Abstracts

We study the temporal evolution of the limit order book shape on the “macroscopic” time scale (e.g. minutes). To gain useful analytical insights, we consider a scaling regime where the price tick size goes to. In this regime, we develop a first-order approximation for the sample path behavior of order book shape. Using a martingale method, we show that a pair of (scaled) measure-valued processes, representing the ``empirical sell-side shape'' and ``empirical buy-side shape'' of the order book, converges weakly to a pair of deterministic measure-valued processes in a certain Skorohod space. Moreover, the density profile of the limiting processes can be described by Ordinary Differential Equations whose coefficients are determined by the first-order statistics of order flows. We also perform empirical studies to test our theoretical model against historical order book data from NYSE Arca. The empirical results demonstrate the potential of our model in predicting the order book shape evolution for highly liquid stocks in a relatively stationary environment.