The Mathematics of Arbitrage

3.1 The Complete Case 39

(ii) The optimiserX̂T(x)in (3.19) exists, is unique and satisfies

X̂T(x)=I

(

y

dQ

dP

)

, or, equivalently, y

dQ

dP

=U′(X̂T(x)), (3.20)

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)=EQ

[

V′

(

y

dQ

dP

)]

(3.21)

xu′(x)=EP

[

X̂T(x)U′(X̂T(x))

]

,yv′(y)=EP

[

y

dQ

dP

V′

(

y

dQ

dP

)]

.(3.22)

Proof.Items (i) and (ii) have been shown in the preceding discussion, hence
we only have to show (iii). The formula forv′(y) in (3.21) and immediately
follows by differentiating the relation

v(y)=EP

[

V

(

y

dQ

dP

)]

=

∑N

n=1

pnV

(

yqpnn

)

.

Of course, the formula forv′in (3.22) is an obvious reformulation of the
one in (3.21). But we present both of them to stress their symmetry with the
formulae foru′(x).
The formula foru′in (3.21) translates via the relations exhibited in (ii)
into the identity

y=EP

[

y

dQ

dP

]

,

while the formula foru′(x) in (3.22) translates into

v′(y)y=EP

[

V′

(

y

dQ

dP

)

y

dQ

dP

]

,

which we just have verified to hold true.

Remark 3.1.4.Let us recall the economic interpretation of (3.20)

U′

(

X̂T(x)(ωn)

)

=y

qn

pn

,n=1,...,N.

This equality means that in every possible state of the worldωn,themarginal
utilityU′(X̂T(x)(ωn)) of the wealth of an optimally investing agent at time
Tisproportional to the ratio of the priceqnof the corresponding Arrow se-

curity (^1) {ωn}and the probability of its successpn=P[ωn]. This basic relation
was analyzed in the fundamental work of K. Arrow and allows for a convinc-
ing economic interpretation: consider for a moment the situation where this