230 11 Change of Num ́eraire
(1)fcan be hedged,
(2)there isQinMe(P)such that
EQ[f]=sup{ER[f]|R∈Me(P)}<∞.Proof.(1)implies(2): Iffcan be hedged, then there is an admissible strategy
Hand a real numberx, such thatf=x+(H·S)∞andH·Sis a uniformly
integrable martingale for someQ∈Me(P). For allR∈Me(P)wehavethat
H·Sis a super-martingale and henceER[f]≤x=EQ[f].
(2)implies(1): If we haveEQ[f]=sup{ER[f]|R∈Me(P)}<∞,then
clearly we have that
x=EQ[f]=min{z|∃h∈Ksuch thatz+h≥f}<∞.The duality relation of Sect. 3 now implies that there is an admissible inte-
grandHsuch thatf≤x+(H·S)∞.SinceH·Sis a super-martingale forQ
we have that
x=EQ[f]≤x+EQ[(H·S)∞]≤x
and henceEQ[(H·S)∞] = 0. This implies thatf=x+(H·S)∞and that
H·Sis uniformly integrable forQ. Therefore (H·S)∞is maximal inK.