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 ) = S0 e ( µ − q )T . Because µ > r and F0 = S0 e( 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? a) The forward price, F0 , is given by equation (5.1) as:
F0 = 40e0.1×1 = 44.21 or $44.21. The initial value of the forward contract is zero. 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? Using equation (5.3) the six month futures price is 150e(0.07−0.032)×0.5 = 152.88 or $152.88.
Problem 5.11. Assume that the risk-free interest rate is 9% 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 5% per annum. In other months, dividends are paid at a rate of 2% per annum. Suppose that the value of the index on July 31 is 1,300. What is the futures price for a contract deliverable on December 31 of the same year? 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? The theoretical futures price is
400e(0.10−0.04)×4/12 = 408.08 The actual futures price is only 405. This shows that the index futures price is too low relative to the index. The correct arbitrage strategy is 1. Buy futures contracts 2. Short the shares underlying the index. 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. The settlement prices for the futures contracts are to Sept: 1.0502 Dec: 1.0512 The December 2009 price is about 0.0952% above the September 2009 price. This suggests that the short-term interest rate in the United States exceeded short-term interest rate in the United Japan by about 0.0952% per three months or about 0.38% per year. 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? The theoretical futures price is
0.8000e(0.05−0.02)×2/12 = 0.8040 The actual futures price is too high. This suggests that...