The Mathematics of Arbitrage

6.11 Interpretation of theL∞-Bound in the DMW Theorem 109

Proof.IfQ 0 ∈Ma∩Pkthen for allY∈W 1 we haveuk(Y)=inf{EQ[Y]|
Q∈Pk}≤EQ 0 [Y] = 0. ThereforeW 1 ∩Ok=∅.
Conversely ifW 1 ∩Ok =∅we may separate the vector spaceW 1 from
the convex open setOk. This yields an elementf∈L∞such that for all
Z∈Ok,allY ∈W 1 :E[fY]<EQ[fZ]. These inequalities show thatfis
non-negative and not identically zero. We may normalisefso thatE[f]=1,
hereby defining a probabilityQso thatddQP=f∈L∞.SinceEQ[Z]≥0for
allZ∈Okwe must haveQ∈Pk. AlsoEQ[Y] = 0 for allY∈W 1 and hence
Q∈Ma.

Remark 6.11.2.In [HK 79] M. Harrison and D. Kreps related the concept of
no-arbitrage to the concept of viability which is based on utility considerations
somewhat similar to the above considerations.

Remark 6.11.3.The coherent monetary utility functionukis essentially the
same as the so calledtail expectationwith parameterα=^1 k.
To avoid trivialities let us suppose that (Ω,F,P) is a non-atomic proba-
bility space. ForY∈L^1 (Ω,F,P) define the quantileqα(Y)as

qα=inf{x|P[Y≤x]≥α}.

IfYhas a continuous distribution then we have

uk(Y)=E[Y|Y≤qα],

where the right hand side is called, for obvious reasons, theα-tail expectation

ofY. Indeed, it suffices to considerddQP=k (^1) {Y≤qα}.
In the general case we have to be slightly more careful as it might happen
thatP[Y=qα(Y)]>0. In this case letβ=P[Y<qα]andverifythat
uk(Y)=E[Y|Y<qα]+(α−β)qα.
The above Theorem 6.11.1 tells us that the constantαverifying 1≥k^1 =
α>0 is sufficiently small such thatMa∩Pk=∅iff it is not possible to find an
element (H·S)T∈W 1 which yields a strictly higher utilityuk((H·S)T)than
uk(0) = 0. As a trivial illustration considerα=k= 1: then this statement
boils down to the fact thatSis a martingale underPiff for each (H·S)Twe
have thatu 1 ((H·S)T)=E[(H·S)T]=0.
We refer to [De 00] for a more detailed situation in which an interpretation
in terms of superhedging and risk measures is given.