258 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory
Proof.Suppose thatK·S≥−(a+L·S)whereLis admissible and maximal.
Clearly we have thatK+Lisa-admissible. But at infinity we have that
((K+L)·S)∞≥(L·S)∞and by maximality ofLwe obtain the equality
((K+L)·S)∞=(L·S)∞, which is equivalent to (K·S)∞=0a.s..
In the same way we prove the subsequent result.Proposition 13.2.13.Suppose thatSis a locally bounded semi-martingale
that satisfies the (NFLVR) property. IfKis acceptable and(K·S)∞≥−c
for some positive real constantc, then the strategyKis alreadyc-admissible.
Proof.Takeε>0andlet
T 1 =inf{t|(K·S)t<−c−ε}.We then define
T 2 =inf{t>T 1 |(K·S)t≥−c}.
By assumption we have that on{T 1 <∞}the strategyK (^1) ]]T 1 ,T 2 ]]produces an
outcome (K·S)T 2 −(K·S)T 2 ≥ε. This strategy is easily seen to be acceptable.
Indeed
(K (^1) ]]T 1 ,T 2 ]])·S≥c+ε+(−a−H·S)
for some real numberaand some maximal strategyH. By the previous lemma
we necessarily have that the contingent claim is zero a.s. and henceT 1 =∞
a.s..
We now turn again to the analysis of maximal admissible contingent
claims.
Theorem 13.2.14.IfS is a locally bounded semi-martingale that satisfies
the (NFLVR) property, iffandgare maximal admissible contingent claims,
thenf+gis also a maximal contingent claim. It follows that the setKmaxof
maximal contingent claims is a convex cone.
Proof.Letf=(H^1 ·S)∞andg=(H^2 ·S)∞,whereH^1 andH^2 are maximal
strategies and are respectivelya 1 -anda 2 -admissible. Suppose thatK is a
k-admissible strategy such that (K·S)∞ ≥f+g. From the inequalities
(K−H^2 )·S=K·S−H^2 ·S≥−k−H^2 ·S, it follows thatK−H^2 is
acceptable. Since also
(
(K−H^2 )·S
)
∞≥f≥−a^1 , the Proposition 13.2.13
shows that( K−H^2 isa 1 -admissible. Becausefwas maximal we have that
(K−H^2 )·S
)
∞=fand hence we have that (K·S)∞=f+g. This shows
thatf+gis maximal. Since the setKmaxis clearly closed under multiplication
with positive scalars, it follows that it is a convex cone.
Corollary 13.2.15.IfS is a locally bounded semi-martingale that satisfies
the (NFLVR) property and if(fn) 1 ≤n≤N is a finite sequence of contingent
claims inKsuch that for eachnthere is an equivalent risk neutral measure
Qn∈MewithEQn[fn]=0, then there is an equivalent risk neutral measure
Q∈Mesuch thatEQ[fn]=0for eachn≤N.