The Mathematics of Arbitrage

9.7 Simple Integrands 197

Example 9.7.7.We take a standard Wiener processWwith its natural filtra-
tion (Gt) 0 ≤t≤ 1. Before we define the price processS, we first define a local
martingale of exponential type by:

Lt=

{

exp

(

−(f·W)t−^12

(∫

t

0 f

(^2) (u)du

))

, ift< 1

0 , ift=1.

wherefis the deterministic function defined asf(t)=√ 11 −t.

We define the stopping timeTasT=inf{t|Lt≥ 2 }. The stopped process
LTis a bounded martingale starting at zero. ClearlyLT=2ifT<1and
equals 0 ifT= 0. ThereforeP[T<2] =^12. We now define the price process
by its differential

dSt=

{

dWt+√ 11 −tdt, ift≤T

0 , ift≥T.

The filtration is now defined as (Ft) 0 ≤t≤ 1 =(Gmin(t,T)) 0 ≤t≤ 1. Except for sets
of measure zero, this is also the natural filtration of the processSand of the
stopped Wiener processWT. All local martingales with respect to this filtra-
tion are stochastic integrals with respect to the Wiener process (stopped at
T) (see [RY 91, Theorem 4.2] and stop all the local martingales at the stop-
ping timeT). Girsanov’s formula therefore implies that the only probability
measureQ, absolutely continuous with respect toPand for whichSis a local
martingale, is precisely the measureQdefined through its density onF 1 as
dQ=LTdP. As we shall see,Ssatisfies the property of no-arbitrage(NA).
Important in the proof of this, is the fact that fort<1, the measuresQandP
are equivalent onFt, (the densityLTt is strictly positive). Because the process
Sis continuous the proof thatSsatisfies(NA)reduces to verifying the state-
ment that forHadmissible, (H·S) 1 cannot be almost everywhere positive
without being zero a.s.. TakeHadmissible and suppose that (H·S) 1 ≥0,P-
a.s.. This certainly implies that (H·S) 1 ≥0,Q-a.s.. BecauseSis a continuous
Q-local martingale, we know thatH·Sis a continuousQ-local martingale and
becauseHis admissible forQ,Qbeing absolutely continuous with respect to
P,H·Sis aQ-super-martingale. From this it follows thatEQ[(H·S) 1 ]≤ 0
and by positivity of (H·S) 1 , this in turn implies that (H·S) 1 =0,Q-a.s..
Under the probabilityQ, the processSis a local martingale and hence sat-
isfies(NA)with respect toQ!Foreachε>0, let nowV be the stopping
time defined as inf{t|(H·S)t≥ε}. The integrandK=(1[[ 0,V]]H) is clearly
admissible and (K·S) 1 =0on{V=1}, whereas on{V< 1 }the outcome
isε, i.e. strictly positive. The(NA)property forS(underQ!) implies that
Q[V<1] = 0. In other words the processH·Snever exceedsεQ-a.s.. This
implies (H·S)t≤0,Q-a.s. for allt<1. BecauseQandPare equivalent
onFt,fort<1, this is the same as (H·S)t≤0,P-a.s. for allt<1. From
this and the continuity of the process (H·S) we deduce that (H·S) 1 ≤0,
P-a.s.. This in turn implies that (H·S) 1 =0,P-a.s.. The processStherefore
satisfies(NA)under the probabilityP.