Optimal starting times, stopping times and risk measures for algorithmic trading: target close and implementation shortfall

ABSTRACT We derive explicit recursive formulas for target close (TC) and implementation shortfall (IS) in the Almgren Chriss framework. We explain how to compute the optimal starting and stopping times for IS and TC, respectively, given a minimum trading size. We also show how to add a minimum participation rate constraint (percentage of volume) for both TC and IS. We also study an alternative set of risk measures for the optimization of algorithmic trading curves. We assume a self similar process (eg, Lévy process, fractional Brownian motion or fractal process) and define a new risk measure, the p-variation, which reduces to the variance if the process is a Brownian motion. We deduce the explicit formulas for the TC and IS algorithms under a self-similar process. We show that there is a two-way relationship between self-similar models and a family of risk measures called p-variations. Indeed, it is equivalent to have a self-similar process and calibrate empirically the parameter p for the p-variation, or a Brownian motion and use the p-variation as risk measure instead of the variance. We also show that p can be seen as a fine-tuning parameter that modulates the aggressiveness of the trading protocol: p increases if and only if the TC algorithm starts later and executes faster. Finally, we show how the parameter p of the p-variation can be implied from the optimal starting time of TC. Under this framework p can be viewed as a measure of the joint impact of market impact (ie, liquidity) and volatility.