Mean–variance portfolio selection based on a generalized BNS stochastic volatility model

Abstract

We study a mean–variance portfolio selection problem via optimal feedback control based on a generalized Barndorff-Nielsen and Shephard stochastic volatility model, where an investor trades in a generalized Black–Scholes market. The random coefficients of the market are driven by non-Gaussian Ornstein–Uhlenbeck processes that are independent of the underlying multi-dimensional Brownian motion. Our contribution is to explicitly compute and justify optimal portfolios over an admissible set that is large enough to cover some important classes of strategies such as the class of feedback controls of Markov type. Concretely, the mean–variance efficient portfolios and efficient frontiers are explicitly calculated through the method of generalized linear-quadratic control and explicitly constructed solutions to three integro-partial differential equations under a quite mild condition that only requires one stock whose appreciation-rate process is different from the interest-rate process. Related minimum variance issue is also addressed via our main results.

Keywords:

• mean–variance portfolio selection
• non-Gaussian Ornstein–Uhlenbeck process
• generalized Black–Scholes model
• optimal feedback control
• integro-partial differential equation

2000 AMS Subject Classifications :

• 45K05
• 60H10
• 60H30
• 60K30
• 91B06
• 93E20

Acknowledgements

This project is supported by National Natural Science Foundation of China under grant no. 10971249.