8.2 The Crucial Lemma 131Regarding condition (ii) in the above definition: this is a strong and eco-
nomically convincing requirement to rule out the above discussed doubling
strategy, as well as similar schemes, which try to make a final gain at the
cost of possibly going very deep into the red. Condition (ii) goes back to the
work of Harrison and Pliska [HP 81]: the interpretation is that there is a finite
credit lineMobliging the investor to finance her trading in such a way that
this credit line is respected at all timest≥0.
Definition 8.1.2.LetSbe anRd-valued semi-martingale and let
K={
(H·S)∞
∣
∣
∣Hadmissible and(H·S)∞= lim
t→∞
(H·S)texists a.s.}
,
(8.1)
which forms a convex cone of functions inL^0 (Ω,F,P),and
C={g∈L∞(P)|g≤ffor somef∈K}. (8.2)We say thatSsatisfies the condition ofno free lunch with vanishing risk
(NFLVR), if
C∩L∞+(P)={ 0 },
whereCnow denotes the closure ofCwith respect to the norm topology of
L∞(P).
Comparing the present definition with the notion of “no free lunch”
(NFL), the weak-star topology has been replaced by the topology of uni-
form convergence. Taking up again the discussion following the Kreps-Yan
Theorem 5.2.2, we now find a better economic interpretation:Sallows for
afree lunch with vanishing risk,ifthereisf∈L∞+(P){ 0 }and sequences
(fn)∞n=0=((Hn·S)∞)∞n=0∈K,where(Hn)∞n=0is a sequence of admissible
integrands and (gn)∞n=0satisfyinggn≤fn, such that
lim
n→∞
‖f−gn‖∞=0.In particular the negative parts ((fn)−)∞n=0and ((gn)−)∞n=0tend to zero
uniformly, which explains the term“vanishing risk”.
8.2 The CrucialLemma .....................................
We now come back to the formulation of ageneral version of the fundamental
theorem of asset pricing. We first restrict to the case oflocally bounded pro-
cessesS, as treated in Chap. 9. This technical assumption makes life much
easier.
Theorem 8.2.1.(Corollary 9.1.2) The following assertions are equivalent for
anRd-valued locally bounded semi-martingale modelS=(St)t≥ 0 of a financial
market: