52 3 Utility Maximisation on Finite Probability Spaces
Applying formula (3.29) from Theorem 3.2.1 we find, forω=n,thatdQ̂(y)
dP[n]=y−^1 U′(X̂ 1 (x)(n))=y−^1 U′(x)=(
1+
(1−m)βν^2
2 σ^2∆t+o(∆t))
x−(α−1)xα−^1=1+
(1−m)βν^2
2 σ^2∆t+o(∆t).As expected, the initial endowmentxcancels out so thatQ̂(y)[n]doesnot
depend onyand we shall therefore denote it byQ̂[n]. We find
Q̂[n]=dQ̂
dP[n]P[n]=m+m(1−m)βν^2
2 σ^2∆t+o(∆t)which gives us a rather complete information how this value depends on the
parameters of the model. In fact,Q̂[n] determines alreadyQ̂,alsoforω=g
andω =b. Indeed, the two relationsQ̂[g]+Q̂[n]+Q̂[b]=1aswellas
EQ̂[S 1 −S 0 ] = 0 yield
Q̂[g]=q(
1 −m−m(1−m)βν^2
2 σ^2∆t)
+o(∆t),Q̂[b]=(1−q)(
1 −m−
m(1−m)βν^2
2 σ^2∆t)
+o(∆t),whereqis given by (3.38), which gives
Q̂[g]=^1
2(
(1−m)(
1 −
ν
σ∆t12 )
−
m(1−m)βν
2 σ^2∆t)
+o(∆t),Q̂[b]=^1
2(
(1−m)(
1+
ν
σ∆t12 )
−
m(1−m)βν
2 σ^2∆t)
+o(∆t).Summing up, we have seen that also in the case of the one step trinomial
model all relevant quantities may be calculated explicitly. The fact that we
have chosen the parameterisation of the example such that, forω=n,the
stock price simply does not move, made the determination of the primal so-
lution particularly easy. However, one could also abandon this assumption, at
the cost of somewhat more cumbersome calculations.
Example 3.3.5 (The binomial model (Nperiods)).This model is called
the Cox-Ross-Rubinstein model [CRR 79] and it is extremely popular in fi-
nance, for numerical calculations as well as for pedagogical purposes. It is the
discrete version of the Black-Scholes model.
Using the notation of the one step model, example 3.3.1, above and fixing
N ≥1 we simply concatenate the one step model in a multiplicative way: