Unlock document.
This document is partially blurred.
Unlock all pages and 1 million more documents.
Get Access
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)
