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A fractional reaction–diffusion description of supply and demand

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  • Michael Benzaquen

    (LadHyX - Laboratoire d'hydrodynamique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique)

  • Jean-Philippe Bouchaud

    (SPEC - UMR3680 - Service de physique de l'état condensé - IRAMIS - Institut Rayonnement Matière de Saclay (DRF) - CEA - Commissariat à l'énergie atomique et aux énergies alternatives - Université Paris-Saclay - Université Paris-Saclay - CNRS - Centre National de la Recherche Scientifique)

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 −β with β = 1/2 in the diffusive case, which is inconsistent with market efficiency. In the sub-diffusive case the decay exponent β takes any value in [0, 1/2], and can be tuned to match the empirical value β ≈ 1/4. Numerical simulations confirm our theoretical results. Several extensions of the model are suggested.

Suggested Citation

  • Michael Benzaquen & Jean-Philippe Bouchaud, 2018. "A fractional reaction–diffusion description of supply and demand," Post-Print hal-02323544, HAL.
    • Handle: RePEc:hal:journl:hal-02323544
    DOI: 10.1140/epjb/e2017-80246-9
    Note: View the original document on HAL open archive server: https://hal.science/hal-02323544
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    References listed on IDEAS

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    1. Jean-Philippe Bouchaud & J. Doyne Farmer & Fabrizio Lillo, 2008. "How markets slowly digest changes in supply and demand," Papers 0809.0822, arXiv.org.
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    Cited by:

    1. Johann Lussange & Ivan Lazarevich & Sacha Bourgeois-Gironde & Stefano Palminteri & Boris Gutkin, 2021. "Modelling Stock Markets by Multi-agent Reinforcement Learning," Computational Economics, Springer;Society for Computational Economics, vol. 57(1), pages 113-147, January.
    2. Johann Lussange & Boris Gutkin, 2023. "Order book regulatory impact on stock market quality: a multi-agent reinforcement learning perspective," Papers 2302.04184, arXiv.org.
    3. Qing Tang & Fabio Camilli, 2020. "Variational Time-Fractional Mean Field Games," Dynamic Games and Applications, Springer, vol. 10(2), pages 573-588, June.
    4. Johann Lussange & Stefano Vrizzi & Sacha Bourgeois-Gironde & Stefano Palminteri & Boris Gutkin, 2023. "Stock Price Formation: Precepts from a Multi-Agent Reinforcement Learning Model," Computational Economics, Springer;Society for Computational Economics, vol. 61(4), pages 1523-1544, April.

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