Admati And Pfleiderer-A Theory Of Intraday Patterns - Volume And Price Variability

4 assume there are m discretionary liquidity traders

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Unformatted text preview: where T' < T" < T.4 Assume there are m discretionary liquidity traders and let be the total demand of the jth discretionary liquidity trader (revealed to that trader in period T'). Since each discretionary liquidity trader is risk-neutral, he determines his trading policy so as to minimize his expected cost of trading, subject to the condition that he trades a total of shares by period T’. Until Section 4 we assume that each discretionary liquidity trader only trades once between time T' and time T"; that is, a liquidity trader cannot divide his trades among different periods. Prices for the asset are established in each period by a market maker who stands prepared to take a position in the asset to balance the total demand of the remainder of the market. The market maker is also assumed to be risk-neutral, and competition forces him to set prices so that he earns zero expected profits in each period. This follows the approach in Kyle (1985) and in Glosten and Milgrom (1985).5 3 This assumption is reasonable since the span of time coveted by the T periods in this model is to be taken as relatively short and since our main interests concern the volume of trading and the variability of prices. The nature of our results does not change if a positive discount rate is assumed. 4 In reality. of course, different traders may realize their liquidlty demands at different times, and the time that can elapse before these demands must be satisfied may also be different for different traders. The nature of our results will not change if the model is complicated to capture this. See the discussion in Section 5.1. 5 The model here can be viewed as the limit of a model with a finite number of market makers as the number of market makers grows to Infinity. However, our results do not depend in any important way on the assumption of perfect competition among market makers. The same basic results would obtain in an analogous model with a finite number of market makers, where each market maker announces a (linear) pricing schedule as a function of his own order flow and traders can allocate their trade among different market makers. In such a model, market makers earn positive expected profits. See Kyle (1984). 8 Let be the ith informed trader’s order in period t, be the order of the jth discretionary liquidity trader in that period, and let us denote by the total demand for shares by the nondiscretionary liquidity traders in period t, Then the market maker must purchase shares in period t. The market maker determines a price in period t based on the history of public information, and on the history of order flows, . . . , . The zero expected profit condition implies that the price set in period t by the market maker, satisfies (2) Finally, we assume that the random variables are mutually independent and distributed multivariate normal, with each variable having a mean of zero. 1.2 Equilibrium We will be concerned with the (Nash) equilibria of the trading game that our model defines among traders. Under our assumptions, the market maker has a passive role in the model.7 Two types of traders do make strategic decisions in our model. Informed traders must determine the size of their market order in each period. At time t, this decision is made knowing S t-1, the history of order flows up to period t - 1; A,, the innovations up to t; and the signal, The discretionary liquidity traders must choose a period in [T', T"] in which to trade. Each trader takes the strategies of all other traders, as well as the terms of trade (summarized by the market maker’s price-setting strategy), as given. The market maker, who only observes the total order flow, sets prices to satisfy the zero expected profit condition. We assume that the market maker’s pricing response is a linear function of and In the equilibrium that emerges, this will be consistent with the zero-profit condition. Given our assumptions, the market maker learns nothing in period t from past order flows that cannot be inferred from the public information A,. This is because past trades of the informed traders are independent of and because the liquidity trading in any period is independent of that in any other period. This means that the price set in period t is equal to the expectation of conditional on all public information observed in that period plus an adjustment that reflects the information contained in the current order flow 6 If the price were a function of individual orders, then anonymous traders could manipulate the price by submitting canceling orders. For example, a trader who wishes to purchase 10 shares could submit a purchase order for 200 shares and a sell order for 190 shares. When the price is solely a function of the total order flow, such manipulations are not possible. 7 It is actually possible to think of the market maker also as a player in the game, whose payoff is minus the sum of the squared deviations of the prices from the true payoff. 9 (3) Our not...
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This note was uploaded on 02/22/2013 for the course ECON 101 taught by Professor Svendsson during the Spring '12 term at Stockholm School of Economics.

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