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

Proof of Proposition 4.1
If an informed trader chooses to trade in the continuous market after observing ˜s = +1, his
expected profits are equal to E ˜πi | ˜s = +1 = p − a. Since traders always have the option not
to trade, informed traders trade if and only if their common precision p is greater than a. Let
us first assume that p < 2κ
λ
2κ
. Then the equilibrium of Proposition 2.1, with a = λ , still holds.
Indeed, because a > p, the informed traders choose not to trade in this market. We also know from
Proposition 2.1 that the market-maker can recoup his fixed costs of κ by quoting this ask price.
Since lowering his ask price can only reduce the market-maker’s profits (through lower revenues
and potentially more adverse selection if a dips below p), this is the lowest possible a that has him
break even.
If instead p > 2κ
λ
2κ
, the equilibrium ask price cannot be a = λ , as the market-maker breaks even
on his trades with impatient hedgers but, since p is then larger than a, he loses p − a to each of
the n informed traders who now find it profitable to trade on their information. The market-maker
must therefore set a to a value greater than 2κ
λ and his expected profits are then
E aλ
˜π MM = − (p − a)n − κ.
2
(A.6)
In equilibrium, the market-maker breaks even and so a must make this last quantity equal to zero.
This yields (11). Of course, the conjectured equilibrium only holds if the informed traders and
impatient hedgers choose to trade, that is, if a ≤ p and a ≤ r H with the value of a just derived.
Straightforward manipulations show that the first of these two inequalities is equivalent to p > 2κ
λ ,
which is true in this second part of the proposition. The second inequality reduces to (10). When
it does not hold, the ask price required to break even no longer attracts the impatient traders and
so no ask price in [0, 1] can produce an equilibrium.
Proof of Lemma 4.1
Let us first consider the second part of the result, that is, suppose that in equilibrium the
patient hedgers and informed traders place orders in the conditional market. As mentioned in the
paragraph preceding this lemma, a patient trader is better off when the market clears as long as
(13) is greater than −r L for all ˜m = m that trigger market-clearing. It is easy to show that this
condition is equivalent to φ > p−(rL−c)
p+(rL−c)
reduces to
or, equivalently, to
which, after replacing φ by m(1−λ)
m(1−λ)+2n
m > n p − (r L − c)
(1 − λ)(r L − c)
m(1 − λ)
2
> np − (rL − c)
.
2(rL − c)
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as defined in (12),