138 Mathematics for Finance
can be suitably modified and the equality between the futures and forward
prices remains valid.
In an economy with constant interest ratesrwe obtain a simple structure
of futures prices,
f(t, T)=S(t)er(T−t) (6.11)
if the stock pays no dividends. The futures prices are random, but this is caused
entirely by the randomness of the prices of the underlying asset. If the futures
prices depart from the values given by the above formula, it is a reflection of
the market’s view of future interest rate changes.
Exercise 6.8
Suppose the interest rateris constant. GivenS(0), find the priceS(1)
of the stock after one day such that the marking to market of futures
with delivery in 3 months is zero on that day.This exercise shows an important benchmark for the profitability of a fu-
tures position: An investor who wants to take advantage of a predicted increase
in the price of stock above the risk-free rate should enter into a long futures
position. A short futures position will bring a profit should the stock price go
down or increase below the risk-free rate.
6.2.2 Hedging with Futures
The Basis.One relatively simple way to hedge an exposure to stock price
variations is to enter a forward contract. However, a contract of this kind may
not be readily available, not to mention the risk of default. Another possibility is
to hedge using the futures market, which is well-developed, liquid and protected
from the risk of default.
Example 6.2
LetS(0) = 100 dollars and let the risk-free rate be constant atr= 8%. Assume
that marking to market takes place once a month, the time step beingτ=1/12.
Suppose that we wish to sell the stock after 3 months. To hedge the exposure
to stock price variations we enter into one short futures contract on the stock
with delivery in 3 months. The payments resulting from marking to market are
invested (or borrowed), attracting interest at the risk-free rate. The results for
two different stock price scenarios are displayed below. The column labelled