Optimal Trading in a Dynamic Market

We consider the problem of mean-variance optimal agency execution strategies, when the market liquidity and volatility vary randomly in time. Under specific assumptions for the stochastic processes satisfied by these parameters, we construct a Hamilton-Jacobi-Bellman equation for the optimal cost and strategy. We solve this equation numerically and illustrate optimal strategies for varying risk aversion. These strategies adapt optimally to the instantaneous variations of market quality. A fundamental part of agency algorithmic trading in equities and other asset classes is trade scheduling. Given a trade target, that is, a number of shares that must be bought or sold before a fixed time horizon, trade scheduling means planning how many shares will be bought or or sold by each time instant between the beginning of trading and the horizon. Grinold and Kahn (1995) and Almgren and Chriss (2000) suggested that the optimal trajectory could be determined by balancing market impact cost, which leads toward slow trading, against volatility risk which pushes toward rapid completion of the order. This framework leads to an “efficient frontier,” in which the trade schedule is selected from a one-parameter family based on a risk aversion parameter that must be specified by the trading client. While other factors such as anticipated price drift, serial correlation or other short-term signals, and daily patterns, are certainly important, this fundamental “arrival price” framework has proven remarkably robust and useful in designing practical trading systems. A fundamental assumption of most of this work has been that the market parameters, such as liquidity and volatility, are constant or at least have known predictable profiles. This assumption is reasonably accurate for large-cap US stocks. Under that assumption, optimal strategies are static; that is, the trade schedule can be determined before trading starts and is not modified by the new information revealed by price moves during trading. (Almgren and Chriss (2000) ∗New York University Courant Institute and Quantitative Brokers, [email protected]