Justificación

The earlier literature on the modelling of the limit order book focuses on the trade- off between immediate execution of the market order and the better price (but uncertain execution) of the limit order. Cohen et al. (1981) develop a “gravitational pull” model to explain when a trader would submit a limit order as opposed to a market order. The trader’s choice between a limit and market order strategy depends on the balancing of the relative costs of price improvement and execution risk associated with using a limit order vis-à-vis the use of a market order. As spreads narrow, the benefits of a better price associated with using a limit order decreases, causing more traders to place market orders. However, as more traders use market orders instead of limit orders, the spread is likely to widen and increase the attractiveness of the limit order.

Another earlier paper to consider the open limit order book is Glosten (1994). He analyses an “idealized electronic open limit order book” under fairly general conditions (Glosten, 1994, p.1127). One of the important assumptions of his model and also of subsequent models relates to the ability to trade on private information. As in Kyle (1985), Glosten (1994) assumes traders can submit orders of any quantity. However, the orders are not batched but arrive one at a time. In the limit order book market considered, competing individuals determine the terms of trade. The electronic limit order book is modelled as a publicly visible screen providing bids and offers, each specifying a price and a quantity. The source of bids and offers is a large population of risk neutral “patient traders”. These liquidity suppliers are thought of as “patient” or “value” traders in that their only interest in trading is expected profit. Glosten suggests that it might be reasonable to think of this population as consisting of managers of reasonably large institutional and individual portfolios.

In the presence of adverse selection, the limit order book exhibits a positive bid-ask spread. The possibility of trading with an informed trader increases the probability of losses to the trader, who places an offer to sell at the lowest price or an offer to buy

at the highest price. Market orders traded against the book pick off the limit orders at their limit prices. The market orders are presumed to be placed by risk averse traders after some rational optimisation process and possibly by informed traders. A limit buy (sell) order trader can expect to lose if the order executes upon the arrival of an informed trader with a valuation below (above) the limit price, and can expect to gain if the order executes upon the arrival of a liquidity trader. Traders will not choose to place a limit order unless the expected gain from transacting with a liquidity trader exceeds the expected loss from transacting with an informed trader. While Glosten’s (1994) model provides the equilibrium price schedule in the open limit order book, it does so by assuming the existence of two distinct classes of trader: traders who place limit orders and those who place market orders. The analysis does not model the trader’s choice to trade via limit order or market order.

In an extension of Glosten’s analysis, Handa and Schwartz (1996) consider the choice faced by an investor who wishes to buy or sell a share of a risky asset. The investor can choose to trade via a market order and demand liquidity from the market or use a limit order and supply liquidity to the market. The choice depends critically on the probability of the limit order trading against an informed trader versus a liquidity trader. The model assumes that any transaction price change caused by the arrival of a liquidity trader is temporary and reversible while any change due to the arrival of an informed trader is permanent and irreversible. A limit order trader finds trading with an informed trader is undesirable but trading with a liquidity trader is desirable. In addition to the adverse selection problem, limit orders suffer from the risk of non-execution. If the limit order fails to trade, the trader has to decide whether to trade at the prevailing transaction price using a market order or forego trading. The act of not trading, obviously, has cost implications.

In comparing the performance of executed limit orders and market orders, Handa and Schwartz (1996) find that the differential limit order returns conditional upon execution are consistently positive and increase steadily for orders placed further behind the market. They suggest that limit orders are associated with higher returns because a sufficient proportion of limit order executions occur due to liquidity driven price changes and that prices tend to rebound over relatively short investment horizons. The question arises as to why we do not observe an abundance of limit orders. Handa and Schwartz (1996) argue that it is because of the positive cost of

non-execution. Using market-adjusted returns, they find that differential limit order returns conditional on execution are positive and those conditional on non-execution are negative compared to market orders.4 They suggest that an eager trader could find limit orders costly due to the non-execution and would choose to use market orders instead. However, a patient trader can avoid the cost of non-execution by simply not trading if the limit order does not execute.

While Handa and Schwartz (1996) analyse the rationale and profitability of limit order trading, they do not explicitly model the investor’s decision to trade via a market or limit order. Foucault (1999) incorporates an investor’s decision to trade via a limit order or market order, and develops a simple model in which the mix between limit and market orders can be characterised in equilibrium. Trading occurs in Foucault’s model because of differences in valuation of the stock. It assumes that changes in the valuation of the stock are driven by public information and not by private information.

A limit order trader suffers from two risks: (1) execution risk and (2) the winner’s curse. The probability that a limit order is not executed gives rise to execution risk. On the other hand, the winner’s curse is associated with the order being “picked off” when the value of the stock changes and the limit order placed has not been amended to reflect the change in value. The bid-ask spread on the limit order book is determined by the trade-off between the two risks. The primary finding in Foucault’s paper is that the volatility of the asset is a main determinant of the mix between limit and market orders. As volatility increases, the probability of being “picked off” before the limit order trader has a chance to amend his order increases; thus limit order traders ask for a larger compensation for providing liquidity. Limit order traders have to post higher ask prices and lower bid prices relative to their reservation prices in markets with higher volatility. Thus, market orders become less attractive and traders use more limit orders instead of market orders.

Handa, Schwartz and Tiwari (1998) use a model similar to Foucault (1999) to describe the limit order book. They suggest the “economics that drive the order- driven markets are intricate, and their viability is not obvious”. Their model

4 _{Cho and Nelling (2000), in a subsequent paper, find that the probability of execution decreases as the}

describes two economic forces that drive trading: (1) a liquidity event and (2) an information event.5 An information event is the advent of news that affects all investors’ assessments of a security’s share value while a liquidity event is unique to the individual investor. An example of the latter is a cash flow expenditure resulting in the investor’s need to sell his shares.

Traders who submit limit orders always lose if only an information event occurs. Buy limit orders will execute only if bearish news occurs, while ask limit orders will execute only if the news is bullish. However, limit order traders profit from liquidity events where the arrival of liquidity-motivated sell (buy) market orders causes share prices to fall (rise) temporarily. After the liquidity event, prices tend to revert to their previous levels. As a result, buy (sell) limit order placers profit from the execution. The mean reversion of prices after liquidity events is associated with short-period price volatility. This accentuated short-period volatility offsets the cost of information events to limit order placers, enticing traders who are patient to place limit orders.

The order-driven market achieves a balance between limit and market order traders when the accentuated short-period volatility is just sufficient to compensate the marginal investor for placing a limit order. Conversely, the non-execution of limit orders makes it costly for impatient traders, inducing them to place market orders. Handa et al. (1998) suggest that unlike market makers the objective of limit order traders is to implement a portfolio decision. Limit order traders are not obliged to provide a two-sided market. The provision of liquidity and gaining the spread for providing immediacy is a secondary effect when placing a limit order.

In a subsequent paper, Handa, Schwartz and Tiwari (2003) extend Foucault (1999) by focusing on the bid-ask spread in a limit order market where the proportion of buyers and sellers is free to vary without restriction. In Foucault’s model, the proportion of buyers and sellers is restricted to 0.5 although this restriction was relaxed in the special case considered. They also introduce adverse selection to the model by incorporating privately informed traders. The model assumes a single risky asset that trades in a continuous market environment where there are two groups of

traders in the market, one placing a high value to the asset and the other attaching a low value to the asset. A proportion of these investors is privately informed and assumed to trade using only market orders. The uninformed traders have a choice of market or limit orders.

The bid-ask spread derived in the model is a function of (1) the adverse selection cost, (2) the differences in valuation among groups of investors and (3) the proportion of investors in each of the groups. The difference in valuation causes the spread to exist even in the absence of asymmetric information. The spread is shown to widen with an increase in adverse selection costs or differences in valuation. The spread is widest when the proportion of investors in the two groups (with differences in valuation) is equal and minimised when the proportion is close to zero or one. Handa et al. (2003) suggest the results can be explained intuitively. The imbalance in the type of trader creates a competitive environment where traders on the crowded side compete with each other to gain priority. For example, traders place more aggressive buy limit orders if there are many buyers. The assumption here is that traders on the sell side do not shift their supply schedule. Handa et al. (2003) test their theory using CAC40 Index stocks from the Paris Bourse and find support for their model. The model was rejected at the 5% significance level for only 22% of the firms in the sample, which the authors argue is encouraging given the complexity of spread determination and the number of factors affecting spreads.