Limit order market analysis and modelling: on an universal cause for over-diffusive prices

We briefly review data analysis of the Island order book, part of NASDAQ, which suggests a framework to which all limit order market models should comply. Using a simple exclusion particle model, we argue that short-time price over-diffusion in limit order markets is due to the non-equilibrium of order placement, cancellation and execution rates, which is an inherent feature of real limit order markets. By contrast with usual data on volume and transaction price of a set of stocks or foreign exchanges [1,2,3], limit order markets provide instantaneous information about unfilled orders. In the language of Physics, one has access to an additional dimension, the price. The analysis and the modeling of such markets is therefore more complex. Economics literature has focused on the design of limit order markets (see [5,6]), asking for instance why and when limit orders may be preferred by traders [4] and introducing sophisticated models based on equilibrium prices (see for instance [7]), while physicists recently begun to analyze data and to play with dynamics stochastic toy-models where the price evolution is a consequence of stochastic order arrival [8,9,10,11,12,13,14,15,16,17]. Some definitions first. A limit order is characterized by three properties: its price p, its size m and its lifetime τ . When it is placed into the order book of a given stock, it publishes the wish to buy (or sell) m units of this stock at the predefined price p. If there is already a sell order (buy order) in the book at a lower or equal price, both orders will be automatically executed, partly or fully, depending on the size of matching opposite order(s). If this is not the case the new order will wait in the book until a compatible opposite order is placed or until its lifetime is over. This implies that at any time, there is a spatial distribution of unfilled buy and sell orders. The best prices are defined as the largest buy price and the lowest sell price. The difference between the two is called the bid-ask spread. As often emphasized, submitting a limit order is a trade-off between the advantage of obtaining a fixed price, Preprint submitted to Elsevier Science 1 February 2008 1 10 10