CHAPTER 20 Spotting and Valuing Options

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1 CHAPTER 20 Spotting and Valuing Options Answers to Practice Questions The six-month call option is more valuable than the six month put option since the upside potential over time is greater than the limited downside potential. a. The put places a floor on value of investment, i.e., less risky than buying stock. b. Benefit from upside, but also lose on the downside. c. Naked option positions are riskier than the underlying asset. Option valuation models equate buy a call with a levered purchase of stock. d. Investor exchanges uncertain upside changes in stock price for certain option premiums. Less risky than purchase of the stock. e. Safe investment. f. From put-call parity, this is equivalent (for European options) to buy bond. Therefore, this is a safe investment. g. Another naked, high-risk position. a. The payoffs from the two options, at expiration will be: 1

2 Option value Share price Taking into account the money that must be repaid at expiration, the net position will be: Option value Share price 2

3 b. Using the relationship that: we find: Value of call + PV (EX) = Value of put + Share price Value of put (EX = 150) - Value of put (EX = 50) - PV (150-50) is equal to: Value of call (EX = 150) - Value of call (EX = 50) Thus, one combination that gives the same set of payoffs is: Buy a call with an exercise price of $150 Sell a call with an exercise price of $50 Similarly, another combination with the same set of payoffs is: Buy a put with an exercise price of $150 Buy a share of stock Borrow the present value of $150 Sell a call with an exercise price of $50 Statement (b) is the only correct one. The appropriate diagram is in Figure 20-2 in the text. We make use of the conversion formula: Value of call + PV (EX) = Value of put + Share price a. - Share price = Value of put - Value of call - PV (EX) This implies that to replicate a short sale, you would purchase a put, sell a call, and borrow the present value of the exercise price. b. - PV (EX) = Value of call - Value of put - Share price This implies that to replicate a bond, you would sell a call, buy a put, and buy the stock. 3

4 a. Use the fundamental relationship for European options: Value of call + PV of exercise price = Value of put + Share price Solve for the value of the put: Value of put = Value of call + PV of exercise option - Share price Thus, to replicate a put, you would buy a call with an exercise price of $100 and the same maturity, invest the present value of the exercise price in a 26-week risk-free security, and sell the stock short. b. Using the above relationship, we know that the European put will sell for: 8 + (100 / 1.05) - 90 = $13.24 (Note that an American put will sell for slightly more.) a. Straddle Payoff 100 Payoff to put Payoff to call 100 Share price 4

5 Butterfly Payoff Buy call, EX = Payoff to call, EX = 120; buy two Share price Payoff to sell call, EX = 110; sell two The buyer of the straddle profits if the stock price moves substantially in either direction; hence, the straddle is a bet on high variability. The buyer of the butterfly profits if the stock price doesn t move very much, and hence, this is a bet on low variability. Imagine two stocks, each price at $100. You have an at-the-money option to buy each at $100. Stock A s price now falls to $50 and Stock B s rises to $150. The value of your portfolio is now: Value Call on A 0 Call on B 50 Total $50 Now compare this with the value of an at-the-money call to buy a portfolio with equal holdings of A and B. Since the average change in the prices of the two stocks is zero, the call expires worthless. This is an example of a general rule: An option on a portfolio is less valuable than a portfolio of options on the individual stocks because in the later case, you can choose which options to exercise. 5

6 CHAPTER 21 Real Options Answers to Practice Questions a. A five-year American call option on oil. The initial exercise price is $32 a barrel, but the exercise price rises by 5 percent per year. b. An American put option to abandon the restaurant at an exercise price of $5 million. The restaurant s current value is $700,000 / r. The annual standard deviation of the changes in the value of the restaurant as a going concern is 15 percent. c. A put option as in (b), except that the exercise price should be interpreted as $5 million in real estate value plus the present value of the future fixed costs avoided by closing down the restaurant. Thus, the exercise price is 5,000,000 + (300,000 / 1.0) = $8,000,000. Note: The underlying asset is now PV (revenue variable cost), with annual standard deviation of 10.5 percent. d. The phrase if technical and economic conditions permit and the suggestion that supply can be expanded without affecting price are somewhat obscure. However, Eurotunnel has an option to make a followon investment, which expires in the year At that point, other companies may be permitted to build the drive-through link, and their costs will determine the prices that could be charged by Eurotunnel. S tan dard deviation time =.15 PV Asset value (Exercise price) = / 1.12 =.952 From Appendix Table 6, the call value is approximately:. 039 (1.7) =.0663 million, or $66,300 Note: The 20 percent past appreciation rate is not relevant. 6

7 As in Question 4, but the asset value for the Black-Scholes formula is reduced by: PV (rents) =.15 / 1.12 =.134 million. to give an asset value of: = $1.57 million. Thus: S tandard deviation time =.15 PV Asset value (Exercise price) = / 1.12 =.879 From Appendix Table 6, the call value is approximately:. 017 (1.57) =.0270 million, or $27,000 a. The possible prices of Buffelhead stock and the associated call options values (shown in parentheses) are: 220 (?) 110 (?) 440 (?) 55 (0) 220 (55) 880 (715) Let p equal the pseudo-probability of a rise in the stock price. Then, if investors are risk-neutral, p (1.00) + (1 - p) (-.50) =.10 p =.4 If the stock price after 6 months is $110, the call option will be worth: [.4 (55) +.6 (0) ] / 1.1 = $20 Similarly, if the stock price is $440 after 6 months, the call option will be worth: [.4 (715) +.6 (55) ] / 1.1 = $290 Thus, the value today of the call option is: 7

8 [.4 (290) +.6 (20) ] / 1.1 = $ b. If the price rises to $440: Delta = = If the price falls to $110: 55 0 Delta = = c. The option delta is 1.0 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. d. If the stock price is $110 at 6 months, the option delta is one-third; therefore, in order to replicate the stock, we buy three calls and lend: Initial Stock Stock Outlay Price = 55 Price = 220 Buy 3 calls Lend PV (55) Is equivalent to: Buy stock a. The following tree shows stock prices with option values in parentheses: With dividend Ex-dividend (1.82) 220 (117.7) (291.5) 42.5 (0) 170 (5) (42.5) 830 (665 We calculate option values as follows: 8

9 1. Option values after 6 months if not exercised: and (.4 5) + (.6 0) = (.4 665) + ( ) = If the stock price at month 6 is $110, it clearly would not pay to exercise the option. Similarly, if the stock price is $440 in month 6, it does not pay to exercise the call immediately (in this case, the call is worth = 275). 2. Working back to month 0 gives: ( ) + ( ) Option value = = b. If the option were European, it would not be possible to exercise early to get the dividend. However, because here we choose not to exercise early, in this case, there is no difference between the values of an American and a European option. The following tree (see Problem 10) shows stock prices with a 1-year call option values in parentheses: (20) 440 (290) 55 (0) 220 (55) 880 (715) 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), and with a price of $440, it is better to hold on to the call (value = $290). Working back to month 0 gives: (4. 290) + (.6 55) Option value = = $

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