This chapter adds learning to heterogeneity and interaction: agents are firms who do not only adaptively survive a system's changes, they also aim to improve their condition with anticipatory and reflexive capabilities. Agents shape the macroeconomic environment, adapt to its changes and also make decisions to fulfill this goal. From this perspective, the main aim of agents is to survive and to improve their capacity of withstanding changes in the environment.

This chapter provides another example of the usefulness of the ME approach as it demonstrates how the ME is able to mimic the behaviour of a dynamical system with a large number of equations, which cannot be analytically treated.

The structure of the chapter is as follows. Section 5.1 introduces the model and frames it within the literature, stressing the relevance of the reflexivity and anticipatory theory for a macro-model with learning agents. Section 5.2 presents the ABM model, detailing the behavioural rules for firms. Section 5.3 illustrates how the ME is built from the behavioural assumptions in order to make inferences based on the ABM. Section 5.4 provides the results of the simulations, contrasting the ABM and the ME solution. Finally, Section 5.5 offers some concluding remarks.

Introduction

This chapter considers agents that are not only adaptive but, to some extent, also able to think or, more generally, to learn, particularly with reference to the theory of George Soros (2013) on reflexive systems and of Robert Rosen (1985) on anticipatory systems.

The reflexivity principle applies only to systems of thinking agents and, briefly, it states that interpretations of reality have a deep influence on the behaviour of the agents participating in that reality. Such influence is so effective that changes in behaviours may modify the reality, creating uncertainty. As an example: if investors believe that markets are efficient, then that belief will change the way they invest, which in turn will change the nature of the markets in which they are participating (though not necessarily making them more efficient) (Soros, 2013, p.310).

The synthesis then comes with the human uncertainty principle: The uncertainty associated with fallibility and reflexivity is inherent in the human condition. To make this point, I lump together the two concepts as the human uncertainty principle (Soros, 2013, p.310).

We list below some examples of orders that exist in different exchanges, along with short

descriptions:

1. • Market order: A market order is an order to buy or sell an asset at the bid or offer price currently available in the marketplace.

2. • Limit order: A limit order is an order to buy or sell a contract at a specified price or better.

3. • Good till date order (GTD): An order that remains in the marketplace until it is executed or until the market closes on the date specified.

4. • Fill or kill order (FOK): An order that must be executed as a complete order immediately, otherwise it is cancelled.

5. • Market on close order (MOC): A market order submitted to be executed as close to the closing price as possible.

6. • Market on open order (MOO): A market order to be executed when the market opens.

7. • Limit on close order (LOC): A limit order to be executed as a market order at the closing price if the closing price is equal to or better than the submitted limit price.

8. • Limit on open order (LOO): A limit order to be executed as a market order when the the market opens if the opening price is equal to or better than the limit price.

9. • Stop order: An order converted to a market buy or sell order once a specified stop price is attained or penetrate

10. • Pegged to market order: An order that is pegged to buy on the best offer and sell on the best bid.

11. • Market to limit order: an order that is sent in as a market order to be executed at the current best price. If the entire order is not immediately executed at the market price, the remainder of the order is resubmitted as a limit order with the limit price set to the price at which the original order was executed as a market order.

12. • Discretionary order: An order that allows the broker to delay the execution at her discretion to try and get a better price.

13. • Iceberg order: An order whose (generally large) large volume is only partially disclosed. Iceberg orders belong to the category of “hidden orders”, whereby investors wishing to hide large-size orders, can do so by applying the “hidden” attribute to a large volume order and hide the submitted quantity from the market.

This chapter is dedicated to a review of agent-based models of limit order books, that is, models depicting, at the individual agent level, possibly from a statistical point of view, the interactions that lead to a transaction on a financial market. Far from being exhaustive, the survey is based on selected models that we feel are representative of some important, specific trends in agent-based modelling.

Although known, at least partly, for a long time – Mandelbrot (1963) gives a reference for a paper dealing with non-normality of price time series in 1915, followed by several others in the 1920's – some stylized facts of asset returns (heavy tails, volatility clustering, etc.) have often been left aside when modelling financial markets. They were even often referred to as “anomalous” characteristics, as if observations failed to comply with theory. Much has been done these past twenty years in order to address this challenge and provide new models that can reproduce these facts. These recent developments have been built on top of early attempts at modelling mechanisms of financial markets with agents. Stigler (1964), investigating some rules of the SEC1, or Garman (1976), investigating double-auction microstructure, belong to those historical works. It seems that the first modern attempts at that type of models were made in the field of behavioural finance. This field aims at improving financial modelling based on the psychology and sociology of the investors. Models are built with agents who can exchange shares of stocks according to exogenously defined utility functions reflecting their preferences and risk aversions. LeBaron (2006a,b)shows that this type of modelling offers good flexibility for reproducing some of the stylized facts and provides a review of that type of model. However, although achieving some of their goals, these models suffer from many drawbacks: First, they are very complex, and it may be a very difficult task to identify the role of their numerous parameters and the types of dependence on these parameters; second, the chosen utility functions do not necessarily reflect what is observed on the mechanisms of a financial market

A sensible change in modelling appears with much simpler models implementing only well-identified and presumably realistic “behaviour”: Cont and Bouchaud (2000) uses noise traders that are subject to “herding”, i.e., formation of random clusters of traders sharing the same view on the market. The idea is used in Raberto et al.

When physicists became convinced that matter was not continuous but made from atoms, new ideas on old subjects started flourishing. Not only well-known macroscopic laws (thermodynamics, hydrodynamics) became better understood and bolstered by a more fundamental underlying reality, but a host of spectacular and often unexpected effects were rationalized, in particular collective emergent phenomena phase transitions, superconductivity, avalanches, etc. Similarly, after decades of mathematical finance devoted to the study of effective low frequency models of markets (chiefly based on variations on the Brownian motion), the increasing availability of high frequency data now allows a comprehensive study of price formation and of the microstructure of supply and demand. A new era of financial modelling is opening up, with the hope of addressing a hitherto neglected yet crucial aspect of price dynamics: feedback effects that can lead to market anomalies, instabilities and crashes. Instead of considering the market as an inert, reliable measurement apparatus that merely reveals the fundamental value of assets without influencing it, the empirical study of the order book reveals that markets do generate their own dynamics. New intuitions about market dynamics are necessary. New fascinating statistical regularities are collected and modelled, in particular using numerical simulations of agent based models. New analytical tools are being built to account for these observations. The final goal is, much as in physics, to understand the emergent phenomena and replace ad-hoc models of prices by micro-founded ones where jumps, fat-tails and clustered volatility would have a clear origin. This is important on many counts: while the intellectual endeavour is of course exciting in itself, its offshoots will deeply influence the way we think about market regulation in the wake of high-frequency trading, and the models we use for financial engineering (from derivative pricing to algorithmic trading and optimal execution).

Limit order books offers a much needed, broad review of a field that has literally exploded in the last 20 years, where researchers from economics, financial mathematics, physics, computer science, etc. compete and confront. This diversity is well illustrated by the content of the present book that covers a very wide ground, from empirical facts to advanced mathematical techniques and numerical simulation tools. It will be a very useful and inspiring entry point for all scientists, engineers, regulators and traders interested in understanding how financial markets really work at the basic level.

An experimental approach to the study of limit order books lies in the availability of data. Most of the results presented in this book - in any case, those we have produced ourselves - use the Thomson Reuters Tick History (TRTH) database. All exchange-traded assets worldwide are present in the TRTH database, where they are identified by their Reuters Identification Code (RIC). Similar to most historical databases directly provided by the exchanges, the TRTH data come into the form of two separate files, a trade file recording all transactions, and an event file recording every change in the limit order book. Some very specific information, such as traders’ identities, cannot be publicly disclosed for obvious confidentiality reasons, but in theory, one could reconstruct the sequence of order arrivals of all types using this trade and event files.

After explaining the algorithm used for the processing of limit order book data, we describe the specific data sets that have been used at various places in this book. That way, our results can be reproduced, extended and, of course, challenged, based on the very same data sets we have used.

Limit Order Book Data Processing

Because one cannot distinguish market orders from cancellations just by observing changes in the limit order book (the “event” file), and since, the timestamps of the “trade” and “event” files are asynchronous, we use a matching procedure to reconstruct the order book events.

In a nutshell, we proceed as follows for each stock and each trading day:

1. Parse the “event” file to compute order book state variations:

1. • If the variation is positive (volume at one or more price levels has increased), then label the event as a limit order.

2. • If the variation is negative (volume at one or more price levels has decreased), then label the event as a “likely market order”.

3. • If no variation—this happens when there is just a renumbering in the field “Level” that does not affect the state of the book—do not count an event.

2. Parse the “trade” file and for each trade:

1. • Compare the trade price and volume to likely market orders whose timestamps are in, where tTris the trade timestamp and Δt is a predefined, market-dependent time window.

Introduction

In this chapter, we present empirical studies that shed some new light on the dependency structure of order arrival times, in particular, on the mutual and self-excitations of limit and market orders and, to a lesser extent, of cancellation orders. Examples for various assets and markets (mostly equities, but also bond and index futures) are provided. These empirical studies lay the ground for the advanced mathematical models studied in Chapters 8 and 9.

Re-introducing Physical Time

As seen in Chapter 2, the Poisson hypothesis for the arrival times of orders of different kinds does not stand under careful scrutiny. However, the study of arrival times of orders in an order book has not been a primary focus in the first attempts at order book modelling. Toy models leave this dimension aside when trying to understand the complex dynamics of an order book. In many order-driven market models (Cont and Bouchaud, 2000; Lux and Marchesi, 2000; Alfi et al. 2009a), and in some order book models as well (Preis et al., 2006), a time step in the model is an arbitrary unit of time during which many events may happen. We may call that clock aggregated time. In most order book models (Challet and Stinchcombe, 2001; Mike and Farmer, 2008), one order is simulated per time step with given probabilities, i.e. these models use the clock known as event time. In the simple case where these probabilities are time-homogeneous and independent of the state of the model, such a time treatment is equivalent to the assumption that order flows are homogeneous Poisson processes. A likely reason for the use of event time in order book modelling – leaving aside the fact that models can be sufficiently complicated without 30 Limit Order Books adding another dimension – is that many puzzling empirical observations can be made in event time (e.g. autocorrelation of the signs of limit and market orders) or in aggregated time (e.g. volatility clustering) (see Chapter 2).

However, it is clear that physical time has to be taken into account for the modelling of a realistic order book model. For example, market activity varies widely, and intraday seasonality is often observed as a well known U-shaped pattern. Even for a short time scale model – a few minutes, a few hours – durations of orders (i.e. time intervals between orders) are very broadly distributed.

Introduction

Elaborating on the results presented in Section 2.6, this chapter focuses on the shape of limit order books and the influence of the size of incoming orders. It confirms the theoretical findings of the models studied in Chapters 6 and 7.

Methodology

We use the order book data described in Appendix B.3. All movements on the first 10 limits of the ask side and the bid side of the order book are available, which allows us to reconstruct the evolution of the first limits of the order book during the day. Each trading day is divided into 12 thirty-minute intervals from 10am to 4pm. We obtain T = 391 intervals for each stock (three and a half days of trading are missing or incomplete in our dataset: January 15th, the morning of January 21st, February 18th and 19th). For each interval t = 1,..., T, and for each stock k = 1,..., 14, we compute the total number of limit orders Lk(t) and market orders Mk(t), and the average volume (in euros) of limit orders VkL (t) and market orders Vkm(t). Table 3.1 gives the average number of orders and their volumes (the overline denotes the average over the time intervals:, and similarly for other quantities). The lowest average activity is observed on UBIP.PA and LAGA.PA (which are the only two stocks in the sample with less than 200 market orders and 4000 limit orders in average). The highest activity is observed on BNPP.PA (which is the only stock with more than 600 market orders and 6000 limit orders on average). The smallest sizes of orders are observed on AIRP.PA (108.25 and 205.4 for market and limit orders), and the largest sizes are observed on AXA.PA (535.6 and 876.6 for market and limit orders).

We also compute the average cumulative order book shape at 1 to 10 ticks from the best opposite side. Denoting by PB(t) and PA(t) the best bid and ask prices at time t, the average cumulative depth Bki (t) is the cumulative quantity available in the order book for stock k in the price range {PB(t) + 1,..., PB(t) + i} (in ticks) for ask limit orders, or in the price range {PA(t)- i,..., PA(t)- 1} for bid limit orders, averaged over time during interval t.