Partial liquidation under reference-dependent preferences

Vicky Henderson,Jonathan Muscat

doi:10.1007/s00780-020-00421-8

Abstract

We propose a multiple optimal stopping model where an investor can sell a divisible asset position at times of her choosing. Investors have S-shaped reference-dependent preferences, whereby utility is defined over gains and losses relative to a reference level and is concave over gains and convex over losses. For a price process following a time-homogeneous diffusion, we employ the constructive potential-theoretic solution method developed by Dayanik and Karatzas (Stoch. Process. Appl. 107:173–212, 2003). As an example, we revisit the single optimal stopping model of Kyle et al. (J. Econ. Theory 129:273–288, 2006) to allow partial liquidation. In contrast to the extant literature, we find that the investor may partially liquidate the asset at distinct price thresholds above the reference level. Under other parameter combinations, the investor sells the asset in a block, either at or above the reference level.

Highlights

Prospect theory was proposed by Kahneman and Tversky [22] and extended by Tversky and Kahneman [33]
Optimal stopping models employing reference-dependent preferences have been developed in order to understand the dynamic behaviour of individuals with such preferences and to see to what extent the theory can be used to explain both experimental and empirically observed behaviour
We extend the model of Kyle et al [24] to consider the question of partial liquidation of assets

Summary

Introduction

Prospect theory was proposed by Kahneman and Tversky [22] and extended by Tversky and Kahneman [33]. We propose an infinite-horizon optimal stopping model where an investor with S-shaped reference-dependent preferences can sell her divisible asset position at times of her choosing in the future. Reference-dependence is a long-standing explanation of why individual investors tend to sell winners too early and ride losers too long, a behaviour called the disposition effect (Shefrin and Statman [31], Odean [28]) In this vein, Kyle et al [24], Henderson [14], Barberis and Xiong [4] and Ingersoll and Jin [20] contribute optimal stopping models for an investor with reference-dependent preferences under differing assumptions. We focus in this paper on reference-dependent S-shaped preferences in the absence of probability weighting and extend the literature in the direction of holding a quantity of asset rather than just one unit

The partial liquidation problem

Reference-dependent preferences

The price process

The general problem

Piecewise exponential utility and drifting Brownian motion

Discussion and conclusions