108156.pdf

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.