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

is given by
E
ũj | ˜rj = rL =
1 1
(−c) + φ(−c) + (1 − φ)(−rL − p)
2 2
= 1 + φ 1 − φ
(−c) + (−rL − p),
2
2
(13)
where we have used the fact that a hedger whose order is unfilled suffers a cost of r L for not hedging
as well as an adverse selection cost of p. Because patient hedgers have the option not to trade (and
realize a sure utility loss of r L), they benefit from market-clearing if and only if (13) exceeds −r L.
Because −r L − p < −r L < −c, this will be the case when φ (the probability that an order on the
heavier side of the conditional market gets crossed when this market clears) is close to one, that is,
when n is small and/or m is large. This is intuitive: for a given precision p, adverse selection in the
conditional market is minimal when this market is populated with only a small relative number of
informed traders. As n gets larger relative to m however, the likelihood that a hedger’s order goes
unfilled as the market clears increases; unless the informed traders’ informational advantage (p) is
small relative to the patient hedgers’ potential gain from trade (r L − c), patient hedgers then prefer
the conditional market not to clear for this value of ˜m = m. This leads to the following result
about the potential viability of the conditional market.
Lemma 4.1 There is no equilibrium in which patient hedgers and informed traders use the condi-
tional market unless
n p − (r L − c) < (1 − λ)(r L − c). (14)
If such an equilibrium exists, market-clearing improves the welfare of patient hedgers with probability
one if and only if the volume condition satisfies
n p − (r L − c)
2(r L − c)
< ψ <
1 − λ
. (15)
2
Condition (14) is the analogue of condition (10), which prevents market breakdowns when the
continuous market operates alone. That is, when the conditional market attracts patient hedgers
and informed traders, (14) is necessary for it to be viable. Clearly, this will happen when the number
of informed traders (n) and the precision of their information (p) are small enough. Condition (15)
ensures that the patient hedgers are better off every time the conditional market clears. Although
this condition is sufficient to make these traders use the conditional market in the first place, it is
not necessary. Indeed, the patient traders will use the conditional market if it makes them better
off on average, and not necessarily if they are better off for every realization of ˜m that triggers
market-clearing (i.e., for every ˜m ≥ 2ψ
1−λ ). Still, the point remains that a larger volume condition
reduces the adverse selection problem of hedgers in the conditional market by reducing the average
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