Discounted Optimal Stopping for Maxima in Diffusion Models with Finite Horizon

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  • Pavel V. Gapeev

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.

Suggested Citation

  • Pavel V. Gapeev, 2006. "Discounted Optimal Stopping for Maxima in Diffusion Models with Finite Horizon," SFB 649 Discussion Papers SFB649DP2006-057, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  • Handle: RePEc:hum:wpaper:sfb649dp2006-057
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    File URL: http://sfb649.wiwi.hu-berlin.de/papers/pdf/SFB649DP2006-057.pdf
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    References listed on IDEAS

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

    1. Yerkin Kitapbayev, 2015. "The British Lookback Option with Fixed Strike," Applied Mathematical Finance, Taylor & Francis Journals, vol. 22(3), pages 238-260, July.

    More about this item

    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;

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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