Unformatted text preview: ion release and the magnitude of the
nondiscretionary liquidity trading in each period. Various patterns can
easily be obtained by making the appropriate assumptions about these
exogenous variables. Since our main interest in this article is to examine
the effects of traders’ strategic behavior on prices and volume, we wish to
abstract from these other determinants. If the rate at which information
becomes public is constant and the magnitude of nondiscretionary liquidity trading is the same in all periods, then any patterns that emerge are
due solely to the strategic behavior of traders. We therefore assume in this
section that var( ) = g var( δ t ) = 1, and var(
t) = φ for all t. Setting var
to be constant over time guarantees that public information arrives at a
constant rate. [The normalization of var( ) to 1 is without loss of generality.]
Before presenting our results on the behavior of prices and trading
volume, it is important to discuss how volume should be measured. Suppose that there are k traders with market orders given by
Assume that the
are independently and normally distributed, each with
mean 0 Let
The total volume of
trade (including trades that are “crossed” between traders) is max
The expected volume is
13 (7)
where σ i, is the standard deviation of
One may think that var , the variance of the total order flow, is appropriate for measuring the expected volume of trading. This is not correct.
Since ii, is the net demand presented to the market maker, it does not
include trades that are crossed between traders and are therefore not met
by the market maker. For example, suppose that there are two traders in
period t and that their market orders are 10 and 16, respectively (i.e.,
the first trader wants to purchase 10 shares, and the second trader wants
to sell 16 shares). Then the total amount of trading in this period is 16
shares, 10 crossed between the two traders and 6 supplied by the market
maker
= 6 in this case). The parameter var
, which is represented
by the last term in Equation (7), only considers the trading done with the
market maker. The other terms measure the expected volume of trade
across traders. In light of the above discussion, we will focus on the following measures of trading volume, which identify the contribution of
each group of traders to the total trading volume: In words,
measure the expected volume of trading of the informed
traders and the liquidity traders, respectively, and
measures the
expected trading done by the market maker. The total expected volume,
Vt, is the sum of the individual components. These measures are closely
related to the true expectation of the actual measured volume.10
Proposition 1 asserts that a typical equilibrium for our model involves
the concentration of all discretionary liquidity trading in one period. Let
10 Our measure of volume is proportional to the actual expected volume if there is exactly one nondiscretionary liquidity trader; otherwise, the trading crossed between these traders will not be counted, and
will be lower than the true contribution of the liquidity traders. This presents no problem for our
analysis, however, since the amount of this trading In any period is Independent of the strategic behavior
of the other traders. 14 this period be denoted by t*. Note that if we assume that nt, var
are independent of t, then t* can be any period in [T',
T"].
The, following result summarizes the equilibrium patterns of trading
volume in our model.
Proposition 2. I n an equilibrium in which all discretionary liquidity
trading occurs in period t*,
2.
2.
3.
Proof. Part 1 is trivial, since there is more liquidity trading in t* than in
other periods. To prove part 2, note that
(12)
Thus, an increase in Ψ t, the total variance of liquidity trading, decreases
λ t, and increases the informed component of trading. Part 3 follows immediately from parts 1 and 2. n
This result shows that the concentration of liquidity trading increases
the volume in the period in which it occurs not only directly through the
actual liquidity trading (an increase in V) but also indirectly through the
additional informed trading it induces (an increase in
This is an
example of trading generating trading. An example that illustrates this
phenomenon is presented following the next result.”
We now turn to examine two endogenous parameters related to the price
process. The first parameter measures the extent to which prices reveal
private information, and it is defined by
(13)
The second is simply the variance of the price change:
(14)
Prposition 3. Assume that nt, = n for every t. Then Proof. It is straightforward to show that in general
(15)
11 Note that the amount of informed trading is independent of the precision of the signal that informed
traders observe. This is due to the assumed risk neutrality of informed traders. 15 (16)
The result follows since both Rt, and Qt, are independent of Ψ t , and nt, =
n. n
As observed in Kyle (1984, 1985), the amount of private information
revealed by the price is independent of the total variance of liquidity
trading. Thus,...
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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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