154 9 Fundamental Theorem of Asset Pricing
In [DS 94a] counter-examples are given which show that in the above corol-
lary one can only assert the existence of a measureQunder whichSis alocal
martingale. Even if the variablesSt, are uniformly bounded inLpfor some
p>1, this does not imply thatSis a martingale. On the other hand we
do not know whether the hypothesis of local boundedness is essential for the
corollary to hold. There is some hope that the condition is superfluous but
at present this remains an open question.†In the discrete time case the local
boundedness assumption is not needed as shown in [S 94].
The proof of Theorem 9.1.1 is quite technical and will be the subject of
Sect. 9.4. The rest of the paper is organised as follows. Sect. 9.2 deals with
definitions, notation and results of general nature. In Sect. 9.3 we examine
the property(NFLVR)and we prove that under this condition, the limit
(H·S)∞= limt→∞(H·S)texists almost surely for admissible integrands. The
fifth section is devoted to the study of the set of local martingale measures.
Here we give a new characterisation of a complete market. It turns out that
if each local martingale measure that is absolutely continuous with respect to
the original measure, is already equivalent to the original measure, then the
market is complete and there is only one equivalent (absolutely continuous)
local martingale measure. These results are related to results in [AS 94, J 92].
We also show that the framework of admissible integrands allows to formulate
a general duality theorem (Theorem 9.5.8). In Sect. 9.6 we investigate the
relation between the no free lunch with vanishing risk(NFLVR)property
and the no free lunch with bounded risk(NFLBR)property. In the case of
an infinite horizon the latter property permits to restrict to strategies that
are of bounded support. They have a more intuitive interpretation since they
only require ’planning’ up to a bounded time. In Sect. 9.7 we introduce the
no free lunch properties(NFLVR),(NFLBR)and(NFL)stated in terms of
simple strategies. It is shown that in the case of continuous price processes
one can avoid the use of general integrands and restrict oneself to simple
integrands. The result generalises the main theorem of [D 92] in the case of
a finite dimensional price process. The relation between the no free lunch
with vanishing risk property for simple integrands and the semi-martingale
property is also investigated in Sect. 9.7. We also give examples that show
that the use of simple integrands is not enough to obtain a general theorem
and relate the present results to previous ones, in particular to [D 92, S 94].
Appendix 9.8 contains some technical lemmas already used in [S 94]. We state
versions which are more general and provide somewhat easier proofs.
†Note added in this reprint: The answer to this question is given in Chap. 14 below.
In fact the notion of a local martingale measure has to be replaced by the notion
of a sigma-martingale measure. This is precisely the theme of Chap. 14 below.