A generalized birth–death stochastic model for high-frequency order book dynamics

Abstract

We use a generalized birth–death stochastic process to model the high-frequency dynamics of the limit order book, and illustrate it using parameters estimated from Level II data for a stock on the London Stock Exchange. A new feature of this model is that limit orders are allowed to arrive in multiple sizes, an important empirical feature of the order book. We can compute various quantities of interest without resorting to simulation, conditional on the state of the order book, such as the probability that the next move of the mid-price will be upward, or the probability, as a function of order size, that a limit ask order will be executed before a downward move in the mid-price. This generalizes the successful model of Cont et al. [Oper. Res., 2010, 58, 549–563] by means of a new technical approach to computing the distribution of first passage times.

Keywords:

• Limit order book
• Birth-death processes
• High frequency data
• First passage times
• Automated trading

Notes

†High-frequency data used for this paper was provided by the Chair of Quantitative Finance, Ecole Centrale Paris, France. http://fiquant.mas.ecp.fr