CHAPTER 17
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.…
b. The buyer of a put option pays money for the right to sell....
d. The seller of a put option receives money for the obligation to buy….
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 owner the right
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 $10,
a riskless $5 profit.
6. The prices of both the call and the put option should increase. The higher level of downside risk still
that the asset will finish in the money.
7. False. The value of a call option depends on the total variance of the underlying asset, not just the
8. The call option will sell for more since it provides an unlimited profit opportunity, while the potential
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
should be measured and reported. They currently are not, at least not in a systematic fashion.
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
option to increase (or decrease in the case of a put) in value. For example, imagine an out-of-the–
to expiration would obviously increase its value.
12. An increase in volatility acts to increase both call and put values because the greater volatility increases
13. A put option is insurance since it guarantees the policyholder will be able to sell the asset for a specific
price. Consider homeowners insurance. If a house burns down, it is essentially worthless. In essence,
14. The equityholders of a firm financed partially with debt can be thought 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 remaining assets.
However, if the value of the firm is less than the face value of debt when it matures, the firm must
15. Since you have a large number of stock options in the company, you have an incentive to accept the
second project, which will increase the overall risk of the company and reduce the value of the firm’s
debt. However, accepting the risky project will increase your wealth, as the options are more valuable
16. Rearranging the put-call parity formula, we get: S – PV(E) = C – P. Since we know that the stock price
and exercise price are the same, assuming a positive interest rate, the left hand side of the equation
must be greater than zero. This implies the price of the call must be higher than the price of the put in
17. Rearranging the put-call parity formula, we get: S – PV(E) = C – P. If the call and the put have the
same price, we know C – P = 0. This must mean the stock price is equal to the present value of 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 calls are in the money. The intrinsic value of the calls is $2.
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 call
October put, with a positive cash inflow today.
2. a. Each contract is for 100 shares, so the total cost is:
b. If the stock price at expiration is $105, the payoff is:
Payoff = 10(100)($105 – 95)
Payoff = $10,000
If the stock price at expiration is $112, the payoff is:
Payoff = $17,000
c. Remembering that each contract is for 100 shares of stock, the cost is:
Cost = 10(100)($9.12)
Cost = $9,120
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:
If the stock price at expiration is $89, the position will have a value of:
d. At a stock price of $87 the put is in the money. As the writer, you will make:
Net profit = $9,120 – 10(100)($95 – 87)
Net profit = $1,120
At a stock price of $103 the put is out of the money, so the writer will make the initial cost:
3. Using put-call parity and solving for the put price, we get:
$47 + P = $50e–(.039)(3/12) + $2.61
4. Using put-call parity and solving for the call price we get:
5. Using put-call parity and solving for the stock price we get:
6. Using put-call parity, we can solve for the risk-free rate as follows:
$94.13 + 7.05 = $90e–R(4/12) + 11.74
$89.44 = $90e–R(4/12)
7. Using the Black-Scholes option pricing model to find the price of the call option, we find:
12/3
Putting these values into the Black-Scholes model, we find the call price is:
C = $68(.6376) – ($65e–.04(.25))(.5466)
Using put-call parity, the put price is:
8. Using the Black-Scholes option pricing model to find the price of the call option, we find:
d1 = [ln($96/$100) + (.03 + .492/2) (2/12)]/(.49
) = –.0807
d2 = –.0807 – (.49
) = –.2808
N(d1) = .4678
N(d2) = .3894
Putting these values into the Black-Scholes model, we find the call price is:
9. The delta of a call option is N(d1), so:
d1 = [ln($59/$55) + (.05 + .432/2) (9/12)]/(.43
)12/9(
) = .4754
N(d1) = .6828
For a call option the delta is .6828. For a put option, the delta is:
10. Using the Black-Scholes option pricing model, with a ‘stock’ price of $1,450,000 and an exercise price
of $1,600,000, the price you should receive is:
d1 = [ln($1,450,000/$1,600,000) + (.05 + .252/2) (12/12)]/(.25
12/12
) = –.0688
d2 = –.0688 – (.25
12/12
) = –.3188
N(d1) = .4726
Putting these values into the Black-Scholes model, we find the call price is:
C = $1,450,000(.4726) – ($1,600,000e–.05(1))(.3750)
P = $1,600,000e–.05(1) + 114,587.94 – 1,450,000
12. Using the Black-Scholes option pricing model to find the price of the call option, we find:
d1 = [ln($68/$70) + (.04 + .342/2) (6/12)]/(.34
)12/6(
) = .0828
Putting these values into the Black-Scholes model, we find the call price is:
C = $68(.5330) – ($70e–.04(.50))(.4374)
C = $6.23
P = $70e–.04(.50) + 6.23 – 68
P = $6.85
a. The intrinsic value of each option is:
Call intrinsic value = Max[S – E, 0] = $0
Call option value = Intrinsic value + Time value
$6.23 = $0 + TV
TV = $6.23
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.
13. The stock price can either increase 13 percent, or decrease 13 percent. The stock price at expiration
will either be:
Stock price increase = $58(1 + .13) = $65.54
Stock price decrease = $58(1 – .13) = $50.46
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[$65.54 – 55, 0] = $10.54
Payoff if stock price decreases = Max[$50.46 – 55, 0] = $0
Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall)
.04 = (ProbabilityRise)(.13) + (1 – ProbabilityRise)(–.13)
ProbabilityRise = .6538
And the probability of a stock price decrease is:
ProbabilityFall = 1 – .6538 = .3462
So, the risk neutral value of a call option will be:
15. If the exercise price is equal to zero, the call price will equal the stock price, which is $75.
16. If the standard deviation is zero, d1 and d2 go to +∞, so N(d1) and N(d2) go to 1, which gives us:
C = SN(d1) – EN(d2)e–Rt
C = $68(1) – $65(1)e–.04(6/12)
C = $4.29
negative infinity so N(d2) goes to 0. In this case, the call price is equal to the stock price, which is $35.
18. We can use the Black-Scholes model to value the equity of a firm. Using the asset value of $16,100 as
equity is:
d1 = [ln($16,100/$15,000) + (.06 + .322/2) 1]/(.32
1
) = .5687
d2 = .5687 – (.32
1
) = .2487
N(d1) = .7152
N(d2) = .5982
Putting these values into the Black-Scholes model, we find the equity value is:
Equity = $16,100(.7152) – ($15,000e–.06(1))(.5982)
Equity = $3,064.54
The value of the debt is the firm value minus the value of the equity, so:
Debt = $13,035.46
19. a. We can use the Black-Scholes model to value the equity of a firm. Using the asset value of
d1 = [ln($16,900/$15,000) + (.06 + .502/2) 1]/(.50
1
) = .6085
d2 = .6085 – (.50
1
) = .1085
N(d1) = .7286
N(d2) = .5432
Putting these values into the Black-Scholes model, we find the equity value is:
EquityA = $16,900(.7286) – ($15,000e–.06(1))(.5432)
EquityA = $4,639.36
The value of the debt is the firm value minus the value of the equity, so:
1
d2 = 1.0212 – (.23
1
) = .7912
N(d1) = .8464
N(d2) = .7856
Putting these values into the Black-Scholes model, we find the equity value is:
EquityB = $17,400(.8464) – ($15,000e–.06(1))(.7856)
EquityB = $3,630.15
c. Yes. If the same group of investors have equal stakes in the firm as bondholders and stock–
holders, then total firm value matters and Project B should be chosen, since it increases the value
d. Stockholders may have an incentive to take on riskier, less profitable projects if the firm is
leveraged; the higher the firm’s debt load, all else the same, the greater is this incentive.