A Monte-Carlo method for portfolio optimization under partially observed stochastic volatility

In this paper we implement an algorithm for the optimal selection of a portfolio of stock and risk-free asset under the stochastic volatility (SV) model with discrete observation and trading. The SV model extends the classical Black-Scholes model (1973) by allowing the noise intensity (volatility) to be random. The main assumption is that the portfolio manager has discrete access to the continuous-time stock prices; as a consequence the volatility is not observed directly. In this partial information situation, one cannot hope for an arbitrarily accurate estimate of the stochastic volatility. Using instead a new type of optimal stochastic filtering, and its associated particle method due to del Moral, Jacod, and Protter (1990), our algorithm, of the "smart" Monte-Carlo-type, approximates the new Hamilton-Jacobi-Bellman equation that is required for solving the stochastic control problem that is defined by the portfolio optimization question.