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

The additional informed trading is just sufficient to

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Unformatted text preview: despite the concentration of trading in t*, Qt. = Qt for all t. The intuition behind this is that although there is more liquidity trading in period t* , there is also more informed trading, as we saw in Proposition 2. The additional informed trading is just sufficient to keep the information content of the total order flow constant. Proposition 3 also says that the variance of price changes is the same when n informed traders trade in each period as it is when there is no informed-trading. [When there is no informed trading, Pt - Pt-1 = δ t, so Rt= var( δ t) = 1 for all t.] With some informed traders, the market gets information earlier than it would otherwise, but the overall rate at which information comes to the market is unchanged. Moreover, the variance of price changes is independent of the variance of liquidity trading in period t. As will be shown in the next section, these results change if the number of informed traders is determined endogenously. Before turning to this analysis, we illustrate the results of this section with an example. Example (continued). Consider again the example introduced in Section 1.2. Recall that in the equilibrium we discussed, both of the discretionary liquidity traders trade in period 3. Table 2 shows the effects of this trading on volume and price behavior. The volume-of-trading measure in period 3 is V3 = 13.14, while that in the other periods is only 4.73. The difference is only partly due to the actual trading of the liquidity traders. Increased trading by the three informed traders in period 3 also contributes to higher volume. As the table shows, both Q, and R, are unaffected by the increased liquidity trading. With three informed traders, three quarters of the private information is revealed through prices no matter what the magnitude of liquidity demand. 2. Endogenous Information Acquisition In Section 1 the number of informed traders in each period was taken as fixed. We now assume, instead, that private information is acquired at some cost in each period and that traders acquire this information if and only if their expected profit exceeds this cost. The number of informed traders is therefore determined as part of the equilibrium. It will be shown that endogenous information acquisition intensifies the result that trading is concentrated in equilibrium and that it alters the results on the distribution and informativeness of prices. 16 Table 2 Effects of discretionary liquidity trading on volume and price behavior when the number of informed traders is constant over time A four-period example, with nt = 3 informed traders In each period. For t = 1, 2, 3, 4, the table gives λ t, the market-depth parameter; Vt, a measure of total trading volume; a measure of the Informed-trading volume; a measure of liquidity trading volume; a measure of the trading volume of the market maker; Q,, 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. Let us continue_to assume that public information arrives at a constant rate and that var( δ t ) = 1 and var = g for all t. Let c be the cost of observing in period t, where var = φ. We assume that This will guarantee that in equilibrium at least one trader is informed in each period. We need to determine the equilibrium number of informed traders in period t12 Define to be the expected trading profits of an informed trader (over one period) when there are n, informed traders in the market and the total variance of all liquidity trading is Ψ t. Let λ (nt Ψ ,) be the equilibrium value of λ t, under these conditions. (Note that these functions are the same in all periods.) The total expected cost of the liquidity traders is Since each of the nt, informed traders submits the same market order, they divide this amount equally. Thus, from Lemma 1 we have It is clear that a necessary condition for an equilibrium with n informed traders is otherwise, the trading profits of informed traders do not cover the cost of acquiring the information. Another condition for an equilibrium with nt informed traders is that no additional trader has incentives to become informed. We will discuss two models of entry. One approach is to assume that a potential entrant cannot make his presence known (that is, he cannot credibly announce his presence to the rest of the market). Under this assumption, a potential entrant takes the strategies of all other traders and the market maker as given and assumes that they will continue to behave 12 Note that we are assuming that the precision of the information, measured by the parameter together with the cost of becoming Informed, are constant over time. If the precision of the signal varied across periods, then there might also be a different cost to acquiring different signals. We would then need to specify a cost function for signals as a function of their precision. 17 Table 3 Expected trading profits of informed traders when the variance of liquidity demand is 6 For some possible number of Informed traders, n, the table gives π (n, 6), the expected profits of each of the info...
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