48 3 Utility Maximisation on Finite Probability Spaces
Coming back to the case of generalα∈]−∞,1[{ 0 },weobtainfrom
(3.37) that the optimal investment in the stock equalŝh=̂kxwhere the
constant̂k=̂k(u, d, α) given by (3.37) satisfies 0<̂k<∞. It may be seen
as a very elementary version of Merton’s result (see [M 90] and example 3.3.5
below), that for the Black-Scholes model and power utility, it is optimal to
always invest a constant proportion of your wealth, where the constant may
be calculated explicitly as a function of the parameters of the model and the
utility function. Observe that this constant may very well be bigger than one,
in which case one goes short in the bond.
We now specialise to the case that ̃u=σ∆t(^12)
+ν∆tandd ̃=−σ∆t
(^12)
+ν∆t
for some ∆t>0, which corresponds to the notation in the Black-Scholes model
below (Sect. 4.4). We determine the different quantities up to the relevant
termsofpowersof∆t:
q=
−d ̃
̃u−d ̃
=
σ∆t(^12)
−ν∆t
2 σ∆t
1
2
=
1
2
(
1 −
ν
σ∆t12 )
, (3.38)
1 −q=̃u
̃u−d ̃=
σ∆t(^12)
+ν∆t
2 σ∆t
12 =
1
2
(
1+
ν
σ∆t12 )
,
cV=1
2
(
(2q)β+(2(1−q))β)
(3.39)
=
1
2
(
1 −β
ν
σ∆t(^12)
- β(β−1)
2
ν^2
σ^2
∆t
+1 +β
ν
σ
∆t
(^12)
β(β−1)
2
ν^2
σ^2
∆t
)
=1+
β(β−1)ν^2
2 σ^2∆t+o(∆t),cU=c^1 V−α=(
1+
β(β−1)ν^2
2 σ^2∆t+o(∆t)) 1 −^1 β=1−
βν^2
2 σ^2∆t+o(∆t). (3.40)For the optimal investment̂hwe obtain
̂h=x[
c−V^1 (2q)1α− (^1) − 1
]
̃u−^1 (3.41)=x[(
1 −
β(β−1)ν^2
2 σ^2∆t)(
1 −
ν(β−1)
σ∆t(^12)
)
− 1
]
·σ−^1 (∆t)−(^12)
+O
(
∆t12 )
=
xν(1−β)
σ^2+O
(
∆t