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A fractional reaction–diffusion description of supply and demand

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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.

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Authors and Affiliations

  1. Ladhyx, UMR CNRS 7646, École Polytechnique, 91128, Palaiseau Cedex, France

    Michael Benzaquen

  2. Capital Fund Management, 23 rue de l’Université, 75007, Paris, France

    Michael Benzaquen & Jean-Philippe Bouchaud

Authors
  1. Michael Benzaquen
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  2. Jean-Philippe Bouchaud
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Corresponding author

Correspondence to Jean-Philippe Bouchaud.

Additional information

Contribution to the Topical Issue “Continuous Time Random Walk Still Trendy: Fifty-year History, Current State and Outlook”, edited by Ryszard Kutner and Jaume Masoliver.

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Benzaquen, M., Bouchaud, JP. A fractional reaction–diffusion description of supply and demand. Eur. Phys. J. B 91, 23 (2018). https://doi.org/10.1140/epjb/e2017-80246-9

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  • DOI: https://doi.org/10.1140/epjb/e2017-80246-9

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