Price Dynamics in a Markovian Limit Order Market

Abstract

We propose a simple stochastic model for the dynamics of a limit order book, in which arrivals of market orders, limit orders, and order cancellations are described in terms of a Markovian queueing system. Price dynamics are endogenous and result from the execution of market orders against outstanding limit orders. Through its analytical tractability, the model allows us to obtain analytical expressions for various quantities of interest, such as the distribution of the duration between price changes, the distribution and autocorrelation of price changes, and the probability of an upward move in the price, conditional on the state of the order book. We study the diffusion limit of the price process and express the volatility of price changes in terms of parameters describing the arrival rates of buy and sell orders and cancellations. These analytical results provide some insight into the relation between order flow and price dynamics in limit order markets.