A fractional reaction–diffusion description of supply and demand

Michael Benzaquen () and Jean-Philippe Bouchaud ()
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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

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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.

Date: 2018-02
New Economics Papers: this item is included in nep-cmp
Note: View the original document on HAL open archive server: https://hal.science/hal-02323544
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Citations: View citations in EconPapers (4)

Published in The European Physical Journal B: Condensed Matter and Complex Systems, 2018, 91 (2), ⟨10.1140/epjb/e2017-80246-9⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-02323544

DOI: 10.1140/epjb/e2017-80246-9

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