Optimal Portfolio Problem Using Entropic Value at Risk: When the Underlying Distribution is Non-Elliptical

This paper is devoted to study the optimal portfolio problem. Harry Markowitz's Ph.D. thesis prepared the ground for the mathematical theory of finance. In modern portfolio theory, we typically find asset returns that are modeled by a random variable with an elliptical distribution and the notion of portfolio risk is described by an appropriate risk measure. In this paper, we propose new stochastic models for the asset returns that are based on Jumps- Diffusion (J-D) distributions. This family of distributions are more compatible with stylized features of asset returns. On the other hand, in the past decades, we find attempts in the literature to use well-known risk measures, such as Value at Risk and Expected Shortfall, in this context. Unfortunately, one drawback with these previous approaches is that no explicit formulas are available and numerical approximations are used to solve the optimization problem. In this paper, we propose to use a new coherent risk measure, so-called, Entropic Value at Risk(EVaR), in the optimization problem. For certain models, including a jump-diffusion distribution, this risk measure yields an explicit formula for the objective function so that the optimization problem can be solved without resorting to numerical approximations.