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The order book as a queueing system: average depth and influence of the size of limit orders

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  • I. Muni Toke

Abstract

We study the analytical properties of a one-side order book model in which the flows of limit and market orders are Poisson processes and the distribution of lifetimes of cancelled orders is exponential. Although simplistic, the model provides an analytical tractability that should not be overlooked. Using basic results for birth-and-death processes, we build an analytical formula for the shape (depth) of a continuous order book model which is both founded on market mechanisms and very close to empirically tested formulas. We relate this shape to the probability of execution of a limit order, highlighting a law of conservation of the flows of orders in an order book. We then extend our model by allowing random sizes of limit orders, thereby allowing the study of the relationship between the size of the incoming limit orders and the shape of the order book. Our theoretical model shows that, for a given total volume of incoming limit orders, the fewer limit orders are submitted (i.e. the larger the average size of these limit orders), the deeper is the order book around the spread. This theoretical relationship is finally empirically tested on several stocks traded on the Paris stock exchange.

Suggested Citation

  • I. Muni Toke, 2015. "The order book as a queueing system: average depth and influence of the size of limit orders," Quantitative Finance, Taylor & Francis Journals, vol. 15(5), pages 795-808, May.
  • Handle: RePEc:taf:quantf:v:15:y:2015:i:5:p:795-808
    DOI: 10.1080/14697688.2014.963654
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    Citations

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    Cited by:

    1. Ben Hambly & Jasdeep Kalsi & James Newbury, 2018. "Limit order books, diffusion approximations and reflected SPDEs: from microscopic to macroscopic models," Papers 1808.07107, arXiv.org, revised Jun 2019.
    2. Ioane Muni Toke, 2016. "Reconstruction of Order Flows using Aggregated Data," Post-Print hal-01705074, HAL.
    3. M. Derksen & B. Kleijn & R. de Vilder, 2019. "Clearing price distributions in call auctions," Papers 1904.07583, arXiv.org, revised Nov 2019.
    4. Zijian Shi & Yu Chen & John Cartlidge, 2021. "The LOB Recreation Model: Predicting the Limit Order Book from TAQ History Using an Ordinary Differential Equation Recurrent Neural Network," Papers 2103.01670, arXiv.org.
    5. Ioane Muni Toke, 2017. "Stationary Distribution Of The Volume At The Best Quote In A Poisson Order Book Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(06), pages 1-22, September.
    6. Clinet, Simon & Yoshida, Nakahiro, 2017. "Statistical inference for ergodic point processes and application to Limit Order Book," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1800-1839.
    7. Xiaofei Lu & Fr'ed'eric Abergel, 2018. "Order-book modelling and market making strategies," Papers 1806.05101, arXiv.org.
    8. Marcello Rambaldi & Emmanuel Bacry & Fabrizio Lillo, 2016. "The role of volume in order book dynamics: a multivariate Hawkes process analysis," Papers 1602.07663, arXiv.org.
    9. Ioane Muni Toke, 2017. "Stationary Distribution Of The Volume At The Best Quote In A Poisson Order Book Model," Post-Print hal-01705085, HAL.
    10. Ioane Muni Toke, 2016. "Reconstruction of Order Flows using Aggregated Data," Papers 1604.02759, arXiv.org.
    11. Ioane Muni Toke & Nakahiro Yoshida, 2017. "Modelling intensities of order flows in a limit order book," Post-Print hal-01705080, HAL.
    12. Lahmiri, Salim & Bekiros, Stelios, 2020. "Nonlinear analysis of Casablanca Stock Exchange, Dow Jones and S&P500 industrial sectors with a comparison," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 539(C).
    13. Frank Kelly & Elena Yudovina, 2018. "A Markov Model of a Limit Order Book: Thresholds, Recurrence, and Trading Strategies," Mathematics of Operations Research, INFORMS, vol. 43(1), pages 181-203, February.
    14. Kyle Bechler & Michael Ludkovski, 2017. "Order Flows and Limit Order Book Resiliency on the Meso-Scale," Papers 1708.02715, arXiv.org.

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