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A Mathematical Approach to Order Book Modeling

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  • Frederic Abergel
  • Aymen Jedidi
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    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.

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    File URL: http://arxiv.org/pdf/1010.5136
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1010.5136.

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    Date of creation: Oct 2010
    Date of revision: Mar 2013
    Handle: RePEc:arx:papers:1010.5136

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    Web page: http://arxiv.org/

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    Cited by:
    1. Aurélien Alfonsi & Pierre Blanc, 2014. "Dynamic optimal execution in a mixed-market-impact Hawkes price model," Working Papers hal-00971369, HAL.
    2. Aymen Jedidi & Frédéric Abergel, 2013. "Stability and price scaling limit of a Hawkes-process based order book model," Working Papers hal-00821607, HAL.
    3. Aur\'elien Alfonsi & Pierre Blanc, 2014. "Dynamic optimal execution in a mixed-market-impact Hawkes price model," Papers 1404.0648, arXiv.org.

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