Portfolio Optimization under Partial Information: Stochastic Volatility in a Hidden Markov Model

We consider a multi-stock market model where prices satisfy a stochastic differential equation (SDE) with instantaneous rates of return modeled as an unobserved continuous time, finite state Markov chain. The investor wishes to maximize the expected utility of terminal wealth but only the prices are available to him for his investment decisions. Thus we have a hidden Markov model (HMM) for the stock returns. Extending the results in [9] to stochastic volatility we obtain explicit optimal trading strategies in terms of the unnormalized filter of the drift process. We propose a simple volatility model in which the volatility is a function of the filter for the drift process. When applied to historical prices, the optimal strategies clearly outperform the strategies based on constant volatility.