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Beyond the Square Root: Evidence for Logarithmic Dependence of Market Impact on Size and Participation Rate

Elia Zarinelli

LIST S.p.A., Via Carducci 20, I-34127 Trieste, Italy

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Michele Treccani

Mediobanca S.p.A., Piazzetta E. Cuccia 1, 20121 Milano, Italy

QUANTLab, Via Pietrasantina 123, 56122 Pisa, Italy

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J. Doyne Farmer

Institute for New Economic Thinking at the Oxford Martin School and Mathematical Institute, Eagle House, Walton Well Rd., Oxford OX2 6ED United Kingdom

Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA

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, and

Fabrizio Lillo

QUANTLab, Via Pietrasantina 123, 56122 Pisa, Italy

Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA

E-mail Address: [email protected]

Corresponding author.

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https://doi.org/10.1142/S2382626615500045Cited by:67 (Source: Crossref)

Abstract

We make an extensive empirical study of the market impact of large orders (metaorders) executed in the US 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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