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CHAPTER 22
OPTIONS AND CORPORATE FINANCE
Answers to Concept Questions
1. A call option confers the right, without the obligation, to buy an asset at a given price on or before a
given date. A put option confers the right, without the obligation, to sell an asset at a given price on
or before a given date. You would buy a call option if you expect the price of the asset to increase.
2. a. The buyer of a call option pays money for the right to buy....
b. The buyer of a put option pays money for the right to sell....
3. An American option can be exercised on any date up to and including the expiration date. A
European option can only be exercised on the expiration date. Since an American option gives its
4. The intrinsic value of a call is Max[S – E, 0]. The intrinsic value of a put is Max[E – S, 0]. The
5. The call is selling for less than its intrinsic value; an arbitrage opportunity exists. Buy the call for
6. The prices of both the call and the put option should increase. The higher level of downside risk still
8. The call option will sell for more since it provides an unlimited profit opportunity, while the
10. The reason they don’t show up is that the U.S. government uses cash accounting; i.e., only actual
cash inflows and outflows are counted, not contingent cash flows. From a political perspective, they
11. Increasing the time to expiration increases the value of an option. The reason is that the option gives
the holder the right to buy or sell. The longer the holder has that right, the more time there is for the
12. An increase in volatility acts to increase both call and put values because the greater volatility
13. A put option is insurance since it guarantees the policyholder will be able to sell the asset for a
14. The equityholders of a firm financed partially with debt can be thought of as holding a call option on
the assets of the firm with a strike price equal to the debt’s face value and a time to expiration equal
to the debt’s time to maturity. If the value of the firm exceeds the face value of the debt when it
Let VL = the value of a firm financed with both debt and equity
FV(debt) = the face value of the firm’s outstanding debt at maturity
If VL < FV(debt) If VL > FV(debt)
Notice that the payoff to equityholders is identical to a call option of the form Max(0, ST – K), where
15. Since you have a large number of stock options in the company, you have an incentive to accept the
16. Rearranging the put–call parity formula, we get: S – PV(E) = C – P. Since we know that the stock
17. Rearranging the put–call parity formula, we get: S – PV(E) = C – P. If the call and the put have the
18. A stock can be replicated using a long call (to capture the upside gains), a short put (to reflect the
Solutions to Questions and Problems
NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple
steps. Due to space and readability constraints, when these intermediate steps are included in this
solutions manual, rounding may appear to have occurred. However, the final answer for each problem is
found without rounding during any step in the problem.
Basic
1. a. The value of the call is the stock price minus the present value of the exercise price, so:
b. The value of the call is the stock price minus the present value of the exercise price, so:
c. The value of the put option is $0 since there is no possibility that the put will finish in the
2. a. The calls are in the money. The intrinsic value of the calls is $3.
b. The puts are out of the money. The intrinsic value of the puts is $0.
c. The Mar call and the Oct put are mispriced. The call is mispriced because it is selling for less
than its intrinsic value. If the option expired today, the arbitrage strategy would be to buy the
3. a. Each contract is for 100 shares, so the total cost is:
b. If the stock price at expiration is $140, the payoff is:
If the stock price at expiration is $125, the payoff is:
c. Remembering that each contract is for 100 shares of stock, the cost is:
The maximum gain on the put option would occur if the stock price goes to $0. We also need to
subtract the initial cost, so:
d. At a stock price of $103 the put is in the money. As the writer, you will make:
At the breakeven, you would recover the initial cost of $4,700, so:
For terminal stock prices above $105.30, the writer of the put option makes a net profit
(ignoring transaction costs and the effects of the time value of money).
4. a. The value of the call is the stock price minus the present value of the exercise price, so:
b. Using the equation presented in the text to prevent arbitrage, we find the value of the call is:
5. a. The value of the call is the stock price minus the present value of the exercise price, so:
b. Using the equation presented in the text to prevent arbitrage, we find the value of the call is:
6. Using put–call parity and solving for the put price, we get:
7. Using put–call parity and solving for the call price we get:
8. Using put–call parity and solving for the stock price we get:
9. Using put–call parity, we can solve for the risk-free rate as follows:
10. Using the Black–Scholes option pricing model to find the price of the call option, we find:
4/12
d2 = .0555 – (.54
√
4/12
) = –.2563
Putting these values into the Black–Scholes model, we find the call price is:
Using put–call parity, the put price is:
11. Using the Black–Scholes option pricing model to find the price of the call option, we find:
5/12
d2 = .3156 – (.53
√
5/12
) = –.0265
Putting these values into the Black–Scholes model, we find the call price is:
12. The delta of a call option is N(d1), so:
.75
13. Using the Black–Scholes option pricing model, with a ‘stock’ price of $1,300,000 and an exercise
price of $1,450,000, the price you should receive is:
12/12
Putting these values into the Black–Scholes model, we find the call price is:
14. Using the call price we found in the previous problem and put–call parity, you would need to pay:
15. Using the Black–Scholes option pricing model to find the price of the call option, we find:
d1 = [ln($83/$80) + (.06 + .472/2) (6/12)] / (.47
√
(6/12 )
) = .3672
6/12
Putting these values into the Black–Scholes model, we find the call price is:
Using put–call parity, we find the put price is:
a. The intrinsic value of each option is:
b. Option value consists of time value and intrinsic value, so:
Call option value = Intrinsic value + Time value
Put option value = Intrinsic value + Time value
c. The time premium (theta) is more important for a call option than a put option; therefore, the time
premium is, in general, larger for a call option.
16. The stock price can either increase 15 percent, or decrease 15 percent. The stock price at expiration
will either be:
The payoff in either state will be the maximum stock price minus the exercise price, or zero, which
is:
To get a 15 percent return, we can use the following expression to determine the risk-neutral
probability of a rise in the price of the stock:
And the probability of a stock price decrease is:
17. The stock price increase, decrease, and option payoffs will remain unchanged since the stock price
change is the same. The new risk neutral probability of a stock price increase is:
And the probability of a stock price decrease is:
Intermediate
19. If the standard deviation is zero, d1 and d2 go to +∞, so N(d1) and N(d2) go to 1. So:
20. If the standard deviation is infinite, d1 goes to positive infinity so N(d1) goes to 1, and d2 goes to
21. We can use the Black–Scholes model to value the equity of a firm. Using the asset value of $21,700
as the stock price, and the face value of debt of $20,000 as the exercise price, the value of the firm’s
equity is:
Putting these values into the Black–Scholes model, we find the equity value is:
The value of the debt is the firm value minus the value of the equity, so:
22. a. We can use the Black–Scholes model to value the equity of a firm. Using the asset value of
$22,900 (the $21,700 current value of the assets plus the $1,200 project NPV) as the stock price,
and the face value of debt of $20,000 as the exercise price, the value of the firm if it accepts
Project A is:
d1 = [ln($22,900/$20,000) + (.05 + .552/2) 1] / (.55
√
1
) = .6121
Putting these values into the Black–Scholes model, we find the equity value is:
The value of the debt is the firm value minus the value of the equity, so:
And the value of the firm if it accepts Project B is:
d1 = [ln($23,300/$20,000) + (.05 + .342/2) 1] / (.34
√
1
) = .7662
1
Putting these values into the Black–Scholes model, we find the equity value is:
The value of the debt is the firm value minus the value of the equity, so:
b. Although the NPV of Project B is higher, the equity value with Project A is higher. While NPV
represents the increase in the value of the assets of the firm, in this case, the increase in the value
c. Yes. If the same group of investors have equal stakes in the firm as bondholders and stock-
d. Stockholders may have an incentive to take on riskier, less profitable projects if the firm is
