Homework #6 Suggested Solutions

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1 JEM034 Corporate Finance Winter Semester 2017/2018 Instructor: Olga Bychkova Homework #6 Suggested Solutions Problem 1. (22) Buffelhead s stock price is $220 and could halve or double in each six month period (equivalent to a standard deviation of 98%). A one year American call option on Buffelhead has an exercise price of $165. The interest rate is 20% a year. (a) What is the value of the Buffelhead call? (b) Now calculate the option delta for the second six months if (i) the stock price rises to $440 and (ii) the stock price falls to $110. (c) How does the call option delta vary with the level of the stock price? Explain intuitively why. (Hint: Disscuss the cases when option delta is equal to one or zero.) (a) The possible prices of Buffelhead stock and the associated call option values (shown in parentheses) are: Let p equal the probability of a rise in the stock price. Then, if investors are risk neutral: p (1 p) ( 50) = 10 p = 0.4. If the stock price in month 6 is $110, then the option will not be exercised so that it will be worth: 0.4 $ $0 = $20. Similarly, if the stock price is $440 in month 6, then, if it is exercised, it will be worth $440 $165 = $275. If the option is not exercised, it will be worth: 0.4 $ $55 = $290. Therefore, the call option will not be exercised, so that its value today is: 0.4 $ $20 = $ (b) Delta = spread of option prices spread of stock prices. 1

2 (i) If the price rises to $440: Delta = = 1. (ii) If the price falls to $110: Delta = = (c) The option delta is 1 when the call is certain to be exercised and is zero when it is certain not to be exercised. If the call is certain to be exercised, it is equivalent to buying the stock with a partly deferred payment. So a one dollar change in the stock price must be matched by a one dollar change in the option price. At the other extreme, when the call is certain not to be exercised, it is valueless, regardless of the change in the stock price. Problem 2. (23) Suppose that you own an American put option on Buffelhead stock (see the previous problem) with an exercise price of $220. (a) Calculate the value of the put. (b) Now compare the value with that of an equivalent European put option. (a) The possible prices of Buffelhead stock and the associated American put option values (shown in parentheses) are: Let p equal the probability of a rise in the stock price. Then, if investors are risk neutral: p (1 p) ( 50) = 10 p = 0.4. If the stock price in month 6 is $110, and if the American put option is not exercised, it will be worth: 0.4 $ $165 = $90. On the other hand, if it is exercised after 6 months, it is worth $110. Thus, the investor should exercise the put early. Similarly, if the stock price in month 6 is $440, and if the American put option is not exercised, it will be worth $0. On the other hand, if it is exercised after 6 months, it will cost the investor $220. The investor should not exercise early. Finally, the value today of the American put option is: 0.4 $ $110 = $60. 2

3 (b) Unlike the American put in part (a), the European put can not be exercised prior to expiration. We noted in part (a) that, if the stock price in month 6 is $110, the American put would be exercised because its value if exercised (i.e., $110) is greater than its value if not exercised (i.e., $90). For the European put, however, the value at that point is $90 because the European put can not be exercised early. Therefore, the value of the European put is: 0.4 $ $90 = $ Problem 3. (24) Recalculate the value of the Buffelhead call option (see problem 1), assuming that the option is American and that at the end of the first six months the company pays a dividend of $25. (Thus the price at the end of the year is either double or half the ex dividend price in month 6.) How would your answer change if the option were European? The following tree shows stock prices, with option values in parentheses: The option values in month 6, if the option is not exercised, are computed as follows: 0.4 $ $0 = $1.82, 0.4 $ $42.5 = $265. If the stock price in month 6 is $110, then it would not pay to exercise the option. If the stock price in month 6 is $440, then the call is worth: $440 $165 = $275. Therefore, the option would be exercised at that time. Working back to month 0, we find the option value as follows: Option value = 0.4 $ $1.82 = $ If the option were European, it would not be possible to exercise early. Therefore, if the price rises to $440 at month 6, the value of the option is $265, not $275 as in the case for the American option. Therefore, in this case, the value of the European option is less than the value of the American option. The value of the European option is computed as follows: 0.4 $ $1.82 Option value = = $

4 Problem 4. (25) Suppose that you have an American option that allows you to sell Buffelhead stock (see problem 1) in month 6 for $165 (put option) or to buy it in month 12 for $165 (call option). What is the value of this unusual option? The following tree (see problem 10) shows stock prices, with the values for the option in parentheses: The put option is worth $55 in month 6 if the stock price falls and $0 if the stock price rises. Thus, with a 6 month stock price of $110, it pays to exercise the put (value = $55). With a price in month 6 of $440, the investor would not exercise the put since it would cost $275 to exercise. The value of the option in month 6, if it is not exercised, is determined as follows: 0.4 $ $55 = $290. Therefore, the month 0 value of the option is: Option value = 0.4 $ $55 = $ Problem 5. (21.20) Other things equal, which of these American options are you most likely to want to exercise early? (a) A put option on a stock with a large dividend or a call on the same stock. (b) A put option on a stock that is selling below exercise price or a call on the same stock. (c) A put option when the interest rate is high or the same put option when the interest rate is low. Illustrate your answer with examples. (a) The call option. (You would delay the exercise of the put until after the dividend has been paid and the stock price has dropped.) (b) The put option. (You never exercise a call if the stock price is below exercise price.) (c) The put when the interest rate is high. (You can invest the exercise price.) Problem 6. (21.21) Is it better to exercise a call option on the with dividend date or on the ex dividend date? How about a put option? Explain. When you exercise a call, you purchase the stock for the exercise price. Naturally, you want to maximize what you receive for this price, and so you would exercise on 4

5 the with dividend date in order to capture the dividend. When you exercise a put, your gain is the difference between the price of the stock and the amount you receive upon exercise, i.e., the exercise price. Therefore, in order to maximize your profit, you want to minimize the price of the stock and so you would exercise on the ex dividend date. Problem 7. (22.4) You own a parcel of vacant land. You can develop it now, or wait. (a) What is the advantage of waiting? (b) Why might you decide to develop the property immediately? (a) You learn more about land prices and best use of the land. (b) By developing immediately, you capture rents immediately. Problem 8. (22.13) You have an option to purchase all of the assets of the Overland Railroad for $2.5 billion. The option expires in nine months. You estimate Overland s current (month 0) present value (PV) as $2.7 billion. Overland generates after tax free cash flow (FCF) of $50 million at the end of each quarter (i.e., at the end of each three month period). If you exercise your option at the start of the quarter, that quarter s cash flow is paid out to you. If you do not exercise, the cash flow goes to Overland s current owners. In each quarter, Overland s PV either increases by 10% or decreases by 9.09%. This PV includes the quarterly FCF of $50 million. After the $50 million is paid out, PV drops by $50 million. Thus the binomial tree for the first quarter is (figures in millions): The risk free interest rate is 2% per quarter. (a) Build a binomial tree for Overland, with one up or down change for each three month period (three steps to cover your nine month option). (b) Suppose you can only exercise your option now, or after nine months (not at month 3 or 6). Would you exercise now? (c) Suppose you can exercise now, or at month 3, 6, or 9. What is your option worth today? Should you exercise today, or wait? 5

6 (a) The values in the binomial tree below are the ex dividend values, with the option values shown in parentheses. (b) The option values in the binomial tree above are computed using the risk neutral method. Let p equal the probability of a rise in asset value. Then, if investors are risk neutral: p 10 + (1 p) ( 9.09) = 2 p = If, for example, asset value at month 6 is $3,162 (this is the value after the $50 cash flow is paid to the current owners), then the option value will be: $ $ = $711. Similarly, we can compute the option values at other tree nodes. Therefore, you should not exercise the option now because its value if not exercised ($327) is greater than its value if exercised ($200). (c) If you exercise the option early, it is worth the with dividend value less $2,500. For example, if you exercise in month 3 when the with dividend value is $2,970, the option would be worth: $2, 970 $2, 500 = $470. Since the option is worth $491 if not exercised, you are better off keeping the option open. At each point before month 9, the option is worth more unexercised than exercised. There is one minor exception to this conclusion: if the option is exercised at month 6 when asset value is $3,212, then the option value is: $3, 212 $2, 500 = $712, which is greater than the option value if it s not exercised ($711). Therefore, the option value at this node is $712. Due to rounding, this difference does not affect any option value computations at other nodes. Therefore, you should wait rather than exercise today. The value of the option today is $327, as shown in the binomial tree above. 6