IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1704.02638.html
   My bibliography  Save this paper

A fractional reaction-diffusion description of supply and demand

Author

Listed:
  • Michael Benzaquen
  • Jean-Philippe Bouchaud

Abstract

We suggest that the broad distribution of time scales in financial markets could be a crucial ingredient to reproduce realistic price dynamics in stylised Agent-Based Models. We propose a fractional reaction-diffusion model for the dynamics of latent liquidity in financial markets, where agents are very heterogeneous in terms of their characteristic frequencies. Several features of our model are amenable to an exact analytical treatment. We find in particular that the impact is a concave function of the transacted volume (aka the "square-root impact law"), as in the normal diffusion limit. However, the impact kernel decays as $t^{-\beta}$ with $\beta=1/2$ in the diffusive case, which is inconsistent with market efficiency. In the sub-diffusive case the decay exponent $\beta$ takes any value in $[0,1/2]$, and can be tuned to match the empirical value $\beta \approx 1/4$. Numerical simulations confirm our theoretical results. Several extensions of the model are suggested.

Suggested Citation

  • Michael Benzaquen & Jean-Philippe Bouchaud, 2017. "A fractional reaction-diffusion description of supply and demand," Papers 1704.02638, arXiv.org, revised Aug 2017.
    • Handle: RePEc:arx:papers:1704.02638
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1704.02638
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Jean-Philippe Bouchaud & J. Doyne Farmer & Fabrizio Lillo, 2008. "How markets slowly digest changes in supply and demand," Papers 0809.0822, arXiv.org.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Gianbiagio Curato & Jim Gatheral & Fabrizio Lillo, 2014. "Optimal execution with nonlinear transient market impact," Papers 1412.4839, arXiv.org.
    2. Juan C. Henao-Londono & Sebastian M. Krause & Thomas Guhr, 2021. "Price response functions and spread impact in correlated financial markets," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 94(4), pages 1-20, April.
    3. Sabrina Camargo & Silvio M. Duarte Queiros & Celia Anteneodo, 2013. "Bridging stylized facts in finance and data non-stationarities," Papers 1302.3197, arXiv.org, revised May 2013.
    4. Nicolas Huth & Frédéric Abergel, 2012. "The times change: multivariate subordination, empirical facts," Post-Print hal-00620841, HAL.
    5. Anirban Chakraborti & Ioane Muni Toke & Marco Patriarca & Frédéric Abergel, 2011. "Econophysics review: II. Agent-based models," Post-Print hal-00621059, HAL.
    6. B. Tóth & F. Lillo & J. D. Farmer, 2010. "Segmentation algorithm for non-stationary compound Poisson processes," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 78(2), pages 235-243, November.
    7. Ban Zheng & François Roueff & Frédéric Abergel, 2014. "Ergodicity and scaling limit of a constrained multivariate Hawkes process," Post-Print hal-00777941, HAL.
    8. Thibault Jaisson, 2014. "Market impact as anticipation of the order flow imbalance," Papers 1402.1288, arXiv.org.
    9. Bence Toth & Imon Palit & Fabrizio Lillo & J. Doyne Farmer, 2011. "Why is order flow so persistent?," Papers 1108.1632, arXiv.org, revised Nov 2014.
    10. Stephan Grimm & Thomas Guhr, 2019. "How spread changes affect the order book: comparing the price responses of order deletions and placements to trades," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 92(6), pages 1-11, June.
    11. Jacopo Rocchi & Enoch Yan Lok Tsui & David Saad, 2017. "Emerging interdependence between stock values during financial crashes," PLOS ONE, Public Library of Science, vol. 12(5), pages 1-15, May.
    12. Chávez-Casillas, Jonathan A. & Figueroa-López, José E., 2017. "A one-level limit order book model with memory and variable spread," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2447-2481.
    13. 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.
    14. F. Caccioli & M. Marsili & P. Vivo, 2009. "Eroding market stability by proliferation of financial instruments," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 71(4), pages 467-479, October.
    15. Paolo Barucca & Fabrizio Lillo, 2017. "Behind the price: on the role of agent's reflexivity in financial market microstructure," Papers 1708.07047, arXiv.org.
    16. Campbell, Michael J., 2022. "Heavy-tailed distributions of volume and price-change resulting from strategy coordination and decision noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 607(C).
    17. Li-Xin Wang, 2014. "Dynamical Models of Stock Prices Based on Technical Trading Rules Part II: Analysis of the Models," Papers 1401.1891, arXiv.org, revised Feb 2016.
    18. Andre Cardoso Barato & Iacopo Mastromatteo & Marco Bardoscia & Matteo Marsili, 2011. "Impact of meta-order in the Minority Game," Papers 1112.3908, arXiv.org, revised Nov 2012.
    19. B. Tóth & Z. Eisler & F. Lillo & J. Kockelkoren & J.-P. Bouchaud & J.D. Farmer, 2012. "How does the market react to your order flow?," Quantitative Finance, Taylor & Francis Journals, vol. 12(7), pages 1015-1024, May.
    20. Rama Cont & Adrien De Larrard, 2011. "Price dynamics in a Markovian limit order market," Papers 1104.4596, arXiv.org.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:1704.02638. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.