Market impact models choosing static and adaptive strategies for the optimal execution of portfolio transactions

Abstract

Volume Weighted Average Price, or in short terms VWAP has taken a leading role in algorithm trading for a long time until a signi ficant move occurred towards implementation shortfall algorithms or arrival price algorithms. These algorithms, however, enable to create the trade-off between the market impact cost of fast execution and the volatility risk due to slow execution which is essential for determining optimal trade schedules. Almgren and Chriss (2000) choose execution algorithms that are path-independent (also called static) in such a way that their trade schedules are deterministic and do not change the speed of trading in respect to price motions. In this context, models with market impacts are differentiated which are given linear and non-linear with different results observable leading to exact solutions on the one hand and to slightly approximative solutions on the other.<br />In the mean-variance framework, an optimal execution strategy is reached when the expected cost is minimized for a defi ned maximum level of variance or conversely. Following this framework, an improvement for the static exe- cution strategies can be provided due to adaptive trading. This can already be shown by simple strategies that update only once during trading at some intermediary time in order to react to stock price movements until that mo- ment. These so-called single-update strategies thus offer much better results than the static trajectories from Almgren and Chriss (2000) as either lower expected cost for the same level of variance or lower variance for the same expected cost is obtained.<br />Using optimal dynamic strategies instead even improves the fi rst results of the single-update strategies. To any requested degree of precision, they are computable with an application of the dynamic programming technique.<br />This approach simpli es the determination of optimal dynamic strategies to a series of single-period convex constrained optimization problems. It shows that the resulting strategies are "aggressive-in-the-money" which implicates an increase in execution speed when the price moves towards the trader's favor. As a consequence, parts of the trading gains are used to reduce the risk that has already come up.