The Mathematics of Arbitrage

3.3 The Binomial and the Trinomial Model 49

In the special caseα=^12 ,β=−1, this yields

̂h=^2 xν

σ^2

+O

(

∆t

12 )

.

We observe from (3.41) that, for fixedν>0,σ>0, the factor

̂k=(1−β)ν

σ^2

(3.42)

ranges between 0 and∞asβruns through ]−∞,1[{ 0 }(the caseβ=0
corresponding to the logarithmic utilityU(x) = log(x)).
For the optimal portfolioX̂ 1 (x) we thus find

X̂ 1 (x)=

⎧

⎨

⎩

x

(

1+ν(βσ−1)∆t

12 )

+O(∆t),forω=g,

x

(

1 −ν(βσ−1)∆t

12 )

+O(∆t),forω=b.

(3.43)

Regarding the logarithmic and exponential utility we only report the re-
sults and leave the derivation, which is entirely analogous to the above, as
exercises.

Example 3.3.2.Under the same assumptions onSas in 3.3.1, but with let-
tingU(x) = ln(x), we obtainV(y)=−ln(y)−1,

v(y)=−ln(y)−1+c 1 ,

wherec 1 =−ln 2−^12 ln(q(1−q)), so that

u(x) = ln(x)+c 1.

For the optimal investment, we obtain

X̂ 1 (x)=

{

x

2 q, forω=g,

x

2(1−q),forω=b,

so thatX̂ 1 (x)=x+̂h∆S 1 , where the optimal trading strategŷhis given by

̂h=x^1 −^2 q

2 q ̃u

=x

̃u+d ̃

− 2 d ̃ ̃u

. (3.44)

Example 3.3.3.Using again the assumptions onSas in 3.3.1, but letting
nowU(x)=−exp(−x), we obtainV(y)=y(ln(y)−1) and

v(y)=V(y)−c 2 y, u(x)=−exp(−(x−c 2 ))

wherec 2 =−[qlnq+(1−q)ln(1−q) + ln 2].