- Risky Assets 49
Figure 3.1 Tree of price movements in Example 3.2Exercise 3.1
Sketch a tree representing the scenarios and price movements in Exam-
ple 3.1.Exercise 3.2
Suppose that the stock price on any given day can either be 5% higher or
4% lower than on the previous day. Sketch a tree representing possible
stock price movements over the next three days, given that the price
today is $20. How many different scenarios can be distinguished?3.1.1 Return............................................
It proves convenient to describe the dynamics of stock pricesS(n)intermsof
returns. We assume that the stock pays no dividends.
Definition 3.1
Therate of return,orbrieflythereturnK(n, m) over a time interval [n, m](in
fact [mτ, nτ]), is defined to be the random variable
K(n, m)=S(m)−S(n)
S(n).
The return over a single time step [n− 1 ,n] will be denoted byK(n), that is
K(n)=K(n− 1 ,n)=S(n)−S(n−1)
S(n−1),
which implies that
S(n)=S(n−1)(1 +K(n)). (3.1)