Unformatted text preview: an subtract
, from the total
order flow in period 1. This reveals
which is informative about
the discretionary liquidity demand in period 2.
Since the terms of trade in period 2 depend on the order flow in period
1, a trader who is informed in both periods will take into account the effect
that his trading in the first period will have on the profits he can earn in
the second period. This complicates the analysis considerably. To avoid
these complications and to focus on the behavior of discretionary liquidity
traders, we assume that no trader is informed in more than one period.
Suppose that the price in period 1 is given by where and that the price in period 2 is given by
where Note that the form of the price is the same in the two periods, but the
order flow in the second period has been modified to reflect the prediction
of the discretionary liquiditytrading component based on the order flow
in the first period and the realization of
Let γ be the coefficient in the
regression of
Then it can be shown that the problem
each discretionary liquidity trader faces, taking the strategies of all other
traders and the market maker as given, is to choose α to minimize
The solution to this problem is to set
(31)
26 Given that discretionary liquidity traders allocate their trades in this fashion, the market maker sets λ1 and λ2 so that his expected profit in each
period (given all the information available to him) is zero. It is easy to
show that in equilibrium λ t, and β t are given by Lemma 1, with and
While it can be shown that this model has an equilibrium, it is generally
impossible to find the equilibrium in closed form. We now discuss two
limiting cases, one in which the nondiscretionary liquidity component
vanishes and one in which it is infinitely noisy; we then provide examples
in which the equilibrium is calculated numerically.
Consider first the case in which most of the liquidity trading is nondiscretionary. This can be thought of as a situation in which g →∞. In this
situation the market maker cannot infer anything from the information
available in the second period about the liquidity demand in that period.
It can then be shown that γ→0, so that past order flows are uninformative
to the market maker. Moreover, For example, if n1 = n2, , then α→ 1/2. Not surprisingly, this is the solution
we would obtain if we assumed that the price in each period is independent
of the previous order flow. When discretionary liquidity trading is a small
part of the total liquidity trading, we do not obtain a concentratedtrading
equilibrium.
Now consider the other extreme case, in which
In this
case almost all the liquidity trading is discretionary, and therefore the
market maker can predict with great precision the liquidity component of
the order flow in the second period, given his information. It can be shown
that in the limit we get α = 1, so that all liquidity trading is concentrated
in the first period. Note that since there is no liquidity trading in the second
period, λ2→∞; thus, in a model with endogenous information acquisition
we will get n2 = 0 and there will be no trade in the second period.15
In general, discretionary liquidity traders have to take into account the
fact that the market maker can infer their demands as time goes on. This
causes their trades to be more concentrated in the earlier periods, as is
illustrated by the two examples below. Note that, unlike the concentration
result in Proposition 1, it now matters whether trading occurs at time T'
15 Note that if indeed there is no trading by either the informed or the liquidity traders, then λ is undetermined,
If we interpret it as a regression coefficient in the regression of
However, with no liquidity
trading the market maker must refuse to trade. This is equivalent to setting λ t ,to infinity. 27 Table 6
Volume and price behavior when discretionary liquidity traders allocate trading across several
periods A fourperiod example in which the number of informed traders, nt, is determined endogenously, assuming
that the cost of information is 0.13 and that liquidity traders can allocate their trade in different periods
between 2:00 P.M. and 4:00 P.M. For t = 1, 2, 3, 4, the table gives λ t, the marketdepth parameter; Vt, a
measure of total trading volume; VtI, a measure of the informedtrading volume; VtL, a measure of liquiditytrading volume; VtM, a measure of the trading volume of the market maker; Qt, 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. or later; the different trading periods are not equivalent from the point of
view of the discretionary liquidity traders. This will have implications when
information acquisition is endogenous. Consider the following two examples.
In the first example we make all the parametric assumptions made in
our previous examples, except that now we allow the discretionary liquidity
traders A and B to alloca...
View
Full Document
 Spring '12
 Svendsson
 Economics, Variance, Financial Markets, Order theory

Click to edit the document details