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• Print publication year: 2016
• Online publication date: July 2016

# 2 - Statistical Properties of Limit Order Books: A Survey

from PART ONE - EMPIRICAL PROPERTIES OF ORDERDRIVEN MARKETS

## Summary

Introduction

The computerization of financial markets in the second half of the 1980s provided empirical scientists with easier access to extensive data on order books. Biais et al. (1995) is an early study of the data flows on the newly (at that time) computerized Paris Bourse. Many subsequent papers offer complementary empirical findings and modelling perspectives, e.g., Gopikrishnan et al. (2000), Challet and Stinchcombe (2001), Maslov and Mills (2001), Bouchaud et al. (2002), Potters and Bouchaud (2003). In this chapter, we present a summary of some fundamental empirical facts. Basic statistical properties of limit order books, which can be observed from real data, are described and studied. Many variables crucial to a fine modelling of order flows and dynamics of order books are studied: Time of arrival of orders, placement of orders, size of orders, shape of order books, etc.

The markets we are dealing with are order-driven markets with no official market maker, in which orders are submitted in a double auction and executions follow price/time priority. In order to make the results we present both self-contained and reproducible, the statistics have been computed directly using our own database. The set of data that we have used in this chapter is detailed in Appendix B, which contains the precise description of all the data sets used throughout this book.

Time of Arrivals of Orders

We compute the empirical distribution for interarrival times – or durations – of market orders for the stock BNP Paribas using the data set described in Appendix B.2. The results are plotted in Figs 2.1 and 2.2, both in linear and log scale. It is clearly observed that the exponential fit is not a good one. We check however that the Weibull distribution fit is potentially a very good one. Weibull distributions have been suggested for example in Ivanov et al. (2004). Politi and Scalas (2008) also obtain good fits with q-exponential distributions.

In the Econometrics literature, these observations of non-Poisson arrival times have given rise to a large trend of modelling of “irregular” financial data. Engle and Russell (1997) and Engle (2000) have introduced autoregressive condition duration or intensity models that may help modelling these processes of orders’ submission (see Hautsch (2004) for a textbook treatment).