Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model

Jiannan Zhang,Ping Chen,Shuanming Li,Zhuo Jin

doi:10.3934/jimo.2019133

Abstract

<p style='text-indent:20px;'>This paper investigates a continuous-time mean-variance portfolio selection problem based on a log-return model. The financial market is composed of one risk-free asset and multiple risky assets whose prices are modelled by geometric Brownian motions. We derive a sufficient condition for open-loop equilibrium strategies via forward backward stochastic differential equations (FBSDEs). An equilibrium strategy is derived by solving the system. To illustrate our result, we consider a special case where the interest rate process is described by the Vasicek model. In this case, we also derive the closed-loop equilibrium strategy through the dynamic programming approach.

Highlights

Mean-variance portfolio selection problem can be date back to [15], where a single-period model is investigated
We find that our solutions meet the conventional investment wisdom, that is, rich and young people invest more in risky assets and there is no short sale in the long run
This paper studies the open-loop and closed-loop equilibrium strategies for a mean-variance portfolio selection problem based on a logreturn approach

Summary

Introduction

Mean-variance portfolio selection problem can be date back to [15], where a single-period model is investigated. The contribution of this paper is that we consider the open-loop equilibrium strategy for the mean-variance portfolio selection through the log-return instead of the wealth level in a continuous-time setting. To show how it works, we derive both the open-loop and closed-loop equilibrium strategies for a special case where the interest rate process follows the Vasicek model. We will present the closed-loop equilibrium strategy for the log-return mean-variance portfolio selection problem. Using the approach derived by [4], the optimal strategy is obtained by a system of extended HJB equation for a special case where the interest rate process follows the Vasicek model.

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