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

It is easy to show that the variance of the total

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Unformatted text preview: almost all parameters for which an equilibrium exists, only such concentrated-trading equilibria exist. We now show that, when an equilibrium exists, the basic nature of the results we derived in the previous sections do not change when informed traders have diverse information. We continue to assume that ϕ t = ϕ for all t. Consider first the trading volume. It is easy to show that the variance of the total order flow of the informed traders is given by (22) This is clearly increasing in Ψ t and in nt . Since informed traders are diversely informed, there will generally be some trading within the group of informed traders. (For example, if a particular informed trader draws an extreme signal, his position may have an opposite sign to that of the aggregate position of informed traders.) Thus, VtI, the measure of trading volume by informed traders, will be greater than the expression in Equation (22). The amount of trading within the group of informed traders is clearly an increasing function of nt. Thus, this strengthens the effect of concentrated trading on the volume measures: more liquidity trading leads to more informed traders, which in turn implies an even greater trading volume. The basic characteristics of the price process are also essentially unchanged in this model. First consider the informativeness of the price, as measured by With diverse information it can be shown that (23) As in Kyle (1984), an increase in the number of informed traders increases the informativeness of prices. This is due in part to the increased competition among the informed traders. It is also due to the fact that more information is gathered when more traders become informed. This second effect was not present in the model with common private information. The implications of the model remain the same as before: with endogenous information acquisition, prices will be more informative in periods with higher liquidity trading (i.e., periods in which the discretionary liquidity traders trade). In the model with diverse private information, the behavior of Rt (the variance of price changes) is very similar to what we saw in the model 24 with common information. It can be shown that (24) As before, if nt = n for all t, then Rt = 1, and Rt > 1 if and only if n t > n t-1. 4. The Allocation of Liquidity Trading In the analysis so far we have assumed that the discretionary liquidity traders can only trade once, so that their only decision was the timing of their single trade. We now allow discretionary liquidity traders to allocate their trading among the periods in the interval [T', T"], that is, between the time their liquidity demand is determined and the time by which it must be satisfied. Since the model becomes more complicated, we will illustrate what happens in this case with a simple structure and by numerical examples. Suppose that T' = 1 and T" = 2, so that discretionary liquidity traders can allocate their trades over two trading periods. Suppose that there are n1, informed traders in period 1 and n2, informed traders in period 2 and that the informed traders obtain perfect information (i.e., they observe at time t). Each discretionary liquidity trader must choose a, the proportion of the liquidity demand that is satisfied in period 1. The remainder will be satisfied in period 2. Discretionary liquidity trader j therefore trades shares in period 2. shares in period 1 and To obtain some intuition, suppose that the price function is as given in the previous sections; that is, (25) where λ t is given by Lemma 1. Note that the price in period t depends only on the order flow in period t. In this case the discretionary liquidity trader’s problem is to minimize. the cost of liquidity trading, which is given by It is easy to see that this is minimized by setting For example, if λ1 = λ2, then the optimal value of a is 1/2. Thus, if each price is independent of previous order flows, the cost function for a liquidity trader is convex, and so discretionary liquidity traders divide their trades among different periods. It is important to note that the optimal a is independent of . This means that all liquidity traders will choose the same a. If the above argument were correct, it would seem to upset our results on the concentration of trade. However, the argument is flawed, since the assumption that each price is independent of past order flows is no longer 25 appropriate. Recall that the market maker sets the price in each period equal to the conditional expectation of given all the information available to him at the time. This includes the history of past order flows. In the models of the previous sections, there is no payoff-relevant information that is not revealed by the public information in past order flows in period t. This is no longer true here, since past order flows enable the market maker to forecast the liquidity component of current order flows. This improves the precision of his prediction of the informed-trading component, which is relevant to future payoffs. Specifically, since the information that informed traders have in period 1 is revealed to the market maker in period 2, the market maker c...
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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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