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

This complicates the analysis considerably to avoid

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Unformatted text preview: an subtract , from the total order flow in period 1. This reveals which is informative about the discretionary liquidity demand in period 2. Since the terms of trade in period 2 depend on the order flow in period 1, a trader who is informed in both periods will take into account the effect that his trading in the first period will have on the profits he can earn in the second period. This complicates the analysis considerably. To avoid these complications and to focus on the behavior of discretionary liquidity traders, we assume that no trader is informed in more than one period. Suppose that the price in period 1 is given by where and that the price in period 2 is given by where Note that the form of the price is the same in the two periods, but the order flow in the second period has been modified to reflect the prediction of the discretionary liquidity-trading component based on the order flow in the first period and the realization of Let γ be the coefficient in the regression of Then it can be shown that the problem each discretionary liquidity trader faces, taking the strategies of all other traders and the market maker as given, is to choose α to minimize The solution to this problem is to set (31) 26 Given that discretionary liquidity traders allocate their trades in this fashion, the market maker sets λ1 and λ2 so that his expected profit in each period (given all the information available to him) is zero. It is easy to show that in equilibrium λ t, and β t are given by Lemma 1, with and While it can be shown that this model has an equilibrium, it is generally impossible to find the equilibrium in closed form. We now discuss two limiting cases, one in which the nondiscretionary liquidity component vanishes and one in which it is infinitely noisy; we then provide examples in which the equilibrium is calculated numerically. Consider first the case in which most of the liquidity trading is nondiscretionary. This can be thought of as a situation in which g →∞. In this situation the market maker cannot infer anything from the information available in the second period about the liquidity demand in that period. It can then be shown that γ→0, so that past order flows are uninformative to the market maker. Moreover, For example, if n1 = n2, , then α→ 1/2. Not surprisingly, this is the solution we would obtain if we assumed that the price in each period is independent of the previous order flow. When discretionary liquidity trading is a small part of the total liquidity trading, we do not obtain a concentrated-trading equilibrium. Now consider the other extreme case, in which In this case almost all the liquidity trading is discretionary, and therefore the market maker can predict with great precision the liquidity component of the order flow in the second period, given his information. It can be shown that in the limit we get α = 1, so that all liquidity trading is concentrated in the first period. Note that since there is no liquidity trading in the second period, λ2→∞; thus, in a model with endogenous information acquisition we will get n2 = 0 and there will be no trade in the second period.15 In general, discretionary liquidity traders have to take into account the fact that the market maker can infer their demands as time goes on. This causes their trades to be more concentrated in the earlier periods, as is illustrated by the two examples below. Note that, unlike the concentration result in Proposition 1, it now matters whether trading occurs at time T' 15 Note that if indeed there is no trading by either the informed or the liquidity traders, then λ is undetermined, If we interpret it as a regression coefficient in the regression of However, with no liquidity trading the market maker must refuse to trade. This is equivalent to setting λ t ,to infinity. 27 Table 6 Volume and price behavior when discretionary liquidity traders allocate trading across several periods A four-period example in which the number of informed traders, nt, is determined endogenously, assuming that the cost of information is 0.13 and that liquidity traders can allocate their trade in different periods between 2:00 P.M. and 4:00 P.M. For t = 1, 2, 3, 4, the table gives λ t, the market-depth parameter; Vt, a measure of total trading volume; VtI, a measure of the informed-trading volume; VtL, a measure of liquiditytrading volume; VtM, a measure of the trading volume of the market maker; Qt, a measure of the amount of private information revealed In the price; and Rt, the variance of the price change from period t - 1 to period t. or later; the different trading periods are not equivalent from the point of view of the discretionary liquidity traders. This will have implications when information acquisition is endogenous. Consider the following two examples. In the first example we make all the parametric assumptions made in our previous examples, except that now we allow the discretionary liquidity traders A and B to alloca...
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