The Mathematics of Arbitrage

46 3 Utility Maximisation on Finite Probability Spaces

We assume w.l.o.g. thatu ̃≥−d ̃; we do so — mainly for notational conve-
nience — in order to ensure thatEP[S 11 ]≥S^10 , so that the optimal portfolios
calculated below always have a long position in the stockS^1 .If ̃u<−d ̃we
obtain analogous results, but the position in the stock will be short.

Lettingq= −

d ̃

̃u−d ̃=

r−d

u−dand definingQ[g]=qandQ[b]=1−q=

̃u

̃u−d ̃=

u−r

u−dwe obtain the unique martingale measureQfor the processS.

Consider the utility functionU(x)=x

α
α forα∈]−∞,1[{^0 }with con-
jugate functionV(y)=−y

β

β,whereα−1=(β−1)

− (^1) , i.e.,β= α
α− 1 .We
note that the case of logarithmic and exponential utility (which correspond
— after proper renormalisation — toα=0andtoα=−∞) are similar (see
3.3.2 and 3.3.3 below).
Fixing the initial endowmentx>0 we want to solve the utility maximi-
sation problem (3.1) by applying the duality theory developed above. Well,
this is shooting with canons on pigeons, but we find it instructive to do some
explicit calculations exemplifying the abstract formulae.
The dual value function
v(y)=E

[

V

(

y

dQ

dP

)]

,y> 0 ,

equals

v(y)=

1

2

V(y 2 q)+

1

2

V(y2(1−q))

=cVV(y),

where

cV=

1

2

(

(2q)β+(2(1−q))β

)

.

Forβ<0 (which corresponds toα∈]0,1[) we have, forq=^12 ,thatcV> 1
by Jensen and the strict convexity ofy→yβ. Similarly, forβ∈]0,1[ (which
corresponds toα<0) we havecV<1(orcV=1inthecaseq=^12 ). In any
casev(y)≥V(y), as this must hold.

To calculate the primal value functionu(x)weusethewell-knownand
easily verified fact that, given a constantc>0 and two conjugate functions
U(x)andV(y), the functioncU

(x

c

)

is conjugate tocV(y). Hence

u(x)=cVU

(

x

cV

)

=c^1 V−αU(x)=cUU(x), (3.34)

where

cU=c^1 V−α=

(

1

2

(

(2q)β+(2(1−q))β

)

) 1 −α

. (3.35)

For fixedx>0, we obtain as critical Lagrange multiplier̂y(x)=u′(x)=
cUU′(x)sothat