Portfolio rebalancing error with jumps and mean reversion in asset prices

Abstract

We analyze the error between a discretely rebalanced portfolio and its continuously rebalanced counterpart in the presence of jumps or mean-reversion in the underlying asset dynamics. With discrete rebalancing, the portfolio’s composition is restored to a set of fixed target weights at discrete intervals; with continuous rebalancing, the target weights are maintained at all times. We examine the difference between the two portfolios as the number of discrete rebalancing dates increases. With either mean reversion or jumps, we derive the limiting variance of the relative error between the two portfolios. With mean reversion and no jumps, we show that the scaled limiting error is asymptotically normal and independent of the level of the continuously rebalanced portfolio. With jumps, we show that the scaled relative error cannot converge to a normal distribution, though asymptotic normality can be recovered if jumps are smaller at higher rebalancing frequencies. For both the mean-reverting and jump-diffusion cases, we derive “volatility adjustments” to improve the approximation of the discretely rebalanced portfolio by the continuously rebalanced portfolio, based on on the limiting covariance between the relative rebalancing error and the level of the continuously rebalanced portfolio. These results are based on strong approximation results for jump-diffusion processes.

Keywords

• Jump-diffusion processes
• portfolio analysis
• strong approximation
• Ito-Taylor expansion