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Unformatted text preview: f the company takes on the riskier project, the company is less likely to be able to pay off its bondholders. Since the risk of default increases if the new project is undertaken, the value of the company’s debt decreases. Bondholders would prefer the company to undertake the more conservative project. 33. a. Going back to the chapter on dividends, the price of the stock will decline by the amount of the dividend (less any tax effects). Therefore, we would expect the price of the stock to drop when a dividend is paid, reducing the upside potential of the call by the amount of the dividend. The price of a call option will decrease when the dividend yield increases. b. Using the Black-Scholes model with dividends, we get: d1 = [ln($106/$100) + (.05 – .02 + .502/2) × .5] / (.50 × d2 = .3840 – (.50 × N(d1) = .6495 N(d2) = .5121 C = $106–(.02)(.5)(.6495) – ($100e–.05(.5))(.5121) = $18.21 .5 ) = .3840 .5 ) = .0305 462 34. a. Going back to the chapter on dividends, the price of the stock will decline by the amount of the dividend (less any tax effects). Therefore, we would expect the price of the stock to drop when a dividend is paid. The price of put option will increase when the dividend yield increases. b. Using put-call parity to find the price of the put option, we get: $106e–.02(.5) + P = $100e–.05(.5) + 18.21 P = $10.80 35. N(d1) is the probability that “z” is less than or equal to N(d1), so 1 – N(d1) is the probability that “z” is greater than N(d1). Because of the symmetry of the normal distribution, this is the same thing as the probability that “z” is less than N(–d1). So: N(d1) – 1 = –N(–d1). 36. From put-call parity: P = E × e-Rt + C – S Substituting the Black-Scholes call option formula for C and using the result in the previous question produces the put option formula: P P P P = E × e-Rt + C – S = E × e-Rt + S ×N(d1) – E × e-Rt ×N(d2) – S = S ×(N(d1) – 1) + E × e-Rt ×(1 – N(d2)) = E × e-Rt ×N(–d2) – S × N(–d1) 37. Based on Black-Scholes, the call option is worth $50! The reason is that present value of the exercise price is zero, so the second term disappears. Also, d1 is infinite, so N(d1) is equal to one. The problem is that the call option is European with an infinite expiration, so why would you pay anything for it since you can never exercise it? The paradox can be resolved by examining the price of the stock. Remember that the call option formula only applies to a non-dividend paying stock. If the stock will never pay a dividend, it (and a call option to buy it at any price) must be worthless. 38. The delta of the call option is N(d1) and the delta of the put option is N(d1) – 1. Since you are selling a put option, the delta of the portfolio is N(d1) – [N(d1) – 1]. This leaves the overall delta of your position as 1. This position will change dollar for dollar in value with the underlying asset. This position replicates the dollar “action” on the underlying asset. 463 CHAPTER 23 OPTIONS AND CORPORATE FINANCE: EXTENSIONS AND APPLICATIONS
Answers to Concepts Review and Critical Thinking Questions 1. One of the purposes to give stock options to CEOs (instead of cash) is to tie the performance of the firm’s stock with the compensation of the CEO. In this way, the CEO has an incentive to increase shareholder value. Most businesses have the option to abandon under bad conditions and the option to expand under good conditions. Virtually all projects have embedded options, which are ignored in NPV calculations and likely leads to undervaluation. As the volatility increases, the value of an option increases. As the volatility of coal and oil increases, the option to burn either increases. However, if the prices of coal and oil are highly correlated, the value of the option would decline. If coal and oil prices both increase at the same time, the option to switch becomes less valuable since the company will likely save less money. The advantage is that the value of the land may increase if you wait. Additionally, if you wait, the best use of the land other than sale may become more valuable. The company has an option to abandon the mine temporarily, which is an American put. If the option is exercised, which the company is doing by not operating the mine, it has an option to reopen the mine when it is profitable, which is an American call. Of course, if the company does reopen the mine, it has another option to abandon the mine again, which is an American put. Your colleague is correct, but the fact that an increased volatility increases the value of an option is an important part of option valuation. All else the same, a call option on a venture that has a higher volatility will be worth more since the upside potential is greater. Even though the downside is also greater, with an option, the downside is irrelevant since the option will not be exercised and will expire worthless no matter how low the asset falls. With a put option, the reverse is true in that the option becomes more valuable the further the asset falls, and if the asset increases in value, the option is allowed to expire. Real option analysis is not a technique that can be applied in isolation. The value of the asset in real option analysis is calculated...
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