The Mathematics of Arbitrage

3.1 The Complete Case 37

We now apply these general facts about the Legendre transform to calcu-
lateΨ(y). Using definition (3.9) of the conjugate functionVand (3.5), formula
(3.7) becomes:

Ψ(y)=

∑N

n=1

pnV

(

yqpnn

)

+yx

=EP

[

V

(

y

dQ

dP

)]

+yx.

Denoting byv(y) the dual value function

v(y):=EP

[

V

(

y

dQ

dP

)]

=

∑N

n=1

pnV

(

yqpnn

)

,y> 0 , (3.14)

the functionvhas the same qualitative properties as the functionVlisted in
Definition 3.1.1, since it is a convex combination ofV calculated on linearly
scaled arguments.
Hence by (3.10), (3.11), and (3.12) we find, for fixedx∈dom(U), a unique
̂y=ŷ(x)>0 such thatv′(̂y(x)) =−x, which is therefore the unique minimiser
to the dual problem

Ψ(y)=EP

[

V

(

y

dQ

dP

)]

+yx=min!

Fixing the critical valueŷ(x), the concave function

(ξ 1 ,...,ξN)→L(ξ 1 ,...,ξN,ŷ(x))

defined in (3.5) assumes its unique maximum at the point (̂ξ 1 ,...,̂ξN)satis-
fying

U′(̂ξn)=ŷ(x)pqnn or, equivalently, ξ̂n=I

(

ŷ(x)qpnn

)

,

so that we have

inf

y> 0

Ψ(y) = inf

y> 0

(v(y)+xy) (3.15)

=v(̂y(x)) +xŷ(x)

=L(ξ̂ 1 ,...,ξ̂N,̂y(x)).

Note that the ξ̂n are in the interior of dom(U), for 1 ≤ n ≤ N,so
thatLis continuously differentiable at (ξ̂ 1 ,...,ξ̂N,̂y(x)), which implies that
the gradient of L vanishes at (̂ξ 1 ,...,̂ξN,̂y(x)) and, in particular, that
∂
∂yL(ξ^1 ,...,ξN,y)|(ξ̂ 1 ,...,̂ξN,ŷ(x))= 0. Hence we infer from (3.4) and̂y(x)>0,
that the constraint (3.3) is binding, i.e.,