9.6 No Free Lunch with Bounded Risk 189The Proposition 9.6.3 allows us to obtain a sharpening of the main theorem
of [S 94, Theorem 1.6]. We leave the economic interpretation to the reader.
Proposition 9.6.4.Let(Sn)nbe a locally bounded adapted stochastic process
for the discrete time filtration(Fn)n. If there does not exist an equivalent local
martingale measure forSthen at least one of the following two conditions must
hold:
(1)Sfails (NA) for general admissible integrands, i.e. there is an admissible
integrandHsuch that(H·S)∞≥ 0 a.s. andP[(H·S)∞>0]> 0.
(2)Sfails (NFLVR) for elementary integrals, i.e. there is a sequence(Hn)n
of elementary integrals such that(Hn·S)≥−n−^1 and(Hn·S)∞tends
almost surely to a functionf:W→[0,∞]withP[f>0]> 0.
Proof.For discrete time processes, elementary integrands and general inte-
grands with bounded support are the same. Therefore ifSsatisfies both con-
ditions (1) and (2), then by Proposition 9.6.3,Salso satisfies(NFLVR)for
general integrands. The main Theorem 9.1.1 now asserts thatSadmits an
equivalent local martingale measure. The proposition is the contraposition of
this statement.
The following example shows that in general the no free lunch with van-
ishing risk property for admissible integrands with bounded support does not
imply the no free lunch with vanishing risk property for general admissible in-
tegrands! As Proposition 9.6.3 indicates there should be arbitrage for general
integrands.
Example 9.6.5.We give the example in discrete time. The extension to contin-
uous time processes is obvious. The set Ω is the compact space of all sequences
of−1or+1:{− 1 ,+1}N.Theσ-algebrasGnof the filtration are defined as
the smallestσ-algebras making the firstnco-ordinates measurable. On Ω we
put two measuresPandQ. The measurePis defined as the Haar mea-
sure, this is the only measure such that the co-ordinatesrnare a sequence
of independent, identically distributed variables withP[rn=±1] =^12 .The
measureQis defined as^12 (P+δa), whereδais the Dirac measure giving all
its mass to the elementa, the sequence identically 1. Definefas the vari-
ablef=− (^1) {a}+ (^1) Ω{a}.ClearlyEQ[f] = 0. Define now the processSnby
Sn=EQ[f|Gn]. Theσ-algebrasFnof the filtration are defined as the small-
estσ-algebras making theS 1 ,...,Snmeasurable, i.e. the natural filtration of
S.Theσ-algebraFis generated by the sequence (Sn)n. It is easy to see that
onF,Sadmits only one equivalent martingale measure, namelyQ. We will
now consider the processSunder the measureP.Oneachσ-algebraFnthe
two measures,PandQ,areequivalent.SupposenowthatHnis a sequence of
boundedly supported predictable integrands such thatgn=(Hn·S)∞≥−n^1
almost everywhere for the measureP.Foreachnthere isknbig enough such
thatgnis measurable forFn. Therefore alsoQ
[
gn≥−n^1]
=1foreachn.
SinceQis a martingale measure forSit follows thatEQ[gn]=0andthat