The Mathematics of Arbitrage

50 3 Utility Maximisation on Finite Probability Spaces

For the optimal investment, we obtain

X̂ 1 (x)=

{

x+̂hu, ̃forω=g,

x+̂hd, ̃forω=b,

where the optimal trading strategŷhis given by

̂h=−c^2 +ln(2q)

u ̃

=

1

u ̃−d ̃

ln

(

u ̃

−d ̃

)

. (3.45)

Note that in this casêhdoes not depend on the initial endowmentx.

Example 3.3.4 (The trinomial model (one-period)).We now analyze
the simplest model of an incomplete market where we add to the possibilities
“good” and “bad” a third possibility “neutral”. The probability space Ω now
consists of three points, Ω ={g, n, b}whereP[n]=m,P[g]=P[b]=^1 − 2 m
andm∈]0,1[ is a parameter still kept free. We defineS(already in discounted
terms and dropping the notation for the bond) byS 0 =1and

S 1 =

⎧

⎨

⎩

1+u, ̃ifω=g,

1 , ifω=n,

1+d, ̃ifω=b,

where 1 + ̃u> 1 >1+d> ̃ 0, similarly as above. Again we assume ̃u≥−d ̃.
This model may be viewed as an embryonic version of a stochastic volatil-
ity model: the determination of the value of the random variableS 1 can be
interpreted as the result of two consecutive steps (taking place, however, at
the same timet= 1). First one performs an experiment describing an event
which happens or not with probabilitymand 1−mrespectively. According
to the outcome of this event the volatility is “low” or “high”. If it is “low”
— in the present embryonic example simply zero — the stock price does not
change; if it is “high”, a fair coin is tossed similarly as in the binomial model
to determine whether the stock price moves to 1 +u ̃or 1 +d ̃.
We now again consider power, logarithmic and exponential utility and
want to apply theorem 3.2.1 to the present situation. One way to solve the
portfolio optimisation problem is to proceed similarly as in the complete case
above: first solve the dual problem and then derive the primal problem by
using (3.29). This is possible but — in contrast to the complete situation
— we now would have to solve an optimisation problem to obtain the dual
solution.
In our specific example, it will be more convenient to solve the primal
problem directly and then to deduce the solution of the dual problem via
(3.29). The solutionQ̂(y) of the dual problem then allows for an interpretation
as a pricing functional (see Remark 3.2.2).

Here is the crucial observation to reduce the present example to the case of
3.3.1 above: the optimal strategŷhobtained in (3.37) (resp. (3.44) and (3.45)