Comparison Between Mean-Variance and Monotone Mean-Variance Preferences Under Jump Diffusion and Stochastic Factor Model

This paper compares the optimal investment problems based on monotone mean-variance (MMV) and mean-variance (MV) preferences in a Lévy market with an untradable stochastic factor. It is an open question proposed by Trybuła and Zawisza. Using the dynamic programming and Lagrange multiplier methods, we get the Hamilton-Jacobi-Bellman-Isaacs (HJBI) and Hamilton-Jacobi-Bellman (HJB) equations corresponding to the two investment problems. The equations are transformed into a new-type parabolic equation, from which the optimal strategies under both preferences are derived. We prove that the two optimal strategies and value functions coincide if and only if an important market assumption holds. When the assumption is violated, MMV investors act differently from MV investors. Thus, we conclude that the difference between continuous-time MMV and MV portfolio selections is due to the discontinuity of the market. In addition, we derive the efficient frontier and analyze the economic impact of the jump diffusion risky asset. We also provide empirical evidence to demonstrate the validity of the assumption in real financial markets.