The Benefits of Volume-Conditional Order-Crossing - Singapore ...

Also, any equilibrium in which both venues are used will have the (1 − λ) ˜m patient hedgers use
the conditional market, with half of these traders placing buy orders and half placing sell orders.
Thus, given a volume condition of ψ, the conditional market clears with probability
(1 − λ) ˜m
Pr
≥ ψ = 1 −
2
2ψ
≡ 1 − ˆm, (A.1)
1 − λ
When it does clear, the total number of traders (number of shares being exchanged) in this market
is given by
(1 − λ)E ˜m | ˜m ≥ ˆm = (1 − λ)
1 + ˆm
. (A.2)
2
Thus, under the conjectured equilibrium, the market-maker collects a from an average of λE ˜m = λ
2
impatient traders in the continuous market, and c with probability 1 − ˆm from an average of
1+ ˆm (1 − λ) 2
patient traders in the conditional market. His expected profits are therefore given by
E λ
˜π MM = a
2
In equilibrium, he breaks even with
+ c(1 − ˆm)(1 − λ)1 + ˆm
2
− κ. (A.3)
a = 2κ − (1 − ˆm2 )(1 − λ)c
, (A.4)
λ
which reduces to (8) after ˆm is replaced by 2ψ
1−λ .
We need to verify that patient traders do not prefer trading in the continuous market to trading
in the conditional market. A sufficient condition for this is that a > r L. Since 2κ > r L and c ≤ r L,
we have
a = 2κ − (1 − ˆm2 )(1 − λ)c
λ
> r L − (1 − ˆm 2 )(1 − λ)r L
λ
≥ r L − (1 − λ)r L
λ
= r L,
and so this condition holds (and this establishes that a > c as well). We also need to verify that
the impatient traders find it worthwhile to use the continuous market (i.e., a ≤ r H), and prefer that
market to the conditional market. Since 2κ < r Hλ, we have
a = 2κ − (1 − ˆm2 )(1 − λ)c
λ
< 2κ
λ < r H,
so indeed the impatient hedgers benefit from trading in the continuous market. Their expected
utility from doing so is −a. The expected utility of an impatient trader who deviates from the
equilibrium and instead places an order in the conditional market, which clears with probability
1 − ˆm, is
E ũj | ˜rj = r H
= (1 − ˆm)(−c) + ˆm(−rH).
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