A steady-state model of the continuous double auction

A model of the continuous double auction is constructed and analysed. Given the underlying supply and demand functions, the analysis yields steady-state probability distributions for the best ask, best bid and transaction prices. Under fairly general assumptions it is found that these prices are confined to a clearly defined window. Expressions are also obtained for the depth of the order book at arbitrary prices, and for the expected time-to-execution of a given order. These can be used to calculate the optimal order price for a trader with a specified level of impatience, to determine when a market order is preferable to a limit order, and hence in some cases to detect the presence of irrational or ill-informed traders in the market. It is conjectured that, in a market of rational and well-informed traders, the two sides of the order book should be statistically independent. The most efficient markets are generally those in which there is active competition between buyers as well as sellers. If the traded goods are completely standardized, a trader competes by declaring the price and quantity of goods she is willing to buy or sell. For such a declaration to be credible, it must constitute a firm offer. A continuously operating market for standardized goods in which both buyers and sellers can announce offers to trade specified quantities at specified prices, and can also initiate trades by accepting such offers, is known as a ‘continuous double auction’ (CDA). This mechanism provides a natural multilateral generalization of the familiar process of haggling. It has been found in laboratory experiments to give very rapid convergence to a competitive equilibrium, as discussed by Smith (1962) and Smith et al (1982), and also to yield extremely efficient allocations. A number of relevant studies are surveyed by Friedman (1993). The CDA mechanism is especially well-suited to the needs of modern financial markets, and is the basis of nearly all the automated trading systems that have been implemented in such markets over the last two decades. Such systems utilize an electronic ‘order book’, in which unexecuted or partially executed orders are stored and displayed while awaiting execution. Countless versions of the CDA mechanism exist, each characterized by the types of orders that may be submitted and the way in which these are handled. In practice most automated exchanges operate primarily as limit order markets, and accept orders of the following basic types: • Limit orders—orders to buy a specified quantity at a price not exceeding some specified maximum, or to sell a specified quantity at a price not less than a specified minimum. If it can be matched with any of the opposing orders already queued in the order book, a new limit order is immediately executed against the most competitive of these at the best available price. Otherwise, it joins the queue of unexecuted sells or buys in the order book and remains there until it is either cancelled or else executed against a subsequent order. • Market orders—orders to buy or sell a specified quantity at the best price currently available. A market order is executed immediately and as fully as possible. Any unexecuted part may then be converted to a limit order at 1469-7688/03/050385+20$30.00 © 2003 IOP Publishing Ltd PII: S1469-7688(03)60324-6 385 H Luckock QUANTITATIVE FI N A N C E the same price (as on the Paris Bourse), or else executed at the next best available price (as on the Australian Stock Exchange). Although some exchanges utilize hybrid systems, an increasing number (including the major exchanges in Paris, Tokyo, Toronto, Hong Kong, Sydney, Mumbai and Stockholm) function as pure limit order markets. Glosten (1994) argues that exchanges operating in this manner are not only good providers of liquidity in extreme situations, but are also immune to competition from other exchanges. In a limit order market, transaction prices are determined by the interaction of incoming orders with the order book. An understanding of this interaction is therefore required to properly model price formation and the stochastic properties of price fluctuations. This has motivated several empirical studies of real limit order markets, including those by Biais et al (1995), Hamao and Hasbrouck (1995), Maslov and Mills (2001), Bouchaud et al (2002), and Potters and Bouchaud (2002). A tractable model of the order book would be of considerable value, but the development of such a model has been hindered by the inherent complexity of limit order markets. Most investigations reported in the finance literature, such as those by Chakravarty and Holden (1995) and Parlour (1998), have focused on the strategies available to traders and market makers but provide limited insight into the statistical properties of the order book. Instead of attempting to anticipate how traders will behave, the approach taken here is to start by supposing that their combined effect is to generate flows of buy and sell orders with known price distributions. The next step is to determine as much as possible about the resulting price distribution of the orders stored in the book, and the distribution of transaction prices. This is similar to the approach used by Garman (1976) although he did not attempt to develop a solvable model (except for the simple case in which all orders are made at a single price). It is also related to the approach used by Domowitz and Wang (1994), though their results are invalid due to an error in the proof of their main theorem (as discussed in appendix B). The final step is to partially close the model by investigating the way in which the properties of the order book can, in turn, influence the order flow. This is done by identifying optimal order placement strategies for traders with specified levels of impatience, taking into account the cost of execution delays. If all traders are assumed to be rational and informed, then some collection of these optimal strategies should be able to reproduce the order flow assumed at the outset. If not, then the postulated form of the order flow can be rejected as inconsistent with trader rationality. In this way it is possible to eliminate various plausible models of order flow, such as those occurring in markets with price-independent elasticities of supply and demand. (Hopefully it may also be possible to find a general characterization of markets that are consistent with trader rationality, although this has not yet been achieved.) As with all theoretical models, certain assumptions are made in order to render the problem tractable. These are described in detail in section 1, and are summarized in essence by the statement that no price-sensitive information enters the market or diffuses within it during the period of interest. Briefly, the situation envisaged is one in which large numbers of investors trade single units of an asset over an unbroken period, during which there are no external events affecting the underlying supply and demand. All traders have access to full historical data but, due to the high frequency of trades and delays in accessing data and transmitting orders, cannot respond quickly enough to take advantage of temporary anomalies that might appear in the order book. Instead they place anticipatory limit orders and market orders, guided by their knowledge of the statistical properties of the order book. Other researchers have considered models with some of these features. For example, Mendelson (1982) used renewal theory to analyse a clearing house (sealed bid double auction) mechanism under assumptions similar to those described above, and an additional assumption of price-independent order arrival rates. More recently, Maslov (2000) has used computer simulations to study a model market in which traders randomly submit market or limit orders for fixed quantities of shares, each new limit order price being offset from the current market price by a random variable with a uniform distribution. The models of Challet and Stinchcombe (2001) and of Daniels et al (2001) are somewhat similar except that limit order prices are offset from the opposing quote. The latter model, which assumes that limit orders arrive with the same frequency at every possible price, has been analysed in some detail by Smith et al (2002) who also used numerical simulations to estimate quantities of interest such as the expected density of orders in the book and their expected times-to-execution for a variety of parameter choices. In all these models, price movements are driven by purely by technical considerations, without reference to any fundamentals. Chiarella and Iori (2002) have constructed a somewhat more realistic (and hence more complex) model, in which traders employ a variety of pricing strategies that can take into account both technical and fundamental factors. As this model appears not to be amenable to theoretical analysis, conclusions must be inferred from simulation data obtained with specific parameter choices and so it is difficult to judge the extent of their generality. The model of Foucault et al (2001) is also similar in some respects to that presented here. In particular, these authors assume that all orders are for a single unit; that traders cannot respond to detailed information about the state of the order book; and that orders cannot be cancelled. However, they make an additional assumption that new limit orders must narrow the spread by at least one tick, so that the order book can never contain more than one order at a given price. This forces all traders to compete against each other for priority, and denies patient traders the possibility of seeking better prices by placing less competitive orders. Foucault et al also assume that transaction prices are confined within a fixed window, bounded above by a price at which outside investors stand ready to sell unlimited quantities of the asset, and bounded below by a price at