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

The reciprocal of t is kyles market depth parameter

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Unformatted text preview: ation conforms with that in Kyle (1984, 1985). The reciprocal of λ t, is Kyle’s market-depth parameter, and it plays an important role in our analysis. The main result of this section shows that in equilibrium there is a tendency for trading to be concentrated in the same period. Specifically, we will show that equilibria where all discretionary liquidity traders trade in the same period always exist and that only such equilibria are robust to slight changes in the parameters. Our analysis begins with a few simple results that characterize the equilibria of the model. Suppose that the total amount of discretionary liquidity if the jth discretionary liquidity demands in period t is where trader trades in period t a nd where otherwise. Define that is, Ψ t is the total variance of the liquidity trading in period t. (Note that Ψ t must be determined in equilibrium since it depends on the trading positions of the discretionary liquidity traders.) The following lemma is proved in the Appendix. Lemma 1. If the market maker follows a linear pricing strategy, then in equilibrium ,each informed trader i submits at time t a market order of where (4) The equilibrium value of λ t is given by (5) This lemma gives the equilibrium values of A, and β t for a given number of informed traders and a given level of liquidity trading. Most of the comparative statics associated with the solution are straightforward and intuitive. Two facts are important for our results. First, λ t, is decreasing in Ψ t, the total variance of liquidity trades. That is, the more variable are the liquidity trades, the deeper is the market. Less intuitive is the fact that λ t, is decreasing in nt, the number of informed traders. This seems surprising since it would seem that with more informed traders the adverse selection problem faced by the market maker is more severe. However, informed traders, all of whom observe the same signal, compete with each other, 10 and this leads to a smaller λ t. This is a key observation in the next section, where we introduce endogenous entry by informed traders.8 When some of the liquidity trading is discretionary, Ψ t, is an endogenous parameter. In equilibrium each discretionary liquidity trader follows the trading policy that minimizes his expected transaction costs, subject to meeting his liquidity demand We now turn to the determination of this equilibrium behavior. Recall that each trader takes the value of λ t (as well as the actions of other traders) as given and assumes that he cannot influence it. The cost of trading is measured as the difference between what the liquidity trader pays for the security and the security’s expected value. Specifically, the expected cost to the jth liquidity trader of trading at time t ∈ [T', T"] is (6) Substituting for -and using the fact that where T are independent of (which is the information of discretionary liquidity trader j )-the cost simplifies to Thus, for a given set of λ t, t ∈ [T', T"], the expected cost of liquidity trading is minimized by trading in that period t* ∈ [T', T"] in which A, is the smallest. This is very intuitive, since λ t, measures the effect of each unit of order flow on the price and, by assumption, liquidity traders trade only once. Recall that from Lemma 1, λ t, is decreasing in Ψ t. This means that if in equilibrium the discretionary liquidity trading is particularly heavy in a particular period t, then λ t, will be set lower, which in turn makes discretionary liquidity traders concentrate their trading in that period. In sum, we obtain the following result. Proposition 1. There always exist equilibria in which all discretionary liquidity trading occurs in the same period. Moreover, only these equilibria are robust in the sense that if for some set of parameters there exists an equilibrium in which discretionary liquidity traders do not trade in the same period, then for an arbitrarily close set of parameters [e.g., by perturbing the vector of variances of the liquidity demands Yj), the only possible equilibria involve concentrated trading by the discretionary liquidity traders. 8 More intuition for why λ t, is decreasing in nt, can be obtained from statistical inference. Recall that A, is the regression coefficient in the forecast of given the total order flow . The order flow can be written as represents the total trading position of the informed traders and û is the position of the liquidity traders with As the number of informed traders increases, a increases. For a given level of a, the market maker sets λ t equal to This is an Increasing function of a if and only if which in this model occurs if and only if nt ≤ 1. We an think of the market maker’s inference problem in two pans: first he uses to predict then he sales this down by a factor of 1/a to obtain his prediction of The weight placed upon in predicting is always increasing in a, but for a large enough value of a the scaling down by a factor of l/a ev...
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