A MATHEMATICAL APPROACH TO ORDER BOOK MODELING

Frédéric Abergel () and Aymen Jedidi ()
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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

International Journal of Theoretical and Applied Finance (IJTAF), 2013, vol. 16, issue 05, 1-40

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.

Keywords: Limit order book; agent-based modeling; order flow; bid-ask spread; Markov chain; stochastic stability; FCLT; geometric mixing (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (56)

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DOI: 10.1142/S0219024913500258

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