11.4 Hedging and Change of Num ́eraire 229(1) (H·S)∞is maximal inK
(2)there isQ∈Me(P)such thatEQ[(H·S)∞]=0
(3)there isQ∈Me(P)such thatH·Sis aQ-uniformly integrable martingale.
The theorem also allows us to give a characterisation of strict local martin-
gales as studied by Elworthy, Li and Yor, [ELY 99]. They define a strict local
martingale as a local martingale that is not a uniformly integrable martingale.
Corollary 11.4.7.LetS=Lbe a strictly positive locally bounded local mar-
tingale such thatL∞> 0 a.s.. Let
Me(P)={
Q
∣
∣
∣
∣
Qis equivalent toP
and the processLis aQ-local martingale}
.
The processL^1 satisfies the (NA) property with respect to general admissible
integrands if and only ifLis a uniformly integrable martingale for someQin
Me(P).
Remark 11.4.8.From Schachermayer [S 93] (see Chap. 10 for an easier exam-
ple) it follows that under the assumptions of the corollary, the processLneed
not be a uniformly integrable martingale under all elements ofMe(P).
Remark 11.4.9.In the case thatR=L^11 equals the Bessel(3) process with its
natural filtration, stopped at time 1, we have thatL^1 is a local martingale
forP. This is the only candidate for a martingale measure and hence we
deduce thatRhas arbitrage with respect to general admissible integrands.
The preceding corollary is a generalisation of this phenomenon to the case
thatMe(P) is not a singleton, see also Sect. 11.2.
Definition 11.4.10.IfSis a locally bounded semi-martingale that satisfies
the (NFLVR) property with respect to general admissible integrands, then we
say that a positive random variable (or contingent claim)fcan be hedged if
there isx∈Rand a maximal elementh∈Ksuch thatf=x+h.
There is a good reason to require the use of maximal elements. Ifhis not
maximal then there is a maximal elementg∈K,g=hsuch thatg≥h.An
investor who would try to hedgefby using an admissible strategy, would be
better off to use a strategy that gives her the outcomeginstead ofh.Starting
with the same initial investmentx, she will obtain something better thanf
and sinceg>hon a set of positive measure, she will be strictly better off in
some cases. In such a case the contingent claimfis not the result of a good
optimal hedging policy.
The following theorem is due to Ansel-Stricker [AS 94] and, independently,
to Jacka [J 92]. They proved it usingH^1 - BMOduality. We shall see that it
is also a consequence of the characterisation of maximal elements.
Theorem 11.4.11.IfS is a locally bounded semi-martingale that satisfies
the (NFLVR) property with respect to general admissible integrands then for
a random variablef≥ 0 , the following are equivalent: