When S 2 =0, arbitrageurs liquidate their position at a gain at t=2, and
hold cash until t=3. In this case, W=a(D 1 ∗V/p 1 +F 1 −D 1 )+(1−a)F 1.
When S 2 =S, in contrast, arbitrageurs third-period funds are given by W=
(V/p 2 )∗[a{D 1 ∗p 2 /p 1 +F 1 −D 1 }+(1−a)F 1 ]. Arbitrageurs then maximize:
(7)2.Performance-Based Arbitrage and Market EfficiencyBefore analyzing the pattern of prices in our model, we specify what the
benchmarks are. The first benchmark is efficient markets, in which arbi-
trageurs have access to all the capital they want. In this case, since noise
trader shocks are immediately counteracted by arbitrageurs, p 1 =p 2 =V.
An alternative benchmark is one in which arbitrageurs resources are lim-
ited, but PBA is inoperative, that is, arbitrageurs can always raise F 1. Even
if they lose money, they can replenish their capital up to F 1. In this case,
p 1 =V−S 1 +F 1 and p 2 =V−S+F 1. Prices fall one for one with noise
trader shocks in each period. This case corresponds most closely to the ear-
lier models of limited arbitrage. There is one final interesting benchmark in
this model, namely the case of a=1. This is the case in which arbitrageurs
cannot replenish the funds they have lost, but do not suffer withdrawals be-
yond what they have lost. We will return to this special case below.
The first-order condition to the arbitrageur’s optimization problem is
given by:
(8)with strict inequality holding if and only if D 1 =F 1 , and equality holding if
D 1 <F 1. The first term of Eq. (8) is an incremental benefit to arbitrageurs
from an extra dollar of investment if the market recovers at t=2. The sec-
ond term is the incremental loss if the price falls at t=2 before recovering
at t=3, and so they have foregone the option of being able to invest more
in that case. Condition (8) holds with a strict equality if the risk of price
deterioration is high enough, and this deterioration is severe enough, that
arbitrageurs choose to hold back some funds for the option to invest more
at time 2. On the other hand, Eq. (8) holds with a strict inequality if qis
low, if p 1 is low relative to V(S 1 is large), if p 2 is not too low relative to p 1
()11 10
12
12−−
+−
q ≥
V
pq
p
pV
pEW q aDV
pFD aFqV
paDp
pFD aF=− + −
+−
+
+−
+−
() ()()1111
111 1212
111 1∗∗∗THE LIMITS OF ARBITRAGE 87