Optimal Stopping Under Probability Distortion

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 * We are grateful for comments from seminar and conference participants at Oxford, ETH, University The main part of this author's work was carried out when he was a Nomura Fellow at the Nomura Centre 1 This is the Pre-Published Version.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".

Stopping Behaviors of Naive and Non-Committed Sophisticated Agents When They Distort Probability

“…Finally, a sophisticated agent who is able to commit simply solves the problem once at t = 0 and then sticks to the corresponding stopping plan. The problem of the last type, the so-called pre-committed agent, is actually a static (instead of dynamic) problem and has been solved in various contexts and, in the case of optimal stopping under probability distortion, by Xu and Zhou (2013). The goal of this paper is to study the behaviors of the first two types of agents.…”

“…This is not only because, in the one-dimensional case, pre-committed stopping laws have been obtained rather thoroughly by Xu and Zhou (2013) on which a naïve strategy depends, but also because the fixed-point iteration (2.11) turns out to be much more manageable and does converge to an equilibrium, due to a key technical result (Lemma 3.1) that holds only for a one-dimensional process.…”

“…µ = 1 2 σ 2 ), S t = se σBt for t ≥ 0. If the supremum sup{U(x) : x > 0} is not attained, then Xu and Zhou (2013), p. 255, shows that an optimal stopping time of (4.2) does not exist. In this case, the naïve agent never stops; recall (2.5).…”

“…Pre-committed strategies (i.e. those of Type 2's) have been investigated and obtained thoroughly for geometric Brownian motions by Xu and Zhou (2013) 2 , which are conceptually and technically useful for the present paper.…”

“…We then apply this result to the stopping problem of a geometric Brownian motion and a RDU type of payoff functional involving probability distortion, whose pre-committed stopping strategies have been studied thoroughly by Xu and Zhou (2013). Besides characterizing the stopping behaviors of the naïve and the sophisticated, we are particularly interested in the connection and potential transformation among the three type of agents: the naïve agent in effect solves a pre-committed problem at every state and time who, interestingly, may turn himself into a sophisticate if he carries out several levels (sometimes just one level) of the strategic reasoning.…”