CHAPTER 23
OPTIONS AND CORPORATE FINANCE:
EXTENSIONS AND APPLICATIONS
Answers to Concepts Review and Critical Thinking Questions
1. One of the purposes to giving stock options to CEOs (instead of cash) is to tie the performance of the
4. As the volatility increases, the value of an option increases. As the volatility of coal and oil
increases, the option to burn either increases. However, if the prices of coal and oil are highly
5. The advantage is that the value of the land may increase if you wait. Additionally, if you wait, the
6. The company has an option to abandon the mine temporarily, which is an American put. If the option
is exercised, which the company is doing by not operating the mine, it has an option to reopen the
7. Your colleague is correct, but the fact that increased volatility increases the value of an option is an
important part of option valuation. All else the same, a call option on a venture that has higher
volatility will be worth more since the upside potential is greater. Even though the downside is also
8. Real option analysis is not a technique that can be applied in isolation. The value of the asset in real
9. Insurance is a put option. Consider your homeowner’s insurance. If your house were to burn down,
10. In a market with competitors, you must realize that the competitors have real options as well. The
decisions made by these competitors may often change the payoffs for your company’s options. For
example, the first entrant into a market can often be rewarded with a larger market share because 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 inputs to the Black–Scholes model are the current price of the underlying asset (S), the
strike price of the option (E), the time to expiration of the option in fractions of a year (t), the
variance (2) of the underlying asset, and the continuously-compounded risk-free interest rate
d1 = [ln(S/E) + (R + 2/2)(t) ] / (2t)1/2
√
5
d2 = .9019 – (.61
√
5
) = –.4621
Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative
infinity to d2, respectively. Doing so:
Now we can find the value of each option, which will be:
Since the option grant is for 25,000 options, the value of the grant is:
b. Because he is risk-neutral, you should recommend the alternative with the highest net present
c. If he is risk-averse, he may or may not prefer the stock option package to the immediate bonus.
Even though the stock option package has a higher value, he may not prefer it because it is
2. The total compensation package consists of an annual salary in addition to 20,000 at-the-money
stock options. First, we will find the present value of the salary payments. Since the payments occur
at the end of the year, the payments can be valued as a three-year annuity, which will be:
Next, we can use the Black–Scholes model to determine the value of the stock options. Doing so, we
find:
Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative
infinity to d2, respectively. Doing so:
Now we can find the value of each option, which will be:
Since the option grant is for 20,000 options, the value of the grant is:
The total value of the contract is the sum of the present value of the salary, plus the option value, or:
3. Since the contract is to sell up to 5 million gallons, it is a call option, so we need to value the
contract accordingly. Using the binomial mode, we will find the value of u and d, which are:
u = 1.3634
This implies the percentage increase if gasoline increases will be 36.34 percent, and the percentage
decrease if prices fall will be –26.66 percent. So, the price in three months with an up or down move
will be:
The option is worthless if the price decreases. If the price increases, the value of the option per
gallon is:
Next, we need to find the risk neutral probability of a price increase or decrease, which will be:
And the probability of a price decrease is:
The contract will not be exercised if gasoline prices fall, so the value of the contract with a price
decrease is zero. So, the value per gallon of the call option contract will be:
This means the value of the entire contract is:
4. When solving a question dealing with real options, begin by identifying the option-like features of
the situation. First, since the company will exercise its option to build if the value of an office
building rises, the right to build the office building is similar to a call option. Second, an office
building would be worth $53.2 million today. This amount can be viewed as the current price of the
underlying asset (S). Third, it will cost $55 million to construct such an office building. This amount
state model to value the option to build on the land. First, we need to find the return of the land if the
value rises or falls. The return will be:
Now we can find the risk-neutral probability of a rise in the value of the building as:
Value of building (millions) Value of real call option with a strike of $55 (millions)
Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall)
So, the probability of a fall is:
Using these risk-neutral probabilities, we can determine the expected payoff of the real option at
expiration.
Since this payoff will occur 1 year from now, it must be discounted at the risk-free rate in order to
find its present value, which is:
Therefore, the right to build an office building over the next year is worth $2,033,908.21 today.
5. When solving a question dealing with real options, begin by identifying the option-like features of
the situation. First, since the company will only choose to drill and excavate if the price of oil rises,
the right to drill on the land can be viewed as a call option. Second, since the land contains 435,000
Third, since the company will not drill unless the price of oil in one year will compensate its
excavation costs, these costs can be viewed as the real option’s strike price (E). Finally, since the
winner of the auction has the right to drill for oil in one year, the real option can be viewed as having
a time to expiration (t) of one year. Using the Black–Scholes model to determine the value of the
option, we find:
Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative
infinity to d2, respectively. Doing so:
Now we can find the value of the call option, which will be:
This is the maximum bid the company should be willing to make at auction.
Intermediate
6. When solving a question dealing with real options, begin by identifying the option-like features of
the situation. First, since Sardano will only choose to manufacture the steel rods if the price of steel
falls, the lease, which gives the firm the ability to manufacture steel, can be viewed as a put option.
The amount received can be viewed as the put option’s strike price (E). Third, since the project
requires Sardano to purchase 500 tons of steel and the current price of steel is $690 per ton, the
Finally, since Sardano must decide whether to purchase the steel or not in six months, the firm’s real
option to manufacture steel rods can be viewed as having a time to expiration (t) of six months. In
order to calculate the value of this real put option, we can use the Black–Scholes model to determine
Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative
infinity to d2, respectively. Doing so:
Now we can find the value of call option, which will be:
Now we can use put–call parity to find the price of the put option, which is:
This is the most the company should be willing to pay for the lease.
7. In one year, the company will abandon the technology if the demand is low since the value of
abandonment is higher than the value of continuing operations. Since the company is selling the
technology in this case, the option is a put option. The value of the put option in one year if demand
is low will be:
Of course, if demand is high, the company will not sell the technology, so the put will expire
worthless. We can value the put with the binomial model. In one year, the percentage gain on the
project if the demand is high will be:
Now we can find the risk-neutral probability of a rise in the value of the technology as:
So, a probability of a fall is:
Using these risk-neutral probabilities, we can determine the expected payoff of the real option at
expiration. With high demand, the option is worthless since the technology will not be sold, and the
value of the technology with low demand is the $900,000 we calculated previously. So, the value of
the option to abandon is: