Realization Utility with Path-Dependent Reference Points

In this paper, we propose a model to study the impact of path-dependent reference points on optimal trading strategies of a realization utility investor, where the reference points are initial purchase prices adjusted dynamically by subsequent paper gains and losses. By homogeneity, this optimal trading problem can be reformulated as a new one with a unique state variable $x$ (wealth-reference ratio), which can be solved analytically. Importantly, we find that the introduction of path-dependent reference points can generate two interesting effects: (a) a discount effect, i.e., a constant subjective discount factor in the original problem becomes stochastic in the new problem; and (b) a mean-reverting effect, i.e., the state variable $x$ follows a mean-reverting process. These two effects offset each other in the state of paper gain ($x>1$) while getting reinforced in the state of paper loss ($x<1$), leading to a disposition effect even if a subjective discount factor in the original problem is zero. Finally, the model can be easily extended to incorporate other factors such as asymmetric adaptation of reference points, jump risks in the underlying stocks, and liquidation shocks.