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Order Book Dynamics in Liquid Markets: Limit Theorems and Diffusion Approximations

40 Pages Posted: 20 Feb 2012 Last revised: 5 Oct 2012

Rama Cont

Imperial College London; CNRS; Norges Bank Research

Adrien de Larrard

Université Paris VII Denis Diderot

Date Written: February 1, 2012


We propose a model for the dynamics of a limit order book in a liquid market where buy and sell orders are submitted at high frequency. We derive a functional central limit theorem for the joint dynamics of the bid and ask queues and show that, when the frequency of order arrivals is large, the intraday dynamics of the limit order book may be approximated by a Markovian jump-diffusion process in the positive orthant, whose characteristics are explicitly described in terms of the statistical properties of the underlying order flow and shown to depend only on the first and second moments of the order flow. This result allows to obtain tractable analytical approximations for various quantities of interest, such as the probability of a price increase or the distribution of the duration until the next price move, conditional on the state of the order book.

Our results allow for a wide range of distributional assumptions and temporal dependence in the order flow and apply to a wide class of stochastic models proposed for order book dynamics, including models based on Poisson point processes, self-exciting point processes and models of the ACD-GARCH family.

Keywords: limit order book, limit order market, queueing systems, heavy traffic limit, functional central limit theorem, diffusion limit, high-frequency data, market microstructure, point process, limit order market

Suggested Citation

Cont, Rama and de Larrard, Adrien, Order Book Dynamics in Liquid Markets: Limit Theorems and Diffusion Approximations (February 1, 2012). Available at SSRN: or

Rama Cont (Contact Author)

Imperial College London ( email )

London, SW7 2AZ
United Kingdom


CNRS ( email )

Laboratoire de Probabilites & Modeles aleatoires
Universite Pierre & Marie Curie (Paris VI)
Paris, 75252


Norges Bank Research ( email )

P.O. Box 1179
Oslo, N-0107

Adrien De Larrard

Université Paris VII Denis Diderot ( email )

2, place Jussieu
Paris, 75005

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