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

in c for any a. As such, we know that the specialist will set c to its highest possible value, that is,
c = r L.
As a result, the specialist’s problem amounts to maximizing (A.8) with respect to a when c = r L.
Differentiation with respect to a yields the first-order condition for this problem, 23
λ
2 − r 1
L(1 − λ) ˆma,rL = 0 (A.9)
rH − rL which, after solving for a, yields (17). Using this value for a along with c = rL, we find that ˆma,c
reduces to λ(rH−rL)
(1−λ) ˆma,c
, so that ψ = 2(1−λ)rL 2 reduces to (18). Clearly, as conjectured, the ask price in
(17) is larger than rL, so that the patient hedgers are not attracted by the continuous market. As
such, it only remains to verify that a ≤ r H so that indeed impatient traders choose to trade in the
continuous market. Straightforward manipulations show that this is the case when (16) is satisfied.
Otherwise, the left-hand side of (A.9) is strictly positive for all value of a below r H, and so the
specialist would like to increase a as much as possible without chasing the impatient hedgers away
by setting a = r H. With this value for a and with c = r L, we have ˆma,c = 1. Thus the conditional
market never clears, and its presence has no effect on the specialist’s expected profits.
23 The second-order condition is easy to verify.
32