Optimal Portfolio Selection for an Investor with Asymmetric Attitude to Gains and Losses

The description of Cumulative Prospect Theory (CPT) includes three important parts: a value function over outcomes, v(⋅ ); a weighting function over cumulative probabilities, w(⋅ ); CPT-utility as unconditional expectation of the value function v under probability distortion w. In this paper we consider the problem of choosing an CPT-investor’s portfolio in the case of complete market. The problem of finding the optimal portfolio for CPT-investor is to maximize the unconditional expectation of the value function v under probability distortion w over terminal consumption, subject to budget constraint on initial wealth. We find the optimal payoffs for CPT-investor for the classic Black-Scholes environment assuming that there are a single lognormally distributed stock and a risk free bond. We compare the optimal payoffs of CPT-investor with the optimal payoffs of the investor that maximizes expected power utility over terminal payoffs, subject to budget constraint on initial wealth.