3.3 The Binomial and the Trinomial Model 53the increments are assumed to be independent. To be formal, let (εt)Nt=1be
i.i.d. Bernoulli random variables defined on some (Ω,F,P)sothatP[εt=
1] =P[εt=−1] =^12 ,fort=1,···,N. The reason why we now use the letter
Ninstead of the previously usedTwill become apparent after (3.46) below.
We denote byFttheσ-algebra generated by (εn)tn=1.LetS 0 = 1 and, for
t=1,···,N,defineStinductively by
St={
St− 1 (1 +u ̃)ifεt=1,
St− 1 (1 +d ̃)ifεt=− 1.LettingU(x)=xα
α,forsomefixedα∈]−∞,1[{^0 }, we again want to
determine the optimal investment strategy and other related quantities.
Our aim is to maximise the expected utility of terminal wealthX̂N(x), i.e.
E
[
U
(
x+∑N
n=1hn∆Sn)]
→max!where (hn)Tn=1runs through all predictable processes.
To do so, we define, fort=0,···,N, the conditional value functions
ut(x)=sup{
E
[
U
(
x+∑N
n=t+1hn∆Sn)∣
∣
∣
∣∣Ft]}
where the supremum is taken over all collections (hn)Nn=t+1of (Fn− 1 )Nn=t+1-
measurable functions. In general ut(x) will depend on ω ∈ ΩinanFt-
measurable way; but in the present easy example, the i.i.d. assumption on
the returns
(
St
St− 1)N
t=1implies, thatut(x) does not depend onω∈Ω.
In fact, it is straightforward to calculateut(x) by backward induction on
t=N,···,0: fort=N, we obviously have
uN(x)=U(x),and fort=N−1 we are just in the situation of the one step model 3.3.1, so
that we find
uN− 1 (x)=cUU(x)
wherecU=
(
1
2(
(2q)β+(2(1−q))β)) 1 −α
, as we have computed in (3.35).
Let us take a closer look why this is indeed the case: the reader might
object, that in the present example the valueSN− 1 of the stock at timeN− 1
as well as the possible gain ̃uSN− 1 resp. lossdS ̃N− 1 , depend onω∈Ωinan
FN− 1 -measurable way, while in the one step example 3.3.1 we hadS 0 =1,
and the possible gains ̃uand lossesd ̃were also deterministic. But, of course,
this difference is only superficial: if̂h∈Rdenotes the optimal investment
in the stockS 0 in the one step example, we now have to choose the optimal