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Brownian trading excursions and avalanches

Author

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  • Friedrich Hubalek
  • Paul Kruhner
  • Thorsten Rheinlander

Abstract

We study a parsimonious but non-trivial model of the latent limit order book where orders get placed with a fixed displacement from a center price process, i.e.\ some process in-between best bid and best ask, and get executed whenever this center price reaches their level. This mechanism corresponds to the fundamental solution of the stochastic heat equation with multiplicative noise for the relative order volume distribution. We classify various types of trades, and introduce the trading excursion process which is a Poisson point process. This allows to derive the Laplace transforms of the times to various trading events under the corresponding intensity measure. As a main application, we study the distribution of order avalanches, i.e.\ a series of order executions not interrupted by more than an $\varepsilon$-time interval, which moreover generalizes recent results about Parisian options.

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  • Friedrich Hubalek & Paul Kruhner & Thorsten Rheinlander, 2017. "Brownian trading excursions and avalanches," Papers 1701.00993, arXiv.org.
    • Handle: RePEc:arx:papers:1701.00993
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    References listed on IDEAS

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    1. Delattre, Sylvain & Robert, Christian Y. & Rosenbaum, Mathieu, 2013. "Estimating the efficient price from the order flow: A Brownian Cox process approach," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2603-2619.
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    3. Łukasz Kruk, 2003. "Functional Limit Theorems for a Simple Auction," Mathematics of Operations Research, INFORMS, vol. 28(4), pages 716-751, November.
    4. Bence Toth & Yves Lemperiere & Cyril Deremble & Joachim de Lataillade & Julien Kockelkoren & Jean-Philippe Bouchaud, 2011. "Anomalous price impact and the critical nature of liquidity in financial markets," Papers 1105.1694, arXiv.org, revised Nov 2011.
    5. Rama Cont & Adrien de Larrard, 2013. "Price Dynamics in a Markovian Limit Order Market," Post-Print hal-00552252, HAL.
    6. Sylvain Delattre & Christian Y. Robert & Mathieu Rosenbaum, 2013. "Estimating the efficient price from the order flow: a Brownian Cox process approach," Papers 1301.3114, arXiv.org, revised Apr 2013.
    7. Christian Yann Robert & Sylvain Delattre & Mathieu Rosenbaum, 2013. "Estimating the efficient price from the order flow: A Brownian Cox process approach," Post-Print hal-02006747, HAL.
    8. Frederic Abergel & Aymen Jedidi, 2010. "A Mathematical Approach to Order Book Modeling," Papers 1010.5136, arXiv.org, revised Mar 2013.
    9. Devroye, Luc, 2009. "On exact simulation algorithms for some distributions related to Jacobi theta functions," Statistics & Probability Letters, Elsevier, vol. 79(21), pages 2251-2259, November.
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    Cited by:

    1. Horst, Ulrich & Kreher, Dörte, 2019. "A diffusion approximation for limit order book models," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4431-4479.

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