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
2. a. The buyer of a call option pays money for the right to buy….
3. An American option can be exercised on any date up to and including the expiration date. A European
5. The call is selling for less than its intrinsic value; an arbitrage opportunity exists. Buy the call for $10,
6. The prices of both the call and the put option should increase. The higher level of downside risk still
7. False. The value of a call option depends on the total variance of the underlying asset, not just the
systematic variance.
9. The value of a call option will increase, and the value of a put option will decrease.
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 would
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
13. A put option is insurance since it guarantees the policyholder will be able to sell the asset for a specific
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
matures, the firm will pay off the debtholders in full, leaving the equityholders with the firm’s
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 price
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
downside losses) and a T-bill (to reflect the time value component – the “wait” factor).
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:
C0 = $74 – $70/1.034
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 money.
The intrinsic value is also $0.
2. a. The calls are in the money. The intrinsic value of the calls is $3.
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 call
for $2.80, exercise it and pay $80 for a share of stock, and sell the stock for $83. A riskless profit
of $.20 results. The October put is mispriced because it sells for less than the July put. To take
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:
Payoff = 10(100)($140 – 110)
Payoff = $30,000
c. Remembering that each contract is for 100 shares of stock, the cost is:
Cost = 10(100)($4.70)
Cost = $4,700
d. At a stock price of $103 the put is in the money. As the writer, you will make:
Net loss = $4,700 – 10(100)($110 – 103)
Net loss = –$2,300
At a stock price of $132 the put is out of the money, so the writer will make the initial cost:
4. a. The value of the call is the stock price minus the present value of the exercise price, so:
C0 = $68 – $60/1.04
5. a. The value of the call is the stock price minus the present value of the exercise price, so:
C0 = $66 – $40/1.05
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:
$57.30 + 2.65 = $55e–R(2/12) + $5.32
$54.63 = $55e–R(2/12)
10. Using the Black-Scholes option pricing model to find the price of the call option, we find:
d1 = [ln($86/$90) + (.05 + .482/2) (4/12)]/(.48
12/4
) = .0347
Putting these values into the Black-Scholes model, we find the call price is:
C = $86(.5138) – ($90e–.05(4/12))(.4042)
C = $8.41
11. Using the Black-Scholes option pricing model to find the price of the call option, we find:
d1 = [ln($58/$55) + (.04 + .532/2) (5/12)]/(.53
12/5
) = .3750
Putting these values into the Black-Scholes model, we find the call price is:
C = $58(.6462) – ($55e–.04(5/12))(.5131)
12. The delta of a call option is N(d1), so:
d1 = [ln($76/$80) + (.05 + .492/2) .75]/(.49
.75
) = .1797
N(d1) = .5713
For a call option the delta is .5713. For a put option, the delta is:
13. Using the Black-Scholes option pricing model, with a ‘stock’ price of $1,100,000 and an exercise price
of $1,325,000, the price you should receive is:
N(d2) = .2730
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:
N(d1) = .6387
N(d2) = .5091
Putting these values into the Black-Scholes model, we find the call price is:
C = $62(.6387) – ($60e–.06(.50))(.5091)
C = $9.96
16. The stock price can either increase 16 percent or decrease 16 percent. The stock price at expiration
will either be:
Stock price increase = $77(1 + .16) = $89.32
Stock price decrease = $77(1 – .16) = $64.68
The payoff in either state will be the maximum stock price minus the exercise price, or zero, which
is:
Payoff if stock price increases = Max[$89.32 – 75, 0] = $14.32
Payoff if stock price decreases = Max[$64.68 – 75, 0] = $0
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:
Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall)
18. If the exercise price is equal to zero, the call price will equal the stock price, which is $75.
19. If the standard deviation is zero, d1 and d2 go to +∞, so N(d1) and N(d2) go to 1. So:
21. We can use the Black-Scholes model to value the equity of a firm. Using the asset value of $11,200 as
the stock price, and the face value of debt of $10,000 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:
22. a. We can use the Black-Scholes model to value the equity of a firm. Using the asset value of $12,400
(the $11,200 current value of the assets plus the $1,200 project NPV) as the stock price, and the
face value of debt of $10,000 as the exercise price, the value of the firm if it accepts Project A is:
d1 = [ln($12,400/$10,000) + (.05 + .552/2) 1]/(.55
1
) = .7570
N(d2) = .5820
Putting these values into the Black-Scholes model, we find the equity value is:
EquityA = $12,400(.7755) – ($10,000e–.05(1))(.5820)
N(d1) = .8516
N(d2) = .7590
Putting these values into the Black-Scholes model, we find the equity value is:
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
23. We can use the Black-Scholes model to value the equity of a firm. Using the asset value of $16,900 as
the stock price, and the face value of debt of $15,000 as the exercise price, the value of the firm’s
equity is:
d1 = [ln($16,900/$15,000) + (.05 + .532/2) 1]/(.53
1
) = .5844
Putting these values into the Black-Scholes model, we find the equity value is:
Equity = $16,900(.7205) – ($15,000e–.05(1))(.5217)
Equity = $4,733.13
24. a. The combined value of equity and debt of the two firms is:
Equity = $2,544.31 + 4,733.13 = $7,277.45
b. For the new firm, the combined market value of assets is $28,100, and the combined face value of
debt is $25,000. Using Black-Scholes to find the value of equity for the new firm, we find:
d1 = [ln($28,100/$25,000) + (.05 + .292/2) 1]/(.29
1
) = .7205
d2 = .7205 – (.29
1
) = .4305
1
c. The change in the value of the firm’s equity is:
Equity value change = $5,627.54 – 7,277.45
d. In a purely financial merger, when the standard deviation of the assets declines, the value of the
equity declines as well. The shareholders will lose exactly the amount the bondholders gain. The
25. a. Using the Black-Scholes model to value the equity, we get:
d1 = [ln($7,100,000/$8,000,000) + (.06 + .392/2) 10]/(.39
01
) = 1.0064
b. The value of the debt is the firm value minus the value of the equity, so:
c. Using the equation for the PV of a continuously compounded lump sum, we get:
$2,916,715.42 = $8,000,000e–R(10)
d. The new value of assets is the current asset value plus the project NPV. Using the Black-Scholes
model to value the equity, we get:
d1 = [ln($8,200,000/$8,000,000) + (.06 + .392/2) 10]/(.39
01
) = 1.1232
d2 = 1.1232 – (.39
01
) = –.1101
e. The value of the debt is the firm value minus the value of the equity, so:
Debt = $8,200,000 – 5,125,644.67
Debt = $3,074,355.33
26. a. In order to solve a problem using the two-state option model, we first need to draw a stock price
tree containing both the current stock price and the stock’s possible values at the time of the
option’s expiration. Next, we can draw a similar tree for the option, designating what its value
will be at expiration given either of the two possible stock price movements.
Price of stock
Call option price with a strike of $65
1 year
1 year
$73
$13
=Max($73 – 60, 0)
$49
$0
=Max($49 – 60, 0)
expression to determine the risk-neutral probability of a rise in the price of the stock:
Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall)
Risk-free rate = (ProbabilityRise)(ReturnRise) + (1 – ProbabilityRise)(ReturnFall)
.025 = (ProbabilityRise)(.1774) + (1 – ProbabilityRise)(–.2097)
ProbabilityRise = .6063, or 60.63%
b. Yes, there is a way to create a synthetic call option with identical payoffs to the call option
described above. In order to do this, we will need to buy shares of stock and borrow at the risk–
free rate. The number of shares to buy is based on the delta of the option, where delta is defined
as: