The Mathematics of Arbitrage

3.3 The Binomial and the Trinomial Model 55

Let us have a closer look at the constant̂k: the leading term (1−σ 2 β)ν is
proportional toνand inversely proportional toσ^2 , which is economically
intuitive. As regards (1−β)=(1−α)−^1 we observe that this quantity becomes
arbitrary large whenαtends to 1. In particular, forα<1 sufficiently close
to 1, the proportion̂kof the wealth held in the stock is bigger than one, and
therefore the position in the bond is negative (“short”).

Regarding the constantcNU ≈exp

(

−βν

(^2) T
2 σ^2

)

we observe that the right hand

side is bigger than one forβ∈]−∞,0[, which corresponds toα∈]0,1[. This
makes sense, as in this caseU(x)=x
α
α takes positive values, so that the
value functionu(x) defined in (3.46) increases withN.Ifβ ∈]0,1[, which

corresponds toα∈]−∞,0[, then exp

(

−βν

(^2) T
2 σ^2

)

is less than 1. This too does

make sense economically: forα∈]−∞,0[ the utility functionU(x)=x

α
α takes
negative values so that again we find that the utility functionu(x) in (3.46)
increases withN.
Looking at the constants appearing in exp

(

−βν

(^2) T
2 σ^2

)

, the roles ofν^2 ,σ^2

andTare quite intuitive. Somewhat more puzzling is the role ofβ,orrather

−β: while it is intuitive that forα →1 the factor exp

(

−βν

(^2) T
2 σ^2

)

tends to

infinity

(

β=αα− 1

)

so that the problem degenerates forα→1, the behaviour

is less intuitive forα→0: in this caseβ→0toosothatexp

(

−βν

(^2) T
2 σ^2

)

tends

to 1, i.e., in the limit there seems no difference betweenU(x)=x

α
α and the
corresponding value functionu(x). On the other hand limα→ 0 x

α− 1
α =ln(x)
so that — after proper normalisation — the utility maximisation problem
remains meaningful also asαtends to 0. The point is, that one has to be
careful about these renormalisations in the limit, which involves also a term
of orderα−^1.
A good way of dealing with these issues is to recall that the multiplicative
termcNU in (3.46) pertains to theutility scaleof the investor. We shall trans-
form it to thewealth scaleof the investor. We follow B. de Finetti’s idea of
“certainty equivalent”: consider the following two possibilities for an investor.
Either she holds an initial endowmentx>0 and is allowed to invest in stock
andbondintheabovemodeluptotimeT=N∆t; or she holds an initial en-
dowmentwUNx, where the letterwstands for “wealth”, and is only allowed to
invest into the bond (where its value simply remains constant) up to timeT.
If her goal is to maximise expected utility at timeT, what is the value of the
constantwNU for which the agent is indifferent between these two possibilities.
Using (3.46) this leads to the equation

cNUU(x)=U

(

wNUx

)

so thatxcancels out (as expected) and we obtain

(

cNU

)α^1

=wNU