186 9 Fundamental Theorem of Asset Pricing
Remark 9.5.10.Let us recall that the dual ofL∞isba(Ω,F,P), the space of
all bounded, finitely additive measures on theσ-algebraF, absolutely contin-
uous with respect toP. We can try to define the set of all finitely additive
measures that can be considered as local martingale measures forS.Itisnot
immediately clear how this can be done in a canonical way. But, if we define
Mba(P)={Q|Q∈ba,Q[Ω] = 1,EQ[h]≤0 for allhinC},thenitiseasy
to see, via the equality in Theorem 9.5.8, thatM(P)isσ(ba, L∞)-dense in
Mba(P). In other wordsMba(P)istheσ(L∞,L^1 )-closure ofM(P)inthe
spaceba(Ω,F,P), the dual ofL∞. This is of course the good definition of
Mba(P). We remark that the setChas to be used and not just the setW^0.
Indeed Example 9.5.6 shows that the setM(P) is not necessarilyσ(ba, L∞)-
dense in the set{Q|Qfinitely additive, positive,Q[Ω] = 1 andEQ[h]=
0 for allhinW^0 }. To see this, we observe that the functionfdefined in Ex-
ample 9.5.6 is not in the norm closure ofx+W^0 −L∞+ for anyx<1. By the
Hahn-Banach theorem there is a finitely additive positive probabilityQsuch
thatEQ[f]=1andEQ[h] = 0 for allhinW^0. Because supQ∈M(P)EQ[f]=^12
this elementQcannot be in theσ(ba, L∞)-closure of the setM(P). This sug-
gests that the “good” definition of such finitely additive measures should use
the inequalityEQ[h]≤0 for allhinCand not only for allhinW^0 −L∞+.
9.6 No Free Lunch with Bounded Risk
In this section we will compare the property of no free lunch with vanishing risk
(NFLVR)with the previously used property of no free lunch with bounded risk
(NFLBR). This property was used in a series of papers: [MB 91, D 92, S 94].
The property(NFLBR)is a generalisation of the property(NFLVR). To define
this property we need some more notation. ByCwe denoted the closure of
Cwith respect to the norm topology ofL∞,byC
∗
we will denote the weak-
star-closure ofC.ThesetC ̃is the set of all limits of weak-star converging
sequencesof elements ofC. Although the fact that a convex set inL∞is weak-
star-closed if and only if it is sequentially closed for the weak-star topology, the
closure of a convex set cannot necessarilybe obtained by taking all limits of
sequences. (In [B 32, Annexe th ́eor`eme 1] one can find for eachk,examplesof
convex sets such that afterkiterations of taking weak-star limits of sequences,
the weak-star-closure is not obtained, but afterk+ 1 iterations the closure is
found.) Therefore in general, there is a difference betweenC
∗
andC ̃and the
use of nets is essential to find the weak-star-closure ofC.
Definition 9.6.1.IfSis a semi-martingale then we say thatSsatisfies the
property
(i) no free lunch with bounded risk(NFLBR) ifC ̃∩L∞+={ 0 },
(ii)no free lunch(NFL) ifC
∗
∩L∞+={ 0 }.