This paper examines the problem of choosing the optimal portfolio for an investor with asymmetric attitude to gains and losses described in the cumulative prospect theory (CPT) of A. Tversky and D. Kahneman. We consider the problem of finding the portfolio that maximizes CPT-utility over the set of all feasible portfolios along the mean–variance efficient frontier under some conditions on the stochastic behavior both of the portfolio price and the discount factor. It should be noted that since the value function of the CPT-utility is S-shaped and not-differentiable in the origin, numerical approach to the optimization of CPT-utility is not trivial. We introduce a model with stochastic behavior of both the portfolio price and a discount factor. Following routing procedure for Black-Scholes environment we find the closed-form solution for CPT-utility in a simple particular case. Nevertheless, it turned out that in the general case the closed-form solution can not be obtained. Therefore, we use computational methods for studying the properties of CPT-utility. The paper shows that this approach to the optimal portfolio choice for CPT-investor might be promising.