The Mathematics of Arbitrage

44 3 Utility Maximisation on Finite Probability Spaces

(ii) The optimisersX̂T(x)andQ̂(y)in (3.27) and (3.28) exist, are unique,
Q̂(y)∈Me(S)and satisfy

X̂T(x)=I

(

y

dQ̂(y)

dP

)

,y

dQ̂(y)

dP

=U′(X̂T(x)), (3.29)

wherex∈dom(U)andy> 0 are related viau′(x)=yor, equivalently,
x=−v′(y).
(iii)The following formulae foru′andv′hold true:

u′(x)=EP[U′(X̂T(x))],v′(y)=EQ̂

[

V′

(

yd

Q̂(y)

dP

)]

(3.30)

xu′(x)=EP[X̂T(x)U′(X̂T(x))],yv′(y)=EP

[

yd

Q̂(y)

dP V

′

(

yd

Q̂(y)

dP

)]

.(3.31)

Remark 3.2.2.Let us again interpret the formulae (3.30), (3.31) foru′(x) sim-
ilarly as in Remark 3.1.4 above. In fact, the interpretations of these formulae
as well as their derivation remain exactly the same in the incomplete case.
But a new and interesting phenomenon arises when we pass to the variation
of the optimal pay-off functionX̂T(x) by a small unit of an arbitrary pay-off
functionf∈L∞(Ω,F,P). Similarly as in (3.25) we have the formula

EQ̂(y)[f]u′(x) = lim

h→ 0

EP[U(X̂T(x)+hf)−U(X̂T(x))]

h

, (3.32)

the only difference being thatQhas been replaced byQ̂(y) (recall thatxand
yare related viau′(x)=y).
The remarkable feature of this formula is that it does not only pertain to
variations of the formf=x+(H·S)T, i.e, contingent claims attainable at
pricex, but to arbitrary contingent claimsf, for which — in general — we
cannot derive the price from no-arbitrage considerations.
The economic interpretation of formula (3.32) is the following: the pricing
rulef→EQ̂(y)[f] yields precisely those prices at which an economic agent
with initial endowmentx, utility functionUand investing optimally, is indif-
ferent of first order towards adding a (small) unit of the contingent claimf
to her portfolioX̂T(x).
In fact, one may turn this around: this was done by M. Davis [D 97] (com-
pare also the work of Sir J.R. Hicks [H 86] and L. Foldes [F 90]): one maydefine
Q̂(y) by (3.32), verify that this is indeed an equivalent martingale measure
forSand interpret this pricing rule as “pricing by marginal utility”, which is,
of course, a classical and basic paradigm in economics.
Let us give a proof for (3.32) (under the hypotheses of Theorem 3.2.1).
One possible strategy for the proof, which also has the advantage of providing
a nice economic interpretation, is the idea of introducing “fictitious securities”
as developed in [KLSX 91]: fixx∈dom(U)andy=u′(x)andlet(f^1 ,...,fk)
be finitely many elements ofL∞(Ω,F,P) such that the spaceK={(H·S)T|