Simple model of a limit order-driven market

We introduce and study a simple model of a limit order-driven market. Traders in this model can either trade stock (or any other risky asset for that matter) at the market price or place a limit order, i.e., an instruction to buy (sell) a certain amount of the stock if its price falls below (raises above) a prede ned level. The choice between these two options is purely random (there are no strategies involved), and the execution price of a limit order is determined simply by o setting the most recent market price by a random amount. Numerical simulations of this model revealed that despite such minimalistic rules the price pattern generated by this model has such realistic features as “fat” tails of the probability distribution of price uctuations, characterized by a crossover between two power law exponents, long range correlations of the volatility, and a non-trivial Hurst exponent of the price signal. c © 2000 Elsevier Science B.V. All rights reserved. In recent years a considerable e ort was invested in high-quality statistical analysis of short-time uctuations in prices of various risky assets. 1 The unifying feature of all these assets is that their price is determined by law of supply and demand while the asset being traded on an open market. These studies resulted in a discovery of robust and to a certain degree universal features of such uctuations, and triggered theoretical studies aimed at explaining or simply mimicking these observations. The list of empirical facts that need to be addressed by any successful theory or model is: (i) The histogram of short time-lag increments of market price has a very peculiar non-Gaussian shape with a sharp maximum and broad wings [3]. The current consensus about the functional form of this distribution is that up to a certain point it follows a Pareto–Levy distribution, with the exponent of its power law E-mail address: maslov@bnl.gov (S. Maslov) 1 A general discussion of the phenomenology of market prices can be found in two recent books by Bouchaud and Potters [1] (a draft of its English translation can be downloaded from http:==www.sciencenance.fr=book.html), and Mantegna and Stanley [2]. 0378-4371/00/$ see front matter c © 2000 Elsevier Science B.V. All rights reserved. PII: S 0378 -4371(00)00067 -4 572 S. Maslov / Physica A 278 (2000) 571–578 tail 1 + 1 ∼ 2:4 − 2:7, after which it crosses over either to a steeper power law with an exponent 1 + 2 ∼ 3:7 − 4:3 [4–6], or, as was reported in earlier studies [7,8], to an exponential decay. In both cases this crossover ensures a nite variance (second moment) of the distribution. (ii) When viewed on time scales less than several trading days, the graph of price vs. time appears to have a Hurst exponent H ' 0:6–0.7 [3,8], di erent from an ordinary uncorrelated random walk value HRW = 0:5. (iii) The volatility (the second moment of price uctuations) exhibits correlated behavior. It is manifested in clustering of volatility, i.e., the presence of regions of unusually high amplitude of uctuations separated by relatively quiet intervals, visible with a “naked eye” in the graph of price increment vs. time. These clustering e ects determine the shape of the autocorrelation function of volatility as a function of time, which was shown to decay as a power law with a very small exponent ' 0:3–0.4 and no apparent cuto [7,9]. There are several approaches to modeling market mechanics. In one type of models price uctuations result from trading activity of conscious agents, whose decisions to buy or sell are dictated by strategies they follow. These strategies evolve in time (often according to some Darwinian rules) and give rise to a slowly changing uctuation pattern. There is little doubt that the evolution and dynamics of investor’s strategies and beliefs in uence the long term behavior of real market prices. For example, if some company could not keep up with the competition, sooner or later investors would realize it, and in the long-term its stock price would go down. However, it is unclear how does it in uence the properties of stock price uctuations at very short timescales, which do not allow time for traders to update their strategies or for a company to change its pro le. Another problem with models explaining short time price uctuations in terms of strategy evolution is that they inevitably lead their creators to shaky grounds of speculations about relevant and irrelevant psychological motivations of a “typical” trader in a highly heterogeneous trader population. The remarkable universality of the general features of price uctuations in markets of di erent types of risky assets such as stocks and their derivatives, foreign currencies, and commodities (say, cotton or oil) makes one to suspect that in fact psychological factors play little role in determining their short time properties, and leads one to try to look for a simpler mechanism giving rise to these features. In this work we do a rst step in this direction by introducing and numerically studying a simple market model, where a nontrivial price pattern arises not due to the evolution of trading strategies, but rather as a consequence of trading rules themselves, and to the way in which supply and demand determine the price on an open market. Before we proceed with formulating the rules of our model we need to de ne several common market terms. A trader on an open market is usually allowed to place a so-called “limit order to sell (buy)”, which is an instruction to automatically sell (buy) a particular amount of the traded asset, which we for simplicity will be simply referring to as stock, if its market price would raise higher (or drop lower for a limit buy order) than the predetermined threshold. This threshold is sometimes referred to as S. Maslov / Physica A 278 (2000) 571–578 573 the execution price of the limit order. In many modern markets, known to economists as order-driven markets,2 limit orders placed by ordinary traders constitute the major source of the liquidity of the market. It means that a request to immediately buy or sell a particular amount of stock at the best market price, or “market order”, is lled by matching it to the best unsatis ed limit order in the limit order book. To illustrate how orders are executed in an order-driven market let us consider the following simple example: suppose one trader (trader #1) has submitted a limit order to sell 1000 shares of the stock of a company X, provided that its price would exceed $20=share. Subsequently, another trader (trader #2) has submitted a limit order to sell 2000 shares of X if the price would exceed $21=share. Finally, a third trader decides to buy 2000 shares of X at the market price. In the absence of other limit orders his order will be lled as follows: he will buy 1000 shares from trader #1 at $20=share and 1000 shares from trader #2 at $21=share. After this transaction the limit order book would contain only one partially lled limit order to sell, that of trader #2 to sell 1000 shares of X at $21=share. Traders in our model can either trade stock at the market price or place a limit order to sell or buy. To simplify the rules of our toy market, traders are allowed to trade only one unit (lot) of stock in each transaction. That makes all limit and market orders to be of the same size. The empirical study of limit-order trading at the ASX can be used to partially justify this simpli cation. In this work it was observed that limit orders mostly come in sizes given by round numbers such as 1000 shares and (to a lesser extent) 10 000 shares. Unlike many other market models, we do not x the number of traders. Instead, at each time step a “new” trader appears out of a “trader pool” and attempts to make a transaction. With equal probabilities this new trader is a seller or a buyer. He then performs one of the following two actions: • with probability qlo he places a limit order to sell=buy. • otherwise (with probability 1− qlo) he trades (sells or buys) at the market price. The rule of execution of a market order in our model is particularly simple. Since all orders are of the same size, a market order is simply lled with the best limit order (i.e., the highest bid among limit orders to buy and the lowest ask among limit orders to sell), which is subsequently removed from the limit order book. This transaction performed at the execution price of the best limit order sets a new value of the market price p(t). To complete the de nition of the rules one needs to specify how a trader who selected to place a new limit order decides on its execution price. Traders in our model do this in a very “non-strategic” way by simply o setting the price of the last transaction performed on the market (current market price p(t)), by a random number . This positive random number is drawn each time from the same probability distribution P( ). A new limit order to sell is placed above the current price at p(t)+ , while a 2 A nice introduction to the mechanics of limit-order driven markets in general, and the Australian Stock Exchange in particular, by W. Yang can be found at http:==www.af.ecel.uwa.edu.au=acc n=WorkingPapers= abstracts=99-98.htm. 574 S. Maslov / Physica A 278 (2000) 571–578 new limit order to buy – below it at p(t)− . This way ranges of limit orders to sell and to buy never overlap, i.e., there is always a positive gap between highest bid and lowest ask prices. This “random o set” rule constitutes a reasonable rst-order approximation to what may happen in real order-driven markets and is open to modi cations if it fails a reality check. The most obvious variants of this rule, which we plan to study in the near future, are (i) A model where each trader has his individual distribution P( ). This modi cation would allow for the coexistence of “patient” traders who do not care very much about when their order will be executed or if it will be executed at all, and can therefore select large and poc