Determination of Forward and Futures Prices: Practice Questions
Determination of Forward and Futures Prices: Practice Questions
Determination of Forward and Futures Prices: Practice Questions
Practice Questions
Problem 5.8.
Is the futures price of a stock index greater than or less than the expected future value of the
index? Explain your answer.
The futures price of a stock index is always less than the expected future value of the index. This
follows from Section 5.14 and the fact that the index has positive systematic risk. For an
alternative argument, let µ be the expected return required by investors on the index so that
E ( ST ) = S 0e ( µ −q )T . Because µ > r and F0 = S0e ( r −q )T , it follows that E ( ST ) > F0 .
Problem 5.9.
A one-year long forward contract on a non-dividend-paying stock is entered into when the stock
price is $40 and the risk-free rate of interest is 10% per annum with continuous compounding.
a) What are the forward price and the initial value of the forward contract?
b) Six months later, the price of the stock is $45 and the risk-free interest rate is still 10%. What
are the forward price and the value of the forward contract?
b) The delivery price K in the contract is $44.21. The value of the contract, f , after six
months is given by equation (5.5) as:
f = 45 − 44.21e−0.1×0.5
= 2 .95
i.e., it is $2.95. The forward price is:
45e0.1×0.5 = 47.31
or $47.31.
Problem 5.10.
The risk-free rate of interest is 7% per annum with continuous compounding, and the dividend
yield on a stock index is 3.2% per annum. The current value of the index is 150. What is the six-
month futures price?
The futures contract lasts for five months. The dividend yield is 2% for three of the months and
5% for two of the months. The average dividend yield is therefore
1
(3 × 2 + 2 × 5) = 3. 2 %
5
The futures price is therefore
1300e(0.09− 0.032)×0.4167 = 1, 331.80
or $1331.80.
Problem 5.12.
Suppose that the risk-free interest rate is 10% per annum with continuous compounding and that
the dividend yield on a stock index is 4% per annum. The index is standing at 400, and the
futures price for a contract deliverable in four months is 405. What arbitrage opportunities does
this create?
Problem 5.13.
Estimate the difference between short-term interest rates in Japan and the United States on
August 4, 2009 from the information in Table 5.4.
Problem 5.14.
The two-month interest rates in Switzerland and the United States are 2% and 5% per annum,
respectively, with continuous compounding. The spot price of the Swiss franc is $0.8000. The
futures price for a contract deliverable in two months is $0.8100. What arbitrage opportunities
does this create?