Optimal Control of Execution Costs

We derive dynamic optimal trading strategies that minimize the expected cost of trading a large block of equity over a fixed time horizon. Specifically, given a fixed block $\overline{S}$ of shares to be executed within a fixed finite number of periods $T$, and given a price-impact function that yields the execution price of an individual trade as a function of the shares traded and market conditions, we obtain the optimal *sequence* of trades or quot;best execution strategyquot; as a function of market conditions---closed-form expressions in some cases---that minimizes the expected cost of executing $\overline{S}$ within T periods. Our analysis is extended to the portfolio case in which price impact *across* stocks can have an important effect on the total cost of trading a portfolio. We also discuss generalizations to other price impact functions, imposing constraints, and algorithms for performing the optimization numerically. (The text quot;$\overline{S}$quot; comes from a text-processor called TeX and stands for a mathematical symbol which is an upper case S with a bar over it.)