56 Mathematics for Finance
Condition 3.1
The one-step returnsK(n) on stock are identically distributed independent
random variables such that
K(n)={
u with probabilityp,
d with probability 1−p,at each time stepn,where− 1 <d<uand 0<p<1.
This condition implies that the stock priceS(n) can move up or down by a
factor 1 +uor 1 +dat each time step. The inequalities− 1 <d<uguarantee
that all pricesS(n) will be positive ifS(0) is.
Letrbe the return on a risk-free investment over a single time step of
lengthτ.
Condition 3.2
The one-step returnron a risk-free investment is the same at each time step
and
d<r<u.
The last condition describes the movements of stock prices in relation to
risk-free assets such as bonds or cash held in a bank account. The inequalities
d<r<uare justified because of Proposition 1.1 in Chapter 1 (which will be
generalised in Proposition 4.2).
SinceS(1)/S(0) = 1+K(1),Condition 3.1 implies that the random variable
S(1) can take two different values,
S(1) =
{
S(0)(1 +u) with probabilityp,
S(0)(1 +d) with probability 1−p.Exercise 3.12
How many different values do the random variablesS(2) andS(3) take?
What are these values and the corresponding probabilities?The values ofS(n) along with the corresponding probabilities can be found
for anynby extending the solution to Exercise 3.12. In ann-steptreeofstock
prices each scenario (or path through the tree) with exactlyiupward andn−i
downward price movements produces the same stock priceS(0)(1 +u)i(1 +