198 9 Fundamental Theorem of Asset Pricing
We now give some more examples motivating the introduction of gen-
eral integrands. As seen in the above theorems and examples, the case of
continuous processes can essentially be reduced to simple integrands. The
following examples show that for general semi-martingales the no free lunch
with bounded risk(NFLBR)property for simple integrands is not sufficient
to imply the existence of an equivalent local martingale measure.
The examples are very similar in nature; the problems arise from the fact
that the jumps do not occur at an increasing sequence (Tn)n≥ 1 ofpredictable
stopping times (a case already solved in [S 94]). In our examples the jumps
occur at anincreasing sequence ofaccessible stopping times, similarly as in
Example 9.7.5. The first example of this kind is an unbounded process but it
contains all the ingredients and the general idea. The second example of this
kind gives a bounded process. Of course the price to pay is the use of more
technique.
Example 9.7.8.The first example uses the processX introduced in Exam-
ple 9.7.5. The semi-martingaleSwe will need, is defined asSt=Xt+t.The
processSis now a special semi-martingale and again ifHis simple predictable
withH·Sbounded from below thenH= 0. ThereforeStrivially satisfies
thenofreelunch(NFL)property with simple integrands. If, however, we put
H= (^1) ([0,1]\Q)×Ω(sell before each rational and buy back immediately after
it) we have (H·S)t=t(for 0≤t≤1) and this violates(NA)for general
integrands. IfQwere an equivalent local martingale measure for the process
S, then becauseH= (^1) ([0,1]\Q)×Ωis bounded,H·Sis also a local martingale
(see [P 90, Theorem 2.9]). This is absurd.
The previous example has at least one disadvantage: the processSis un-
bounded. The next example overcomes this problem. This time we will work
with a doubly indexed sequence of Rademacher variables (rn,m)n≥ 1 ,m≥ 1 , i.e.
variables with distributionP[rn,m =1]=P[rn,m =−1] =^12 ,andwith
a doubly indexed sequence of variables (φn,m)n≥ 1 ,m≥ 1 with the property
P[φn,m =1]=2−(n+m)andP[φn,m =0]=1− 2 −(n+m). We also need
a sequence of Brownian motionsWnstarting at 0. All these variables and
processes are supposed to be independent. The rationals in ]0,1[ are again
enumerated as (qn)n≥ 1. We first define theL^2 -martingalesYmas:
Ytm=
∑
n;qn≤tφn,mrn,m.The Borel-Cantelli implies, as in Example 9.7.5, that eachYmis piecewise
constant. We define the stopping timeTmas:
Tm=min(
inf{t||Wtm|=morYtm=0}, 1)
.
We make the crucial observation that