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Call Option
If the stock price rises to $73: Payoff = $13
If the stock price falls to $49: Payoff = $0
Delta Shares
If the stock price rises to $73: Payoff = (.54)($73) = $39.54
c. Since the cost of the stock purchase is $33.58 to purchase .54 of a share and $25.89 is borrowed,
the total cost of the synthetic call option is:
27. a. In order to solve a problem using the two-state option model, we first 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
Put option price with a strike of $60
Today
6 months
Today
6 months
Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall)
Risk-free rate = (ProbabilityRise)(ReturnRise) + (1 – ProbabilityRise)(ReturnFall)
The risk-free rate over the next six months must be used in the order to match the timing of the
expected stock price change. Since the risk-free rate per annum is 5 percent, the risk-free rate
over the next six months is 2.47 percent (= 1.051/2 –1), so.
Since this payoff occurs 6 months from now, we must discount it at the risk-free rate in order to
find its present value, which is:
PV(Expected payoff at expiration) = $5.43/1.051/2
PV(Expected payoff at expiration) = $5.30
b. Yes, there is a way to create a synthetic put option with identical payoffs to the put option
described above. In order to do this, we need to short shares of the stock and lend at the risk-free
rate. The number of shares that should be shorted is based on the delta of the option, where delta
is defined as:
Therefore, the first step in creating a synthetic put option is to short .63 of a share of stock. Since
the stock is currently trading at $57 per share, the amount received will be $36.00 (= .63 × $57)
as a result of the short sale. In order to determine the amount to lend, compare the payoff of the
actual put option to the payoff of delta shares at expiration.
Put option
If the stock price rises to $67: Payoff = $0
c. Since the short sale results in a positive cash flow of $36.00 and we will lend $41.30, the total
cost of the synthetic put option is:
28. a. The company would be interested in purchasing a call option on the price of gold with a strike
price of $1,380 per ounce and 3 months until expiration. This option will compensate the
company for any increases in the price of gold above the strike price and places a cap on the
b. In order to solve a problem using the two-state option model, first draw a price tree containing
both the current price of the underlying asset and the underlying asset’s possible values at the
time of the option’s expiration. Next, 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 gold
Call option price with a strike of $1,380
Today
3 months
Today
3 months
$1,445
$65
=Max($1,445 – 1,380,
0)
0)
The price of gold is $1,280 per ounce today. If the price rises to $1,445, the company will exercise
its call option for $1,380 and receive a payoff of $65 at expiration. If the price of gold falls to
$1,130, the company will not exercise its call option, and the firm will receive no payoff at
expiration. If the price of gold rises, its return over the period is 12.89 percent (= $1,445/$1,280
– 1). If the price of gold falls, its return over the period is –11.72 percent (= $1,130/$1,280 – 1).
Use the following expression to determine the risk-neutral probability of a rise in the price of
gold:
c. 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, the company will need to buy gold and borrow at the risk-
free rate. The amount of gold to buy is based on the delta of the option, where delta is defined as:
Delta = (Swing of option)/(Swing of price of gold)
Since the call option will be worth $65 if the price of gold rises and $0 if it falls, the swing of the
call option is $65 (= $65 – 0). Since the price of gold will either be $1,445 or $1,130 at the time
of the option’s expiration, the swing of the price of gold is $315 (= $1,445 – 1,130). Given this
information, the delta of the call option is:
Delta = (Swing of option)/(Swing of price of gold)
Delta = $65/$315
Delta = .21
Call Option
If the price of gold rises to $1,445: Payoff = $65
If the price of gold falls to $1,130: Payoff = $0
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
N(d1) = .4270
N(d2) = .2955
Putting these values into the Black-Scholes model, we find the call price is:
of $55. Doing so, we find:
Price of put option with $55 strike price:
d1 = [ln($68/$55) + (.07 + .502/2) (6/12)]/(.50
6 /12
) = .8759
Putting these values into the Black-Scholes model, we find the call price is:
C = $68(.8095) – ($55e–.07(6/12))(.6993)
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:
d1 = [ln($59,000/$90,000) + (.05 + .602/2) 2]/(.60
2
) = .0445
d2 = .0445 – (.60
2
) = –.8041
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:
Value of risky bond = $81,435.37 – 35,824.80
Value of risky bond = $45,610.57
d. The value of the debt with five years to maturity at the risk-free rate is:
PV = $90,000 e–.05(5)
PV = $70,092.07
Putting these values into Black-Scholes:
Equity = $59,000(.7062) – ($90,000e–.05(5))(.2121)
Equity = $26,802.68
And using put-call parity, the price of the put option is:
Put = $90,000e–.05(5) + $26,802.68 – $59,000
Put = $37,894.75
31. a. Using the equation for the PV of a continuously compounded lump sum, we get:
PV = $40,000 e–.06(5)
PV = $29,632.73
b. Using the Black-Scholes model to value the equity, we get:
d1 = [ln($38,000/$40,000) + (.06 + .502/2) 5]/(.50
5
) = .7815
Putting these values into Black-Scholes:
Equity = $38,000(.7827) – ($40,000e–.06(5))(.3682)
Equity = $18,832.56
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:
Value of risky bond = $29,632.73 – 10,465.29
Value of risky bond = $19,167.44
d. Using the equation for the PV of a continuously compounded lump sum, we get:
PV = $40,000 e–.06(5)
PV = $29,632.73
Putting these values into Black-Scholes:
Equity = $38,000(.8041) – ($40,000e–.06(5))(.3137)
Equity = $21,258.89
And using put-call parity, the price of the put option is:
e. From c and d, bondholders lose: $16,741.11 – 19,167.44 = –$2,426.34
From b and d, stockholders gain: $21,258.89 – 18,832.56 = $2,426.34
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
expiration equal to the debt’s time to maturity, the value of the company’s equity equals a call
option with a strike price of $150 million and 1 year until expiration.
In order to value this option using the two-state option model, first draw a tree containing both
Today
1 year
Today
1 year
$198
$48
=Max($198 – 150, 0)
If the project is successful and the company’s value rises, the percentage increase in value over
the period is 21.47 percent (= $198/$163 – 1). If the project is unsuccessful and the company’s
value falls, the percentage decrease in value over the period is –26.38 percent (= $120/$163 – 1).
We can determine the risk-neutral probability of an increase in the value of the company as:
Using these risk-neutral probabilities, we can determine the expected payoff to the equityholders’
call option at expiration, which will be:
Expected payoff at expiration = (.6976)($48,000,000) + (.3024)($0)
Expected payoff at expiration = $33,483,076.92
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:
c. The market value of the firm’s debt is $131,707,404.74. The present value of the same face
amount of riskless debt is $140,186,915.89 (= $150,000,000/1.07). 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. In other
d. The value of the company today is $163 million. It will either increase to $217 million or decrease
to $105 million in one year as a result of the new project. If the firm’s value increases to $217
million, the equityholders will exercise their call option, and they will receive a payoff of $67
million at expiration. However, if the firm’s value decreases to $105 million, the equityholders
will not exercise their call option, and they will receive no payoff at expiration.
Value of company (in millions)
Equityholders’ call option price with a strike of $150
(in millions)
Today
1 year
Today
1 year
$217
$67
=Max($217 – 150, 0)
Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall)
Risk-free rate = (ProbabilityRise)(ReturnRise) + (1 – ProbabilityRise)(ReturnFall)
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:
Therefore, the current value of the firm’s equity is $38,805,657.54.
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:
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:
d1 = [ln($108/$105) + (.05 – .02 + .352/2) .5]/(.35
.5
) = .2982
34. 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 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
36. From put-call parity:
P = E × e–Rt + C – S
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 anything
for it since you can never exercise it? The paradox can be resolved by examining the price of the stock.
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
