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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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    20. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2018. "Volatility is rough," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 933-949, June.
    21. Jonathan Donier & Julius Bonart, 2014. "A Million Metaorder Analysis of Market Impact on the Bitcoin," Papers 1412.4503, arXiv.org, revised Sep 2015.
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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 Jan 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 Jul 2023.
    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.
    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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