9.1 Introduction 153stricted in order to allow the definition of integralsH·Sfor more general trad-
ing strategies.Shas to be a semi-martingale to realise this. This is precisely
the content of the Bichteler-Dellacherie theorem (see [P 90]). It turns out that
this is not really a restriction. From the work of [FS 91, AS 93], we know that
no free lunch conditions stated with simple integrands, imply that a cadlag
adapted process is a special semi-martingale. (A process is called cadlag, if
almost every trajectory admits left limits and is right continuous). We refer
to Sect. 9.7 of this paper for a general version of this result, adapted to our
framework. The second difficulty arises from the fact that doubling-like strate-
gies have to be excluded. This may be done by using the concept ofadmissible
integrandsH, requiring that the processH·Sis uniformly bounded from be-
low, a concept going back to [HP 81] and developed in [D 92, MB 91, S 94].
The concept of admissible integrand is a mathematical formulation of the
requirement that an economic agent’s position cannot become too negative,
a practice sometimes referred to as “your friendly broker calls for extra mar-
gin”. The third problem is to make sure that (H·S)∞= limt→∞(H·S)thas
a meaning. We shall see that this problem has a very satisfactory solution if
one restricts to admissible integrands.
The condition ofno free lunch with vanishing risk (NFLVR)can now be
described as follows. There should be no sequence of final payoffs of admissible
integrands,fn=(Hn·S)∞such that the negative partsfn−tend to 0uniformly
und such thatfntends almost surely to a [0,∞]-valued functionf 0 satisfying
P[f 0 >0]>0. We will give a detailed discussion of this property below
in Sect. 9.3. For the time being let us remark that the property(NFLVR)is
different from the previously considered concept of no free lunch with bounded
risk in the sense that we require that the risk taken, the lower bounds on the
processes (Hn·S), tend to zero uniformly. In the property(NFLBR)one only
requires that this risk is uniformly bounded below und that the variablesfn−
tend to zero in probability. The main theorem of the paper can now be stated
as:
Theorem 9.1.1.LetSbe a bounded real-valued semi-martingale. There is an
equivalent martingale measure forSif und only ifSsatisfies (NFLVR).
One implication in the above theorem is almost trivial: if there is an equiv-
alent martingale measure forSthen it is easy to see thatSsatisfies(NFLVR),
see the first part of the proof in the beginning of Sect. 9.4. The interesting
aspect of Theorem 9.1.1 lies in the reverse implication: the (economically
meaningful) assumption(NFLVR)guarantees the existence of an equivalent
martingale measure forSund thus opens the way to the wide range of appli-
cations from martingale theory.
lf the processSis only a locally bounded semi-martingale we still obtain
the following partial result:
Corollary 9.1.2.Let S be a locally bounded real-valued semi-martingale.
There is an equivalent local martingale measure forSif and only ifSsat-
isfies (NFLVR).