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Optimal stopping under probability distortion

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  • Zuo Quan Xu
  • Xun Yu Zhou

Abstract

We formulate an optimal stopping problem for a geometric Brownian motion where the probability scale is distorted by a general nonlinear function. The problem is inherently time inconsistent due to the Choquet integration involved. We develop a new approach, based on a reformulation of the problem where one optimally chooses the probability distribution or quantile function of the stopped state. An optimal stopping time can then be recovered from the obtained distribution/quantile function, either in a straightforward way for several important cases or in general via the Skorokhod embedding. This approach enables us to solve the problem in a fairly general manner with different shapes of the payoff and probability distortion functions. We also discuss economical interpretations of the results. In particular, we justify several liquidation strategies widely adopted in stock trading, including those of "buy and hold", "cut loss or take profit", "cut loss and let profit run" and "sell on a percentage of historical high".

Suggested Citation

  • Zuo Quan Xu & Xun Yu Zhou, 2011. "Optimal stopping under probability distortion," Papers 1103.1755, arXiv.org, revised Feb 2013.
  • Handle: RePEc:arx:papers:1103.1755
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    File URL: http://arxiv.org/pdf/1103.1755
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    Cited by:

    1. Zuo Quan Xu, 2018. "Quantile optimization under derivative constraint," Papers 1803.02546, arXiv.org.
    2. Henderson, Vicky & Hobson, David & Tse, Alex S.L., 2017. "Randomized strategies and prospect theory in a dynamic context," Journal of Economic Theory, Elsevier, vol. 168(C), pages 287-300.
    3. Zuo Quan Xu, 2014. "A Note on the Quantile Formulation," Papers 1403.7269, arXiv.org, revised Apr 2014.
    4. repec:eee:insuma:v:77:y:2017:i:c:p:111-118 is not listed on IDEAS
    5. Paul Viefers & Philipp Strack, 2014. "Too Proud to Stop: Regret in Dynamic Decisions," Discussion Papers of DIW Berlin 1401, DIW Berlin, German Institute for Economic Research.
    6. Zuo Quan Xu, 2013. "A New Characterization of Comonotonicity and its Application in Behavioral Finance," Papers 1311.6080, arXiv.org, revised Jun 2014.

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