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

Assume that the ith informed trader observes in

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Unformatted text preview: is section we discuss an alternative formulation of the model, in which informed traders observe different signals as in Kyle [1984]. The basic results about trading and volume patterns or price behavior do not change. However, the analysis of endogenous information acquisition is somewhat different. Assume that the ith informed trader observes in period t the signal and assume that the are independently and identically distributed with variance ϕ. Note that as n increases, the total amount of private information increases as long as ϕ > 0. The next result, which is analogous to Lemma 1 for the case of identical private signals, gives the equilibrium parameters for a given level of liquidity trading and a given number of informed traders. (The proof is a simple modification of the proof of Lemma 1 and is therefore omitted). Lemma 3. Assume that nt, informed traders trade in period t and that each observes an independent signal where and for all i. Let Ψ t, be the total variance of the liquidity trading in period t. Then (18) The ith informed trader submits market order t with in each period (19) Note that, as in the case of identical signals, λ t , is decreasing in Ψ t,. This immediately implies that Proposition 1 still holds in the model with diverse signals. Thus, if the number of informed traders is exogenously specified, the only robust equilibria are those in which trading by all discretionary liquidity traders is concentrated in one period. Recall that the results when information acquisition is endogenous were based on the observation that when there are more informed traders, they compete more aggressively with each other. This is favorable to the liquidity traders in that λ t, is reduced, intensifying the effects that lead to concentrated trading. However, when informed traders observe different pieces of information, an increase in their number also means that more private information is actually generated in the market as a whole. Indeed, unlike the case of identical signals an increase in nt, can now lead to an increase in λ t. It is straightforward to show that (with ϕ t = ϕ for all t as before) 22 (20) If the information gathered by informed traders is sufficiently imprecise, an increase in nt, will increase λ t. An increase in nt has two effects. First, it increases the degree of competition among the informed traders and this tends to reduce λ t. Second, it increases the amount of private information represented in the order flow. This generally tends to increase λ t. For large values of ϕ and small values of nt, an increase in nt has a substantial effect on the amount of information embodied in the order flow and this dominates the effect of an increase of competition. As a result, λ t increases. The discussion above has implications for equilibrium with endogenous information acquisition. In general, since the profits of each informed trader are increasing in Ψ, there would be more informed traders in periods in which discretionary liquidity traders trade more heavily. When signals are identical, this strengthens the incentives of discretionary liquidity traders to trade in these periods, since it lowers the relevant λ t further. Since in the diverse information case λ t can actually increase with an increase in nt, the argument for concentrated trading must be modified. Assume for a moment that nt, is a continuous rather than a discrete parameter. Consider two periods, denoted by H and L . In period H, the variance of liquidity trading is high and equal to Ψ H; in period L, the variance of liquidity trading is low and equal to Ψ L. Let nH (respectively nL) be the number of traders acquiring information in period H (respectively L ). To establish the viability of the concentrated-trading equilibrium, we need to show that with endogenous information acquisition, . If n is continuous, then endogenous information acquisition implies that profits must be equal across periods: Since Ψ H > Ψ L, it follows that nH > nL. To maintain equality between the profits with nH > nL, it is necessary that . Since it follows that Thus, if n were continuous, the value of λ would always be lower in periods with more liquidity trading, and the concentrated-trading equilibria would always be viable. These equilibria would also be generic as in Proposition 1. The above is only a heuristic argument, establishing the existence of concentrated-trading equilibria with endogenous information acquisition in the model with diverse information. Since nt is discrete, we cannot assert 23 that in equilibrium the profits of informed traders are equal across periods. This may lead to the nonexistence of an equilibrium for some parameter values, as we show in the Appendix. It can be shown, however, that • An equilibrium always exists if the variance of the discretionary liquidity demand is sufficiently high. • If an equilibrium exists, then an equilibrium in which trading is concentrated exists. Moreover, for...
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