The Mathematics of Arbitrage

13.2 Maximal Admissible Contingent Claims 257

Remark 13.2.7.The corollary also shows that iff=(H·S)∞is maximal
admissible, then the strategy that producesfis uniquely determined in the
sense that any other admissible strategyKthat producesfnecessarily satis-
fiesH·S=K·S. The following definitions therefore make sense.

Definition 13.2.8.IfH is an admissible strategy such thatf=(H·S)∞
is a maximal admissible contingent claim, then we say thatHis a maximal
admissible strategy.

Definition 13.2.9.We say that a strategyKis acceptable if there is a posi-
tive numbera and a maximal admissible strategyLsuch that(K·S) ≥
−(a+(L·S)).

Remark 13.2.10.If we takeabig enough, the processV =a+L·Sstays
bounded away from zero and can be used as a new num ́eraire. Under this new
currency unit, the processK·S,whereKis acceptable, has to be replaced
by the processKV·S. The latter process is a stochastic integral with respect
to the process

(S

V,

1

V

)

, more precisely, see Chap. 11 for the details of this
calculation,KV·S=(K,(K·S)−−KS−)·

(S

V,

1

V

)

=K′·

(S

V,

1

V

)

remains bigger
than a constant, i.e. the strategyK′=(K,(K·S)−−KS−) is admissible.
Another way of saying thatKis acceptable, is to say thatK′is admissible
in a new num ́eraire. In Chap. 11 we proved that the only num ́eraires that do
not destroy the no-arbitrage properties are the num ́eraires given by maximal
strategies. The definition of acceptable strategies is therefore very natural.
The outcomes of acceptable strategies are the num ́eraire invariant version of
the outcomes of admissible strategies.

Lemma 13.2.11.IfSis a locally bounded semi-martingale that satisfies the
(NFLVR) property and ifKis acceptable thenlimt→∞(K·S)texists a.s..

Proof.Suppose thatK·S≥−(a+L·S)whereLis admissible and maxi-
mal. Clearly we have thatK+Lisa-admissible and hence by the results of
Chap. 9 limt→∞((K+L)·S)texists a.s.. Because limt→∞(L·S)texists a.s.,
we necessarily have that limt→∞(K·S)talso exists a.s..

The set of outcomes of acceptable strategies, which is a convex cone inL^0 ,
is denoted by
J={(K·S)∞|Kacceptable}.
We now prove some elementary properties of acceptable contingent claims.
Most of these properties are generalisations of no-arbitrage concepts for ad-
missible contingent claims.

Proposition 13.2.12.Suppose thatSis a locally bounded semi-martingale
that satisfies the (NFLVR) property. IfKis acceptable and if(K·S)∞≥ 0 ,
then(K·S)∞=0.