2.4 Pricing by No-Arbitrage 25view ofQ∗∼Pimplies thatf−π(f)≡g;inotherwordsfis attainable at
priceπ(f). This in turn implies thatEQ[f]=π(f) for allQ∈Me(S), and
thereforeIis reduced to the singleton{π(f)}.
Hence, ifπ(f)<π(f),π(f) cannot belong to the intervalI,whichis
therefore open on the right hand side. Passing fromfto−f, we obtain the
analogous result for the left hand side ofI, which is therefore equal toI=
]π(f),π(f)[.
The argument in the proof of the preceding theorem can be recast to yield
the following duality theorem. The reader familiar with the duality theory of
linear programming will recognise the primal-dual relation.
Theorem 2.4.2 (Superreplication).Assume thatSsatisfies (NA). Then,
forf∈L∞, we have
π(f)=sup{EQ[f]|Q∈Me(S)}
=max{EQ[f]|Q∈Ma(S)}
=min{a|there existsk∈K, a+k≥f}.Proof.As shown in the previous proof we havef−π(f)∈Cand hence
f=π(f)+g, for someg∈C
=π(f)+k−h, for somek∈Kandh∈L∞+
≤π(f)+k, for somek∈K.This shows thatπ(f)≥inf{a|there existsk∈K, a+k≥f}.
Let nowa<π(f). We will show that there is no elementk∈Kwith
a+k≥f. This shows thatπ(f) = inf{a|there existsk∈K, a+k≥f}and
moreover establishes that the infimum is a minimum. Sincea<π(f)thereis
Q∈Me(S) withEQ[f]>a. But this implies that for allk∈Kwe have that
EQ[a+k]=a<EQ[f], in contradiction to the relationa+k≥f.
Remark 2.4.3.Theorem 2.4.2 may be rephrased in economic terms: in order
to superreplicatef, i.e., to finda∈RandH∈Hs.t.a+(H·S)T≥f,we
need at least an initial investmentaequal toπ(f).
We now give a conditional version of the duality theorem that allows us
to use initial investments that are not constant and to possibly use the infor-
mationF 0 available at timet= 0. This is relevant when the initialσ-algebra
F 0 is not trivial.
Theorem 2.4.4.Let us assume thatSsatisfies (NA). Denote byMe(S,F 0 )
the set of equivalent martingale measuresQ∈Me(S)so thatQ|F 0 =P.
Then, forf∈L∞, we have
sup{EQ[f|F 0 ]|Q∈Me(S,F 0 )}
=min{h|h isF 0 -measurable and there existsg∈Ksuch thath+g≥f}.