The Mathematics of Arbitrage

9.7 Simple Integrands 195

to be independent. The countable set of rationals in the interval ]0,1[ is enu-
merated as (qn)n≥ 1. Because

∑

P[φn=1]<∞the Borel-Cantelli lemma
tells us that for almost allω∈Ω there are only a finite number of natural
numbersnsuch thatφn=1.
The stochastic processXdefined as

Xt=

∑

n;qn≤t

φnγn

is therefore right continuous, even piecewise constant (by the above Borel-
Cantelli argument). The natural filtration generated by this process is there-
fore right continuous (see [P 90, Theorem 25] for a proof that can be adapted
to this case) and so is the filtration augmented with the negligible sets. The
filtration so constructed therefore satisfies the usual conditions.
Take nowF :[0,1] → Ra continuous function of unbounded varia-
tion, e.g.F(t)=tsin

( 1

t

)

.LetnowSt =Xt+F(t). It is easy to verify
thatX is anL^2 -martingale and hence it is a semi-martingale. IfS were
a semi-martingale thenF would also be a semi-martingale. This, however,
implies thatF is of bounded variation. We conclude thatSis not a semi-
martingale. We will now show thatSsatisfies the(NFL)property for sim-
ple integrands. This certainly implies thatSsatisfies the(NFLVR)property
for simple integrands and it shows that the local boundedness condition in
Theorem 9.7.2 is not superfluous. To verify the(NFL)property with simple

integrands, let us start with an integrandH=f (^1) ]]T,T′]] whereT≤T′are
two stopping times andfisFT-measurable. We will show thatH·Sis not
uniformly bounded from below unlessH= 0. Suppose on the contrary that
P[{T<T′}∩{f> 0 }]>0 (the case{f> 0 }is similar). Taketreal and
qnrational such thatt<qnandP[{f> 0 }∩{T ≤t}∩{qn≤T′}]>0.
Becausef isFT-measurable,t<qnandT′is a stopping time we obtain
thatA ={f> 0 }∩{T ≤t}∩{qn≤T′}∈Fqn− and hence is indepen-
dent ofφnγn. Becauseφnγnis unbounded from below (and from above for
the other similar case) we obtain thatP[A∩{φnγn <−K}]>0 for all
K>0. It is now easy to see that this implies thatH·S is unbounded
from below. It also follows that the only simple integrandHfor whichH·S
is bounded from below is the zero integrand. Since there are no admissi-
ble simple integrands, the(NFL)property with simple integrands is trivially
satisfied!

Theorem 9.7.2 and the Main Theorem 9.1.1 allow us to strengthen the
main theorem in [D 92]. The theorem shows that in the case of continuous
price processes and finite horizon, the condition (d) in [D 92], an equivalent
form of the no free lunch with bounded risk for simple integrands, can be
relaxed. The case of infinite horizon is already treated in Example 9.6.5. By
using the techniques developed in [S 93] one can translate this example into
an example whereSis a continuous process.