- Risky Assets 55
Exercise 3.11
Suppose that the time step is taken to be three months,τ=1/4, and
the quarterly returnsK(1),K(2),K(3),K(4) are independent and iden-
tically distributed. Find the expected quarterly returnE(K(1)) and the
expected annual returnE(K(0,4)) if the expected returnE(K(0,3))
over three quarters is 12%.Remark 3.4
In the case of logarithmic returns additivity extends to expected returns, even
if the one-step returns are not independent. Namely
E(k(n, m)) =E(k(n+1))+E(k(n+2))+···+E(k(m)).This is because the expectation of a sum of random variables is the sum of
expectations.
Remark 3.5
In practice it is difficult to estimate the probabilities and returns in each sce-
nario, needed to compute the expected return. What can readily be computed
is the average return over a past period. The result can be used as an estimate
for the expected future return. For example, if the stock prices on the last
10 consecutive days were $98,$100,$99,$95,$88,$82,$89,$98,$101,$105, then
the average of the resulting nine daily returns would be about 0.77%. However,
the average of the last four daily returns would be about 6.18%. (We use log-
arithmic returns because of their additivity.) This shows that the result may
depend heavily on the choice of data. Using historical prices for prediction is a
complex statistical issue belonging toEconometrics, which is beyond the scope
of this book.
3.2 Binomial Tree Model......................................
We shall discuss an extremely important model of stock prices. On the one
hand, the model is easily tractable mathematically because it involves a small
number of parameters and assumes an identical simple structure at each node
of the tree of stock prices. On the other hand, it captures surprisingly many
features of real-world markets.
The model is defined by the following conditions.