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

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  • Mauricio Labadie
  • Charles-Albert Lehalle

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.

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File URL: http://arxiv.org/pdf/1205.3482
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Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 1205.3482.

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Date of creation: May 2012
Date of revision: Dec 2013
Publication status: Published in Journal of Investment Strategies, Volume 3, Issue 2, 2014
Handle: RePEc:arx:papers:1205.3482

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Web page: http://arxiv.org/

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  1. Sophie Laruelle & Charles-Albert Lehalle & Gilles Pagès, 2009. "Optimal split of orders across liquidity pools: a stochastic algorithm approach," Working Papers hal-00422427, HAL.
  2. Charles-Albert Lehalle, 2013. "Market Microstructure Knowledge Needed for Controlling an Intra-Day Trading Process," Papers 1302.4592, arXiv.org.
  3. Guéant, Olivier & Lehalle, Charles-Albert & Tapia, Joaquin Fernandez, 2011. "Dealing with the Inventory Risk," Economics Papers from University Paris Dauphine 123456789/7390, Paris Dauphine University.
  4. Robert Almgren, 2003. "Optimal execution with nonlinear impact functions and trading-enhanced risk," Applied Mathematical Finance, Taylor & Francis Journals, vol. 10(1), pages 1-18.
  5. Muller, Ulrich A. & Dacorogna, Michel M. & Olsen, Richard B. & Pictet, Olivier V. & Schwarz, Matthias & Morgenegg, Claude, 1990. "Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis," Journal of Banking & Finance, Elsevier, vol. 14(6), pages 1189-1208, December.
  6. Olivier Gu\'eant & Charles-Albert Lehalle & Joaquin Fernandez Tapia, 2011. "Dealing with the Inventory Risk. A solution to the market making problem," Papers 1105.3115, arXiv.org, revised Aug 2012.
  7. Rama Cont & Marc Potters & Jean-Philippe Bouchaud, 1997. "Scaling in stock market data: stable laws and beyond," Papers cond-mat/9705087, arXiv.org.
  8. Xu, Zhaoxia & Gençay, Ramazan, 2003. "Scaling, self-similarity and multifractality in FX markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 323(C), pages 578-590.
  9. Bertsimas, Dimitris & Lo, Andrew W., 1998. "Optimal control of execution costs," Journal of Financial Markets, Elsevier, vol. 1(1), pages 1-50, April.
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