The Mathematics of Arbitrage

14.3 One-period Processes 291

In the above arguments we did suppose that (E,E,π) is complete. Now
we drop this assumption. In that case we first complete the space (E,E,π)by
replacing theσ-algebraEbyE ̃generated byEand all the null sets. We then
obtain anE ̃-measurable mappingF ̃ηwhich can, by modifying it on a set of
measure zero, be replaced by anE-measurable mappingFηsuch thatπalmost
surelyFη=F ̃η.

Remark 14.3.6.We have not striven for maximal generality in the formulation
of Lemma 14.3.5: for example, we could replace the probability measuresFη
by finite non-negative measures onRd.InthiscasewemayobtaintheGηin
such a way that the total massGη(Rd)equalsFη(Rd),π-almost surely.

To illustrate the meaning of the Crucial Lemma we note in the spirit of
[MB 91] which shows in particular the limitations of the no-arbitrage-theory
when applied e.g. to Gaussian models for the stock returns in finite discrete
time.

Proposition 14.3.7.Let(St)Tt=0be an adaptedRd-valued process based on
(Ω,F,(Ft)Tt=0,P)such that for every predictable process(ht)Tt=1we have that
(h·S)T=

∑T

t=1ht∆Stis unbounded from above and from below as soon as
(h·S)T≡ 0. For example, this assumption is satisfied if theFt− 1 -conditional
distributions of the jumps∆Stare non-degenerate and normally distributed
onRd.
Then, forε> 0 , there is a measureQ∼P,‖Q−P‖<ε, such thatSis
aQ-martingale.
As a consequence, the set of equivalent martingale measures is dense with
respect to the variation norm in the set ofP-absolutely continuous measures.

Proof.Suppose first thatT = 1. Contrary to the setting of the motivating
example at the beginning of this section we do not assume thatF 0 is trivial.
Let (E,E,π)be(Ω,F 0 ,P) and denote by (Fω)ω∈ΩtheF 0 -conditional dis-
tribution of ∆S 1 =S 1 −S 0. The assumption of Lemma 14.3.5 is (trivially)
satisfied as by hypothesis theF 0 -measurable functionsx(ω) such thatP-a.s.
we have (xω,y)≥− 1 ,Fω-a.s., satisfy (xω,y)=0,Fω-a.s., forP-a.e.ω∈Ω.
Chooseε(ω)≡ε>0 and findGηas in the lemma. To translate the change
of the conditional distributions of ∆S 1 into a change of the measureP, find
Y:Ω×Rd→R+,

Y(ω, x)=

dGω

dFω

(x),x∈Rd,ω∈Ω

such that, forP-a.e.ω∈Ω,Y(ω, .) is a version of the Radon-Nikod ́ym deriva-
tive ofGωwith respect toFω,andsuchthatY(., .)isF 0 ⊗B(Rd)-measurable.
Letting
dQ̂
dP

(ω)=Y(ω,∆S 1 (ω))