196 9 Fundamental Theorem of Asset Pricing
Theorem 9.7.6.
(a)IfS :[0,1]×Ω→ Ris a continuous semi-martingale having the no-
arbitrage property (NA), ifS satisfies (NFLVR) for simple integrands,
thenShas an equivalent local martingale measure.†
(b)IfS:R+×Ω→Ris a continuous semi-martingale having the no-arbitrage
property (NA), ifSsatisfies the no free lunch with bounded risk property
(NFLBR) for simple integrands, thenShas an equivalent local martingale
measure.†
Proof.LetHnbe a sequence of general admissible integrands. Suppose that
gn=(Hn·S) 1 ≥−εnwhereεntends to zero. By(NA)the integrandHnis now
εn-admissible. We have to prove thatgntends to 0 in probability, which will
prove part (a) in view of the Main Theorem 9.1.1. From the theory of stochas-
tic integration (see [CMS 80]) we deduce that there are simple integrandsLn
such thatP[sup 0 ≤t≤ 1 |(Ln·S)t−(Hn·S)t|≥εn]≤εn.Foreachnwe define the
stopping timeTnas inf{t|(Ln·S)t<− 2 εn}. ClearlyP[Tn<1]≤εn.Since
the processSis continuous, the random variableshn=(Ln·S)Tnare bounded
below by− 2 εnand are therefore results of 2εn-admissible simple integrands.
BecauseP[Tn<1]≤εnandP[sup 0 ≤t≤ 1 |(Ln·S)t−(Hn·S)t|≥εn]≤εnthe
sequencehn−gntends to 0 in probability. From the property no free lunch
with vanishing risk(NFLVR)for simple integrands we deduce thathnand
hencegntend to 0 in probability. Therefore the property no free lunch with
vanishing risk property(NFLVR)is satisfied and by the Main Theorem 9.1.1
there is an equivalent martingale measure.
For the second part we refer to [S 94, Sect. 5, Proposition 5.1].
The above theorem seems to indicate that for continuous processes simple
integrands are sufficient to describe no-arbitrage conditions. This is not true in
general. The Bes^3 (1)-process, (Rt) 0 ≤t≤ 1 gives a counter-example. This process
can be seen as the Euclidean norm of a three dimensional Brownian motion
starting at the point (1, 0 ,0) ofR^3. It plays a major role in the theory of con-
tinuous martingales and Brownian motion, see [RY 91] for details. The process
Rsatisfies the no-arbitrage(NA)property for simple integrands but fails the
no-arbitrage(NA)property for general integrands. We refer to [DS 95c] for
the details. The inverse of this process,L=R−^1 , a local martingale, has been
used in [DS 94a].
As a general question one might ask whether for continuous processes the
no-arbitrage(NA)property for general integrands is sufficient for the existence
of an equivalent local martingale measure. The following example shows that
this is not true.
†Note added in this reprint: In the original paper [DS 94] the hypothesis thatS
satisfies(NA)for general admissible integrands was forgotten but used in the
proof! Several people (including ourselves) have noted this. A counter-example
for a process satisfying(NFLVR)for simple integrands but not satisfying(NA)for
general admissible integrands is the Bessel process in dimension 3 (see [DS 95c]).