Unformatted text preview: te their trades across periods 2, 3, and 4. If information acquisition is endogenous and if the cost of perfect information is
c = 0.13, then we obtain the equilibrium parameters given in Table 6. In
this example, each discretionary liquidity trader j trades about
in period 3, and
in period 4. Note that the measure
of liquidity-trading volume is highest in period 2 and then falls off in
periods 3 and 4. Three informed traders are present in each of the periods
except period 2, when it is profitable for a fourth to enter. The behavior
of prices is therefore similar to that when traders could only time their
In the second example, illustrated in Table 7, we assume that there is
less nondiscretionary liquidity trading. Specifically, we set the variance of
nondiscretionary liquidity trading to be 0.1. With the cost of information
at c = 0.04 and with endogenous information acquisition, we obtain pronounced patterns. For example, there are 11 informed traders in period 2
and three informed traders in each of the other periods. Liquidity trading
is much heavier in period 2 as well, and the patterns of the volume and
price behavior are very pronounced. In this example, each discretionary liquidity trader j trades
in period 2,
in period 3, and
in period 4.
In this section we discuss a number of additional extensions of our basic
model. We show that the main conclusions of the model do not change
28 Table 7
An example of pronounced patterns of volume and price behavior when discretionary liquidity
traders allocate trading across several periods The same example as in Table 6, except that the variance of nondiscretionary liquidity trading is lower
(0.1). The cost of information is assumed to be c = 0.04. in more general settings. This indicates that our results are robust to a
variety of models.
5.1 Different timing constraints for liquidity traders
For simplicity, we have assumed so far that the demands of all the discretionary liquidity traders are determined at the same time and must be
satisfied within the same time span. In reality, of course, different traders
may realize their liquidity demands at different times, and the time that
can elapse before these demands must be satisfied may also be different
for different traders. Our results can be extended to this more general case,
and their basic nature remains unchanged.
For example, suppose that there are three discretionary liquidity traders,
A, B, and C, whose demands have the variances 5, 1, and 7, respectively.
Suppose that trader A realizes his liquidity demand at 9:00 A.M. and must
satisfy it by 2:00 P.M. that day. Trader B realizes his demand at 11:00 A.M.
and must satisfy it by 4:00 P.M., and trader C realizes his demand at 2:30
P.M. and must satisfy it by 10:00 A.M. on the following day. If each of these
traders trades only once to satisfy his liquidity demands, then it is an
equilibrium that traders A and C trade at the same time between 9:00 A.M.
and 10:00 A.M. (e.g., 9:30 A.M.) and that trader B trades sometime between
11:00 A.M. and 4:00 P.M.
Now suppose that the variance of Bs demand is 9 instead of 1. Then the
equilibrium described above is possible only if trader B trades before
2:30 P.M.; otherwise, trader C would prefer to trade at the same time that
B trades rather than at the same time that A trades, and the equilibrium
would break down. Two other equilibrium patterns exist in this situation.
In one, traders B and C trade at the same time between 2:30 P.M. and 4:00
P.M. (e.g., 3:00 P.M.), and trader A trades sometime between 9:00 A.M. and
2:00 P.M. In another equilibrium, traders A and B trade at the same time
between 11:00 A.M. and 2:00 P.M. (e.g., 11:30 A.M.), and trader C trades
sometime between 4:00 P.M. and 10:00 A.M. of the next morning. All these
equilibria involve trading patterns in which two of the traders trade at the
same time. If informed traders can enter the market, then their trading
would also be concentrated in the periods with heavier liquidity trading.
29 Thus, we obtain trading patterns similar to those discussed in the simple
5.2 Risk-averse liquidity traders
We now ask whether our results change if, instead of assuming that all
traders are risk-neutral, it is assumed that some traders are risk-averse. We
focus on the discretionary liquidity traders, since their actions are the prime
determinants of the equilibrium trading patterns we have identified. In the
discussion below we continue to assume that informed traders and the
market maker are risk-neutral. (A model in which these traders are also
risk-averse is much more complicated and is therefore beyond the scope
of this article.)
A risk-averse liquidity trader, say trader j, is concerned with more than
the conditional expectation of
given his own demand
he submits market orders, the price at which he trades is uncertain. In
those periods in which a large amount of liquidity trading takes place, the
variance of the order flow is higher. One may think that since this will
make the price more variable,...
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