Optimal control of execution costs for portfolios

We derive dynamic optimal trading strategies that minimize the expected cost of trading blocks of securities over a xed time horizon. Given xed blocks si of shares of stock i to be traded within a nite number of periods T , i = 1; : : : ; n, and given price-impact functions that yield the execution price of an individual trade as a function of the shares of stock i traded and current market conditions, we obtain the optimal sequence of trades as a function of market conditions|closed-form expressions in some cases|that minimizes the expected cost of executing si within T periods. We also propose an approximation algorithm for incorporating constraints, a particularly important extension in practice. To illustrate the practical relevance of our methods, we apply them to a hypothetical portfolio of 25 stocks by estimating their price-impact functions using historical trade data from 1996 and deriving the optimal execution strategies for the portfolio, and by performing several Monte Carlo simulation experiments. This research is part of the MIT Laboratory for Financial Engineering's Transactions Costs Project. We are grateful to Investment Technology Group, the National Science Foundation (Grant Nos. DMI{9610486 and SBR{9709976), and the sponsors of the LFE for nancial support. We thank Dave Cushing, Chris Darnell, Robert Ferstenberg, Rohit D'Souza, John Heaton, Leonid Kogan, Bruce Lehmann, Greg Peterson, Jiang Wang, and seminar participants at Columbia, Harvard, London Business School, MIT, Northwestern, the NYSE Conference on Best Execution, the Society of Quantitative Analysts, and Yale for helpful comments and discussion. Boeing Professor of Operations Research, MIT Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology, E53-359, Cambridge, MA 02142{1347. Portfolio Strategist, Long Term Capital Management, One East Weaver St., Greenwich, CT 06831. Harris & Harris Group Professor, MIT Sloan School of Management and Operations Research Center, Cambridge, MA 02142{1347.