Dynamics of limit order books: queueing models, diffusion limits and stochastic PDEs

The advent of electronic trading in financial markets has led to a market landscape in which buyers and sellers by submitting orders through a central limit order book, where orders are matched and executed according to time and price priority. The wide range of frequencies involved – from microseconds to days – requires a consistent description of market dynamics across time scales.

Based on a detailed empirical study of high frequency order flow in equity and futures markets, we propose a multi-scale stochastic model for dynamics of price and order flow in a limit order market, which captures the coexistence of high frequency and low frequency order flow and examines the consequences of this heterogeneity on intraday price dynamics, volatility and liquidity.

We then use probabilistic limit theorems to derive the dynamics of the order book and market price at various time scales.
Starting from a description of the order book as a multi-class spatial queueing system at the highest (micro- or milli-second) frequency, we show that over intermediate time scales — seconds — the dynamics of the active queues may be described as a diffusion in a wedge with discontinuous reflection at the boundary, while the market price follows a jump process driven by the boundary local time of this diffusion.

Over longer time scales, the effective dynamics of the order book may be described as a stochastic moving boundary problem while the market price follows a diffusion in a random environment defined by the order book. We will emphasise how asymptotics across time scales provides insights into the relations between supply, demand, liquidity and volatility in limit order markets.