256 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory
The(NA)property can be rephrased as the property that 0 is maximal
inK. It is clear that ifSsatisfies the no-arbitrage property, then the fact
thatfis maximal inKaalready implies thatfis maximal inK. Indeed if
g=(H·S)∞∈Kandg≥fa.s., theng≥−a. From Proposition 9.3.6 it then
follows thatgisa-admissible and hence the maximality offinKaimplies
thatg=fa.s..
Definition 13.2.2.A maximal admissible contingent claim is a maximal
element ofK. The set of maximal admissible contingent claims is denoted
byKmax. The set of maximala-admissible contingent claims is denoted by
Kmaxa.
The proof of the Theorem 13.1.5 uses the following intermediate results,
see Sect. 9.4:
Theorem 13.2.3.IfSis a locally bounded semi-martingale and if(fn)n≥ 1 is
asequenceinK 1 ,then
(1)there is a sequence of convex combinationsgn∈conv{fn,fn+1,...}such
thatgntends in probability to a functiong, taking finite values a.s.,
(2)there is a maximal contingent claimhinK 1 such thath≥ga.s..
Corollary 13.2.4.Under the hypothesis of Theorem 13.2.3, maximal contin-
gent claims of the closureL^0 -closureK 1 ofK 1 , are already inK 1 .ByL^0 -
closure we mean the closure with respect to convergence in measure.
Using a change of num ́eraire technique, the following result was proved in
Chap. 11. We refer also to Ansel-Stricker [AS 94] for an earlier proof of the
equivalence of (2) and (3) below.
Theorem 13.2.5.IfSis a locally bounded semi-martingale that satisfies the
(NFLVR) property then for a contingent claimf∈Kthe following are equiva-
lent
(1)fis maximal admissible,
(2)there is an equivalent local martingale measureQ ∈Me such that
EQ[f]=0,
(3)iff=(H·S)∞for some admissible strategyH,thenH·Sis a uniformly
integrable martingale for someQ∈Me.
Corollary 13.2.6.Suppose that the hypothesis of theorem 13.2.5 is valid. If
fis maximal admissible andf=(H·S)∞for some admissible strategyH,
then for every stopping timeT, the contingent claim(H·S)Tis also maximal.
Proof.Iffis maximal andf =(H·S)∞whereH isa-admissible, then
there isQ∈Mesuch thatEQ[f] = 0, i.e.EQ[(H·S)∞] = 0. BecauseHis
admissible, the processH·Sis, see [AS 94], aQ-local martingale and hence
aQ-super-martingale. BecauseEQ[(H·S)∞] = 0, we necessarily have that
H·Sis aQ-uniformly integrable martingale. It follows thatEQ[(H·S)T]=0
and consequently (H·S)Tis maximal.