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No‐arbitrage implies power‐law market impact and rough volatility

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  • Paul Jusselin
  • Mathieu Rosenbaum

Abstract

Market impact is the link between the volume of a (large) order and the price move during and after the execution of this order. We show that in a quite general framework, under no‐arbitrage assumption, the market impact function can only be of power‐law type. Furthermore, we prove this implies that the macroscopic price is diffusive with rough volatility, with a one‐to‐one correspondence between the exponent of the impact function and the Hurst parameter of the volatility. Hence, we simply explain the universal rough behavior of the volatility as a consequence of the no‐arbitrage property. From a mathematical viewpoint, our study relies, in particular, on new results about hyper‐rough stochastic Volterra equations.

Suggested Citation

  • Paul Jusselin & Mathieu Rosenbaum, 2020. "No‐arbitrage implies power‐law market impact and rough volatility," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1309-1336, October.
    • Handle: RePEc:bla:mathfi:v:30:y:2020:i:4:p:1309-1336
    DOI: 10.1111/mafi.12254
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    References listed on IDEAS

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    1. Stephen J. Hardiman & Nicolas Bercot & Jean-Philippe Bouchaud, 2013. "Critical reflexivity in financial markets: a Hawkes process analysis," Papers 1302.1405, arXiv.org, revised Jun 2013.
    2. Emmanuel Bacry & Iacopo Mastromatteo & Jean-Franc{c}ois Muzy, 2015. "Hawkes processes in finance," Papers 1502.04592, arXiv.org, revised May 2015.
    3. Weibing Huang & Charles-Albert Lehalle & Mathieu Rosenbaum, 2015. "Simulating and Analyzing Order Book Data: The Queue-Reactive Model," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(509), pages 107-122, March.
    4. Emmanuel Bacry & Thibault Jaisson & Jean--François Muzy, 2016. "Estimation of slowly decreasing Hawkes kernels: application to high-frequency order book dynamics," Quantitative Finance, Taylor & Francis Journals, vol. 16(8), pages 1179-1201, August.
    5. Rama Cont & Sasha Stoikov & Rishi Talreja, 2010. "A Stochastic Model for Order Book Dynamics," Operations Research, INFORMS, vol. 58(3), pages 549-563, June.
    6. Mathias Pohl & Alexander Ristig & Walter Schachermayer & Ludovic Tangpi, 2017. "The amazing power of dimensional analysis: Quantifying market impact," Papers 1702.05434, arXiv.org, revised Sep 2017.
    7. Stephen Hardiman & Nicolas Bercot & Jean-Philippe Bouchaud, 2013. "Critical reflexivity in financial markets: a Hawkes process analysis," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 86(10), pages 1-9, October.
    8. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2018. "Volatility is rough," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 933-949, June.
    9. Eric Smith & J Doyne Farmer & Laszlo Gillemot & Supriya Krishnamurthy, 2003. "Statistical theory of the continuous double auction," Quantitative Finance, Taylor & Francis Journals, vol. 3(6), pages 481-514.
    10. Jonathan Donier & Julius Bonart, 2014. "A Million Metaorder Analysis of Market Impact on the Bitcoin," Papers 1412.4503, arXiv.org, revised Sep 2015.
    11. Thibault Jaisson, 2015. "Market impact as anticipation of the order flow imbalance," Quantitative Finance, Taylor & Francis Journals, vol. 15(7), pages 1123-1135, July.
    12. Thibault Jaisson & Mathieu Rosenbaum, 2013. "Limit theorems for nearly unstable Hawkes processes," Papers 1310.2033, arXiv.org, revised Mar 2015.
    13. Bacry, E. & Delattre, S. & Hoffmann, M. & Muzy, J.F., 2013. "Some limit theorems for Hawkes processes and application to financial statistics," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2475-2499.
    14. Jim Gatheral, 2010. "No-dynamic-arbitrage and market impact," Quantitative Finance, Taylor & Francis Journals, vol. 10(7), pages 749-759.
    15. Giulia Livieri & Saad Mouti & Andrea Pallavicini & Mathieu Rosenbaum, 2018. "Rough volatility: Evidence from option prices," IISE Transactions, Taylor & Francis Journals, vol. 50(9), pages 767-776, September.
    16. Omar El Euch & Mathieu Rosenbaum, 2019. "The characteristic function of rough Heston models," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 3-38, January.
    17. V. Filimonov & D. Sornette, 2015. "Apparent criticality and calibration issues in the Hawkes self-excited point process model: application to high-frequency financial data," Quantitative Finance, Taylor & Francis Journals, vol. 15(8), pages 1293-1314, August.
    18. C. Gomes & H. Waelbroeck, 2015. "Is market impact a measure of the information value of trades? Market response to liquidity vs. informed metaorders," Quantitative Finance, Taylor & Francis Journals, vol. 15(5), pages 773-793, May.
    19. Potters, Marc & Bouchaud, Jean-Philippe, 2003. "More statistical properties of order books and price impact," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(1), pages 133-140.
    20. Emmanuel Bacry & Sylvain Delattre & Marc Hoffmann & Jean-François Muzy, 2013. "Some limit theorems for Hawkes processes and application to financial statistics," Post-Print hal-01313994, HAL.
    21. 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, arXiv.org, revised Nov 2011.
    22. Omar Euch & Masaaki Fukasawa & Mathieu Rosenbaum, 2018. "The microstructural foundations of leverage effect and rough volatility," Finance and Stochastics, Springer, vol. 22(2), pages 241-280, April.
    23. Rama Cont & Adrien de Larrard, 2013. "Price Dynamics in a Markovian Limit Order Market," Post-Print hal-00552252, HAL.
    24. Nicky J. Welton & Howard H. Z. Thom, 2015. "Value of Information," Medical Decision Making, , vol. 35(5), pages 564-566, July.
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    Cited by:

    1. Roncalli, Thierry & Cherief, Amina & Karray-Meziou, Fatma & Regnault, Margaux, 2021. "Liquidity Stress Testing in Asset Management - Part 2. Modeling the Asset Liquidity Risk," MPRA Paper 108295, University Library of Munich, Germany.
    2. Timoth'ee Fabre & Ioane Muni Toke, 2024. "Neural Hawkes: Non-Parametric Estimation in High Dimension and Causality Analysis in Cryptocurrency Markets," Papers 2401.09361, arXiv.org, revised Nov 2024.
    3. Carsten Chong & Marc Hoffmann & Yanghui Liu & Mathieu Rosenbaum & Gr'egoire Szymanski, 2022. "Statistical inference for rough volatility: Central limit theorems," Papers 2210.01216, arXiv.org, revised Jun 2024.
    4. Paul Gassiat, 2022. "Weak error rates of numerical schemes for rough volatility," Papers 2203.09298, arXiv.org, revised Feb 2023.
    5. Eduardo Abi Jaber & Nathan De Carvalho, 2023. "Reconciling rough volatility with jumps," Papers 2303.07222, arXiv.org, revised Sep 2024.
    6. Mathieu Rosenbaum & Jianfei Zhang, 2022. "On the universality of the volatility formation process: when machine learning and rough volatility agree," Papers 2206.14114, arXiv.org.

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