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CHAPTER 22 -
29. To construct the collar, the investor must purchase the stock, sell a call option with a high strike
price, and buy a put option with a low strike price. So, to find the cost of the collar, we need to find
the price of the call option and the price of the put option. We can use Black–Scholes to find the
price of the call option, which will be:
Price of call option with $75 strike price:
√
(6/12 )
Putting these values into the Black–Scholes model, we find the call price is:
Now we can use Black–Scholes and put–call parity to find the price of the put option with a strike
price of $45. Doing so, we find:
Price of put option with $45 strike price:
√
(6/12 )
Putting these values into the Black–Scholes model, we find the call price is:
Rearranging the put–call parity equation, we get:
The investor will buy the stock, sell the call option, and buy the put option, so the total cost is:
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Challenge
30. a. Using the equation for the PV of a continuously compounded lump sum, we get:
b. Using the Black–Scholes model to value the equity, we get:
2
Putting these values into Black–Scholes:
And using put–call parity, the price of the put option is:
c. The value of a risky bond is the value of a risk-free bond minus the value of a put option on the
firm’s equity, so:
Using the equation for the PV of a continuously compounded lump sum to find the return on debt,
we get:
d. The value of the debt with five years to maturity at the risk-free rate is:
Using the Black–Scholes model to value the equity, we get:
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5
Putting these values into Black–Scholes:
And using put–call parity, the price of the put option is:
The value of a risky bond is the value of a risk-free bond minus the value of a put option on the
firm’s equity, so:
Using the equation for the PV of a continuously compounded lump sum to find the return on debt,
we get:
$25,873.85 = $75,000e–R(5)
The value of the debt declines because of the time value of money, i.e., it will be longer until
shareholders receive their payment. However, the required return on the debt declines. Under the
31. a. Using the equation for the PV of a continuously compounded lump sum, we get:
b. Using the Black–Scholes model to value the equity, we get:
5
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Putting these values into Black–Scholes:
c. The value of a risky bond is the value of a risk-free bond minus the value of a put option on the
firm’s equity, so:
Using the equation for the PV of a continuously compounded lump sum to find the return on debt,
we get:
d. Using the equation for the PV of a continuously compounded lump sum, we get:
Using the Black–Scholes model to value the equity, we get:
5
Putting these values into Black–Scholes:
And using put–call parity, the price of the put option is:
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The value of a risky bond is the value of a risk-free bond minus the value of a put option on the
firm’s equity, so:
Using the equation for the PV of a continuously compounded lump sum to find the return on debt,
we get:
e. From c and d, bondholders lose: $16,741.11 – 19,167.44 = –$2,426.34
This is an agency problem for bondholders. Management, acting to increase shareholder wealth in
32. a. Since 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
In order to value this option using the two-state option model, first draw a tree containing both
the current value of the firm and the firm’s possible values at the time of the option’s
The value of the company today is $185 million. It will either increase to $213 million or
decrease to $156 million in one year as a result of its new project. If the firm’s value increases
Value of company (in millions)
Equityholders’ call option price with a strike of $175
(in millions)
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CHAPTER 22 -
If the project is successful and the company’s value rises, the percentage increase in value over
the period is 15.14 percent [= ($213 / $185) – 1]. If the project is unsuccessful and the
And the risk-neutral probability of a decline in the company value is:
Using these risk-neutral probabilities, we can determine the expected payoff to the
equityholders’ call option at expiration, which will be:
Since this payoff occurs 1 year from now, we must discount it at the risk-free rate in order to
find its present value. So:
Therefore, the current value of the company’s equity is $26,137,071.65. The current value of
the company is equal to the value of its equity plus the value of its debt. In order to find the
value of company’s debt, subtract the value of the company’s equity from the total value of the
company:
b. To find the price per share, we can divide the total value of the equity by the number of shares
outstanding. So, the price per share is:
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c. The market value of the firm’s debt is $158,862,928.35. The present value of the same face
amount of riskless debt is $163,551,401.87 (= $175,000,000 / 1.07). The firm’s debt is worth
less than the present value of riskless debt since there is a risk that it will not be repaid in full.
d. The value of Strudler today is $185 million. It will either increase to $245 million or decrease
to $135 million in one year as a result of the new project. If the firm’s value increases to $245
million, the equityholders will exercise their call option, and they will receive a payoff of $70
Value of company (in millions)
Equityholders’ call option price with a strike of $175
(in millions)
If the project is successful and the company’s value rises, the increase in the value of the
company over the period is 32.43 percent [= ($245 / $185) – 1]. If the project is unsuccessful
Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall)
So the risk-neutral probability of a decrease in the company value is:
Using these risk-neutral probabilities, we can determine the expected payoff to the
equityholders’ call option at expiration, which is:
Since this payoff occurs 1 year from now, we must discount it at the risk-free rate in order to
find its present value. So:
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The current value of the company is equal to the value of its equity plus the value of its debt. In
order to find the value of the company’s debt, we can subtract the value of the company’s
equity from the total value of the company, which yields:
The riskier project increases the value of the company’s equity and decreases the value of the
company’s debt. If the company takes on the riskier project, the company is less likely to be
33. a. Going back to the chapter on dividends, the price of the stock will decline by the amount of the
dividend (less any tax effects). Therefore, we would expect the price of the stock to drop when a
b. Using the Black–Scholes model with dividends, we get:
.5
34. a. Going back to the chapter on dividends, the price of the stock will decline by the amount of the
b. Using put–call parity to find the price of the put option, we get:
35. N(d1) is the probability that “z” is less than or equal to N(d1), so 1 – N(d1) is the probability that “z”
is greater than N(d1). Because of the symmetry of the normal distribution, this is the same thing as
the probability that “z” is less than N(–d1). So:
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36. From put-call parity:
Substituting the Black–Scholes call option formula for C and using the result in the previous
question produces the put option formula:
37. Based on Black–Scholes, the call option is worth $50! The reason is that the present value of the
exercise price is zero, so the second term disappears. Also, d1 is infinite, so N(d1) is equal to one. The
problem is that the call option is European with an infinite expiration, so why would you pay
38. The delta of the call option is N(d1) and the delta of the put option is N(d1) – 1. Since you are selling
a put option, the delta of the portfolio is N(d1) – [N(d1) – 1]. This leaves the overall delta of your
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