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

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  • Rama Cont
  • Adrien De Larrard
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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 underlying order flow. 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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    File URL: http://arxiv.org/pdf/1202.6412
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1202.6412.

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    Date of creation: Feb 2012
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    Handle: RePEc:arx:papers:1202.6412

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    Web page: http://arxiv.org/

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    1. Ghysels Eric & Jasiak Joanna, 1998. "GARCH for Irregularly Spaced Financial Data: The ACD-GARCH Model," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, De Gruyter, vol. 2(4), pages 1-19, January.
    2. Erhan Bayraktar & Ulrich Horst & Ronnie Sircar, 2007. "A Limit Theorem for Financial Markets with Inert Investors," Papers math/0703831, arXiv.org.
    3. Eric Smith & J Doyne Farmer & Laszlo Gillemot & Supriya Krishnamurthy, 2003. "Statistical theory of the continuous double auction," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 3(6), pages 481-514.
    4. Avellaneda, Marco & Reed, Josh & Stoikov, Sasha, 2011. "Forecasting prices from level-I quotes in the presence of hidden liquidity," Algorithmic Finance, IOS Press, 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," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 2(4), pages 251-256.
    6. 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.
    7. Zhou, Chunsheng, 2001. "An Analysis of Default Correlations and Multiple Defaults," Review of Financial Studies, Society for Financial Studies, Society for Financial Studies, vol. 14(2), pages 555-76.
    8. 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.
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