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Beyond the square root: Evidence for logarithmic dependence of market impact on size and participation rate


  • Elia Zarinelli
  • Michele Treccani
  • J. Doyne Farmer
  • Fabrizio Lillo


We make an extensive empirical study of the market impact of large orders (metaorders) executed in the U.S. equity market between 2007 and 2009. We show that the square root market impact formula, which is widely used in the industry and supported by previous published research, provides a good fit only across about two orders of magnitude in order size. A logarithmic functional form fits the data better, providing a good fit across almost five orders of magnitude. We introduce the concept of an "impact surface" to model the impact as a function of both the duration and the participation rate of the metaorder, finding again a logarithmic dependence. We show that during the execution the price trajectory deviates from the market impact, a clear indication of non-VWAP executions. Surprisingly, we find that sometimes the price starts reverting well before the end of the execution. Finally we show that, although on average the impact relaxes to approximately 2/3 of the peak impact, the precise asymptotic value of the price depends on the participation rate and on the duration of the metaorder. We present evidence that this might be due to a herding phenomenon among metaorders.

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  • Elia Zarinelli & Michele Treccani & J. Doyne Farmer & Fabrizio Lillo, 2014. "Beyond the square root: Evidence for logarithmic dependence of market impact on size and participation rate," Papers 1412.2152,
  • Handle: RePEc:arx:papers:1412.2152

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    References listed on IDEAS

    1. Xavier Gabaix & Parameswaran Gopikrishnan & Vasiliki Plerou & H. Eugene Stanley, 2006. "Institutional Investors and Stock Market Volatility," The Quarterly Journal of Economics, Oxford University Press, vol. 121(2), pages 461-504.
    2. Austin Gerig, 2008. "A Theory for Market Impact: How Order Flow Affects Stock Price," Papers 0804.3818,, revised Jul 2008.
    3. J. Doyne Farmer & Austin Gerig & Fabrizio Lillo & Henri Waelbroeck, 2011. "How efficiency shapes market impact," Papers 1102.5457,, revised Sep 2013.
    4. Michael A. Goldstein & Paul Irvine & Eugene Kandel & Zvi Wiener, 2009. "Brokerage Commissions and Institutional Trading Patterns," Review of Financial Studies, Society for Financial Studies, vol. 22(12), pages 5175-5212, December.
    5. Esteban Moro & Javier Vicente & Luis G. Moyano & Austin Gerig & J. Doyne Farmer & Gabriella Vaglica & Fabrizio Lillo & Rosario N. Mantegna, 2009. "Market impact and trading profile of large trading orders in stock markets," Papers 0908.0202,
    6. Bence Toth & Yves Lemperiere & Cyril Deremble & Joachim de Lataillade & Julien Kockelkoren & Jean-Philippe Bouchaud, 2011. "Anomalous price impact and the critical nature of liquidity in financial markets," Papers 1105.1694,, revised Nov 2011.
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    Cited by:

    1. Gianbiagio Curato & Jim Gatheral & Fabrizio Lillo, 2014. "Optimal execution with nonlinear transient market impact," Papers 1412.4839,
    2. Jonathan Donier & Julius Bonart, 2014. "A Million Metaorder Analysis of Market Impact on the Bitcoin," Papers 1412.4503,, revised Sep 2015.

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