The Mathematics of Arbitrage

200 9 Fundamental Theorem of Asset Pricing

Lemma 9.7.10.If(αm)m≥ 1 is a sequence in]0,1]such thatαm→ 0 fast
enough, thenSsatisfies (NFLBR) for simple integrands.
(By fast enough we mean that for allm 0 we have:
∑

m>m 0

2 m+1m^2 αm<

βm 0

2 m 0

whereβm 0 =exp(− 3 m^50 )).

Proof.For each mnatural number, we know that the processSm, i.e.S
stopped atm, admits an equivalent martingale measureQm. Indeed we can
use a Girsanow transformation to find an equivalent martingale measure such
that forkfixed, the process (Wtk+k^2 t) 0 ≤t≤ 1 stopped atTkis a martingale.
The density of this measure is given by exp(δWTkk−^12 δ^2 Tk)whereδ=−k^2.

This density is bounded above by exp(k^3 )andbelowbyexp(−k^3 −^12 k^4 ).
The density ofQmonFmis therefore bounded below by exp(−

∑m

k=1(k

(^3) +
1
2 k
(^4) ))≥exp(− 2 m (^5) ) and bounded above by exp(∑m
k=1k
(^3) )≤exp(m (^4) ). Under
the measureQmthe processSmis a martingale and hence for eachHthat is
1-admissible,H·Smis aQm-super-martingale (by Theorem 9.2.9) and hence
for each 1-admissible integrand we find
EQm[(H·S)+m]≤EQm[(H·S)−m]
and hence
exp(− 2 m^5 )EP[(H·S)+m]≤exp(m^4 )EP[(H·S)−m]
and
EP[(H·S)−m]≥βmEP[(H·S)+m]forβm=exp(− 3 m^5 ).
We will show that ifαm→0 as announced, the processSsatisfies(NFLBR)
with simple integrands.
Suppose on the contrary thatSdoes not satisfy the(NFLBR)property
for simple integrands. We then chooseHj simple, predictable, 1-admissible
such that (Hj·S)∞tends tof 0 ≥0whereP[f 0 >0]>0. Findm 0 so
thatEP[min(f 0 ,1)]> m^20 .Foreachjwe define the stopping timeUj as
inf{t|(Hj·S)t≥ 1 }and letLj=Hj (^1) [[ 0,Uj]].Foreachjthesimplepredictable
processLjis still 1-admissible and (Lj·S)∞≥min((Hj·S)∞,1), therefore
lim infj→∞(Lj·S)∞≥min(f 0 ,1). EachLjis of the form
∑n
k=1fk^1 ]]Vk− 1 ,Vk]]
wherefkisFTk-measurable andV 0 ≤V 1 ≤...≤Vn<∞are stopping times.
If ]]Vk− 1 ,Vk]]∩]]m− 1 ,m−1+Tm]] is not equivalent to the zero process,
then the probability of a jump betweenVk− 1 andVkis strictly positive by
the same arguments as in Example 9.7.5. Because the jumps ofSare positive
or negative with the same probability we conclude that the downward jump
of (Lj·S) cannot be smaller than−2. (Indeed the process is always bigger
than−1 and is stopped when it hits the level 1). We conclude that also the
positive jump is bounded by 2. Therefore|LjTm∆STm|≤2. We conclude that
|Lj|≤ 2 m+1on ]]m− 1 ,m−1+Tm]]. Because we stopped the process (Lj·S)
when it exceeds the level 1, we see that: