56 3 Utility Maximisation on Finite Probability Spaces
so
wUN=exp(
−
βν^2 T
2 ασ^2)
+o(1)=exp(
ν^2 T
(1−α)2σ^2)
+o(1).In this new scaling the factor 1 −^1 αmakes perfect sense economically: when
α→1, i.e., the investor is less and less risk averse, the factorwNU becomes
large: the investor then appreciates highly the possibility to invest in the
financial market. Ifα→−∞, i.e., the investor is more and more risk averse
and the factorwNUtends to one: in this case the investor has little appreciation
for the possibilities offered by investments into the risky stock.
Note that the functionα→exp(
ν^2 T
2(1−α)σ^2)
is continuous forα∈]−∞,1[ sothat the limiting behaviour asαtends to zero, is not a puzzle any more; in the
caseα= 0 one easily verifies thatwNU=exp
(
ν^2 T
2 σ^2)
indeed yields the certaintyequivalent in the case of the logarithmic utility functionU(x) = log(x).
Example 3.3.6 (The trinomial model (Nperiods)).We now extend the
one-period trinomial model, example 3.3.4, to theNperiod setting similarly
as we just have done for the binomial model.
Let 0<m<1and(ηt)Nt=1 i.i.d. random variables, defined on some
(Ω,F,P) such thatP[ηt=−1] =^1 − 2 m,P[ηt=0]=m,P[ηt=1]=^1 − 2 m.Let
S 0 = 1 and, fort=1,···,Ndefine inductively
St=⎧
⎨
⎩
St− 1 (1 +u ̃),ifηt=1,
0 , ifηt=0,
St− 1 (1 +d ̃),ifηt=− 1.
The analysis of the maximisation of expected utility of terminal wealth
now is entirely analogous to the situation of the binomial model: using the
notation from the preceding examples, the optimal investment strategy again
consists in investing the constant proportion̂k=(1−σβ 2 )ν+O(∆t
(^12)
) of current
wealth found in (3.47) into the stock. For the value functionutri(x) we find
utri(x)=(ctriu)NU(x)
=exp
(
−
(1−m)βν^2 T
2 σ^2)
U(x)+o(1)and for the “certainty equivalent” (wtriU)N, defined analogously as in the pre-
ceding example, we obtain
(wtriU)N=exp(
(1−m)ν^2 T
(1−α)2σ^2)
+o(1).The verifications are simply a combination of the arguments of Exam-
ples 3.3.4.