Mathematical Finance Seminar

November 2, 2000 , 5:30 PM to 7:00 PM

Scott Weiner, Balliol College

The Effect of Stochastic Volatility on Portfolio Optimization with Transaction Costs

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, en- courages 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 el- ements 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 present a numerical procedure to solve the model. We 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. When the risk premium is linear in standard deviation, we show that the optimal strategy is dependent on the state of volatility. We find, however, that following a volatility-independent strategy in this case only marginally a ects the investor's maximum expected long-run growth rate. Finally, we nd that increased transaction activity can occur as a result of the presence of stochastic volatility as well as from subop- timal strategies.