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Order book dynamics in liquid markets: limit theorems and diffusion approximations

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Author Info

  • Rama Cont

    ()
    (LPMA - Laboratoire de Probabilités et Modèles Aléatoires - CNRS : UMR7599 - Université Paris VI - Pierre et Marie Curie - Université Paris VII - Paris Diderot)

  • Adrien De Larrard

    ()
    (LPMA - Laboratoire de Probabilités et Modèles Aléatoires - CNRS : UMR7599 - Université Paris VI - Pierre et Marie Curie - Université Paris VII - Paris Diderot)

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    Abstract

    We propose a model for the dynamics of a limit order book in a liquid market where buy and sell orders are submitted at high frequency. We derive a functional central limit theorem for the joint dynamics of the bid and ask queues and show that, when the frequency of order arrivals is large, the intraday dynamics of the limit order book may be approximated by a Markovian jump-diffusion process in the positive orthant, whose characteristics are explicitly described in terms of the statistical properties of the order flow and only depend on rate of arrival of orders and the covariance structure of order sizes. This result allows to obtain tractable analytical approximations for various quantities of interest, such as the probability of a price increase or the distribution of the duration until the next price move, conditional on the state of the order book. Our results allow for a wide range of distributional assumptions and temporal dependence in the order flow and apply to a wide class of stochastic models proposed for order book dynamics, including models based on Poisson point processes, self-exciting point processes and models of the ACD-GARCH family.

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    Bibliographic Info

    Paper provided by HAL in its series Working Papers with number hal-00672274.

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    Date of creation: 2011
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    Handle: RePEc:hal:wpaper:hal-00672274

    Note: View the original document on HAL open archive server: http://hal.archives-ouvertes.fr/hal-00672274
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    Web page: http://hal.archives-ouvertes.fr/

    Related research

    Keywords: limit order book ; queueing systems ; heavy traffic limit ; functional central limit theorem ; diffusion approximation ; high-frequency data ; market microstructure ; point process;

    This paper has been announced in the following NEP Reports:

    References

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    1. Eric Smith & J. Doyne Farmer & Laszlo Gillemot & Supriya Krishnamurthy, 2002. "Statistical theory of the continuous double auction," Papers cond-mat/0210475, arXiv.org.
    2. Erhan Bayraktar & Ulrich Horst & Ronnie Sircar, 2007. "A Limit Theorem for Financial Markets with Inert Investors," Papers math/0703831, arXiv.org.
    3. Ghysels Eric & Jasiak Joanna, 1998. "GARCH for Irregularly Spaced Financial Data: The ACD-GARCH Model," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 2(4), pages 1-19, January.
    4. Avellaneda, Marco & Reed, Josh & Stoikov, Sasha, 2011. "Forecasting prices from level-I quotes in the presence of hidden liquidity," Algorithmic Finance, IOS Press, vol. 1(1), pages 35-43.
    5. Jean-Philippe Bouchaud & Marc Mezard & Marc Potters, 2002. "Statistical properties of stock order books: empirical results and models," Science & Finance (CFM) working paper archive 0203511, Science & Finance, Capital Fund Management.
    6. Zhou, Chunsheng, 2001. "An Analysis of Default Correlations and Multiple Defaults," Review of Financial Studies, Society for Financial Studies, vol. 14(2), pages 555-76.
    7. Jean-Philippe Bouchaud & Marc Mezard & Marc Potters, 2002. "Statistical properties of stock order books: empirical results and models," Quantitative Finance, Taylor & Francis Journals, vol. 2(4), pages 251-256.
    8. Parameswaran Gopikrishnan & Vasiliki Plerou & Xavier Gabaix & H. Eugene Stanley, 2000. "Statistical Properties of Share Volume Traded in Financial Markets," Papers cond-mat/0008113, arXiv.org.
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    Cited by:
    1. Qinghua Li, 2014. "Facilitation and Internalization Optimal Strategy in a Multilateral Trading Context," Papers 1404.7320, arXiv.org.

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