Continuous-Time Portfolio Optimisation for a Behavioural Investor with Bounded Utility on Gains

TL;DR

This paper investigates continuous-time portfolio optimization for a behavioral investor with bounded utility on gains, establishing conditions for the existence of optimal strategies under Cumulative Prospect Theory.

Contribution

It derives a necessary condition for optimal strategy existence in a continuous-time setting with bounded utility on gains, linking probability distortion and utility functions.

Findings

Necessary condition on probability distortion function for strategy existence

Optimal portfolio exists if distortion function converges sufficiently slowly

Condition is sharp under additional assumptions

Abstract

This paper examines an optimal investment problem in a continuous-time (essentially) complete financial market with a finite horizon. We deal with an investor who behaves consistently with principles of Cumulative Prospect Theory, and whose utility function on gains is bounded above. The well-posedness of the optimisation problem is trivial, and a necessary condition for the existence of an optimal trading strategy is derived. This condition requires that the investor's probability distortion function on losses does not tend to 0 near 0 faster than a given rate, which is determined by the utility function. Under additional assumptions, we show that this condition is indeed the borderline for attainability, in the sense that for slower convergence of the distortion function there does exist an optimal portfolio.