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.