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

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

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Unformatted text preview: rmed traders, assuming that the variance of total liquidity trading is 6; and π (n, 6)/4, the profits of an entrant who assumes that all other traders will use the same equilibrium strategies after he enters as an informed trader. If the cost of information is 0.13, then the equilibrium number of informed traders is n ∈ {3,4,5,6) in the first approach and n = 6 in the second. as if nt traders are informed. Thus we still have λ = λ ( nt, Ψ t). The following lemma gives the optimal market order for an entrant and his expected trading profits under this assumption. (The proof is in the Appendix.) Lemma 2. An entrant into a market with nt, informed traders will trade exactly half the number of shares as the other nt traders for any realization of the signal, and his expected profit will be π (nt, Ψ t )/4. It follows that with this approach nt, is an equilibrium number of informed traders in period t if and only if nt, satisfies If c is large enough, there may be no positive integer nt, satisfying this condition, so that the only equilibrium number of informed traders is zero. However, the assumption that guarantees that this is never the case. In general, there may be several values of nt, that are consistent with equilibrium according to this model. An alternative model of entry by informed traders is to assume that if an additional trader becomes informed, other traders and the market maker change their strategies so that a new equilibrium, with nt + 1 informed traders, is reached. If liquidity traders do not change their behavior, the profits of each informed trader would now become The largest nt, satisfying is the (unique) n satisfying which is the condition for equilibrium under the alternative approach. This is illustrated in the example below. Example (continued). Consider again the example introduced in Section 1.2 (and developed further in Section 1.3). In period 3, when both of the discretionary liquidity traders trade, the total variance of liquidity trading is Ψ3 = 6. Assume that the cost of perfect information is c = 0.13. Table 3 gives π (n, 6) and π (n, 6)/4 as a function of some possible values for n. 13 In fact, the same equilibrium obtains if liquidity traders were assumed to respond to the entry of an informed trader, as will be clear below. 18 Table 4 Expected trading profits of informed traders when the variance of liquidity demand is 3 With c = 0.13, it is not an equilibrium to have only one or two informed traders, for in each of these cases a potential entrant will find it profitable to acquire information. It is also not possible to have seven traders acquiring information since each will find that his equilibrium expected profits are less than c = 0.13. Equilibria involving three to six informed traders are clearly supportable under the first model of entry. Note that n3 = 6 also has the property that π (7, 6) < 0.13 < π (6, 6), so that if informed traders and the market maker (as well as the entrant) change their strategies to account for the actual number of informed traders, each informed trader makes positive profits, and no additional trader wishes to become informed. As is intuitive, a lower level of liquidity trading generally supports fewer informed traders. In period 2 in our example, no discretionary liquidity traders trade, and therefore Ψ2 = g = 1. Table 4 shows that if the cost of becoming informed is equal to 0.13, there will be no more than three informed traders. Moreover, assuming the first model of entry, the lower level of liquidity trading makes equilibria with one or two informed traders viable. To focus our discussion below, we will assume that the number of informed traders in any period is equal to the maximum number that can be supported. With c = 0.13 and Ψ t = 6, this means that nt = 6, and with the same level of cost and Ψ t = 1, we have nt = 3. As noted above, this determination of the equilibrium number of informed traders is consistent with the assumption that an entrant can credibly make his presence known to informed traders and to the market maker. Does endogenous information acquisition change the conclusion of . Proposition 1 that trading is concentrated in a typical equilibrium? We know that with an increased level of liquidity trading, more informed traders will generally be trading. If the presence of more informed traders in the market raises the liquidity traders’ cost of trading, then discretionary liquidity traders may not want to trade in the same period. It turns out that in this model the presence of more informed traders actually lowers the liquidity traders’ cost of trading, intensifying the forces toward concentration of trading. As long as there is some informed trading 19 in every period, liquidity traders prefer that there are more rather than fewer informed traders trading along with them. Of course, the best situation for liquidity traders is for there to be no informed traders, but for nt, > 0, the cost of trading is a decreasing function of n,. The total cost o...
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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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