5.1 A General Framework 75Proof.The proof proceeds by induction onnand yields additionally that the
stopping timesσ 1 andσ 2 canbechosenasσ 1 =τi− 1 andσ 2 =τi,forsome
i≤n.Forn= 1 the statement is obvious sinceH =K will do the job.
So we only check the inductive step. IfP
[
(H·S)τn− 1 < 0]
>0 then we putσ 1 =τn− 1 ,σ 2 =τnandh=hn− 1 χ{(H·S)τn− 1 < (^0) }.ThestrategyK=hχ]]σ 1 ,σ 2 ]]
is an arbitrage opportunity since ({ H·S)τn≥0 a.s. and hence (K·S)τn>0on
(H·S)τn− 1 < 0
}
.If(H·S)τn− 1 =0a.s.thenK=hn− 1 χ]]τn− 1 ,τn]]must give
an arbitrage opportunity. The only remaining case is (H·S)τn− 1 ≥0a.s.and
P
[
(H·S)τn− 1 > 0]
>0. It is here that we apply the inductive hypothesis. Remark 5.1.6.If there is an arbitrage strategy, can we reduce the strategy
even further, e.g. can we takekof the formαχAχ]]σ 1 ,σ 2 ]]whereA∈Fσ 1 and
αis a constant? The following example shows that, ford≥2, this is not the
case.
Letθandηbe independent random variables which are uniformly dis-
tributed on [0,1[ and [0,^12 [ respectively. Define theR^2 -valued process (S 0 ,S 1 )
byS 0 =0andS 1 =e^2 πi(θ+η),whereweidentifyR^2 withC.Theσ-algebra
F 0 is generated byθandF 1 is generated byθandη. This market allows
for arbitrage: indeed lethbe theF 0 -measurableR^2 -valued random variable
h=e^2 πi(θ+
(^14) )
(identifyingR^2 withConce more) so that
(h, S 1 −S 0 )R^2 > 0 , a.s.,
where (·,·)R 2 now is the usual inner product onR^2. On the other hand, it is
easy to verify that for eachhof the formαχA,whereα∈R^2 is a constant
andA∈F 0 ,P[A]>0wehave
P[(h, S 1 −S 0 )R 2 <0]> 0.
We want to prove a version of the fundamental theorem of asset pricing
analogous to Theorem 2.2.7 above. However, things are now more delicate and
the notion of(NAsimple)defined above is not sufficiently strong to imply the
existence of an equivalent local martingale measure:
Proposition 5.1.7.The condition (EMM) of existence of an equivalent lo-
cal martingale measure implies the condition (NAsimple) of no-arbitrage with
respect to simple integrands, but not vice versa.
Proof. (EMM)⇒(NAsimple): this is an immediate consequence of Lemma
5.1.3, noting that forQ∼Pand a non-negative functionf≥0, which does
not vanish almost surely, we haveEQ[f]>0.
(NAsimple)(EMM): we give an easy counter-example which is just an
infinite random walk.
Lettn=1−n+1^1 and define theR-valued processSto start atS 0 =1,
and to be constant except for jumps at the pointstnwhich are defined as
∆Stn=3−nεn,n≥ 1 ,