Xunyu Zhou

Oxford and CUHK

Optimal Stopping under Probability Distortion

Abstract: We formulate an optimal stopping problem 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 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. In particular, we show that the the exit time of an interval (corresponding to the ``cut-loss-or-stop-gain" strategy widely adopted in stock trading) is endogenously optimal for problems with convex distortion functions, including ones where distortion is absent. We also discuss economical interpretations of the results.