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A Mathematical Approach to Order Book Modeling

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  • Frederic Abergel
  • Aymen Jedidi

Abstract

Motivated by the desire to bridge the gap between the microscopic description of price formation (agent-based modeling) and the stochastic differential equations approach used classically to describe price evolution at macroscopic time scales, we present a mathematical study of the order book as a multidimensional continuous-time Markov chain and derive several mathematical results in the case of independent Poissonian arrival times. In particular, we show that the cancellation structure is an important factor ensuring the existence of a stationary distribution and the exponential convergence towards it. We also prove, by means of the functional central limit theorem (FCLT), that the rescaled-centered price process converges to a Brownian motion. We illustrate the analysis with numerical simulation and comparison against market data.

Suggested Citation

  • Frederic Abergel & Aymen Jedidi, 2010. "A Mathematical Approach to Order Book Modeling," Papers 1010.5136, arXiv.org, revised Mar 2013.
  • Handle: RePEc:arx:papers:1010.5136
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    File URL: http://arxiv.org/pdf/1010.5136
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    References listed on IDEAS

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    1. S. P. Sethi & N. A. Derzko & J. P. Lehoczky, 1991. "A Stochastic Extension of the Miller-Modigliani Framework," Mathematical Finance, Wiley Blackwell, vol. 1(4), pages 57-76.
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    Cited by:

    1. Ioane Muni Toke, 2014. "Exact and asymptotic solutions of the call auction problem," Papers 1407.4512, arXiv.org, revised Nov 2014.
    2. Xin Liu & Qi Gong & Vidyadhar G. Kulkarni, 2015. "A Stochastic Model of Order Book Dynamics using Bouncing Geometric Brownian Motions," Papers 1511.04096, arXiv.org, revised Mar 2016.
    3. Ioane Muni Toke & Nakahiro Yoshida, 2016. "Modelling intensities of order flows in a limit order book," Papers 1602.03944, arXiv.org.
    4. Xin Guo & Zhao Ruan & Lingjiong Zhu, 2015. "Dynamics of Order Positions and Related Queues in a Limit Order Book," Papers 1505.04810, arXiv.org, revised Oct 2015.
    5. Jonathan A. Ch'avez-Casillas & Jos'e E. Figueroa-L'opez, 2014. "One-level limit order book models with memory and variable spread," Papers 1407.5684, arXiv.org, revised Mar 2016.
    6. Aurélien Alfonsi & Pierre Blanc, 2016. "Dynamic optimal execution in a mixed-market-impact Hawkes price model," Finance and Stochastics, Springer, vol. 20(1), pages 183-218, January.
    7. repec:eee:spapps:v:127:y:2017:i:8:p:2447-2481 is not listed on IDEAS
    8. Aur'elien Alfonsi & Pierre Blanc, 2014. "Dynamic optimal execution in a mixed-market-impact Hawkes price model," Papers 1404.0648, arXiv.org, revised Jun 2015.
    9. Aymen Jedidi & Frédéric Abergel, 2013. "Stability and price scaling limit of a Hawkes-process based order book model," Working Papers hal-00821607, HAL.
    10. Frédéric Abergel & Aymen Jedidi, 2015. "Long Time Behaviour Of A Hawkes Process-Based Limit Order Book," Post-Print hal-01121711, HAL.
    11. Jos'e E. Figueroa-L'opez & Hyoeun Lee & Raghu Pasupathy, 2017. "Optimal placement of a small order in a diffusive limit order book," Papers 1708.04337, arXiv.org.
    12. Lee, Kyungsub & Seo, Byoung Ki, 2017. "Marked Hawkes process modeling of price dynamics and volatility estimation," Journal of Empirical Finance, Elsevier, vol. 40(C), pages 174-200.
    13. Aurélien Alfonsi & Pierre Blanc, 2016. "Dynamic optimal execution in a mixed-market-impact Hawkes price model," Finance and Stochastics, Springer, vol. 20(1), pages 183-218, January.
    14. Frédéric Abergel, 2015. "Lyapunov function for a Hawkes process-based limit order book," Working Papers hal-01113533, HAL.
    15. repec:wsi:ijtafx:v:20:y:2017:i:06:n:s021902491750039x is not listed on IDEAS
    16. Ulrich Horst & Michael Paulsen, 2015. "A law of large numbers for limit order books," Papers 1501.00843, arXiv.org.
    17. Ioane Muni Toke, 2014. "Exact and asymptotic solutions of the call auction problem," Working Papers hal-01061857, HAL.
    18. repec:hal:wpaper:hal-01121711 is not listed on IDEAS
    19. Ioane Muni Toke, 2015. "Stationary distribution of the volume at the best quote in a Poisson order book model," Papers 1502.03871, arXiv.org.
    20. repec:eee:spapps:v:127:y:2017:i:6:p:1800-1839 is not listed on IDEAS
    21. Ulrich Horst & Wei Xu, 2017. "A Scaling Limit for Limit Order Books Driven by Hawkes Processes," Papers 1709.01292, arXiv.org.
    22. Hua, Jia-Chen & Chen, Lijian & Falcon, Liberty & McCauley, Joseph L. & Gunaratne, Gemunu H., 2015. "Variable diffusion in stock market fluctuations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 419(C), pages 221-233.

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