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Unformatted text preview: ation conforms with that in Kyle (1984, 1985). The reciprocal of
λ t, is Kyle’s marketdepth parameter, and it plays an important role in our
analysis.
The main result of this section shows that in equilibrium there is a
tendency for trading to be concentrated in the same period. Specifically,
we will show that equilibria where all discretionary liquidity traders trade
in the same period always exist and that only such equilibria are robust to
slight changes in the parameters.
Our analysis begins with a few simple results that characterize the equilibria of the model. Suppose that the total amount of discretionary liquidity
if the jth discretionary liquidity
demands in period t is
where
trader trades in period t a nd where
otherwise. Define
that is, Ψ t is the total variance of the liquidity trading in
period t. (Note that Ψ t must be determined in equilibrium since it depends
on the trading positions of the discretionary liquidity traders.) The following lemma is proved in the Appendix. Lemma 1. If the market maker follows a linear pricing strategy, then in
equilibrium ,each informed trader i submits at time t a market order of
where
(4)
The equilibrium value of λ t is given by
(5)
This lemma gives the equilibrium values of A, and β t for a given number
of informed traders and a given level of liquidity trading. Most of the
comparative statics associated with the solution are straightforward and
intuitive. Two facts are important for our results. First, λ t, is decreasing in
Ψ t, the total variance of liquidity trades. That is, the more variable are the
liquidity trades, the deeper is the market. Less intuitive is the fact that λ t,
is decreasing in nt, the number of informed traders. This seems surprising
since it would seem that with more informed traders the adverse selection
problem faced by the market maker is more severe. However, informed
traders, all of whom observe the same signal, compete with each other,
10 and this leads to a smaller λ t. This is a key observation in the next section,
where we introduce endogenous entry by informed traders.8
When some of the liquidity trading is discretionary, Ψ t, is an endogenous
parameter. In equilibrium each discretionary liquidity trader follows the
trading policy that minimizes his expected transaction costs, subject to
meeting his liquidity demand
We now turn to the determination of this
equilibrium behavior. Recall that each trader takes the value of λ t (as well
as the actions of other traders) as given and assumes that he cannot influence it. The cost of trading is measured as the difference between what
the liquidity trader pays for the security and the security’s expected value.
Specifically, the expected cost to the jth liquidity trader of trading at time
t ∈ [T', T"] is
(6)
Substituting for and using the fact that
where
T are independent of
(which is the
information of discretionary liquidity trader j )the cost simplifies to
Thus, for a given set of λ t, t ∈ [T', T"], the expected cost of liquidity trading
is minimized by trading in that period t* ∈ [T', T"] in which A, is the
smallest. This is very intuitive, since λ t, measures the effect of each unit of
order flow on the price and, by assumption, liquidity traders trade only
once.
Recall that from Lemma 1, λ t, is decreasing in Ψ t. This means that if in
equilibrium the discretionary liquidity trading is particularly heavy in a
particular period t, then λ t, will be set lower, which in turn makes discretionary liquidity traders concentrate their trading in that period. In sum,
we obtain the following result.
Proposition 1. There always exist equilibria in which all discretionary
liquidity trading occurs in the same period. Moreover, only these equilibria
are robust in the sense that if for some set of parameters there exists an
equilibrium in which discretionary liquidity traders do not trade in the
same period, then for an arbitrarily close set of parameters [e.g., by perturbing the vector of variances of the liquidity demands Yj), the only
possible equilibria involve concentrated trading by the discretionary liquidity traders. 8 More intuition for why λ t, is decreasing in nt, can be obtained from statistical inference. Recall that A, is the
regression coefficient in the forecast of
given the total order flow . The order flow can be written
as
represents the total trading position of the informed traders and
û is the position of the liquidity traders with
As the number of informed traders increases, a
increases. For a given level of a, the market maker sets λ t equal to
This is an
Increasing function of a if and only if
which in this model occurs if and only if nt ≤ 1.
We an think of the market maker’s inference problem in two pans: first he uses
to predict
then
he sales this down by a factor of 1/a to obtain his prediction of
The weight placed upon
in
predicting
is always increasing in a, but for a large enough value of a the scaling down by a factor
of l/a ev...
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 Spring '12
 Svendsson

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