210 10 Counter-Example
Proof.TakeXas in the preceding theorem and define, through the stochas-
tic logarithm, the processSasdS=dM+d〈M, M〉whereX=E(−M).
The measureQdefined asdQ=X∞Y∞dPis an(ELMM)forS. Obviously
the natural candidate for an(ELMM)suggested by the Girsanov-Maruyama-
Meyer formula, i.e. thedensityX∞, does not define a probability measure.
Remark 10.1.3.If in the previous theorem we replaceSbyE(S), then we can
even obtain a positive price system.
Theorem 10.1.4.There is a processSthat admits an (ELMM) as well as
a hedgeable elementgsuch thatER[g]is not constant on the setMe.
Proof.For the processSwe takeXfrom Theorem 10.1.1. The original measure
Pis an(ELMM)and since there is an(ELMM)QforXsuch thatXbecomes
a uniformly integrable martingale, we necessarily have thatEQ[X∞−X 0 ]=0
and thatX∞−X 0 is maximal. However,EP[X∞−X 0 ]<0.
As for the economic interpretation, let us consider a contingent claimf
that is maximal and such thatER[f]<0=sup{EQ[f]|Q∈Me}for some
R∈Me. Suppose now that a new instrumentTis added to the market and
suppose that the instrumentThas a price at timetequal toER[f|Ft]. The
measureRis still a local martingale measure for the couple (S, T), hence the
financial market described by (S, T) still isarbitrage free— more precisely
it does not admit a free lunch with vanishing risk; but as easily seen the
elementfis no longer maximal in this expanded market. Indeed the element
T∞−T 0 =f−ER[f] dominatesfby the quantity−ER[f]>0! In other
words, before the introduction of the instrumentTthe hedge offas (H·S)∞
may make sense economically, after the introduction ofTit becomes asuicide
strategywhich only an idiot will apply.
Note that an economic agent cannot make arbitrage by going short on
astrategyHthat leads to (H·S)∞=fand by buying the financial instrument
T. Indeed the process−(H·S)+T−T 0 in not bounded below by a constant
and therefore the integrand (−H,1) is not admissible!
On the other hand the maximal elementsfsuch thatER[f] = 0 for all
measuresR∈Mehave a stability property. Whatever new instrumentTwill
be added to the market, as long as the couple (S, T) satisfies the(NFLVR)
property, the elementfwill remain maximal for the new market described
by the price process (S, T). The set of all such elements as well as the space
generated by the maximal elements is the subject of Chap. 13.
Sect. 10.2 of this paper gives an easy example that satisfies the properties
of Theorem 10.1.1. Sect. 10.3 shows that the construction of this example can
be mimicked in most incomplete markets with continuous prices.
10.2 Construction of the Example
We will make use of two independent Brownian motions,BandW, defined
on a filtered probability space (Ω,(Ft) 0 ≤t,P), where the filtrationFis the