Portfolio Optimization and a Factor Model in a Stochastic Volatility Market

Abstract

The aim of this paper is to find optimal portfolio strategies for an n-stock extension of the stochastic volatility model proposed in Ref. [2]. It is a modification of [19], and characterizes the dependence by the use of a factor structure. The idea of a factor structure is that the diffusion components of the stocks contain one Brownian motion that is unique for each stock, and a few Brownian motions that all stocks share. Our model can obtain strong correlations between the returns for different stocks without affecting their marginal distributions. This was not possible in Ref. [19]. Further, the number of model parameters does not grow too fast as the number of stocks n grows. We use dynamic programming to solve Merton’s optimization problem for power utility, with utility drawn from terminal wealth. Optimal strategies for n stocks are obtained. We also discuss how to estimate the model from data. This is illustrated with an example.

Keywords: Stochastic control, Portfolio optimization, Verification theorem, Feynman-Kac formula, Stochastic volatility, Non-Gaussian Ornstein-Uhlenbeck process