Portfolio Selection and Dynamic Behavior in Heston's Stochastic Volatility Model Using a Contingent Claim

Abstract

In the existing literature, little is known about the dynamic behavior of the optimal portfolio in terms of market inputs and arbitrary stochastic factor dynamics in an incomplete market with a stochastic volatility. In this paper, to study the optimal portfolio behavior, we compute and analyze the mean and the variance of the optimal portfolio and of its adjustment speed in terms of market inputs in an incomplete market. The incompleteness arises from the additional source of uncertainty of the volatility in Heston’s stochastic volatility model. We discuss a novel way of completing the incomplete market by introducing an artificial European option, which is then made completely uninteresting to the investor. We then apply the martingale method to the completed market to obtain the optimal portfolio in a closed form. Conducting sensitivity analysis for the mean and the variance of the optimal portfolio process as well as its adjustment speed to the market parameters, we find several interesting investor’s behavioral patterns toward asset price and its volatility shocks. Our results are robust and convergent by the agreement from two simulation methods for different time step increments and the numbers of Monte Carlo simulation paths.