Unformatted text preview: almost all parameters for which an equilibrium exists, only such concentratedtrading equilibria exist.
We now show that, when an equilibrium exists, the basic nature of the
results we derived in the previous sections do not change when informed
traders have diverse information. We continue to assume that ϕ t = ϕ for
all t. Consider first the trading volume. It is easy to show that the variance
of the total order flow of the informed traders is given by
(22)
This is clearly increasing in Ψ t and in nt . Since informed traders are diversely
informed, there will generally be some trading within the group of informed
traders. (For example, if a particular informed trader draws an extreme
signal, his position may have an opposite sign to that of the aggregate
position of informed traders.) Thus, VtI, the measure of trading volume by
informed traders, will be greater than the expression in Equation (22).
The amount of trading within the group of informed traders is clearly an
increasing function of nt. Thus, this strengthens the effect of concentrated
trading on the volume measures: more liquidity trading leads to more
informed traders, which in turn implies an even greater trading volume.
The basic characteristics of the price process are also essentially
unchanged in this model. First consider the informativeness of the price,
as measured by
With diverse information it can be shown
that
(23)
As in Kyle (1984), an increase in the number of informed traders increases
the informativeness of prices. This is due in part to the increased competition among the informed traders. It is also due to the fact that more
information is gathered when more traders become informed. This second
effect was not present in the model with common private information. The
implications of the model remain the same as before: with endogenous
information acquisition, prices will be more informative in periods with
higher liquidity trading (i.e., periods in which the discretionary liquidity
traders trade).
In the model with diverse private information, the behavior of Rt (the
variance of price changes) is very similar to what we saw in the model
24 with common information. It can be shown that
(24)
As before, if nt = n for all t, then Rt = 1, and Rt > 1 if and only if
n t > n t1.
4. The Allocation of Liquidity Trading
In the analysis so far we have assumed that the discretionary liquidity
traders can only trade once, so that their only decision was the timing of
their single trade. We now allow discretionary liquidity traders to allocate
their trading among the periods in the interval [T', T"], that is, between
the time their liquidity demand is determined and the time by which it
must be satisfied. Since the model becomes more complicated, we will
illustrate what happens in this case with a simple structure and by numerical examples.
Suppose that T' = 1 and T" = 2, so that discretionary liquidity traders
can allocate their trades over two trading periods. Suppose that there are
n1, informed traders in period 1 and n2, informed traders in period 2 and
that the informed traders obtain perfect information (i.e., they observe
at time t). Each discretionary liquidity trader must choose a, the proportion
of the liquidity demand
that is satisfied in period 1. The remainder will
be satisfied in period 2. Discretionary liquidity trader j therefore trades
shares in period 2.
shares in period 1 and
To obtain some intuition, suppose that the price function is as given in
the previous sections; that is,
(25)
where λ t is given by Lemma 1. Note that the price in period t depends only
on the order flow in period t. In this case the discretionary liquidity trader’s
problem is to minimize. the cost of liquidity trading, which is given by It is easy to see that this is minimized by setting
For
example, if λ1 = λ2, then the optimal value of a is 1/2. Thus, if each price
is independent of previous order flows, the cost function for a liquidity
trader is convex, and so discretionary liquidity traders divide their trades
among different periods. It is important to note that the optimal a is independent of . This means that all liquidity traders will choose the same a.
If the above argument were correct, it would seem to upset our results
on the concentration of trade. However, the argument is flawed, since the
assumption that each price is independent of past order flows is no longer
25 appropriate. Recall that the market maker sets the price in each period
equal to the conditional expectation of
given all the information available to him at the time. This includes the history of past order flows. In
the models of the previous sections, there is no payoffrelevant information
that is not revealed by the public information
in past order flows
in period t. This is no longer true here, since past order flows enable the
market maker to forecast the liquidity component of current order flows.
This improves the precision of his prediction of the informedtrading component, which is relevant to future payoffs. Specifically, since the information that informed traders have in period 1 is revealed to the market
maker in period 2, the market maker c...
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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.
 Spring '12
 Svendsson

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