23.2. Employee Stock Options Jared Lazarus has just been named the new chief executive officer of BluBell Fitness Centers, Inc. In addition to an annual salary of $375,000, his three-year contract states that his compensation will include 15,000 at-the-money European call options on the company’s stock that expire in three years. The current stock price is $34 per share, and the standard deviation of the returns on the firm’s stock is 74 percent. The company does not pay a dividend. Treasury bills that mature in three years yield a continuously compounded interest rate of 5 percent. Assume that Mr. Lazarus’s annual salary payments occur at the end of the year and that these cash flows should be discounted at a rate of 9 percent. Using the Black–Scholes model to calculate the value of the stock options, determine the total value of the compensation package on the date the contract is signed.

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23.5. Real Options Jet Black is an international conglomerate with a petroleum division and is currently competing in an auction to win the right to drill for crude oil on a large piece of land in one year. The current market price of crude oil is $58 per barrel, and the land is believed to contain 375,000 barrels of oil. If found, the oil would cost $35 million to extract. Treasury bills that mature in one year yield a continuously compounded interest rate of 4 percent, and the standard deviation of the returns on the price of crude oil is 50 percent. Use the Black–Scholes model to calculate the maximum bid that the company should be willing to make at the auction.

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23.7. Real Options Wet for the Summer, Inc., manufactures filters for swimming pools. The company is deciding whether to implement a new technology in its pool filters. One year from now the company will know whether the new technology is accepted in the market. If the demand for the new filters is high, the present value of the cash flows in one year will be $13.4 million. Conversely, if the demand is low, the value of the cash flows in one year will be $7 million. The value of the project today under these assumptions is $11.6 million, and the risk-free rate is 6 percent. Suppose that in one year, if the demand for the new technology is low, the company can sell the technology for $8.2 million. What is the value of the option to abandon?

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23.8. Binomial Model There is a European put option on a stock that expires in two months. The stock price is $58, and the standard deviation of the stock returns is 70 percent. The option has a strike price of $65, and the risk-free interest rate is a 5 percent annual percentage rate. What is the price of the put option today using one-month steps? (Hint: How will you find the value of the option if it can be exercised early? When would you exercise the option early?)

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