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Unformatted text preview: where T' < T" < T.4 Assume there are m discretionary liquidity
traders and let
be the total demand of the jth discretionary liquidity
trader (revealed to that trader in period T'). Since each discretionary liquidity trader is riskneutral, he determines his trading policy so as to
minimize his expected cost of trading, subject to the condition that he
trades a total of
shares by period T’. Until Section 4 we assume that
each discretionary liquidity trader only trades once between time T' and
time T"; that is, a liquidity trader cannot divide his trades among different
periods.
Prices for the asset are established in each period by a market maker
who stands prepared to take a position in the asset to balance the total
demand of the remainder of the market. The market maker is also assumed
to be riskneutral, and competition forces him to set prices so that he earns
zero expected profits in each period. This follows the approach in Kyle
(1985) and in Glosten and Milgrom (1985).5
3 This assumption is reasonable since the span of time coveted by the T periods in this model is to be taken
as relatively short and since our main interests concern the volume of trading and the variability of prices.
The nature of our results does not change if a positive discount rate is assumed.
4
In reality. of course, different traders may realize their liquidlty demands at different times, and the time
that can elapse before these demands must be satisfied may also be different for different traders. The
nature of our results will not change if the model is complicated to capture this. See the discussion in
Section 5.1.
5
The model here can be viewed as the limit of a model with a finite number of market makers as the number
of market makers grows to Infinity. However, our results do not depend in any important way on the
assumption of perfect competition among market makers. The same basic results would obtain in an
analogous model with a finite number of market makers, where each market maker announces a (linear)
pricing schedule as a function of his own order flow and traders can allocate their trade among different
market makers. In such a model, market makers earn positive expected profits. See Kyle (1984). 8 Let
be the ith informed trader’s order in period t,
be the order of
the jth discretionary liquidity trader in that period, and let us denote by
the total demand for shares by the nondiscretionary liquidity traders in
period t, Then the market maker must purchase
shares in period t. The market maker determines a price in period t based
on the history of public information,
and on the history of
order flows,
. . . , . The zero expected profit condition implies that
the price set
in period t by the market maker, satisfies
(2)
Finally, we assume that the random variables
are mutually independent and distributed multivariate normal, with each
variable having a mean of zero.
1.2 Equilibrium
We will be concerned with the (Nash) equilibria of the trading game that
our model defines among traders. Under our assumptions, the market
maker has a passive role in the model.7 Two types of traders do make
strategic decisions in our model. Informed traders must determine the size
of their market order in each period. At time t, this decision is made
knowing S t1, the history of order flows up to period t  1; A,, the innovations up to t; and the signal,
The discretionary liquidity traders
must choose a period in [T', T"] in which to trade. Each trader takes the
strategies of all other traders, as well as the terms of trade (summarized
by the market maker’s pricesetting strategy), as given.
The market maker, who only observes the total order flow, sets prices
to satisfy the zero expected profit condition. We assume that the market
maker’s pricing response is a linear function of and
In the equilibrium
that emerges, this will be consistent with the zeroprofit condition. Given
our assumptions, the market maker learns nothing in period t from past
order flows
that cannot be inferred from the public information A,.
This is because past trades of the informed traders are independent of
and because the liquidity trading in any period is independent
of that in any other period. This means that the price set in period t is
equal to the expectation of conditional on all public information observed
in that period plus an adjustment that reflects the information contained
in the current order flow
6 If the price were a function of individual orders, then anonymous traders could manipulate the price by
submitting canceling orders. For example, a trader who wishes to purchase 10 shares could submit a
purchase order for 200 shares and a sell order for 190 shares. When the price is solely a function of the
total order flow, such manipulations are not possible.
7
It is actually possible to think of the market maker also as a player in the game, whose payoff is minus the
sum of the squared deviations of the prices from the true payoff. 9 (3)
Our not...
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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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