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.