A Markov model of a limit order book: thresholds, recurrence, and trading strategies

Frank Kelly and Elena Yudovina
Mathematics of Operations Research


We formulate an analytically tractable model of a limit order book on short time scales, where the dynamics are driven by stochastic fluctuations between supply and demand. We establish the existence of a limiting distribution for the highest bid, and for the lowest ask, where the limiting distributions are confined between two thresholds. We make extensive use of fluid limits in order to establish recurrence properties of the model. We use our model to analyze various high-frequency trading strategies, and comment on the Nash equilibria that emerge between high-frequency traders when a market in continuous time is replaced by frequent batch auctions.

Keywords: limit order book, queueing, fluid limit, high-frequency trading, Nash equilibrium

arxiv, Cambridge repository
Earlier versions of this work appeared in Elena's PhD thesis Collaborating queues: large service network and a limit order book, 2012, and her paper A simple model of a limit order book, 2012.

The latest version of Jan Swart's paper "Rigorous results for the Stigler-Luckock model for the evolution of an order book" is available from his webpage.