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February 2013 Optimal stopping under probability distortion
Zuo Quan Xu, Xun Yu Zhou
Ann. Appl. Probab. 23(1): 251-282 (February 2013). DOI: 10.1214/11-AAP838


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.”


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Zuo Quan Xu. Xun Yu Zhou. "Optimal stopping under probability distortion." Ann. Appl. Probab. 23 (1) 251 - 282, February 2013.


Published: February 2013
First available in Project Euclid: 25 January 2013

zbMATH: 1286.60038
MathSciNet: MR3059235
Digital Object Identifier: 10.1214/11-AAP838

Primary: 60G40
Secondary: 91G80

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.23 • No. 1 • February 2013
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