Optimal stopping under probability distortion

TL;DR

This paper introduces a new method for solving optimal stopping problems with probability distortion, allowing for flexible payoff and distortion functions, and provides economic interpretations of common trading strategies.

Contribution

It develops a novel approach based on distribution and quantile function optimization, addressing time inconsistency in probability-distorted optimal stopping problems.

Findings

Solution method applicable to various payoff and distortion shapes

Justification of common stock trading liquidation strategies

Framework for deriving optimal stopping times from distributions

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".