Chapter 5: Q16-41I (page 557)
You build a binomial model with one period and assert that over the course of a year thestock price will either rise by a factor of 1.5 or fall by a factor of 2/3. What is your implicitassumption about the volatility of the stock’s rate of return over the next year?
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Show that Black-Scholes call option hedge ratios increase as the stock price increases. Consider a one-year option with exercise price \(50 on a stock with annual standard deviation 20%. The T-bill rate is 3% per year. Find N (d1) for stock prices \)45, \(50, and \)55.
a. Turn to Figure 17.1 and locate the contract on the Standard & Poor’s 500 Index. If the margin requirement is 10% of the futures price times the multiplier of $250, how much must you deposit with your broker to trade the September contract?
b. If the September futures price were to increase to 1,200, what percentage return would you earn on your net investment if you entered the long side of the contract at the price shown in the figure?
c. If the September futures price falls by 1%, what is the percentage gain or loss on your net investment?
Joseph Jones, a manager at Computer Science, Inc. (CSI), received 10,000 shares of company stock as part of his compensation package. The stock currently sells at \(40 a share. Joseph would like to defer selling the stock until the next tax year. In January, however, he will need to sell all his holdings to provide for a down payment on his new house. Joseph is worried about the price risk involved in keeping his shares. At current prices, he would receive \)40,000 for the stock. If the value of his stock holdings falls below \(35,000, his ability to come up with the necessary down payment would be jeopardized.
On the other hand, if the stock value rises to \)45,000, he would be able to maintain a small cash reserve even after making the down payment. Joseph considers three investment strategies:
a. Strategy A is to write January call options on the CSI shares with strike price \(45. These calls are currently selling for \)3 each.
b. Strategy B is to buy January put options on CSI with strike price \(35. These options also sell for \)3 each.
c. Strategy C is to establish a zero-cost collar by writing the January calls and buying the January puts.
Evaluate each of these strategies with respect to Joseph’s investment goals. What are the advantages and disadvantages of each? Which would you recommend?
Joe Finance has just purchased a stock-index fund, currently selling at \(1,200 per share. To protect against losses, Joe plans to purchase an at-the-money European put option on the fund for \)60, with exercise price \(1,200, and three-month time to expiration. Sally Calm, Joe’s financial adviser, points out that Joe is spending a lot of money on the put. She notes that three-month puts with strike prices of \)1,170 cost only $45, and suggests that Joe use the cheaper put.
a. Analyze Joe’s and Sally’s strategies by drawing the profit diagrams for the stock-plus put positions for various values of the stock fund in three months.
b. When does Sally’s strategy do better? When does it do worse?
c. Which strategy entails greater systematic risk?
Consider a stock that will pay a dividend of D dollars in one year, which is when a futures contract matures.
Consider the following strategy: Buy the stock, short a futures contract on the stock, and borrow dollars, where is the current price of the stock.
a. What are the cash flows now and in one year? (Hint: Remember the dividend the stock will pay.)
b. Show that the equilibrium futures price must beto avoid arbitrage.
c. Call the dividend yield d = D / , and conclude that .
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