Problems:
Determination of Forward and Futures Prices
Practice Questions
Problem 1
A one-year long forward contract on a non-dividend-paying stock is entered into when the stock price is $60 and the risk-free rate of interest is 5% 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 $65 and the risk-free interest rate is still 5%. What are the forward price and the value of the forward contract?
Problem 2.
Assume that the risk-free interest rate is 6% per annum with continuous compounding and that the dividend yield on a stock index varies throughout the year. In February, May, August, and November, dividends are paid at a rate of 6% per annum. In other months, dividends are paid at a rate of 2.5% per annum. Suppose that the value of the index on July 31 is 1,500. What is the futures price for a contract deliverable on December 31 of the same year?
Problem 3.
Suppose that the risk-free interest rate is 7% per annum with continuous compounding and that the dividend yield on a stock index is 3% per annum. The index is standing at 500, and the futures price for a contract deliverable in four months is 504. What arbitrage opportunities does this create?
Problem 4.
The two-month interest rates in Switzerland and the United States are, respectively, 2% and 3% per annum with continuous compounding. The spot price of the Swiss franc is $2.0500. The futures price for a contract deliverable in two months is also $2.0500. What arbitrage opportunities does this create?
Problem 5.
The spot price of silver is $27 per ounce. The storage costs are $0.24 per ounce per year payable quarterly in advance. Assuming that interest rates are 5.25% per annum for all maturities, calculate the futures price of silver for delivery in nine months.
Problem 6.
An index is 1,350. The three-month risk-free rate is 4% per annum and the dividend yield over the next three months is 2.2% per annum. The six-month risk-free rate is 4.5% per annum and the dividend yield over the next six months is 2% per annum. Estimate the futures price of the index for three-month and six-month contracts. All interest rates and dividend yields are continuously compounded.
Problem 7.
Suppose the current USD/euro exchange rate is 1.2230 dollar per euro. The six month forward exchange rate is 1.1850. The six month USD interest rate is 2% per annum continuously compounded. Estimate the six month euro interest rate.
Problem 8.
The spot price of oil is $78 per barrel and the cost of storing a barrel of oil for one year is $3, payable at the end of the year. The risk-free interest rate is 5% per annum, continuously compounded. What is an upper bound for the one-year futures price of oil?
Problem 9.
A stock is expected to pay a dividend of $2 per share in two months and in five months. The stock price is $60, and the risk-free rate of interest is 8% per annum with continuous compounding for all maturities. An investor has just taken a short position in a six-month forward contract on the stock.
a) What are the forward price and the initial value of the forward contract?
b) Three months later, the price of the stock is $58 and the risk-free rate of interest is still 8% per annum. What are the forward price and the value of the short position in the forward contract?
Problem 10.
A bank offers a corporate client a choice between borrowing cash at 11% per annum and borrowing gold at 2% per annum. (If gold is borrowed, interest must be repaid in gold. Thus, 200 ounces borrowed today would require 204 ounces to be repaid in one year.) The risk-free interest rate is 9.25% per annum, and storage costs are 0.5% per annum. Discuss whether the rate of interest on the gold loan is too high or too low in relation to the rate of interest on the cash loan. The interest rates on the two loans are expressed with annual compounding. The risk-free interest rate and storage costs are expressed with continuous compounding.