A general framework for modelling limit order book dynamics

Abstract

We present a mathematical framework for modelling the dynamics of limit order books, built on the combination of two modelling ingredients: the order flow -modelled as a general spatial point process- and market clearing, modelled via a deterministic ‘mass transport’ operator acting on distributions of buy and sell orders. At the mathematical level, this corresponds to a natural decomposition of the infinitesimal generator describing the evolution of the limit order book into two operators: the generator of the order flow and the clearing operator. Our model provides a flexible framework for modelling and simulating order book dynamics and studying various scaling limits of discrete order book models. We show that our framework includes previous models as special cases and yields insights into the interaction between the order flow and price dynamics, the use of order book data for prediction of intraday price movements. The framework also allows for model comparison and the study of the order flow. The modular structure of the model is well-adapted to simulation and allows the stochastic model for the order flow and the clearing mechanism to be specified independently. Then, as a simple demonstration, models with different assumptions on the order intensities are compared. The simulation result shows that orders of relatively large size also play an essential role in the evolution of the order book process. We further investigate the asymptotic behaviour of the order book processes, including the fluid scaling and the diffusion scaling. The decomposition relation between the order flow and the order book holds when the ask and bid price do not move. In general, the price processes depend on the order flow and the current state of the order book. We prove that as the tick size becomes small, the ask price and bid price converge to the same limiting process.