The Mathematics of Arbitrage

54 3 Utility Maximisation on Finite Probability Spaces

investment as theFN− 1 -measurable function̂hNdefined bŷhN=
̂h
SN− 1 .Eco-
nomically speaking, the valuêhNSN− 1 of the proportion of the investment in
the stock is constant, while the number̂hNof stocks in the portfolio depends
on the current stock priceSN− 1 : it is simply inversely proportional toSN− 1.

To computeuN− 2 (x), note that this step again is reduced to the analysis of
the one step problem at times{N− 2 ,N− 1 }, where we now have to replace
the original utility functionU(x) by the conditional utility functionuN− 1 (x).
This is just the principle of dynamic programming which reduces to an obvious
fact in the present context. Hence

uN− 2 (x)=sup{E[uN− 1 (x+h∆SN− 1 )]|h∈R}

=c^2 UU(x).

By induction we conclude that

ut(x)=cNU−tU(x),t=0,···,N,

so that we obtain in particular for the value functionu(x)=u 0 (x)

u(x)=cNUU(x). (3.46)
In order to compute the parameters of the optimal investment strategy,
we now assume that there is a fixed horizonT>0 such thatN∆t=Tand
we letN→∞,sothat∆t=TN→0. As above we define ̃u=σ∆t

(^12)
+ν∆t,
d ̃=−σ∆t^12 +μ∆t. We have found in (3.40) that
cU=1−
βν^2
2 σ^2
∆t+o(∆t)
so that
cNU=c
∆Tt
U =exp

(

−

βν^2 T

2 σ^2

)

+o(1).

The optimal investment strategy

(

X̂t(x)

)T

t=0

for initial wealthx>0is

given by

X̂t(x)=x+

∑t

n=1

̂ht∆St

where

̂ht=

X̂t− 1 (x)

St− 1

̂k

and as in (3.42)

̂k=

̂h

x

+O(∆t

(^12)
)=
(1−β)ν
σ^2

(3.47)

is the ratio of the current wealthX̂t− 1 (x) invested into the stock. We thus
find the discrete version of the well-known “Merton-line” investment strategy
[M 90]; the latter applies to the continuous time limit, i.e., the Black-Scholes
model.