Introduction

This chapter somewhat departs from our initial motivation of studying limit order books per se, and addresses the very practical question of the predictability of financial markets based on the information content of limit order books.

Forecasting the market has always been one of the “hottest” topics among market practitioners, and the temptation to identify hopefully profitable signals has never been as high as today. Numerous academic studies aim at identifying some predictive features in the time series of past returns, although many seem to obtain negative results. For instance, it is a well-known stylized fact that there is no evidence of linear correlation between successive returns, see e.g., (Chakraborti et al. 2011a) (Lillo and Farmer, 2004). Such studies seemingly demonstrate the lack of predictive character of the series of past returns, as far as the sign of the next price move is concerned. In that sense, the property generally referred to as the Efficient Market Hypothesis does not seem to be challenged.

Rather intriguingly, several books – some popular amongst finance practitioners – introduce and explain predictive strategies that seem to always make money (see e.g., Murphy, 1999; Vidyamurthy, 2004). But, when backtesting those strategies on realistic samples, the results are often quite disappointing, and the strategies no longer profitable. It is likely that the plague of over-fitting, inherent to many prediction methods, plays a key role in the seemingly good performances published in those books.

However, there exist several ways to actually make better predictions than just using the series of past returns. For instance, Abergel and Politi (2013) exhibit some synthetic baskets that are not traded and therefore, not necessarily arbitrage-free. Based on these baskets, they provide evidence of short-term predictability. More specific to the context of order-driven markets, the use of limit order book data has yielded interesting prediction results (Zheng et al. 2012; Anane et al. 2015; Anane and Abergel, 2015; Cont et al. 2014).

The study presented in this chapter is performed both from an academic and a professional perspective. It is based on an extensive use of market data, inclusive of limit order book data, and aims at identifying signals that can be used as forecasting tools, and studying their performances. Several prediction methods are introduced and systematically benchmarked.

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).

Introduction

In this chapter, we move forward on the mathematical study of zero-intelligence models by deriving some analytical properties of a limit order book under the assumptions introduced in Section 6.2. The model is then extended to the case of random order sizes, thereby allowing to study the relationship between the size of the incoming limit orders and the shape of the order book.

Recall that the shape of the order book is simply the function which for any price gives the number of shares standing in the order book at that price; and the cumulative shape up to price p is the total quantity offered in the order book between the best limit and price p. If the limit order book is ergodic, the (cumulative) shape admits a stationary distribution, and its expectation with respect to this stationary distribution will be simply called the average (cumulative) shape.

Understanding the average shape of an order book and its link to the order flows is not straightforward. The first empirical observations of the shape of an order book stated that, at least for the first limits, the shape of the order book is increasing away from the spread (Biais et al. 1995). With better data, one can complement this view and state that “the average order book has a maximum away from the current bid/ask, and a tail reflecting the statistics of the incoming orders” (Bouchaud et al. 2002; Potters and Bouchaud, 2003), i.e. that the limit order book is hump-shaped. The decrease of the tail of the order book is difficult to estimate because one needs complete data, including limit orders submitted far away from the best quotes, which is often not disclosed by exchanges. A power law decrease (Bouchaud et al. 2002) or an exponential decrease (Gu et al. 2008) or even a whole lognormal shape (Preis et al. 2006) have been suggested.

The hump-shaped order book appears quite easily in simulations, as we will see later in Chapter 9. Studying simulation results, Cont et al. (2010) observe that “the average profile of the order book displays a hump […] that does not result from any fine-tuning of model parameters or additional ingredients such as correlation between order flow and past price moves”, but no explicit link between the average shape and the order flows is made.

The Chair of Quantitative Finance was created at École Centrale Paris, now CentraleSupélec, in 2007. Since its inception, most of its research activities were devoted to the study of high frequency financial data. The interdisciplinary nature of the team, composed of mathematicians, financial engineers, computer scientists and physicists, gave it a special dimension. A sizeable portion of its research efforts has been focused on the characterization and mathematical modelling of limit order books.

Literally at the core of every modern, electronic financial market, the limit order book has triggered a huge amount of research in the past twenty years, marked by the seminal work of Biais et al. (1995) on the empirical analysis of the Paris exchange and revitalized a few years later, in a fascinating manner, by the work of Smith et al. (2003). However, much as this topic is interesting, important and challenging, we realized that there was still no reference book on the subject! We therefore decided to assemble in a single document a survey of the existing literature and our own contributions on limit order books, whether they were pertaining to their statistical properties, mathematical modelling or numerical simulation.

We have tried to follow the intellectual approach of an experimental physicist: empirical data should come first, and only empirical analyses may be considered as a reliable ground for building up any kind of theory. The mathematical modelling follows. Models address the different phenomena that are observed and highlighted, and provide a framework to explain and reproduce these phenomena, and they are studied from theoretical, analytical and numerical perspectives.

The book is thus organized as follows: The first part is devoted to the empirical properties of limit order books; the second part, to their mathematical modelling and the third, to their numerical analysis. The fourth part deals with some advanced topics such as imperfection and predictability. Each part presents a survey of the existing scientific literature, as well as our own contributions.

Significant parts of the material covered in this book have already been presented in bits and pieces in different research and survey articles, in particular Chakraborti et al. (2011a,b); Abergel and Jedidi (2013, 2015); Anane and Abergel (2015); Muni Toke (2015, 2011).

What is a limit order book? It is a device that the vast majority of organized electronic markets (all equity, futures and other listed derivatives markets) use to store in their central computer the list of all the interests of market participants. It is essentially a file in a computer that contains all the orders sent to the market, with their characteristics such as the sign of the order (buy or sell), the price, the quantity, a timestamp giving the time the order was recorded by the market, and a host of various market-dependent information. In other words, the limit order book contains, at any given point in time, on a given market, the list of all the transactions that one could possibly perform on this market. Its evolution over time describes the way the market moves under the influence of its participants. In fact, the study of limit order books can provide deep insight into the understanding of the financial market, which is an excellent example of an evolving “complex system” where the different participants collectively interact to find the best price of an asset. Hence, this field attracts mathematicians, economists, statistical physicists, computer scientists, financial engineers, and many others, besides the practitioners.

A market in which buyers and sellers meet via a limit order book is called an order-driven market. In order-driven markets, buy and sell orders are matched as they arrive over time, subject to some priority rules. Priority is always based on price, and then, in most markets, on time, according to a FIFO (First In, First Out) rule. Such priority rules are enforced in the vast majority of financial markets, although there exist some notable exceptions or variants: For instance, the Chicago Mercantile Exchange (CME) uses for some order books a prorata rule in place of (or together with) time priority. Several different market mechanisms have been studied in the microstructure literature, see for example, Garman (1976); Kyle (1985); Glosten (1994); O'Hara (1997); Biais et al. (1997); Hasbrouck (2007). We will not review these mechanisms in this book [except Garman (1976) in Chapter 5], and rather keep our focus on the almost universal standard of price/time priority.

Investigations over more than a century and the recent availability of electronic databases of income and wealth distribution (ranging from a national sample survey of household assets to the income tax return data available from governmental agencies) have revealed some remarkable features. Irrespective of many differences in culture, history, social structure, indicators of relative prosperity (such as gross domestic product or infant mortality) and, to some extent, the economic policies followed in different countries, the income distribution seems to follow a particular universal pattern, as does the wealth distribution: after an initial rise, the number density of people rapidly decays with their income, the bulk described by a Gibbs or log-normal distribution crossing over at the very high income range (for 5–10% of the richest members of the population) to a power law, as shown in Fig. 1.1. The power law in income and wealth distribution is called the Pareto law, after the Italian sociologist and economist Vilfredo Pareto. The log-normal part is named as the Gibrat law, after the French economist Robert Gibrat. This seems to be a universal feature: from ancient Egyptian society (Abul-Magd 2002) through nineteenth-century Europe (Pareto 1897; Champernowne 1953) to modern Japan (Chatterjee et al. 2005b; Chakrabarti et al. 2006). The same is true across the globe today: from the advanced capitalist economy of USA (Chatterjee et al. 2005b; Kar Gupta 2006a; Richmond et al. 2006) to the developing economy of India (Sinha 2006).

In the earlier chapters, we introduced kinetic exchange models and discussed the mathematics behind them. However, one major point that is missing in the earlier discussions is the choice behaviour of the agents. The outcomes of the stochastic process do not reflect any optimization mechanism on the part of the agents. In this chapter, we will provide a simple economic model which intends to capture the basic features of the kinetic exchange models. We start with some usual assumptions regarding the preference pattern of the agents and the market mechanism. Eventually, it will be shown that the outcomes are exactly the same as those obtained in the kinetic exchange models, thus providing an elementary (but non-unique) way of interpreting the stochastic money evolution equations in economic terms. After that, we will discuss the dynamic features of the asset distribution in the economy if it has time-varying macroeconomic variables. To be precise, we will discuss a possibility of inequality reversal (as has been postulated and discussed in Kuznets (1955, 1965) and Angle (2010)) in the same framework.

Derivation of the basic kinetic exchange model

Following Chakrabarti and Chakrabarti (2009), we consider an N-agent exchange economy in discrete time. At every point of time, exactly two agents are randomly chosen from the pool of N agents, i.e. each agent has equal probability of being chosen for trade. The exact trading mechanism is described below. After they trade, the agents part and leave the market. In the next period again two agents are chosen for trade and the same process is repeated until the distribution of their assets reaches a steady state.

In some of the earlier chapters, we have presented in detail the models which have been inspired from physical theories. In the last chapter, we presented a simple economic model which intends to capture the basic features of the kinetic exchange models. The objective of this chapter is to present analytical discussions on stochastic and related economic models of the distribution of income and wealth.

The economic significance

In order to get an idea about the distributional effects of a particular economic policy it often becomes necessary to have information on the distribution of income. Also inequality based on distribution of income has important effects on development, social outcomes and public finance. The shape of the income distribution in a country enables the policy-makers to get an idea about the amount of tax collection. For instance, in Germany 50.6% (19.7%) of the entire income tax is paid by the top 10% (1%) of the taxpayers (see Merz et al. 2005).

There are economic and non-economic reasons for separate study of the distribution of wealth that cannot be interpreted as human capital, such as educational background. Examples of wealth of this type are financial assets and real properties. Unlike human capital they can be traded in appropriate markets at the time of necessity; for instance, when the flow of regular income reduces (after retirement) and when consumption is likely to increase with an increase in family size. Precautionary motive for saving is also regarded as a major reason for accumulation of wealth. Wealth accumulation is often taken as an indicator of social status. Wealth may be retained as well for the purpose of bequests.

It would be difficult to find any society or country where income or wealth is equally distributed among its people. Socioeconomic inequality is not limited to modern times; it has been a persistent fact, and a constant source of irritation to most, since time immemorial.

The issue of inequality in terms of income and wealth is perhaps the most fiercely debated subject in economics. Economists and philosophers have spent much time on the normative aspects of this problem (Rawls 1971; Scruton 1985; Sen 1999; Foucault 2003). The direct and indirect effects of inequality on society have also been studied extensively. In particular, the effects of inequality on the growth of the economy (Benabou 1994; Aghion et al. 1999; Barro 1999; Forbes 2000) and on the econopolitical scenario (Blau and Blau 1982; Alesina and Rodrik 1992; Alesina and Perotti 1996; Benabou 2000) have attracted major attention. Relatively less emphasis has been put on the sources of the problem itself. There are several nontrivial issues and open questions related to this observation: How are income and wealth distributed? What are the forms of the distributions? Are they universal, or do they depend upon specific conditions of a country? Perhaps the most important question is: if inequality is universal (as some of its gross features indicate), then what is the reason for such universality?