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A Mathematical Approach To Order Book Modeling

Author

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  • FRÉDÉRIC ABERGEL

    (Chaire de Finance Quantitative, Laboratoire de Mathématiques Appliquées aux Systèmes, École Centrale Paris, 92290 Châtenay-Malabry, France)

  • AYMEN JEDIDI

    (Chaire de Finance Quantitative, Laboratoire de Mathématiques Appliquées aux Systèmes, École Centrale Paris, 92290 Châtenay-Malabry, France)

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

  • Frédéric Abergel & Aymen Jedidi, 2013. "A Mathematical Approach To Order Book Modeling," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(05), pages 1-40.
    • Handle: RePEc:wsi:ijtafx:v:16:y:2013:i:05:n:s0219024913500258
    DOI: 10.1142/S0219024913500258
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    References listed on IDEAS

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    1. Hens, Thorsten & Schenk-Hoppe, Klaus Reiner (ed.), 2009. "Handbook of Financial Markets: Dynamics and Evolution," Elsevier Monographs, Elsevier, edition 1, number 9780123742582.
    2. Thakor, Anjan V. & Boot, Arnoud (ed.), 2008. "Handbook of Financial Intermediation and Banking," Elsevier Monographs, Elsevier, edition 1, number 9780444515582.
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