Optimal starting times, stopping times and risk measures for algorithmic trading: Target Close and Implementation Shortfall

TL;DR

This paper develops explicit formulas for optimal trading times and risk measures in algorithmic trading, incorporating self-similar processes and introducing p-variation as a flexible risk metric.

Contribution

It introduces a new framework linking self-similar processes with p-variation risk measures, providing explicit formulas for Target Close and Implementation Shortfall algorithms.

Findings

Explicit recursive formulas for TC and IS in the Almgren-Chriss framework.

A novel risk measure, p-variation, generalizing variance for self-similar processes.

Demonstration of the equivalence between self-similar models and p-variation risk measures.

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, PVol) for both TC and IS. We also study an alternative set of risk measures for the optimisation of algorithmic trading curves. We assume a self-similar process (e.g. Levy 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 formula for the TC and IS algorithms under a self-similar process. We show that there is an equivalence between selfsimilar models and a family of risk measures called p-variations: assuming a self-similar process and calibrating empirically the parameter p for the p-variation yields the same result as assuming a Brownian motion and using the p-variation as risk measure instead of the variance. We also show that p can be seen as a measure of the aggressiveness: 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, and that under this framework p can be viewed as a measure of the joint impact of market impact (i.e. liquidity) and volatility.