Optimal Investment Problem Under Behavioral Setting: A Lagrange Duality Perspective

Abstract

In this paper, we consider the optimal investment problem with both probability distor- tion/weighting and general non-concave utility functions with possibly finite number of inflection points. Our model contains the model under cumulative prospect theory (CPT) as a special case, which has inverse S-shaped probability weighting and S-shaped utility function (i.e. one inflection point). Existing literature have shown the equivalent relationships (strong duality) between the concavified problem and the original one by either assuming the presence of probability weighting or the non-concavity of utility functions, but not both. In this paper, we combine both features and propose a step-wise relaxation method to handle general non-concave utility functions and probability distortion functions. The necessary and sufficient conditions on eliminating the duality gap for the Lagrange method based on the step-wise relaxation have been provided under this circumstance. We have applied this solution method to solve in closed-form several representative examples in mathematical behavioral finance including the CPT model, Value-at-Risk based risk management (VAR-RM) model with probability distortions, Yarri’s dual model and the goal reaching model. We obtain a closed-form optimal trading strategy for a special example of the CPT model, where a “distorted” Merton line has been shown exactly. The slope of the “distorted” Merton line is given by an inflation factor multiplied by the standard Merton ratio, and an interesting finding is that the inflation factor is solely dependent on the probability distortion rather than the non-concavity of the utility function.