978-1259289903 Chapter 17 Solution Manual Part 2

20. We can use the Black-Scholes model to value the equity of a firm. Using the asset value of $23,200

as the stock price, and the face value of debt of $22,000 as the exercise price, the value of the firm’s

equity is:

d1 = [ln($23,200/$22,000) + (.06 + .342/2)  1]/(.34  1) = .5027

d2 = .5027 – (.34 1) = .1627

N(d1) = .6924

N(d2) = .5646

Putting these values into the Black-Scholes model, we find the equity value is:

21. a. Using Black-Scholes model to value the equity, we get:

d1 = [ln($10,700,000/$12,000,000) + (.06 + .472/2)  10]/(.47 01 ) = 1.0697

d2 = 1.0697 – (.47  01 ) = –.4166

N(d1) = .8576

N(d2) = .3385

Putting these values into Black-Scholes:

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:

$3,752,684.33 = $12,000,000e–R(10)

d. Using Black-Scholes model to value the equity, we get:

d1 = [ln($12,000,000/$12,000,000) + (.06 + .472/2)  10]/(.47  01 ) = 1.1468

d2 = 1.1468 – (.47 01 ) = –.3394

N(d1) = .8743

N(d2) = .3671

Putting these values into Black-Scholes:

Equity = $12,000,000(.8743) – ($12,000,000e–.06(10))(.3671)

Debt = $12,000,000 – 8,073,407.10

Debt = $3,926,592.90

Using the equation for the PV of a continuously compounded lump sum, we get:

$3,926,592.90 = $12,000,000e–R(10)

22. 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 $

60

Today

1 year

Today

1 year

$

71

$

11

=Max(0, $

71

–

60

)

$

64

?

$

56

$0

=Max(0, $

56

–

60

)

The stock price today is $64. It will either increase to $71 or decrease to $56 in one year. If the

expiration will be zero.

If the stock price rises, its return over the period is 10.94 percent [= ($71/$64) – 1]. If the stock

price falls, its return over the period is –12.50 percent [= ($56/$64) –1]. We can use the following

expression to determine the risk-neutral probability of a rise in the price of the stock:

Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall)

Using these risk-neutral probabilities, we can now determine the expected payoff of the call

option at expiration. The expected payoff at expiration is:

Expected payoff at expiration = .7680($11) + .2320($0)

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:

Delta = (Swing of option)/(Swing of stock)

Since the call option will be worth $11 if the stock price rises and $0 if it falls, the swing of the

option is $11 (= $11 – 0). Since the stock price will either be $71 or $56 at the time of the option’s

expiration, the swing of the stock is $15 (= $71 – 56). With this information, the delta of the

option is:

Delta = $11/$15

Delta = .7333

However, owning .7333 shares leaves us exactly $11 above the payoff at expiration, regardless

of whether the stock price rises or falls. In order to reduce the payoff at expiration by $41.07, we

should borrow the present value of $41.07 now. In one year, the obligation to pay $41.07 will

reduce the payoffs so that they exactly match those of an actual call option. So, purchase .7333

c. Since the cost of the stock purchase is $46.93 to purchase .7333 shares and $38.93 is borrowed,

the total cost of the synthetic call option is:

Cost of synthetic option = $46.93 – 38.93

Cost of synthetic option = $8.01

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 s

trike of $

75

Today

6 months

Today

6 months

$

83

$0

=Max(0, $

75

–

83

)

$

73

?

$

61

$14

=Max(0, $

75

–

61

)

The stock price today is $73. It will either decrease to $61 or increase to $83 in six months. If the

stock price falls to $61, the put will be exercised and the payoff will be $14. If the stock price

rises to $83, the put will not be exercised, so the payoff will be zero.

If the stock price rises, its return over the period is 13.70 percent [= ($83/$73) – 1]. If the stock

price falls, its return over the period is –16.44 percent [= ($61/$73) –1]. Use the following

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)

Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall)

.0213 = (ProbabilityRise)(.1370) + (1 – ProbabilityRise)(–.1644)

ProbabilityRise = .6160, or 61.60%

Which means the risk-neutral probability of a decrease in the stock price is:

ProbabilityFall = 1 – ProbabilityRise

ProbabilityFall = 1 – .6160

to find its present value, which is:

PV(Expected payoff at expiration) = $5.38/(1.043)1/2

PV(Expected payoff at expiration) = $5.26

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:

Delta = (Swing of option)/(Swing of stock)

Since the put option will be worth $0 if the stock price rises and $14 if it falls, the swing of the

call option is –$14 (= $0 – 14). Since the stock price will either be $83 or $61 at the time of the

option’s expiration, the swing of the stock is $22 (= $83 – 61). Given this information, the delta

of the put option is:

Delta = (Swing of option)/(Swing of stock)

If the stock price rises to $83: Payoff = $0

If the stock price falls to $61: Payoff = $14

Delta shares

If the stock price rises to $83: Payoff = (–.6364)($83) = –$52.82

If the stock price falls to $61: Payoff = (–.6364)($61) = –$38.82

cost of the synthetic put option is:

Cost of synthetic put = $51.72 – 46.45

Cost of synthetic put = $5.26

price of $1,310 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

amount the firm must pay for gold at $1,310 per ounce.

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 gold price movements.

Today

3 months

Today

3 months

$

1,350

$

40

=Max(0, $

1,350

–

1,310

)

$

1,230

?

$

1,120

$0

=Max(0, $

1,120

–

1,310

)

The price of gold is $1,230 per ounce today. If the price rises to $1,350, the company will exercise

its call option for $1,310 and receive a payoff of $40 at expiration. If the price of gold falls to

$1,120, 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 9.76 percent [= ($1,350/$1,230)

– 1]. If the price of gold falls, its return over the period is –8.94 percent [= ($1,120/$1,230) – 1].

Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall)

Risk-free rate = (ProbabilityRise)(ReturnRise) + (1 – ProbabilityRise)(ReturnFall)

The risk-free rate over the next three months must be used in the order to match the timing of the

expected price change. Since the risk-free rate per annum is 4 percent, the risk-free rate over the

next three months is .99 percent [= (1.04)1/4 – 1], so:

.0099 = (ProbabilityRise)(.0976) + (1 – ProbabilityRise)(–.0894)

ProbabilityRise = .5310, or 53.10%

And the risk-neutral probability of a price decline is:

ProbabilityFall = 1 – ProbabilityRise

ProbabilityFall = 1 –.5310

$21.03 today.

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 $40 if the price of gold rises and $0 if it falls, the swing of the

call option is $40 (= $40 – 0). Since the price of gold will either be $1,350 or $1,120 at the time

of the option’s expiration, the swing of the price of gold is $230 (= $1,350 – 1,120). Given this

information the delta of the call option is:

Delta = Swing of option/Swing of price of gold

Delta = $40/$230

Delta = .1739

If the price of gold rises to $1,350: Payoff = .1739($1,350) = $234.78

If the price of gold falls to $1,120: Payoff = .1739($1,120) = $194.78

25. 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

Price of call option with $60 strike price:

d1 = [ln($54/$60) + (.047 + .452/2)  (6/12)]/(.45 )12/6( ) = –.0982

d2 = –.0982 – (.45 12/6 ) = –.4164

N(d1) = .4609

N(d2) = .3386

Putting these values into the Black-Scholes model, we find the call price is:

C = $54(.4609) – ($60e–.047(6/12))(.3386)

C = $5.05

Now we can use Black-Scholes and put-call parity to find the price of the put option with a strike price

of $50. Doing so, we find:

N(d1) = .6825

N(d2) = .5622

Putting these values into the Black-Scholes model, we find the call price is:

C = $54(.6825) – ($50e–.047(6/12))(.5622)

C = $9.40

Rearranging the put-call parity equation, we get:

P = C – S + Ee–Rt

P = $9.40 – 54 + 50e–.047(6/12)