2.2 No-Arbitrage and the FTAP 21After these general observations we pass to the concrete setting of the
coneC⊆L∞(Ω,F,P) of contingent claims super-replicable at price 0. Note
that in our finite dimensional setting this convex cone is closed as it is the
algebraic sum of the closed linear spaceK (a linear space inRNis always
closed) and the closed polyhedral coneL∞−(Ω,F,P) (the verification, that the
algebraic sum of a space and a polyhedral cone inRNis closed, is an easy, but
not completely trivial exercise). We deduce from the bipolar theorem, thatC
equals its bipolarC^00.
Proposition 2.2.9.Suppose thatSsatisfies (NA). Then the polar ofC is
equal tocone(Ma(S)), the cone generated byMa(S),andMe(S)is dense
inMa(S). Hence the following assertions are equivalent for an elementg∈
L∞(Ω,F,P):
(i) g∈C,
(ii) EQ[g]≤ 0 , for allQ∈Ma(S),
(iii)EQ[g]≤ 0 , for allQ∈Me(S).
Proof.The fact that the polarC^0 and the set cone(Ma(S)) coincide, follows
from Lemma 2.2.6 and the observation thatC⊇L∞−(Ω,F,P)andC^0 ⊆
L^1 +(Ω,F,P). Hence the equivalence of (i) and (ii) follows from the bipolar
theorem.
As regards the density ofMe(S)inMa(S) we first deduce from Theorem
2.2.7 that there is at least oneQ∗∈Me(S). For anyQ∈Ma(S)and0<μ≤
1wehavethatμQ∗+(1−μ)Q∈Me(S), which clearly implies the density
ofMe(S)inMa(S). The equivalence of (ii) and (iii) is now obvious.
Similarly we can show the following:Proposition 2.2.10.SupposeSsatisfies (NA). Then forf ∈L∞,thefol-
lowing assertions are equivalent
(i) f∈K, i.e.f=(H·S)Tfor some strategyH∈H.
(ii) For al lQ∈Me(S)we haveEQ[f]=0.
(iii)For al lQ∈Ma(S)we haveEQ[f]=0.
Proof.By Proposition 2.2.4 we have thatf∈Kifff∈C∩(−C). Hence the
result follows from the preceding Proposition 2.2.9.
Corollary 2.2.11.Assume thatSsatisfies (NA) and thatf∈L∞satisfies
EQ[f]=afor allQ∈Me(S),thenf=a+(H·S)Tfor some strategyH.
Corollary 2.2.12 (complete financial markets).For a financial market
Ssatisfying the no-arbitrage condition (NA), the following are equivalent:
(i) Me(S)consists of a single elementQ.
(ii)Eachf∈L∞(Ω,F,P)may be represented as
f=a+(H·S)T for somea∈RandH∈H.