11.4 Hedging and Change of Num ́eraire 227thatY∞−1=(L·X)∞is non-negative and strictly positive on a non-negligible
set. This should produce arbitrage forX.
The second part is proved in a similar way. Suppose thatZallows arbitrage
and thatK is the 1-admissible integrand responsible for it. The outcome
Y∞−1isnowgreaterthanV∞−1, with strict inequality on a non-negligible
set. A contradiction to its maximality.
Corollary 11.4.3.Using the same notation as in the theorem we see thatX
satisfies the (NA) property with respect to general admissible integrands and
V∞− 1 is maximal “forX”ifandonlyifZsatisfies the (NA) property with
respect to general admissible integrands andV^1 ∞− 1 is maximal “forZ”.
Proof.This is a straightforward application of the previous theorem. The only
difference lies in the statement thatV∞−1 is maximal in the set of all outcomes
of admissible integrand and not just in the set of outcomes of 1-admissible
integrands. IfXsatisfies(NA)andV∞−1 is maximal then we can apply both
parts of the theorem. In this case we know, from Sect. 11.2, thatf=V^1 ∞− 1
is maximal in the set of outcomes of all admissible integrands. This proves the
if statement. Theonly if part is the same statement as theif part because
Xis obtained fromZby multiplying withV−^1.
We can now apply the above reasoning to the original setting of this pa-
per. Given a locally bounded semi-martingaleSthat satisfies the(NFLVR)
property with respect to general admissible integrands, we use a process of the
formV=1+H·Sfor the new num ́eraire. IfHis admissible andV∞>0 a.s.,
then we can apply the previous theorem. In this case we certainly have that
the system (S, 1 ,V)hasthe(NA)property with respect to general admissible
integrands. With the assumption thatPwas a local martingale measure for
S, the system (S, 1 ,V) becomes in fact a local martingale forP. The previous
theorem then yields
Theorem 11.4.4.Suppose thatSis a locally bounded semi-martingale that
satisfies the (NFLVR) property with respect to general admissible integrands.
Suppose thatHis admissible and that the processV =1+(H·S)satisfies
f=V∞=1+(H·S)∞> 0 a.s.. Then the following are equivalent:
(1) (H·S)∞is maximal in the setK.
(2)The processS ̃=(VS,V^1 )satisfies (NA) with respect to general admissible
integrands.
(3)There isQ∈Me(P)such thatH·Sis aQ-uniformly integrable martin-
gale.
IfV−^1 is locally bounded then these statements are equivalent to:
(4)The processS ̃has an equivalent local martingale measure.