76 5 The Kreps-Yan Theorem
such that (εn)∞n=1are independent random variables taking the values +1 or
−1 with probabilities
P[εn=1]=1+αn
2, P[εn=−1] =1 −αn
2,
where (αn)∞n=1is a sequence in ]− 1 ,+1[ to be specified below.
Clearly this defines a bounded processS, for which there is a unique mea-
sureQon (Ω,F)=
(
{− 1 , 1 }N,B
(
{− 1 , 1 }N
))
, under whichSis a martingale
in its own filtration (Ft)∞t=0;thismeasureisgivenby
Q[εn=1]=Q[εn=−1] =1
2
,
and (εn)∞n=1are independent underQ.
By a result of Kakutani (see, e.g. [W 91]) we know that Qis either
equivalent to∑ P,orPandQare mutually singular, depending on whether
∞
n=1α
2
n<∞or not.
Taking, for example,αn=^12 , for alln∈N, we have constructed a process
Son (Ω,F,P), for which there is no equivalent (local) martingale measure
Q. On the other hand, it is an easy and instructive exercise to show that,
for simple trading strategies, there are no arbitrage opportunities for the pro-
cessS.
By Lemma 5.1.5 we only need to check for strategies of the formH =
h (^1) ]]σ 1 ,σ 2 ]]. Such a strategy gives the outcomeh(Sσ 2 −Sσ 1 )andhisFσ 1 -
measurable. Suppose that (H·S)∞≥0. Of course we may replacehby sign(h).
Thechoiceof3−nalso yields that on the set
{
σ 1 =1−^1 n}
∩
{
σ 2 ≥ 1 −n+1^1}
thesign ofSσ 2 −Sσ 1 is the same as the sign ofεn. We get that sign
(
h(Sσ 2 −Sσ 1 ))
is on that same set equal to the sign ofhεn. The independence ofεnfrom
F 1 − (^1) nthen gives thath=0on
{
σ 1 =1−^1 n}
∩
{
σ 2 ≥ 1 −n+1^1}
. Combining
all these facts gives thath(Sσ 2 −Sσ 1 ) = 0 a.s..
The example in the above proof shows, why the no-arbitrage condition
(NAsimple)defined in 5.1.4 is too weak: it is intuitively rather obvious that by
a sequence of properly scaled bets on a (sufficiently, i.e.,
∑∞
n=1α2
n=∞)biased
coin one can “produce something like an arbitrage”, while a finite number of
bets (as formalised by Definition 5.1.1) does not suffice to do so.
But here we are starting to move on thin ice, and it will be the crucial
issue to find a mathematically precise framework, in which the above intuitive
insight can be properly formalised.
5.2 No Free Lunch
A decisive step in this direction was done in the work of D. Kreps [K 81],
who realised that the purely algebraic notion of no-arbitrage with respect to
simple integrands has to be complemented with a topological notion: