Dynamic Optimal Mean-Variance Investment with Mispricing in the Family of 4/2 Stochastic Volatility Models

Yumo Zhang

doi:10.3390/math9182293

Abstract

This paper considers an optimal investment problem with mispricing in the family of 4/2 stochastic volatility models under mean–variance criterion. The financial market consists of a risk-free asset, a market index and a pair of mispriced stocks. By applying the linear–quadratic stochastic control theory and solving the corresponding Hamilton–Jacobi–Bellman equation, explicit expressions for the statically optimal (pre-commitment) strategy and the corresponding optimal value function are derived. Moreover, a necessary verification theorem was provided based on an assumption of the model parameters with the investment horizon. Due to the time-inconsistency under mean–variance criterion, we give a dynamic formulation of the problem and obtain the closed-form expression of the dynamically optimal (time-consistent) strategy. This strategy is shown to keep the wealth process strictly below the target (expected terminal wealth) before the terminal time. Results on the special case without mispricing are included. Finally, some numerical examples are given to illustrate the effects of model parameters on the efficient frontier and the difference between static and dynamic optimality.

Highlights

The development of continuous-time stochastic volatility models is deemed crucial in the field of modern finance
The attraction of stochastic volatility models mainly resides in their capacity to explain many stylized facts observed in the financial market such as fat tails, the leverage effect and the volatility smile/skew on implied volatility surfaces
Motivated by the above aspects, within the framework introduced by Pedersen and Peskir [24] to overcome the time inconsistency under mean–variance criterion, in this paper we study a mean–variance portfolio selection problem that takes into consideration the family of 4/2 stochastic volatility models and mispricing simultaneously

Summary

Introduction

The development of continuous-time stochastic volatility models is deemed crucial in the field of modern finance. Motivated by the above aspects, within the framework introduced by Pedersen and Peskir [24] to overcome the time inconsistency under mean–variance criterion, in this paper we study a mean–variance portfolio selection problem that takes into consideration the family of 4/2 stochastic volatility models and mispricing simultaneously. The market model incorporates the 4/2 model and mispricing simultaneously; By making an assumption on the model parameters, a verification theorem is provided to guarantee that the candidate solution to the HJB equation is the optimal value function, and the admissibility of the optimal strategy is verified; We derive both the statically optimal (pre-commitment) and the dynamically optimal (time-consistent) strategies explicitly for the mean–variance problem.

Formulation of the Problem

Solution to the Unconstrained Problem

Static and Dynamic Optimality of the Problem

Numerical Examples

Conclusions