52 Mathematics for Finance
Exercise 3.6
In each of the following three scenarios find the one-step returns, assum-
ing thatK(1) =K(2):
Scenario S(0) S(2)
ω 1 35 41
ω 2 35 32
ω 3 35 28
Exercise 3.7
Given thatK(1) = 10% or−10%, andK(0,2) = 21%, 10% or−1%,
find a possible structure of scenarios such thatK(2) takes at most two
different values.
The lack of additivity is often an inconvenience. This can be rectified by
introducing the logarithmic return on a risky security, motivated by similar
considerations for risk-free assets in Chapter 2.
Definition 3.2
Thelogarithmic returnover a time interval [n, m] (more precisely, [τn,τm]) is
a random variablek(n, m) defined by
k(n, m)=ln
S(m)
S(n).
The one-step logarithmic return will be denoted simply byk(n), that is,
k(n)=k(n− 1 ,n)=ln
S(n)
S(n−1)
,
so that
S(n)=S(n−1)ek(n). (3.2)
The relationship between the returnK(m, n) and the logarithmic return
k(m, n) is obvious by comparing their definitions, namely
1+K(m, n)=ek(m,n).
Because of this we can readily switch from one return to the other.