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Unformatted text preview: cntually dominates, lowering λ t. 11 Proof. Define
that is, the total variance of discretionary
liquidity demands. Suppose that all discretionary liquidity traders trade in
period t and that the market maker adjusts λ t, and informed traders set β t
accordingly. Then the total trading cost incurred by the discretionary traders is λ t(h)h, where λ t(h) is given in Lemma 1 with
Consider the period t* ∈ [T', T"] for which X,(b) is the smallest. (If there
are several periods in which the smallest value is achieved, choose the
first.) It is then an equilibrium for all discretionary traders to trade in t*.
This follows since X,(b) is decreasing in h, so that we must have by the
definition of t*, λ t(0) ≥λ t.(h) for all t ∈ [T', T"]. Thus, discretionary
liquidity traders prefer to trade in period t* .
The above argument shows that there exist equilibria in which all discretionary liquidity trading is concentrated in one period. If there is an
equilibrium in which trading is not concentrated, then the smallest value
of A, must be attained in at least two periods. It is easy to see that any small
change in var
for some j would make the λ t different in different periods,
upsetting the equilibrium. n
Proposition 1 states that concentrated-trading patterns are always viable
and that they are generically the only possible equilibria (given that the
market maker uses a linear strategy). Note that in our model all traders
take the values of λ t as given. That is, when a trader considers deviating
from the equilibrium strategy, he assumes that the trading strategies of
other traders and the pricing strategy of the market maker (i.e., λ t) do not
change.9 One may assume instead that liquidity traders first announce the
timing of their trading and then trading takes place (anonymously), so that
informed traders and the market maker can adjust their strategies according
to the announced timing of liquidity trades. In this case the only possible
equilibria are those where trading is concentrated. This follows because
if trading is not concentrated, then some liquidity traders can benefit by
deviating and trading in another period, which would lower the value of
λ t in that period.
We now illustrate Proposition 1 by an example. This example will be
used and developed further in the remainder of this article.
Example. Assume that T =5 and that discretionary liquidity traders learn
of their demands in period 2 and must trade in or before period 4 (i.e.,
T' = 2 and T" = 4). In each of the first four periods, three informed traders
trade, and we assume that each has perfect information. Thus, each observes
in period t the realization of
. We assume that public information arrives
at a constant rate, with var( δ ) = 1 for all t. Finally, the variance of the
nondiscretionary liquidity trading occurring each period is set equal to 1.
9 Interestingly, when nt = 1 the equilibrium is the same whether the informed trader ties λ t as given or
whether he takes into account the effect his trading policy has on the market maker’s determination of A,.
In other words, in this model the Nash equilibrium in the game between the informed trader and the
market maker is identical to the Stackelberg equilibrium in which the trader takes the market maker’s
response into account. 12 We are interested in the behavior of the discretionary liquidity Faders.
Assume that there are two of these traders, A and B, and let var(YA) = 4
and var( YB) = 1. First assume that A trades in period 2 and B trades in
period 3. Then λ1 = λ4, = 0.4330, λ2, = 0.1936 and λ3, = 0.3061. This cannot
be an equilibrium, since λ2, < λ3, so B will want to trade in period 2 rather
than in period 3. The discretionary liquidity traders take the λ ’s as fixed
and B perceives that his trading costs can be reduced if he trades earlier.
Now assume that both discretionary liquidity traders trade in period 3. In
this case λ1, = λ2, = λ3, = 0.4330 and λ3 = 0.1767. This is clearly a stable
trading pattern. Both traders want to trade in period 3 since λ3, is the
minimal λ t.
1.3 Implications for volume and price behavior
In this section we show that the concentration of trading that results when
some liquidity traders choose the timing of their trades has a pronounced
effect on the volume of trading. Specifically, the volume is higher in the
period in which trading is concentrated both because of the increased
liquidity-trading volume and because of the induced informed-trading volume. The concentration of discretionary liquidity traders does not affect
the amount. of information revealed by prices or the variance of price
changes, however, as long as the number of informed traders is held fixed
and is specified exogenously. As we show in the next section, the results
on price informativeness and on the variance of price changes are altered
if the number of informed traders in the market is determined endogenously.
It is clear that the behavior of prices and of trading volume is determined
in part by the rate of public-informat...
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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