The Mathematics of Arbitrage

3.3 The Binomial and the Trinomial Model 51

in the case of logarithmic or exponential utility) for the binomial model is also
optimal in the present example. Indeed, distinguish the two cases of “low” and
“high” volatility: conditioning on the event that volatility is high, we are in the
situation of the binomial model, so that the trading strategŷh,ascalculated
in 3.3.1, is the unique optimiser. On the other hand, if volatility is low, the
present example is designed such that the volatility vanishes, i.e., the stock
price does not move. Hence in this case the choice of the investmenthdoes not
influence the result ash(S 1 −S 2 ) is zero anyhow. Summing up, we conclude

that the optimal strategŷhobtained for the binomial model is optimal in
the present trinomial model as well. Thus we conclude that̂hgiven by (3.37)
defines the optimal investment also in the present situation

̂h=x

[

c−V^1 (2q)β−^1 − 1

]

̃u−^1 ,

which yields

̂h=x(1−β)ν

σ^2

+O

(

∆t

12 )

if again, we let ̃u=σ∆t

(^12)
+ν∆t,d ̃=−σ∆t
(^12)
+ν∆t,andifcV is defined as
in (3.39).
For the optimal portfolioX̂ 1 (x) we find, similarly as in (3.43)
X̂ 1 (x)=

⎧

⎪⎪

⎨

⎪⎪

⎩

xc−V^1 (2q)β−^1 =x

(

1+ν(1σ−β)∆t

12 )

+O(∆t), forω=g,

x, forω=n

xc−V^1 (2(1−q))β−^1 =x

(

1 −ν(1σ−β)∆t

12 )

+O(∆t),forω=b.

Denoting byutri(x)=suph∈RE[U(x+h(S 1 −S 2 ))] the value function for
the present trinomial model, we still haveutri(x)=ctriUU(x), for some constant
ctriU >0, by the scaling property ofU(x)=x

α
α. The explicit form is given by
ctriU −1=(1−m)

(

cbiU− 1

)

,wherecbiU is given by the constantcU in the
binomial case above (3.34). Indeed, this relation between the constants of the
binomial and trinomial model simply follows from the fact that, conditionally
on the event{ω=n}, which happens with probability (1−m), the trinomial
model coincides with the binomial one.
ExpandingctriU in terms of ∆tyields

ctriU =1−

(1−m)βν^2

2 σ^2

∆t+o(∆t).

We now may calculate the dual measureQ̂(y) via formula (3.29). Fix the
initial endowmentx.Then

utri(x)=ctriUU(x)

so that

y:=

(

utri

)′

(x)=

(

1 −

(1−m)βν^2

2 σ^2

∆t+o(∆t)

)

xα−^1