Optimal Execution Strategy For Institutional Investors

Based on Almgren & Chriss(1999,2000), we study the optimal trading strategy by different degrees in order for the institutional investors, who are risk averse, to get a trade-off between the variance and the expectation of the execution costs. First, we study the optimal execution strategy with incomplete liquidation under the assumption that the asset price follows an arithmetic Brownian motion. Huberman & Stanzl(2001) shows that the permanent impact is the linear function of the liquidation velocity, while the temporary impact has not yet been determined. So we assume that the permanent impact is the linear function of the liquidation velocity, but the temporary impact is the sum of linear function of the liquidation velocity and random shocks. We study the optimal strategy which minimizes the sum of the variance and the expectation of the execution costs. By using the variation method and stochastic theory, an explicit solution is derived in this paper. It is shown that this strategy is a linear combination of the hyperbolic sine function about time, which decreases the expectation of the execution costs dramatically. The strategy is related to all the factors, such as the temporary impact, excess interest rate,stock volatility, risk preference, the position size, but it has no relation to the permanent impact. The optimal trading strategy of the incomplete liquidation is not equivalent to the one of complete liquidation. Since the optimal execution time is endogenous, the optimal execution time and the optimal strategy are studied in the case of different parameters and the cost under optimal liquidation time is also discussed.If the execution time is small, the optimal execution strategy is reasonable, but on the contrary if the execution time is long, it is unreasonable. So we assume that the asset price follows the geometric Brownian motion. Under this situation, the optimal strategy is also studied which minimizes the sum of the variance and the expectation of the execution costs. Utilizing variations, we find that the optimal strategy is a solution to a second-order differential equation and the optimal strategies are given by the numerical solution using the difference method. In the end, the case with one stock is extended into the one with many tradable stocks...