9.6 No Free Lunch with Bounded Risk 187From the definitions and the results of Sect. 9.3 it follows that(NFL)
implies(NFLBR)implies(NFLVR)implies(NA). As regards the notion of
no free lunch(NFL), this was introduced by [K 81] and is at the basis of
subsequent work on the topic. It requires that there should not existf 0 in
L∞+, not identically 0, as well as a net (fα)α of elements inC such that
fα,convergestof 0 in the weak-star topology ofL∞. Because nets are used,
there is no bound on the negative partfα−offα. It is not excluded that
e.g.‖fα−‖∞tends to∞, reflecting the enormous amount of risk taken by the
agent. It is well-known that for bounded cadlag adapted processesS,(NFL)
(even when defined by simple strategies) is equivalent to the existence of an
equivalent martingale measure. See [S 94] for a proof of this theorem which
is essentially due to [K 81, Y 80]. The drawback of this theorem is twofold.
First it is stated in terms of nets, a highly non-intuitive concept. Second it
involves the use of very risky positions. The main theorem of the present
paper remedies this drawback. We therefore focus attention on variants of
the properties(NFLVR). The following characterisation, the proof of which
is almost the same as the proof of Proposition 9.3.7 and Corollary 9.3.8,
was proved in [S 94]. The proof makes essential use of the Banach-Steinhaus
theorem on the boundedness of weak-star convergentsequences.
Proposition 9.6.2.The semi-martingaleSsatisfies the condition (NFLBR)
if and only if for a sequence of 1 -admissible integrands(Hn)n≥ 1 with final
valuesgn=(Hn·S)∞the conditiongn−→ 0 in probability implies thatgn
tends to 0 in probability.
Proof.Suppose thatSsatisfies the property(NFLBR)and let (Hn)n≥ 1 be
a sequence of 1-admissible integrands (Hn)n≥ 1 with final valuesgn=(Hn·
S)∞such thatg−n →0 in probability. Suppose that this sequence does not
tend to 0 in probability. By selecting a subsequence, still denoted by (gn)n≥ 1
we may suppose that there isα>0 such thatP[gn>α]>αfor alln.By
Lemma 9.8.1 we may take convex combinationsfn∈conv{gk;k≥n}that
converge in probability to a functionf. The negative partsfn−still tend to
0 in probability and hencef:Ω→[0,∞]. The functionfsatisfiesP[f>
0]>0. The functionshn=min(fn,1) are in the convex setCand converge
in probability toh=min(f,1). The functionshnare uniformly bounded
by 1 and therefore the convergence in probability implies the convergence
in the weak-star topology ofL∞. The functionhis therefore inC ̃and the
property(NFLBR)now implies thath= 0 almost everywhere. This, however,
is a contradiction toP[f>0]>0.
Suppose conversely thatSsatisfies the announced property. It is clear that
Ssatisfies the no-arbitrage property(NA). Suppose now thathnis a sequence
inCthat converges weak-star toh.Wehavetoprovethath=0almost
everywhere. By the Banach-Steinhaus property on weak-star bounded sets, the
sequencehnis uniformly bounded. Without loss of generality we may suppose
that it is uniformly bounded by 1 and hencehn≥−1 almost surely. Since the
sequencehntends tohweak-star inL∞it certainly converges weakly tohin