Should Stochastic Volatility Matter to the Cost-Constrained Investor?

Abstract

Significant strides have been made in the development of continuous-time portfolio optimization models since Merton (1969). Two independent advances have been the incorporation of transaction costs and time-varying volatility into the investor's optimization problem. Transaction costs generally inhibit investors from trading too often; they force the investor to choose between holding a suboptimal portfolio and reallocating the portfolio to the optimal allocation by incurring a fee. Several models, including Eastham and Hastings (1988) and Davis and Norman (1990), show that the investor can experience periods of passive investment (i.e., periods without transaction activity) as a result of transaction costs. Time-varying volatility, on the other hand, encourages trading activity, as it can result in an evolving optimal allocation of resources, as in Karatzas (1989). We examine the two-asset portfolio optimization problem when both elements are present. We show that the transaction cost framework in Korn (1998) can be extended to include a stochastic volatility process. We then specify a transaction cost model with stochastic volatility, based on Morton and Pliska (1995), and show that when the risk premium is linear in variance, the optimal strategy for the investor is independent of the level of volatility in the risky asset. We call this the Variance Invariance Principle.