Chapter 21
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1.
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Would you expect a $1 increase in a call option’s exercise price to lead to a
decrease in the option’s value of more or less than $1?
The option's value will decrease by less than $1.
Explanation:
A $1 increase in a call option’s exercise price would lead to a decrease in the option’s value of less than $1. The change in the call price would equal $1 only if: (a) there is a 100% probability that the call would be exercised, and
(b) the interest rate is zero.
Worksheet
Difficulty: 1 Basic
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
2.
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Is a put option on a high-beta stock worth more than one on a low-beta stock? The stocks have identical firm-specific risk.
Yes
Explanation:
Holding firm-specific risk constant, higher beta implies higher total stock volatility. Therefore, the value of the put option increases as beta increases.
Worksheet
Difficulty: 1 Basic
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
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All else equal, is a call option on a stock with a lot of firm-specific risk worth more than one on a stock with little firm-specific risk?
The betas of the two stocks are equal.
Yes
Explanation:
Holding beta constant, the stock with a lot of firm-specific risk has higher total volatility. The option on the stock with higher firm-specific risk is worth more.
Worksheet
Difficulty: 1 Basic
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
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4.
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All else equal, will a call option with a high exercise price have a higher or lower hedge ratio than one with a low exercise price?
Higher
Explanation:
A call option with a high exercise price has a lower hedge ratio. This call option is less in the money. Both
d
1
and
N
(
d
1
)
are lower when
X
is higher.
Worksheet
Difficulty: 1 Basic
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
5.
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In each of the following questions, you are asked to compare two options with parameters as given. The risk-free interest rate for
all
cases should be assumed to be 4%. Assume the stocks on which these options are written pay
no dividends.
Required:
a.
Which put option is written on the stock with the lower price?
Put
T
X
σ
Price of Put
A
0.5
50
0.20
$ 10
B
0.5
50
0.25
$ 10
b.
Which put option must be written on the stock with the lower price?
Put
T
X
σ
Price of Put
A
0.5
50
0.20
$ 10
B
0.5
50
0.20
$ 12
c.
Which call option must have the lower time to expiration?
Call
S
X
σ
Price of Call
A
50
50
0.20
$ 12
B
55
50
0.20
$ 10
d.
Which call option is written on the stock with higher volatility?
Call
T
X
S
Price of Call
A
0.5
50
55
$ 10
B
0.5
50
55
$ 12
e.
Which call option is written on the stock with higher volatility?
Call
T
X
S
Price of Call
A
0.5
50
55
$ 10
B
0.5
50
55
$ 7
a. Which put option is written on the stock with the lower price?
A
b. Which put option must be written on the stock with the lower price?
B
c. Which call option must have the lower time to expiration?
B
d. Which call option is written on the stock with higher volatility?
B
e. Which call option is written on the stock with higher volatility?
A
Explanation:
a.
Put A must be written on the stock with the lower price. Otherwise, given the lower volatility of Stock A, Put A would sell for less than Put B.
b.
Put B must be written on the stock with the lower price. This would explain its higher price.
c.
Call B must have the lower time to expiration. Despite the higher price of Stock B, Call B is cheaper than Call A. This can be explained by a lower time to expiration.
d.
Call B must be written on the stock with higher volatility. This would explain its higher price.
e.
Call A must be written on the stock with higher volatility. This would explain its higher price.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
6.
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Adjust credit
for all students.
Reconsider the determination of the hedge ratio in the two-state model where we showed that one-third share of stock would hedge one option. The possible end-of-year stock prices,
uS
0
= $120
(up state) and
dS
0
= $90
(down state).
Required:
a.
What would be the call option hedge ratio for each of the following exercise prices: $120, $110, $100, $90, given the possible end-of-year stock prices,
uS
0
= $120
(up state) and
dS
0
= $90
(down state)?
b.
What do you conclude about the hedge ratio as the option becomes progressively more in the money?
Required A
Required B
Complete this question by entering your answers in the tabs below.
What would be the call option hedge ratio for each of the following exercise prices: $120, $110, $100, $90, given the possible
end-of-year stock prices,
uS
0
= $120
(up state) and
dS
0
= $90
(down state)?
Note: Round your answers to 3 decimal places.
Required A
Required B
$
$
$
$
Exercise
Price
Hedge Ratio
120
0.000
110
0.333
100
0.667
90
1.000
Explanation:
a.
Exercise Price
Hedge Ratio
$ 120
0 ÷ 30 = 0.000
$ 110
10 ÷ 30 = 0.333
$ 100
20 ÷ 30 = 0.667
$ 90
30 ÷ 30 = 1.000
b.
As the option becomes more in the money, the hedge ratio increases to a maximum of 1.0.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
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6.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Reconsider the determination of the hedge ratio in the two-state model where we showed that one-third share of stock would hedge one option. The possible end-of-year stock prices,
uS
0
= $120
(up state) and
dS
0
= $90
(down state).
Required:
a.
What would be the call option hedge ratio for each of the following exercise prices: $120, $110, $100, $90, given the possible end-of-year stock prices,
uS
0
= $120
(up state) and
dS
0
= $90
(down state)?
b.
What do you conclude about the hedge ratio as the option becomes progressively more in the money?
Required A
Required B
Complete this question by entering your answers in the tabs below.
What do you conclude about the hedge ratio as the option becomes progressively more in the money?
Required A
Required B
Conclusion
Increases to a maximum of 1.0
Explanation:
a.
Exercise Price
Hedge Ratio
$ 120
0 ÷ 30 = 0.000
$ 110
10 ÷ 30 = 0.333
$ 100
20 ÷ 30 = 0.667
$ 90
30 ÷ 30 = 1.000
b.
As the option becomes more in the money, the hedge ratio increases to a maximum of 1.0.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
7.
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Consider a 1-year option with exercise price $50, on a stock with annual standard deviation 20%. The T-bill rate is 3% per year.
Find
N
(
d
1
)
for stock prices $45, $50, and $55.
Note: Do not round intermediate calculations. Round your answers to 4 decimal places.
S
N(d1)
$45
0.3910
$50
0.5987
$55
0.7662
Explanation:
Recall:
For example:
S
d
1
N
(
d
1
)
$ 45
−0.2768
0.3910
$ 50
0.2500
0.5987
$ 55
0.7266
0.7662
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
8.
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Data:
S
0
= 100
;
X
= 110; 1 +
r
= 1.10. The two possibilities for
S
T
are 130 and 80.
Required:
a.
The range of
S
is 50 while that of
P
is 30 across the two states. What is the hedge ratio of the put?
b.
Form a portfolio of three shares of stock and five puts. What is the (nonrandom) payoff to this portfolio?
c.
What is the present value of the portfolio?
d.
Given that the stock currently is selling at 100, calculate the put value.
Required A
Required B
Complete this question by entering your answers in the tabs below.
The range of S is 50 while that of P is 30 across the two states. What is the hedge ratio of the put?
Note: Round your answer to 1 decimal place. Negative value should be indicated by a minus sign.
Required A
Required B
Required C
Required D
Hedge ratio
(0.6)
Explanation:
a.
uS
0
= 130
P
u
= 0
dS
0
= 80
P
d
= 30
The hedge ratio is:
H
= (
P
u
−
P
d
) ÷ (
uS
0
−
dS
0
) = (0 − 30) ÷ (130 − 80) = −3/5 = −0.60
b.
Riskless Portfolio
S
T
= 80
S
T
= 130
Buy 3 shares
$ 240
$ 390
Buy 5 puts
150
0
Total
$ 390
$ 390
c.
Present value = $390 ÷ 1.10 = $354.545
d.
The portfolio cost is: 3
S
+ 5
P
= 300 + 5
P
The value of the portfolio is: $354.545
Therefore: 300 + 5
P
= $354.545 →
P
= $54.545 ÷ 5 = $10.91
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
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8.
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Problems?
Adjust credit
for all students.
Data:
S
0
= 100
;
X
= 110; 1 +
r
= 1.10. The two possibilities for
S
T
are 130 and 80.
Required:
a.
The range of
S
is 50 while that of
P
is 30 across the two states. What is the hedge ratio of the put?
b.
Form a portfolio of three shares of stock and five puts. What is the (nonrandom) payoff to this portfolio?
c.
What is the present value of the portfolio?
d.
Given that the stock currently is selling at 100, calculate the put value.
Required A
Required C
Complete this question by entering your answers in the tabs below.
Form a portfolio of three shares of stock and five puts. What is the (nonrandom) payoff to this portfolio?
Required A
Required B
Required C
Required D
$
Nonrandom payoff
390
Explanation:
a.
uS
0
= 130
P
u
= 0
dS
0
= 80
P
d
= 30
The hedge ratio is:
H
= (
P
u
−
P
d
) ÷ (
uS
0
−
dS
0
) = (0 − 30) ÷ (130 − 80) = −3/5 = −0.60
b.
Riskless Portfolio
S
T
= 80
S
T
= 130
Buy 3 shares
$ 240
$ 390
Buy 5 puts
150
0
Total
$ 390
$ 390
c.
Present value = $390 ÷ 1.10 = $354.545
d.
The portfolio cost is: 3
S
+ 5
P
= 300 + 5
P
The value of the portfolio is: $354.545
Therefore: 300 + 5
P
= $354.545 →
P
= $54.545 ÷ 5 = $10.91
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
8.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Data:
S
0
= 100
;
X
= 110; 1 +
r
= 1.10. The two possibilities for
S
T
are 130 and 80.
Required:
a.
The range of
S
is 50 while that of
P
is 30 across the two states. What is the hedge ratio of the put?
b.
Form a portfolio of three shares of stock and five puts. What is the (nonrandom) payoff to this portfolio?
c.
What is the present value of the portfolio?
d.
Given that the stock currently is selling at 100, calculate the put value.
Required B
Required D
Complete this question by entering your answers in the tabs below.
What is the present value of the portfolio?
Note: Round your answer to 2 decimal places.
Required A
Required B
Required C
Required D
$
Present value
354.55
Explanation:
a.
uS
0
= 130
P
u
= 0
dS
0
= 80
P
d
= 30
The hedge ratio is:
H
= (
P
u
−
P
d
) ÷ (
uS
0
−
dS
0
) = (0 − 30) ÷ (130 − 80) = −3/5 = −0.60
b.
Riskless Portfolio
S
T
= 80
S
T
= 130
Buy 3 shares
$ 240
$ 390
Buy 5 puts
150
0
Total
$ 390
$ 390
c.
Present value = $390 ÷ 1.10 = $354.545
d.
The portfolio cost is: 3
S
+ 5
P
= 300 + 5
P
The value of the portfolio is: $354.545
Therefore: 300 + 5
P
= $354.545 →
P
= $54.545 ÷ 5 = $10.91
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
8.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Data:
S
0
= 100
;
X
= 110; 1 +
r
= 1.10. The two possibilities for
S
T
are 130 and 80.
Required:
a.
The range of
S
is 50 while that of
P
is 30 across the two states. What is the hedge ratio of the put?
b.
Form a portfolio of three shares of stock and five puts. What is the (nonrandom) payoff to this portfolio?
c.
What is the present value of the portfolio?
d.
Given that the stock currently is selling at 100, calculate the put value.
Required C
Required D
Complete this question by entering your answers in the tabs below.
Given that the stock currently is selling at 100, calculate the put value.
Note: Round your answer to 2 decimal places.
Required A
Required B
Required C
Required D
$
Put value
10.91
Explanation:
a.
uS
0
= 130
P
u
= 0
dS
0
= 80
P
d
= 30
The hedge ratio is:
H
= (
P
u
−
P
d
) ÷ (
uS
0
−
dS
0
) = (0 − 30) ÷ (130 − 80) = −3/5 = −0.60
b.
Riskless Portfolio
S
T
= 80
S
T
= 130
Buy 3 shares
$ 240
$ 390
Buy 5 puts
150
0
Total
$ 390
$ 390
c.
Present value = $390 ÷ 1.10 = $354.545
d.
The portfolio cost is: 3
S
+ 5
P
= 300 + 5
P
The value of the portfolio is: $354.545
Therefore: 300 + 5
P
= $354.545 →
P
= $54.545 ÷ 5 = $10.91
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
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9.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Data:
S
0
= 100;
X
= 110; 1 +
r
= 1.10. The two possibilities for
S
T
are 130 and 80.
Required:
a-1.
The range of
S
is 50 while that of
C
is 20 across the two states. What is the hedge ratio of the call?
Note: Round your answer to 1 decimal place.
a-2.
Calculate the value of a
call
option on the stock with an exercise price of 110. (Do not use continuous compounding to calculate the present value of
X
in this example because we are using a two-state model here; the
assumed 10% interest rate is an effective rate per period.)
Note: Round your answer to 2 decimal places.
a-1. Hedge ratio
0.4
a-2. Call value
10.91
Explanation:
a-1.
The hedge ratio for the call is:
H
= (
C
u
−
C
d
) ÷ (
uS
0
−
dS
0
)
= (20 − 0) ÷ (130 − 80) = 2 ÷ 5 = 0.40
a-2.
Riskless Portfolio
S
= 80
S
= 130
Buy 2 shares
$ 160
$ 260
Write 5 calls
0
−100
Total
$ 160
$ 160
Present value = $160 ÷ 1.10 = $145.455
The portfolio cost is: 2S − 5
C
= $200 − 5
C
The value of the portfolio is $145.455
Therefore:
C
= $54.545 ÷ 5 = $10.91
Does
P
=
C
+ PV(
X
) −
S
?
10.91 = 10.91 + 110 ÷ 1.10 − 100 = 10.91
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
10.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Use the Black-Scholes formula for the following stock:
Time to expiration
6
months
Standard deviation
50%
per year
Exercise price
$ 50
Stock price
$ 50
Annual interest rate
3%
Dividend
0
Calculate the value of a call option.
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
$
Value of a call option
7.34
Explanation:
N
(
d
1
) =
N
(0.2192) = 0.5868
N
(
d
2
) =
N
(−0.1344) = 0.4466
X
×
e
−rT
= $49.2556
C
= $50 × 0.5868 − $49.2556 × 0.4465 = $7.34
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
11.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Use the Black-Scholes formula for the following stock:
Time to expiration
6
months
Standard deviation
50%
per year
Exercise price
$ 50
Stock price
$ 50
Annual interest rate
3%
Dividend
0
Calculate the value of a put option.
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
$
Value of a put option
6.60
Explanation:
N
(
d
1
) =
N
(0.2192) = 0.5868
N
(
d
2
) =
N
(−0.1344) = 0.4466
P
0
=
Xe
−
rT
[1 −
N
(
d
2
)] −
S
0
[1 −
N
(
d
1
)]
P
0
= $50e
−0.03×0.50
[1 − 0.4466] − $50[1 − 0.5868]
= $6.60
This value is derived from our Black-Scholes, but it also could have been derived using put-call parity:
P
=
C
+ PV(
X
) −
S
0
= $7.34 + $49.26 − $50 = $6.60
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
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12.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Use the Black-Scholes formula for the following stock:
Time to expiration
6 months
Standard deviation
50% per year
Exercise price
$ 50
Stock price
$ 50
Annual interest rate
3%
Dividend
0
Recalculate the value of the call with the following changes:
a. Time to expiration
3 months
b. Standard deviation
25% per year
c. Exercise price
$ 55
d. Stock price
$ 55
e. Interest rate
5%
Select each scenario independently.
Note: Input all amounts as a positive value. Round your answers to 2 decimal places.
$
$
$
$
$
Value of the
Call Option
a.
C falls to
5.14
b.
C falls to
3.88
c.
C falls to
5.40
d.
C rises to
10.54
e.
C rises to
7.56
Explanation:
No further explanation details are available for this problem.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
13.
Award:
10.00
points
Problems?
Adjust credit
for all students.
A call option with
X
= $50 on a stock priced at
S
= $55 sells for $10. Using a volatility estimate of σ = 0.30, you find that
N
(
d
1
) = 0.6
and
N
(
d
2
) = 0.5
. The risk-free interest rate is zero. Is the implied volatility based on the
option price more or less than 0.30?
Is the implied volatility based on the option price more or less than 0.30?
Greater than 0.30
Explanation:
According to the Black-Scholes model, the call option should be priced at
[$55 ×
N
(
d
1
)] − [$50 ×
N
(
d
2
)] = ($55 × 0.6) − ($50 × 0.5) = $8
Since the option sells for more than $8, implied volatility is greater than 0.30.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
14.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Mark Washington, CFA, is an analyst with BIC. One year ago, BIC analysts predicted that the U.S. equity market would experience a slight downturn and suggested delta-hedging the BIC portfolio. U.S. equity markets did indeed
fall, but BIC’s portfolio performance was disappointing, lagging its peer group by nearly 10%. Washington is reviewing the options strategy to determine why the hedged portfolio did not perform as expected.
Which of the following best explains a delta-neutral portfolio? The return on a delta-neutral portfolio is hedged against:
Which of the following
best
explains a delta-neutral portfolio?
small price changes in the underlying asset.
Explanation:
A delta-neutral portfolio is perfectly hedged against small price changes in the underlying asset. This is true both for price increases and decreases. That is, the portfolio value will not change significantly if the asset price
changes by a small amount. However, large changes in the underlying asset will cause the hedge to become imperfect. This means that overall portfolio value can change by a significant amount if the price change in the
underlying asset is large.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
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15.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Mark Washington, CFA, is an analyst with BIC. One year ago, BIC analysts predicted that the U.S. equity market would experience a slight downturn and suggested delta-hedging the BIC portfolio. U.S. equity markets did indeed
fall, but BIC’s portfolio performance was disappointing, lagging its peer group by nearly 10%. Washington is reviewing the options strategy to determine why the hedged portfolio did not perform as expected.
After discussing the concept of a delta-neutral portfolio, Washington determines that he needs to further explain the concept of delta. He draws the value of an option as a function of the underlying stock price. In such a diagram,
the option’s delta is the:
In such a diagram, the option’s delta is the:
slope in the option price diagram.
Explanation:
Delta is the change in the option price for a given instantaneous change in the stock price. The change is equal to the slope of the option price diagram.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
16.
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Problems?
Adjust credit
for all students.
Mark Washington, CFA, is an analyst with BIC. One year ago, BIC analysts predicted that the U.S. equity market would experience a slight downturn and suggested delta-hedging the BIC portfolio. U.S. equity markets did indeed
fall, but BIC’s portfolio performance was disappointing, lagging its peer group by nearly 10%. Washington is reviewing the options strategy to determine why the hedged portfolio did not perform as expected.
Washington considers a put option with a delta of −0.65. If the price of the underlying asset decreases by $6, what is the best estimate of the change in option price?
Note: Round your answer to 2 decimal places.
$
Best estimate price
3.90
Explanation:
The best estimate for the change in price of the option is:
Asset Price
× Delta = −$6 × (−0.65) = $3.90
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
17.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Mark Washington, CFA, is an analyst with BIC. One year ago, BIC analysts predicted that the U.S. equity market would experience a slight downturn and suggested delta-hedging the BIC portfolio. U.S. equity markets did indeed
fall, but BIC’s portfolio performance was disappointing, lagging its peer group by nearly 10%. Washington is reviewing the options strategy to determine why the hedged portfolio did not perform as expected.
BIC owns 51,750 shares of Smith & Oates. The shares are currently priced at $69. A call option on Smith & Oates with a strike price of $70 is selling at $3.50 and has a delta of 0.69. How many call options should be written to
make BIC’s overall position delta-neutral? Should the calls be purchased or written?
Number of call options
75,000
. The calls should be
written
Explanation:
The number of call options necessary to delta hedge is 51,750 ÷ 0.69 = 75,000 options, or
750 options contracts
, each covering 100 shares. Since these are call options, the options should be
sold short
.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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18.
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Problems?
Adjust credit
for all students.
Mark Washington, CFA, is an analyst with BIC. One year ago, BIC analysts predicted that the U.S. equity market would experience a slight downturn and suggested delta-hedging the BIC portfolio. U.S. equity markets did indeed
fall, but BIC’s portfolio performance was disappointing, lagging its peer group by nearly 10%. Washington is reviewing the options strategy to determine why the hedged portfolio did not perform as expected.
BIC owns 51,750 shares of Smith & Oates. The shares are currently priced at $69. A call option on Smith & Oates with a strike price of $70 is selling at $3.50 and has a delta of 0.69. Does the number of call options that BIC
must write to maintain a delta-neutral position increase or decrease if the stock price falls?
Does the number of call options that BIC must write to maintain a delta -neutral position increase or decrease if the stock price falls?
Increase
Explanation:
The number of calls needed to create a delta-neutral hedge is inversely proportional to the delta. The delta decreases when stock price decreases. Therefore, the number of calls necessary would increase if the stock price falls.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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for all students.
Mark Washington, CFA, is an analyst with BIC. One year ago, BIC analysts predicted that the U.S. equity market would experience a slight downturn and suggested delta-hedging the BIC portfolio. U.S. equity markets did indeed
fall, but BIC’s portfolio performance was disappointing, lagging its peer group by nearly 10%. Washington is reviewing the options strategy to determine why the hedged portfolio did not perform as expected.
Which of the following statements regarding the goal of a delta-neutral portfolio is
most
accurate? One example of a delta-neutral portfolio is to combine a:
Which of the following statements regarding the goal of a
delta-neutral portfolio is most accurate?
long position in a stock with a short position in call options so that the value of the portfolio does
not change with changes in the value of the stock.
Explanation:
A delta-neutral portfolio can be created with any of the following combinations: long stock and short calls, long stock and long puts, short stock and long calls, and short stock and short puts.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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According to the Black-Scholes formula, what will be the hedge ratio (delta) of a call option as the stock price becomes infinitely large?
Hedge ratio approaches
1
Explanation:
The hedge ratio approaches one. As
S
increases, the probability of exercise approaches 1.0.
N
(
d
1
)
approaches 1.0.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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According to the Black-Scholes formula, what will be the hedge ratio (delta) of a put option for a very small exercise price?
Value of the hedge ratio
0
Explanation:
The hedge ratio approaches 0. As
X
decreases, the probability of exercise approaches 0.
[
N
(
d
1
) −1]
approaches 0 as
N
(
d
1
)
approaches 1.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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The hedge ratio of an at-the-money call option on IBM is 0.4. The hedge ratio of an at-the-money put option is −0.6. What is the hedge ratio of an at-the-money straddle position on IBM?
Note: Negative answer should be indicated by a minus sign. Round your answer to 1 decimal place.
Hedge ratio
(0.2)
Explanation:
A straddle is a call and a put. The hedge ratio of the straddle is the sum of the hedge ratios of the individual options: 0.4 + (−0.6) = −0.2
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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A collar is established by buying a share of stock for $50, buying a 6-month put option with exercise price $45, and writing a 6-month call option with exercise price $55. On the basis of the volatility of the stock, you calculate that
for a strike price of $45 and expiration of six months,
N
(
d
1
) = 0.60
, whereas for the exercise price of $55,
N
(
d
1
) = 0.35.
Required:
a.
What will be the gain or loss on the collar if the stock price increases by $1?
b.
What happens to the delta of the portfolio if the stock price becomes very large?
c.
What happens to the delta of the portfolio if the stock price becomes very small?
Required A
Required B
Complete this question by entering your answers in the tabs below.
What will be the gain or loss on the collar if the stock price increases by $1?
Note: Round your answer to 2 decimal places.
Required A
Required B
Required C
$
Collar
gain
of
0.25
Explanation:
a.
The delta of the collar is calculated as follows:
Position
Delta
Buy stock
1.00
Buy put,
X
= $ 45
N
(
d
1
) − 1 = −0.40
Write call,
X
= $ 55
−
N
(
d
1
) = −0.35
Total
$ 0.25
If the stock price increases by $1, then the value of the collar increases by $0.25. The stock will be worth $1 more, the loss on the purchased put will be $0.40, and the call written represents a
liability
that increases by $0.35.
b.
If
S
becomes very large, then the delta of the collar approaches zero. Both
N
(
d
1
)
terms approach 1. Intuitively, for very large stock prices, the value of the portfolio is simply the (present value of the) exercise price of the call
and is unaffected by small changes in the stock price.
c.
As
S
approaches zero, the delta also approaches zero: both
N
(
d
1
)
terms approach 0. For very small stock prices, the value of the portfolio is simply the (present value of the) exercise price of the put and is unaffected by
small changes in the stock price.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
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23.
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Problems?
Adjust credit
for all students.
A collar is established by buying a share of stock for $50, buying a 6-month put option with exercise price $45, and writing a 6-month call option with exercise price $55. On the basis of the volatility of the stock, you calculate that
for a strike price of $45 and expiration of six months,
N
(
d
1
) = 0.60
, whereas for the exercise price of $55,
N
(
d
1
) = 0.35.
Required:
a.
What will be the gain or loss on the collar if the stock price increases by $1?
b.
What happens to the delta of the portfolio if the stock price becomes very large?
c.
What happens to the delta of the portfolio if the stock price becomes very small?
Required A
Required C
Complete this question by entering your answers in the tabs below.
What happens to the delta of the portfolio if the stock price becomes very large?
Required A
Required B
Required C
$
Delta of the portfolio approaches
0
Explanation:
a.
The delta of the collar is calculated as follows:
Position
Delta
Buy stock
1.00
Buy put,
X
= $ 45
N
(
d
1
) − 1 = −0.40
Write call,
X
= $ 55
−
N
(
d
1
) = −0.35
Total
$ 0.25
If the stock price increases by $1, then the value of the collar increases by $0.25. The stock will be worth $1 more, the loss on the purchased put will be $0.40, and the call written represents a
liability
that increases by $0.35.
b.
If
S
becomes very large, then the delta of the collar approaches zero. Both
N
(
d
1
)
terms approach 1. Intuitively, for very large stock prices, the value of the portfolio is simply the (present value of the) exercise price of the call
and is unaffected by small changes in the stock price.
c.
As
S
approaches zero, the delta also approaches zero: both
N
(
d
1
)
terms approach 0. For very small stock prices, the value of the portfolio is simply the (present value of the) exercise price of the put and is unaffected by
small changes in the stock price.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
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23.
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Problems?
Adjust credit
for all students.
A collar is established by buying a share of stock for $50, buying a 6-month put option with exercise price $45, and writing a 6-month call option with exercise price $55. On the basis of the volatility of the stock, you calculate that
for a strike price of $45 and expiration of six months,
N
(
d
1
) = 0.60
, whereas for the exercise price of $55,
N
(
d
1
) = 0.35.
Required:
a.
What will be the gain or loss on the collar if the stock price increases by $1?
b.
What happens to the delta of the portfolio if the stock price becomes very large?
c.
What happens to the delta of the portfolio if the stock price becomes very small?
Required B
Required C
Complete this question by entering your answers in the tabs below.
What happens to the delta of the portfolio if the stock price becomes very small?
Required A
Required B
Required C
$
Delta of the portfolio approaches
0
Explanation:
a.
The delta of the collar is calculated as follows:
Position
Delta
Buy stock
1.00
Buy put,
X
= $ 45
N
(
d
1
) − 1 = −0.40
Write call,
X
= $ 55
−
N
(
d
1
) = −0.35
Total
$ 0.25
If the stock price increases by $1, then the value of the collar increases by $0.25. The stock will be worth $1 more, the loss on the purchased put will be $0.40, and the call written represents a
liability
that increases by $0.35.
b.
If
S
becomes very large, then the delta of the collar approaches zero. Both
N
(
d
1
)
terms approach 1. Intuitively, for very large stock prices, the value of the portfolio is simply the (present value of the) exercise price of the call
and is unaffected by small changes in the stock price.
c.
As
S
approaches zero, the delta also approaches zero: both
N
(
d
1
)
terms approach 0. For very small stock prices, the value of the portfolio is simply the (present value of the) exercise price of the put and is unaffected by
small changes in the stock price.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
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These three put options are all written on the same stock. One has a delta of −0.9, one a delta of −0.5, and one a delta of −0.1. Assign deltas to the three puts by filling in this table.
Note: Negative value should be indicated by a minus sign. Round your answers to 1 decimal place.
Put
X
Delta
A
10
(0.1)
B
20
(0.5)
C
30
(0.9)
Explanation:
No further explanation details are available for this problem.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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You are
very
bullish (optimistic) on stock EFG, much more so than the rest of the market. In each question, choose the portfolio strategy that will give you the biggest dollar profit if your bullish forecast turns out to be correct.
Required:
a.
Choice A
: $10,000 invested in EFG calls with X = 50.
Choice B
: $10,000 invested in EFG stock.
b.
Choice A
: 10 EFG call option contracts (for 100 shares each), with X = 50.
Choice B
: 1,000 shares of EFG stock.
a. Choose the bigger dollar profit
Choice A
b. Choose the bigger dollar profit
Choice B
Explanation:
a.
Choice A: Calls have higher elasticity than shares. For equal dollar investments, a call’s capital gain potential is greater than that of the underlying stock.
b.
Choice B: Calls have hedge ratios less than 1.0, so the shares have higher profit potential. For an equal number of shares controlled, the dollar exposure of the shares is greater than that of the calls, and the profit potential is
therefore greater.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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You would like to be holding a protective put position on the stock of XYZ Company to lock in a guaranteed minimum value of $100 at year-end. XYZ currently sells for $100. Over the next year, the stock price will increase by
10% or decrease by 10%. The T-bill rate is 5%. Unfortunately, no put options are traded on XYZ Company.
Required:
a.
Suppose the desired put option were traded. How much would it cost to purchase?
b.
What would have been the cost of the protective put portfolio?
c.
What portfolio position in stock and T-bills will ensure you a payoff equal to the payoff that would be provided by a protective put with
X
= 100? Show that the payoff to this portfolio and the cost of establishing the portfolio
match those of the desired protective put.
Required A
Required B
Complete this question by entering your answers in the tabs below.
Suppose the desired put option were traded. How much would it cost to purchase?
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
Required A
Required B
Required C
$
Cost to purchase
2.38
Explanation:
a.
uS
0
= 110
P
u
= 0
dS
0
= 90
P
d
= 10
The hedge ratio is:
H
= (
P
u
−
P
d
) ÷ (
uS
0
−
dS
0
) = (0 − 10) ÷ (110 − 90) = −1/2
A portfolio comprised of one share and two puts provides a guaranteed payoff of $110, with present value: $110 ÷ 1.05 = $104.76
Therefore:
S
+ 2
P
= $104.76
$100 + 2
P
= $104.76
P
= $2.38
b.
Cost of protective put portfolio with a $100 guaranteed payoff:
= $100 + $2.38 = $102.38
c.
Our goal is a portfolio with the same exposure to the stock as the hypothetical protective put portfolio. Since the put’s hedge ratio is −0.5, the portfolio consists of (1 − 0.5) = 0.5 shares of stock, which costs $50, and the
remaining funds ($52.38) invested in T-bills, earning 5% interest.
This payoff is identical to that of the protective put portfolio. Thus, the stock plus bills strategy replicates both the cost and payoff of the protective put.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
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26.
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Problems?
Adjust credit
for all students.
You would like to be holding a protective put position on the stock of XYZ Company to lock in a guaranteed minimum value of $100 at year-end. XYZ currently sells for $100. Over the next year, the stock price will increase by
10% or decrease by 10%. The T-bill rate is 5%. Unfortunately, no put options are traded on XYZ Company.
Required:
a.
Suppose the desired put option were traded. How much would it cost to purchase?
b.
What would have been the cost of the protective put portfolio?
c.
What portfolio position in stock and T-bills will ensure you a payoff equal to the payoff that would be provided by a protective put with
X
= 100? Show that the payoff to this portfolio and the cost of establishing the portfolio
match those of the desired protective put.
Required A
Required C
Complete this question by entering your answers in the tabs below.
What would have been the cost of the protective put portfolio?
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
Required A
Required B
Required C
$
Cost of the protective put portfolio
102.38
Explanation:
a.
uS
0
= 110
P
u
= 0
dS
0
= 90
P
d
= 10
The hedge ratio is:
H
= (
P
u
−
P
d
) ÷ (
uS
0
−
dS
0
) = (0 − 10) ÷ (110 − 90) = −1/2
A portfolio comprised of one share and two puts provides a guaranteed payoff of $110, with present value: $110 ÷ 1.05 = $104.76
Therefore:
S
+ 2
P
= $104.76
$100 + 2
P
= $104.76
P
= $2.38
b.
Cost of protective put portfolio with a $100 guaranteed payoff:
= $100 + $2.38 = $102.38
c.
Our goal is a portfolio with the same exposure to the stock as the hypothetical protective put portfolio. Since the put’s hedge ratio is −0.5, the portfolio consists of (1 − 0.5) = 0.5 shares of stock, which costs $50, and the
remaining funds ($52.38) invested in T-bills, earning 5% interest.
This payoff is identical to that of the protective put portfolio. Thus, the stock plus bills strategy replicates both the cost and payoff of the protective put.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
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26.
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Problems?
Adjust credit
for all students.
You would like to be holding a protective put position on the stock of XYZ Company to lock in a guaranteed minimum value of $100 at year-end. XYZ currently sells for $100. Over the next year, the stock price will increase by
10% or decrease by 10%. The T-bill rate is 5%. Unfortunately, no put options are traded on XYZ Company.
Required:
a.
Suppose the desired put option were traded. How much would it cost to purchase?
b.
What would have been the cost of the protective put portfolio?
c.
What portfolio position in stock and T-bills will ensure you a payoff equal to the payoff that would be provided by a protective put with
X
= 100? Show that the payoff to this portfolio and the cost of establishing the portfolio
match those of the desired protective put.
Required B
Required C
Complete this question by entering your answers in the tabs below.
What portfolio position in stock and T-bills will ensure you a payoff equal to the payoff that would be provided by a protective
put with X = 100? Show that the payoff to this portfolio and the cost of establishing the portfolio match those of the desired
protective put.
Required A
Required B
Required C
Portfolio
S = 90
S = 110
Buy 0.5 shares
45
55
Invest in T-bills
55
55
Total
100
110
Explanation:
a.
uS
0
= 110
P
u
= 0
dS
0
= 90
P
d
= 10
The hedge ratio is:
H
= (
P
u
−
P
d
) ÷ (
uS
0
−
dS
0
) = (0 − 10) ÷ (110 − 90) = −1/2
A portfolio comprised of one share and two puts provides a guaranteed payoff of $110, with present value: $110 ÷ 1.05 = $104.76
Therefore:
S
+ 2
P
= $104.76
$100 + 2
P
= $104.76
P
= $2.38
b.
Cost of protective put portfolio with a $100 guaranteed payoff:
= $100 + $2.38 = $102.38
c.
Our goal is a portfolio with the same exposure to the stock as the hypothetical protective put portfolio. Since the put’s hedge ratio is −0.5, the portfolio consists of (1 − 0.5) = 0.5 shares of stock, which costs $50, and the
remaining funds ($52.38) invested in T-bills, earning 5% interest.
This payoff is identical to that of the protective put portfolio. Thus, the stock plus bills strategy replicates both the cost and payoff of the protective put.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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for all students.
Suppose that the risk-free interest rate is zero. Would an American put option ever be exercised early?
No
Explanation:
If
r
= 0, never exercise a put early. There is no time value cost to waiting to exercise, but there is a volatility benefit from waiting. To demonstrate rigorously, consider the following portfolio: lend $
X
and short one share of stock.
The cost to establish the portfolio is
(
X
−
S
0
)
. The payoff at time T (with zero interest earnings on the loan) is
(
X
−
S
T
)
. In contrast, a put option has a payoff at time T of
(
X
−
S
T
)
if that value is positive, and zero
otherwise. The put’s payoff is at least as large as the portfolio’s, and therefore, the put must cost at least as much as the portfolio to purchase. Hence,
P
≥ (
X
−
S
0
)
, and the put can be sold for more than the proceeds from
immediate exercise. We conclude that it doesn’t pay to exercise early.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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You are attempting to value a call option with an exercise price of $100 and one year to expiration. The underlying stock pays no dividends, its current price is $100, and you believe it has a 50% chance of increasing to $120 and
a 50% chance of decreasing to $80. The risk-free rate of interest is 10%. Calculate the call option’s value using the two-state stock price model.
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
$
Call option’s value
13.64
Explanation:
Step 1: Calculate the option values at expiration. The two possible stock prices and the corresponding call values are:
uS
0
= $120
C
u
= 20
dS
0
= $80
C
d
= 0
Step 2: Calculate the hedge ratio.
H
= (
C
u
−
C
d
) ÷ (
uS
0
−
dS
0
) = (20 − 0) ÷ ($120 − $80) = 1/2
Therefore, form a riskless portfolio by buying one share of stock and writing two calls. The cost of the portfolio is
S
− 2
C
= $100 − 2
C
Step 3: Show that the payoff for the riskless portfolio equals $80:
Riskless Portfolio
S
= $80
S
= $120
Buy 1 share
$ 80
$ 120
Write 2 calls
0
−40
Total
$ 80
$ 80
Therefore, find the value of the call by solving
$100 − 2
C
= $80 ÷ 1.10
C
= $13.64
Notice that we did not use the probabilities of a stock price increase or decrease. These are not needed to value the call option.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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You are attempting to value a call option with an exercise price of $100 and one year to expiration. The underlying stock pays no dividends, its current price is $100, and you believe it has a 50% chance of increasing to $130 and
a 50% chance of decreasing to $70. The risk-free rate of interest is 10%. Calculate the call option's value using the two-state stock price model.
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
$
Value of the call option
18.18
Explanation:
The two possible stock prices and the corresponding call values are:
uS
0
= $130
C
u
= 30
dS
0
= $70
C
d
= 0
The hedge ratio is:
H
= (
C
u
−
C
d
) ÷ (
uS
0
−
dS
0
) = (30 − 0) ÷ ($130 − $70) = 1/2
Form a riskless portfolio by buying one share of stock and writing two calls. The cost of the portfolio is:
S
− 2
C
= $100 − 2
C
The payoff for the riskless portfolio equals $70:
Riskless Portfolio
S
= $70
S
= $130
Buy 1 share
$ 70
$ 130
Write 2 calls
0
−60
Total
$ 70
$ 70
Therefore, find the value of the call by solving
$100 − 2
C
= $70 ÷ 1.10
C
= $18.18
Here, the value of the call is greater than the value in the lower-volatility scenario.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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You are attempting to value a put option with an exercise price of $100 and one year to expiration. The underlying stock pays no dividends, its current price is $100, and you believe it has a 50% chance of increasing to $120 and
a 50% chance of decreasing to $80. The risk-free rate of interest is 10%. Calculate the value of a put option with exercise price $100.
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
$
Value of a put option
4.55
Explanation:
The two possible stock prices and the corresponding put values are:
uS
0
= $120
P
u
= 0
dS
0
= $80
P
d
= 20
The hedge ratio is:
H
= (
P
u
−
P
d
) ÷ (
uS
0
−
dS
0
) = (0 − 20) ÷ ($120 − $80) = −1/2
Form a riskless portfolio by buying one share of stock and buying two puts. The cost of the portfolio is:
S
+ 2
P
= $100 + 2
P
The payoff for the riskless portfolio equals $120:
Riskless Portfolio
S = $80
S = $120
Buy 1 share
$ 80
$ 120
Buy 2 puts
40
0
Total
$ 120
$ 120
Therefore, find the value of the put by solving
$100 + 2
P
= $120 ÷ 1.10
P
= $4.545
According to put-call parity
P
+
S
=
C
+ PV(
X
)
Our estimates of option value satisfy this relationship:
$4.545 + $100 = $13.636 + ($100 ÷ 1.10) = $104.545
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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XYZ Corporation will pay a $2 per share dividend in two months. Its stock price currently is $60 per share. A European call option on XYZ has an exercise price of $55 and 3-month time to expiration. The risk-free interest rate is
0.5% per month, and the stock’s volatility (standard deviation) = 7% per month. Find the Black-Scholes value of the option. (
Hint
: Try defining one “period” as a month, rather than as a year, and think about the net-of-dividend
value of each share.)
Note: Round your answer to 2 decimal places.
$
Black-Scholes value of the option
6.04
Explanation:
If we assume that the only possible exercise date is just prior to the ex-dividend date, then the relevant parameters for the Black-Scholes formula are:
S
0
= $60
r
= 0.5% per month
X
= $55
σ
= 7%
T
= 2 months
In this case:
C
= $6.04
If instead, one commits to foregoing early exercise, then we reduce the stock price by the present value of the dividends. Therefore, we use the following parameters:
S
0
= $60 −
2
e
−(0.005 × 2)
= $58.02
r
= 0.5% per month
X
= $55
σ
= 7%
T
= 3 months
In this case,
C
= $5.05
The pseudo-American option value is the higher of these two values: $6.04
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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“The beta of a call option on FedEx is greater than the beta of a share of FedEx.” True or false?
True
Explanation:
True. The call option has an elasticity greater than 1.0. Therefore, the call’s percentage rate of return is greater than that of the underlying stock. Hence the FedEx call responds more than proportionately when the FedEx stock
price changes in response to broad market movements. Therefore, the beta of the FedEx call is greater than the beta of FedEx stock.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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“The beta of a call option on the S&P 500 index with an exercise price of 3,430 is greater than the beta
of a call on the index with an exercise price of 3,440.” True or false?
False
Explanation:
False. The elasticity of a call option is higher the more out of the money is the option. (Even though the delta of the call is lower, the value of the call is also lower. The proportional response of the call price to the stock price
increases. You can confirm this with numerical examples.) Therefore, the rate of return of the call with the higher exercise price responds more sensitively to changes in the market index, and therefore it has the higher beta.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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You are holding call options on a stock. The stock’s beta is 0.75, and you are concerned that the stock market is about to fall. The stock is currently selling for $5 and you hold 1 million options (i.e., you hold 10,000 contracts for
100 shares each). The option delta is 0.8. How much of the market-index portfolio must you buy or sell to hedge your market exposure?
Note: Enter your answer in dollar not in millions.
$
Market index portfolio to
sell
3,000,000
Explanation:
If the stock market index increases 1%, the 1 million shares of stock on which the options are written would be expected to increase by
0.75% × $5 × 1 million = $37,500
The options would increase by:
Delta × $37,500 = 0.8 × $37,500 = $30,000
To hedge your market exposure, you must sell $3,000,000 of the market index portfolio so that a 1% change in the index would result in a $30,000 change in the value of the portfolio.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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You are a provider of portfolio insurance and are establishing a 4-year program. The portfolio you manage is worth $100 million, and you hope to provide a minimum return of 0%. The equity portfolio has a standard deviation of
25% per year, and T-bills pay 5% per year. Assume that the portfolio pays no dividends.
Required:
a-1.
What is the delta of the implicit put option conveyed by the portfolio insurance?
a-2.
How much of the portfolio should be sold and placed in bills?
b-1.
What is the delta of the new portfolio falls by 3% on the first day of trading?
b-2.
Complete the following:
Req A1
Req A2
Complete this question by entering your answers in the tabs below.
What is the delta of the implicit put option conveyed by the portfolio insurance?
Note: Do not round intermediate calculations. Negative amount should be indicated by a minus sign. Round your answer to 4
decimal places.
Req A1
Req A2
Req B1
Req B2
The put delta is
(0.2578)
Explanation:
S
= $100; current value of portfolio
X
= $100; floor promised to clients (0% return)
σ = 0.25; volatility
r
= 0.05; risk-free rate
T
= 4 years; horizon of program
a.
Using the Black-Scholes formula, we find that:
d
1
= 0.65,
N
(
d
1
) = 0.7422,
d
2
= 0.15,
N
(
d
2
) = 0.5596
Put value = $10.27
Therefore, total funds to be managed equals $110.27 million: $100 million portfolio value plus the $10.27 million fee for the insurance program.
The put delta is
N
(
d
1) − 1 = 0.7422 − 1 = −0.2578
Therefore, sell off $25.78 million of the equity portfolio, placing the remaining funds in T-bills. The amount of the portfolio in equity is therefore $74.22 million, while the amount in T-bills is: $110.27 million − $74.22 million =
$36.06 million
b.
At the new portfolio value of 97, the put delta is
N
(
d
1
) − 1 = 0.7221 − 1 = −0.2779
This means that you must reduce the delta of the portfolio by
0.2779 − 0.2578 = 0.0201
You should sell an additional 2.01% of the equity position and use the proceeds to buy T-bills. Since the stock price is now at only 97% of its original value, you need to sell
$97 million × 0.0201 = $1.9455 million of stock
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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35.
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for all students.
You are a provider of portfolio insurance and are establishing a 4-year program. The portfolio you manage is worth $100 million, and you hope to provide a minimum return of 0%. The equity portfolio has a standard deviation of
25% per year, and T-bills pay 5% per year. Assume that the portfolio pays no dividends.
Required:
a-1.
What is the delta of the implicit put option conveyed by the portfolio insurance?
a-2.
How much of the portfolio should be sold and placed in bills?
b-1.
What is the delta of the new portfolio falls by 3% on the first day of trading?
b-2.
Complete the following:
Req A1
Req B1
Complete this question by entering your answers in the tabs below.
How much of the portfolio should be sold and placed in bills?
Note: Do not round intermediate calculations. Enter your answer in millions rounded to 2 decimal places.
Req A1
Req A2
Req B1
Req B2
$
T-bills
36.06
million
Explanation:
S
= $100; current value of portfolio
X
= $100; floor promised to clients (0% return)
σ = 0.25; volatility
r
= 0.05; risk-free rate
T
= 4 years; horizon of program
a.
Using the Black-Scholes formula, we find that:
d
1
= 0.65,
N
(
d
1
) = 0.7422,
d
2
= 0.15,
N
(
d
2
) = 0.5596
Put value = $10.27
Therefore, total funds to be managed equals $110.27 million: $100 million portfolio value plus the $10.27 million fee for the insurance program.
The put delta is
N
(
d
1) − 1 = 0.7422 − 1 = −0.2578
Therefore, sell off $25.78 million of the equity portfolio, placing the remaining funds in T-bills. The amount of the portfolio in equity is therefore $74.22 million, while the amount in T-bills is: $110.27 million − $74.22 million =
$36.06 million
b.
At the new portfolio value of 97, the put delta is
N
(
d
1
) − 1 = 0.7221 − 1 = −0.2779
This means that you must reduce the delta of the portfolio by
0.2779 − 0.2578 = 0.0201
You should sell an additional 2.01% of the equity position and use the proceeds to buy T-bills. Since the stock price is now at only 97% of its original value, you need to sell
$97 million × 0.0201 = $1.9455 million of stock
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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You are a provider of portfolio insurance and are establishing a 4-year program. The portfolio you manage is worth $100 million, and you hope to provide a minimum return of 0%. The equity portfolio has a standard deviation of
25% per year, and T-bills pay 5% per year. Assume that the portfolio pays no dividends.
Required:
a-1.
What is the delta of the implicit put option conveyed by the portfolio insurance?
a-2.
How much of the portfolio should be sold and placed in bills?
b-1.
What is the delta of the new portfolio falls by 3% on the first day of trading?
b-2.
Complete the following:
Req A2
Req B2
Complete this question by entering your answers in the tabs below.
What is the delta of the new portfolio falls by 3% on the first day of trading?
Note: Do not round intermediate calculations. Negative amount should be indicated by a minus sign. Round your answer to 4
decimal places.
Req A1
Req A2
Req B1
Req B2
Delta of the portfolio
(0.2779)
Explanation:
S
= $100; current value of portfolio
X
= $100; floor promised to clients (0% return)
σ = 0.25; volatility
r
= 0.05; risk-free rate
T
= 4 years; horizon of program
a.
Using the Black-Scholes formula, we find that:
d
1
= 0.65,
N
(
d
1
) = 0.7422,
d
2
= 0.15,
N
(
d
2
) = 0.5596
Put value = $10.27
Therefore, total funds to be managed equals $110.27 million: $100 million portfolio value plus the $10.27 million fee for the insurance program.
The put delta is
N
(
d
1) − 1 = 0.7422 − 1 = −0.2578
Therefore, sell off $25.78 million of the equity portfolio, placing the remaining funds in T-bills. The amount of the portfolio in equity is therefore $74.22 million, while the amount in T-bills is: $110.27 million − $74.22 million =
$36.06 million
b.
At the new portfolio value of 97, the put delta is
N
(
d
1
) − 1 = 0.7221 − 1 = −0.2779
This means that you must reduce the delta of the portfolio by
0.2779 − 0.2578 = 0.0201
You should sell an additional 2.01% of the equity position and use the proceeds to buy T-bills. Since the stock price is now at only 97% of its original value, you need to sell
$97 million × 0.0201 = $1.9455 million of stock
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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35.
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Problems?
Adjust credit
for all students.
You are a provider of portfolio insurance and are establishing a 4-year program. The portfolio you manage is worth $100 million, and you hope to provide a minimum return of 0%. The equity portfolio has a standard deviation of
25% per year, and T-bills pay 5% per year. Assume that the portfolio pays no dividends.
Required:
a-1.
What is the delta of the implicit put option conveyed by the portfolio insurance?
a-2.
How much of the portfolio should be sold and placed in bills?
b-1.
What is the delta of the new portfolio falls by 3% on the first day of trading?
b-2.
Complete the following:
Req B1
Req B2
Complete this question by entering your answers in the tabs below.
Complete the following:
Note: Do not round intermediate calculations. Enter your answer in millions rounded to 4 decimal places.
Req A1
Req A2
Req B1
Req B2
$
Assuming the portfolio does fall by 3%, the manager should
sell
1.9455
in stock.
Explanation:
S
= $100; current value of portfolio
X
= $100; floor promised to clients (0% return)
σ = 0.25; volatility
r
= 0.05; risk-free rate
T
= 4 years; horizon of program
a.
Using the Black-Scholes formula, we find that:
d
1
= 0.65,
N
(
d
1
) = 0.7422,
d
2
= 0.15,
N
(
d
2
) = 0.5596
Put value = $10.27
Therefore, total funds to be managed equals $110.27 million: $100 million portfolio value plus the $10.27 million fee for the insurance program.
The put delta is
N
(
d
1) − 1 = 0.7422 − 1 = −0.2578
Therefore, sell off $25.78 million of the equity portfolio, placing the remaining funds in T-bills. The amount of the portfolio in equity is therefore $74.22 million, while the amount in T-bills is: $110.27 million − $74.22 million =
$36.06 million
b.
At the new portfolio value of 97, the put delta is
N
(
d
1
) − 1 = 0.7221 − 1 = −0.2779
This means that you must reduce the delta of the portfolio by
0.2779 − 0.2578 = 0.0201
You should sell an additional 2.01% of the equity position and use the proceeds to buy T-bills. Since the stock price is now at only 97% of its original value, you need to sell
$97 million × 0.0201 = $1.9455 million of stock
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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Suppose that call options on ExxonMobil stock with time to expiration 3 months and strike price $90 are selling at an implied volatility of 30%. ExxonMobil stock price is $90 per share, and the risk-free rate is 4%.
Required:
a.
If you believe the true volatility of the stock is 32%, would you want to buy or sell call options?
b.
Now you want to hedge your option position against changes in the stock price. How many shares of stock will you hold for each option contract purchased or sold?
Required A
Required B
Complete this question by entering your answers in the tabs below.
If you believe the true volatility of the stock is 32%, would you want to buy or sell call options?
Required A
Required B
Would you want to buy or sell call options?
Buy call options
Explanation:
a.
Because you believe the calls are underpriced (selling at an implied volatility that is too low), you will buy calls.
b.
Using the true volatility (32%) and time to expiration T = 0.25 years, the hedge ratio for ExxonMobil is
N
(
d
1
) = 0.5567
; short 0.5567 shares for each call you buy.
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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Suppose that call options on ExxonMobil stock with time to expiration 3 months and strike price $90 are selling at an implied volatility of 30%. ExxonMobil stock price is $90 per share, and the risk-free rate is 4%.
Required:
a.
If you believe the true volatility of the stock is 32%, would you want to buy or sell call options?
b.
Now you want to hedge your option position against changes in the stock price. How many shares of stock will you hold for each option contract purchased or sold?
Required A
Required B
Complete this question by entering your answers in the tabs below.
Now you want to hedge your option position against changes in the stock price. How many shares of stock will you hold for
each option contract purchased or sold?
Note: Round your answer to 4 decimal places.
Required A
Required B
Number of shares
0.5567
Explanation:
a.
Because you believe the calls are underpriced (selling at an implied volatility that is too low), you will buy calls.
b.
Using the true volatility (32%) and time to expiration T = 0.25 years, the hedge ratio for ExxonMobil is
N
(
d
1
) = 0.5567
; short 0.5567 shares for each call you buy.
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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Suppose that JPMorgan Chase sells call options on $2.5 million worth of a stock portfolio with beta = 1.5. The option delta is 0.8. It wishes to hedge its resultant exposure to a market advance by buying a market-index portfolio.
Suppose it use market index puts to hedge its exposure. The index at current prices represents $4,000 worth of stock and the contract multiplier is 400.
Required:
a.
How many dollars’ worth of the market-index portfolio should it purchase?
b.
What is the delta of a put option?
c.
Complete the following:
Required A
Required B
Complete this question by entering your answers in the tabs below.
How many dollars’ worth of the market-index portfolio should it purchase?
Note: Do not round intermediate calculations. Enter your answer in dollars and not in millions.
Required A
Required B
Required C
$
Market index portfolio
3,000,000
Explanation:
a.
To calculate the hedge ratio, suppose that the market index increases by 1%. Then the stock portfolio would be expected to increase by:
1% × 1.5 = 1.5% or 0.015 × $2,500,000 = $37,500
Given the option delta of 0.8, the option portfolio would increase by
$37,500 × 0.8 = $30,000
JP Morgan Chase’s liability from writing these options would increase by the same amount. The market index portfolio would increase in value by 1%. Therefore, JP Morgan should purchase $3,000,000 of the market index
portfolio to hedge its position so that a 1% change in the index would result in a $30,000 change in the value of the portfolio.
b.
The delta of a put option is 0.8 − 1 = −0.2
c.
Therefore, for every 1% the market increases, the index will rise by 10 points and the value of the put option contract will change by
Delta × 10 × Contract multiplier = −0.2 × 10 × 400 = −$800
Therefore, JP Morgan should write $30,000 ÷ $800 = 37.5 put contracts.
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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Suppose that JPMorgan Chase sells call options on $2.5 million worth of a stock portfolio with beta = 1.5. The option delta is 0.8. It wishes to hedge its resultant exposure to a market advance by buying a market-index portfolio.
Suppose it use market index puts to hedge its exposure. The index at current prices represents $4,000 worth of stock and the contract multiplier is 400.
Required:
a.
How many dollars’ worth of the market-index portfolio should it purchase?
b.
What is the delta of a put option?
c.
Complete the following:
Required A
Required C
Complete this question by entering your answers in the tabs below.
What is the delta of a put option?
Note: Round your answer to 1 decimal place. Negative amount should be indicated by a minus sign.
Required A
Required B
Required C
Delta
(0.2)
Explanation:
a.
To calculate the hedge ratio, suppose that the market index increases by 1%. Then the stock portfolio would be expected to increase by:
1% × 1.5 = 1.5% or 0.015 × $2,500,000 = $37,500
Given the option delta of 0.8, the option portfolio would increase by
$37,500 × 0.8 = $30,000
JP Morgan Chase’s liability from writing these options would increase by the same amount. The market index portfolio would increase in value by 1%. Therefore, JP Morgan should purchase $3,000,000 of the market index
portfolio to hedge its position so that a 1% change in the index would result in a $30,000 change in the value of the portfolio.
b.
The delta of a put option is 0.8 − 1 = −0.2
c.
Therefore, for every 1% the market increases, the index will rise by 10 points and the value of the put option contract will change by
Delta × 10 × Contract multiplier = −0.2 × 10 × 400 = −$800
Therefore, JP Morgan should write $30,000 ÷ $800 = 37.5 put contracts.
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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for all students.
Suppose that JPMorgan Chase sells call options on $2.5 million worth of a stock portfolio with beta = 1.5. The option delta is 0.8. It wishes to hedge its resultant exposure to a market advance by buying a market-index portfolio.
Suppose it use market index puts to hedge its exposure. The index at current prices represents $4,000 worth of stock and the contract multiplier is 400.
Required:
a.
How many dollars’ worth of the market-index portfolio should it purchase?
b.
What is the delta of a put option?
c.
Complete the following:
Required B
Required C
Complete this question by entering your answers in the tabs below.
Complete the following:
Note: Round your answer to 1 decimal place.
Required A
Required B
Required C
Assuming the 1 percent market movement, JP Morgan should
sell
37.5
put contracts.
Explanation:
a.
To calculate the hedge ratio, suppose that the market index increases by 1%. Then the stock portfolio would be expected to increase by:
1% × 1.5 = 1.5% or 0.015 × $2,500,000 = $37,500
Given the option delta of 0.8, the option portfolio would increase by
$37,500 × 0.8 = $30,000
JP Morgan Chase’s liability from writing these options would increase by the same amount. The market index portfolio would increase in value by 1%. Therefore, JP Morgan should purchase $3,000,000 of the market index
portfolio to hedge its position so that a 1% change in the index would result in a $30,000 change in the value of the portfolio.
b.
The delta of a put option is 0.8 − 1 = −0.2
c.
Therefore, for every 1% the market increases, the index will rise by 10 points and the value of the put option contract will change by
Delta × 10 × Contract multiplier = −0.2 × 10 × 400 = −$800
Therefore, JP Morgan should write $30,000 ÷ $800 = 37.5 put contracts.
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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Suppose you are attempting to value a 1-year expiration option on a stock with volatility (i.e., annualized standard deviation) of σ = 0.40.
What would be the appropriate values for
u
and
d
if your binomial model is set up using:
a.
1 period of 1 year.
b.
4 subperiods, each 3 months.
c.
12 subperiods, each 1 month.
Note: Do not round intermediate calculations. Round your answers to 4 decimal places.
Subperiods
Δt = T/n
u = exp(σ√ Δt)
d = exp(-σ√ Δt)
a.
1
1/1 = 1
1.4918
0.6703
b.
4
1/4 = 0.25
1.2214
0.8187
c.
12
1/12 = 0.0833
1.1224
0.8909
Explanation:
No further explanation details are available for this problem.
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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You are attempting to value a put option with an exercise price of $100 and one year to expiration. The underlying stock pays no dividends, its current price is $100, and you believe it has a 50% chance of increasing to $120 and
a 50% chance of decreasing to $80. The risk-free rate of interest is 10%.
Required:
a.
What will be the payoff to the put,
P
u
, if the stock goes up?
b.
What will be the payoff,
P
d
, if the stock price falls?
c.
What is the weighted average value of the pay off?
Required A
Required B
Complete this question by entering your answers in the tabs below.
What will be the payoff to the put,
P
u
, if the stock goes up?
Required A
Required B
Required C
$
Payoff
0
Explanation:
a.
If the stock price rises, the payoff will be zero as profit from a put is only made when the stock price is less than the exercise price.
b.
If the stock price falls below the $100 exercise price, the payoff will be the difference between the two prices.
c.
Using the risk-neutral shortcut
p
= (1 +
r
f
−
d
) ÷ (
u
−
d
) = (1 + 0.1 − 0.8) ÷ (1.2 − 0.8) = 0.75
After one year, the stock price will have either a 75% chance of rising to $120 (a $0 payoff) or a 25% chance of falling to $80 ($20 payoff). Discounting the weighted average of these payoffs by the risk-free rate gives us
(0.75 × 0 + 0.25 × $20) ÷ (1 + 0.10) = 4.55
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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Problems?
Adjust credit
for all students.
You are attempting to value a put option with an exercise price of $100 and one year to expiration. The underlying stock pays no dividends, its current price is $100, and you believe it has a 50% chance of increasing to $120 and
a 50% chance of decreasing to $80. The risk-free rate of interest is 10%.
Required:
a.
What will be the payoff to the put,
P
u
, if the stock goes up?
b.
What will be the payoff,
P
d
, if the stock price falls?
c.
What is the weighted average value of the pay off?
Required A
Required C
Complete this question by entering your answers in the tabs below.
What will be the payoff,
P
d
, if the stock price falls?
Required A
Required B
Required C
$
Payoff
20
Explanation:
a.
If the stock price rises, the payoff will be zero as profit from a put is only made when the stock price is less than the exercise price.
b.
If the stock price falls below the $100 exercise price, the payoff will be the difference between the two prices.
c.
Using the risk-neutral shortcut
p
= (1 +
r
f
−
d
) ÷ (
u
−
d
) = (1 + 0.1 − 0.8) ÷ (1.2 − 0.8) = 0.75
After one year, the stock price will have either a 75% chance of rising to $120 (a $0 payoff) or a 25% chance of falling to $80 ($20 payoff). Discounting the weighted average of these payoffs by the risk-free rate gives us
(0.75 × 0 + 0.25 × $20) ÷ (1 + 0.10) = 4.55
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
References
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39.
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Problems?
Adjust credit
for all students.
You are attempting to value a put option with an exercise price of $100 and one year to expiration. The underlying stock pays no dividends, its current price is $100, and you believe it has a 50% chance of increasing to $120 and
a 50% chance of decreasing to $80. The risk-free rate of interest is 10%.
Required:
a.
What will be the payoff to the put,
P
u
, if the stock goes up?
b.
What will be the payoff,
P
d
, if the stock price falls?
c.
What is the weighted average value of the pay off?
Required B
Required C
Complete this question by entering your answers in the tabs below.
What is the weighted average value of the pay off?
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
Required A
Required B
Required C
Discounting weighted average
4.55
Explanation:
a.
If the stock price rises, the payoff will be zero as profit from a put is only made when the stock price is less than the exercise price.
b.
If the stock price falls below the $100 exercise price, the payoff will be the difference between the two prices.
c.
Using the risk-neutral shortcut
p
= (1 +
r
f
−
d
) ÷ (
u
−
d
) = (1 + 0.1 − 0.8) ÷ (1.2 − 0.8) = 0.75
After one year, the stock price will have either a 75% chance of rising to $120 (a $0 payoff) or a 25% chance of falling to $80 ($20 payoff). Discounting the weighted average of these payoffs by the risk-free rate gives us
(0.75 × 0 + 0.25 × $20) ÷ (1 + 0.10) = 4.55
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Problems - Algorithmic & Static
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Problems?
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Required:
a.
Calculate the intrinsic value for each of the following call options.
b.
Now assume that the effective annual interest rate is 7.06%, which corresponds to a monthly interest rate of 0.57%. Calculate the present value of each call option’s exercise price and the adjusted intrinsic value for each call
option.
Required A
Required B
Complete this question by entering your answers in the tabs below.
Calculate the intrinsic value for each of the following call options.
Note: Round your answers to 2 decimal places.
Required A
Required B
Company
Time to
Expiration
(months)
Strike
S
0
Intrinsic Value
RJay
1
60
63.07
3.07
RJay
2
70
63.11
0.00
Sell-Mart
5
60
65.80
5.80
Xenon
6
7.50
7.08
0.00
Explanation:
a.
The intrinsic value of a call option equals either (
S
0
− Strike price) or zero, whichever value is greater.
b.
The adjusted intrinsic value equals
S
0
− PV(
X
).
The present value of the exercise price is
PV(
X
) =
X ÷
(1 +
r
)
n
where
r
equals the monthly interest rate and
n
equals the number of months to expiration. The calculations for the Sell-Mart call option with 5 months to expiration are shown below.
PV(
X
) = 60 ÷ (
1.0057)
5
= 58.32
Adjusted intrinsic value = 65.80 − 58.32 = 7.48
Company
Time to
Expiration
(months)
Strike
S
0
Intrinsic
Value
PV(X)
Adjusted
Intrinsic
Value
RJay
1
60.00
63.07
3.07
59.66
3.41
RJay
2
70.00
63.11
0.00
69.21
0.00
Sell-Mart
5
60.00
65.80
5.80
58.32
7.48
Xenon
6
7.50
7.08
0.00
7.25
0.00
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Additional Algorithmic Problems
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Problems?
Adjust credit
for all students.
Required:
a.
Calculate the intrinsic value for each of the following call options.
b.
Now assume that the effective annual interest rate is 7.06%, which corresponds to a monthly interest rate of 0.57%. Calculate the present value of each call option’s exercise price and the adjusted intrinsic value for each call
option.
Required A
Required B
Complete this question by entering your answers in the tabs below.
Now assume that the effective annual interest rate is 7.06%, which corresponds to a monthly interest rate of 0.57%.
Calculate the present value of each call option’s exercise price and the adjusted intrinsic value for each call option.
Note: Round your answers to 2 decimal places.
Required A
Required B
Company
Time to
Expiration
(months)
Strike
S
0
PV(
X
)
Adjusted
Intrinsic Value
RJay
1
60
63.07
59.66
3.41
RJay
2
70
63.11
69.21
0.00
Sell-Mart
5
60
65.80
58.32
7.48
Xenon
6
7.50
7.08
7.25
0.00
Explanation:
a.
The intrinsic value of a call option equals either (
S
0
− Strike price) or zero, whichever value is greater.
b.
The adjusted intrinsic value equals
S
0
− PV(
X
).
The present value of the exercise price is
PV(
X
) =
X ÷
(1 +
r
)
n
where
r
equals the monthly interest rate and
n
equals the number of months to expiration. The calculations for the Sell-Mart call option with 5 months to expiration are shown below.
PV(
X
) = 60 ÷ (
1.0057)
5
= 58.32
Adjusted intrinsic value = 65.80 − 58.32 = 7.48
Company
Time to
Expiration
(months)
Strike
S
0
Intrinsic
Value
PV(X)
Adjusted
Intrinsic
Value
RJay
1
60.00
63.07
3.07
59.66
3.41
RJay
2
70.00
63.11
0.00
69.21
0.00
Sell-Mart
5
60.00
65.80
5.80
58.32
7.48
Xenon
6
7.50
7.08
0.00
7.25
0.00
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Additional Algorithmic Problems
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for all students.
You want to buy a stock that is currently selling for $45. You forecast that in one year, the stock’s price will be either $104 or $8, with equal probabilities. There is a one-year call option on the stock available with an exercise price
of $80. You are able to borrow at a rate of 6.50%. You would like to hedge your stock position using the call option.
Required:
a.
What will be the call’s value if the stock price is $104 in one year? What will be the call’s value if the stock price is $8 in one year?
b.
What is the hedge ratio you should use?
c.
Assume that you can purchase fractional shares of stock. How many shares of stock would you buy?
Required A
Required B
Complete this question by entering your answers in the tabs below.
What will be the call’s value if the stock price is $104 in one year? What will be the call’s value if the stock price is $8 in one
year?
Note: Round your answers to the nearest dollar.
Required A
Required B
Required C
$
$
Call value at $104.00
24
Call value at $8.00
0
Explanation:
a.
If the stock’s price is $104 in one year, the call will be exercised and its value will be $104 − 80 = $24. If the stock price is $8.00 in one year, the call will expire worthless. The value trees are shown below.
$60
$104
Call Price
$24
$8
$0
Stock's Value Call Option's Value
b.
The hedge ratio equals the spread in the option values divided by the spread in the stock values.
H
= (
C
u
−
C
d
) ÷ (
uS
0
−
dS
0
) = (24.00 − 0) ÷ (104.00 − 8.00) = 0.2500
c.
You would purchase 0.2500 shares of stock and sell one call option.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Additional Algorithmic Problems
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Problems?
Adjust credit
for all students.
You want to buy a stock that is currently selling for $45. You forecast that in one year, the stock’s price will be either $104 or $8, with equal probabilities. There is a one-year call option on the stock available with an exercise price
of $80. You are able to borrow at a rate of 6.50%. You would like to hedge your stock position using the call option.
Required:
a.
What will be the call’s value if the stock price is $104 in one year? What will be the call’s value if the stock price is $8 in one year?
b.
What is the hedge ratio you should use?
c.
Assume that you can purchase fractional shares of stock. How many shares of stock would you buy?
Required A
Required C
Complete this question by entering your answers in the tabs below.
What is the hedge ratio you should use?
Note: Round your answer to 4 decimal places.
Required A
Required B
Required C
Hedge ratio
0.2500
Explanation:
a.
If the stock’s price is $104 in one year, the call will be exercised and its value will be $104 − 80 = $24. If the stock price is $8.00 in one year, the call will expire worthless. The value trees are shown below.
$60
$104
Call Price
$24
$8
$0
Stock's Value Call Option's Value
b.
The hedge ratio equals the spread in the option values divided by the spread in the stock values.
H
= (
C
u
−
C
d
) ÷ (
uS
0
−
dS
0
) = (24.00 − 0) ÷ (104.00 − 8.00) = 0.2500
c.
You would purchase 0.2500 shares of stock and sell one call option.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Additional Algorithmic Problems
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41.
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Problems?
Adjust credit
for all students.
You want to buy a stock that is currently selling for $45. You forecast that in one year, the stock’s price will be either $104 or $8, with equal probabilities. There is a one-year call option on the stock available with an exercise price
of $80. You are able to borrow at a rate of 6.50%. You would like to hedge your stock position using the call option.
Required:
a.
What will be the call’s value if the stock price is $104 in one year? What will be the call’s value if the stock price is $8 in one year?
b.
What is the hedge ratio you should use?
c.
Assume that you can purchase fractional shares of stock. How many shares of stock would you buy?
Required B
Required C
Complete this question by entering your answers in the tabs below.
Assume that you can purchase fractional shares of stock. How many shares of stock would you buy?
Note: Round your answer to 4 decimal places.
Required A
Required B
Required C
Shares
0.2500
Explanation:
a.
If the stock’s price is $104 in one year, the call will be exercised and its value will be $104 − 80 = $24. If the stock price is $8.00 in one year, the call will expire worthless. The value trees are shown below.
$60
$104
Call Price
$24
$8
$0
Stock's Value Call Option's Value
b.
The hedge ratio equals the spread in the option values divided by the spread in the stock values.
H
= (
C
u
−
C
d
) ÷ (
uS
0
−
dS
0
) = (24.00 − 0) ÷ (104.00 − 8.00) = 0.2500
c.
You would purchase 0.2500 shares of stock and sell one call option.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Additional Algorithmic Problems
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Use the Black-Scholes formula to find the value of a call option based on the following inputs.
Note: Do not round intermediate calculations. Round your final answer to 2 decimal places.
Stock price
$ 51
Exercise price
$ 64
Interest rate
0.068
Dividend yield
0.04
Time to expiration
0.50
Standard deviation of stock’s returns
0.265
$
Call value
0.66
Explanation:
The Black-Scholes formula is
C
0
= S
0
e
−δT
N
(
d
1
)
− Xe
−rT
N
(
d
2
)
d
1
= ((ln(
S
0
÷
X) +
(
r
− + (
σ
2
÷ 2))
T
) ÷
= (ln(51.00 ÷ 64.00) + (0.068 − 0.04 +
0.265
2
÷ 2) × 0.5) ÷ (0.265 ×
)
= (− 0.23 + 0.03) ÷ 0.187 = −1.043
d
2
=
d
1
− ;
= − 1.043 − 0.265;
= −1.231
N
(
d
1
) is the area under the normal distribution curve up to the value of
d
1
, which can be found from the table in the text, or from the NORMSDIST function in Excel.
N
(
d
1
) =
N
(−1.043) = 0.148
N
(
d
2
) =
N
(−1.231) = 0.109
C
0
= $0.109 ×
e
−(0.04 × 0.5)
× 0.148 − $64.00 ×
e
−(0.068 × 0.5)
× 0.109
= $7.42 − 6.76 = $0.66
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Additional Algorithmic Problems
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Problems?
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for all students.
You observe a premium of $6.66 for a call option on Birdwell Enterprises common stock, which is currently selling for $46. The strike price on the call option is $47. The option has four months to maturity. The stock pays no
dividends. The current risk-free interest rate is 3.50%. What is the implied volatility of the stock?
Note: Round your answer to the nearest whole percent.
Implied volatility
65 %
Explanation:
The standard deviation (implied volatility) that is consistent with the Black-Scholes formula is .7.
Stock price
$ 46
Exercise price
$ 47
Interest rate
0.035
Dividend yield
0
Time to expiration
0.33333
Standard deviation of stock’s returns
s = ?
Based on the formula
C
0
= S
0
e
−δT
N
(
d
1
)
− Xe
−rT
N
(
d
2
)
,
where
d
1
= ((ln(
S
0
÷
X) +
(
r
− + (
σ
2
÷ 2))
T
) ÷
= (ln(46.00 ÷ 47.00) + (0.035 − 0 +
σ
2
÷ 2) × 0.3) ÷ (
σ
×
)
= d
2
= d
1
− σ
= d
1
− σ ×
By trial and error, a standard deviation of 0.7 will make
C
0
= $6.66, so the implied volatility of this stock is 0.65.
Using s = 0.7 yields
d
1
= 0.1614 and
d
2
= −0.2139, so
C
0
=
$
46.00 × e
−(0 × 0.33333)
× 0.5640 − $47.00 × e
−(0.33 × 0.33333)
× 0.4150
= $25.95 − 19.29
= $6.66
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Additional Algorithmic Problems
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44.
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10.00
points
Problems?
Adjust credit
for all students.
Use the Black-Scholes model to find the value for a European put option that has an exercise price of $62 and four months to expiration. The underlying stock is selling for $65 currently and pays an annual dividend of $1.77.
The standard deviation of the stock’s returns is 0.185 and risk-free interest rate is 0.045%.
Note: Round intermediary calculations to 4 decimal places. Round your final answer to 2 decimal places.
$
Put value
1.33
Explanation:
According to the Black-Scholes model,
P
=
Xe
−rT
[
1 − N(d
2
)] −
S
0
e
−δT
[1 − N(d
1
)]
The dividend yield,
, equals $1.77 ÷ $65 = 0.0272 and T = 4 ÷ 12 = .3333.
d
1
= (ln(S
0
÷ X) + (r −
+ (
σ
2
÷ 2)T ) ÷
= (ln(65 ÷ 62) + (0.045 − 0.0272 +
0.185
2
÷ 2) × 0.3333) ÷ (0.185 ×
) = 0.5513
d
2
=
d
1
−
= 0.5513
− 0.185 ×
= 0.4445
N(
d
1
) = 0.7093, N(
d
2
) = 0.6716
P
= 62
e
−(0.045 × 0.3333)
[1 − 0.6716] − 65
e
−0.0272 × 0.3333
[1 − 0.7093] = 1.33
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Additional Algorithmic Problems
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Calculate the elasticity of a call option with a premium of $4.50 and a strike price of $67. The call has a hedge ratio of 0.7, and the underlying stock’s price is currently $59.
Note: Round your answer to 2 decimal places.
Elasticity of the call
9.18
%
Explanation:
If the stock’s price rises by $1, that will be a percentage increase of $1 ÷ $59 = 1.695%.
Because the stock’s price has risen, the call’s premium will increase. In terms of dollars, the call premium will go up by the amount of the hedge ratio times $1, or $0.70. That’s a percentage change of $0.70 ÷ $4.50 = 15.56%.
Elasticity of the call = % Change in call premium ÷ % Change in stock price = 15.56% ÷ 1.695% = 9.18
At a stock price level of $59, for an increase of 1% in the stock’s price, the call option’s value will increase by 9.18%.
Worksheet
Difficulty: 1 Basic
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 21: Option Valuation > Chapter 21 Additional Algorithmic Problems
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1.
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Before expiration, the time value of an in-the-money call option is always:
equal to zero.
positive.
negative.
equal to the stock price minus the exercise price.
None of the options are correct.
The difference between the actual option price and the intrinsic value is called the time value of the option. Time value is always positive before expiration.
References
Multiple Choice
Difficulty: 1 Basic
Before expiration, the time value of an in-the-money put option is always:
equal to zero.
negative.
positive.
equal to the stock price minus the exercise price.
None of the options are correct.
The difference between the actual option price and the intrinsic value is called the time value of the option. Time value is always positive before expiration.
References
Multiple Choice
Difficulty: 1 Basic
Before expiration, the time value of an at-the-money call option is usually:
positive.
equal to zero.
negative.
equal to the stock price minus the exercise price.
None of the options are correct.
The difference between the actual option price and the intrinsic value is called the time value of the option. Time value is usually positive before expiration.
References
Multiple Choice
Difficulty: 1 Basic
Before expiration, the time value of an at-the-money put option is always:
equal to zero.
equal to the stock price minus the exercise price.
negative.
positive.
None of the options are correct.
The difference between the actual option price and the intrinsic value is called the time value of the option. Time value is always positive before expiration.
References
Multiple Choice
Difficulty: 1 Basic
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5.
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At expiration, the time value of an in-the-money call option is always:
equal to zero.
positive.
negative.
equal to the stock price minus the exercise price.
None of the options are correct.
The difference between the actual option price and the intrinsic value is called the time value of the option. Time value is always zero at expiration.
References
Multiple Choice
Difficulty: 1 Basic
At expiration, the time value of an in-the-money put option is always:
equal to zero.
negative.
positive.
equal to the stock price minus the exercise price.
None of the options are correct.
The difference between the actual option price and the intrinsic value is called the time value of the option. Time value is always zero at expiration.
References
Multiple Choice
Difficulty: 1 Basic
At expiration, the time value of an at-the-money call option is always:
positive.
equal to zero.
negative.
equal to the stock price minus the exercise price.
None of the options are correct.
The difference between the actual option price and the intrinsic value is called the time value of the option. Time value is always zero at expiration.
References
Multiple Choice
Difficulty: 1 Basic
At expiration, the time value of an at-the-money put option is always:
equal to zero.
equal to the stock price minus the exercise price.
negative.
positive.
None of the options are correct.
The difference between the actual option price and the intrinsic value is called the time value of the option. Time value is always zero at expiration.
References
Multiple Choice
Difficulty: 1 Basic
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A call option has an intrinsic value of zero if the option is:
at the money, only.
out of the money, only.
in the money, only.
at the money and in the money.
at the money or out of the money.
Intrinsic value can never be negative; thus it is set equal to zero for out of the money and at the money options.
References
Multiple Choice
Difficulty: 1 Basic
A put option has an intrinsic value of zero if the option is:
at the money.
out of the money.
in the money.
at the money and in the money.
at the money or out of the money.
Intrinsic value can never be negative; thus it is set equal to zero for out of the money and at the money options.
References
Multiple Choice
Difficulty: 1 Basic
Prior to expiration,:
the intrinsic value of a call option is greater than its actual value.
the intrinsic value of a call option is always positive.
the actual value of a call option is greater than the intrinsic value.
the intrinsic value of a call option is always greater than its time value.
None of the options are correct.
Prior to expiration, any option will be selling for a positive price, thus the actual value is usually greater than the intrinsic value.
References
Multiple Choice
Difficulty: 2 Intermediate
Prior to expiration,:
the intrinsic value of a put option is greater than its actual value.
the intrinsic value of a put option is always positive.
the actual value of a put option is greater than the intrinsic value.
the intrinsic value of a put option is always greater than its time value.
None of the options are correct.
Prior to expiration, any option will be selling for a positive price, thus the actual value is greater than the intrinsic value.
References
Multiple Choice
Difficulty: 2 Intermediate
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If the stock price increases, the price of a put option on that stock _________, and that of a call option _________.
decreases; increases
decreases; decreases
increases; decreases
increases; increases
does not change; does not change
As stock prices increase, call options become more valuable (the owner can buy the stock at a bargain price). As stock prices increase, put options become less valuable (the owner can sell the stock at a price less than market
price).
References
Multiple Choice
Difficulty: 2 Intermediate
If the stock price decreases, the price of a put option on that stock _________, and that of a call option _________.
decreases; increases
decreases; decreases
increases; decreases
increases; increases
does not change; does not change
As stock prices decrease, call options become less valuable (the owner cannot buy the stock at a bargain price). As stock prices decrease, put options become more valuable (the owner can sell the stock at a price greater than
market price).
References
Multiple Choice
Difficulty: 2 Intermediate
Other things equal, the price of a stock call option is positively correlated with the following factors
except
:
the stock price.
the time to expiration.
the stock volatility.
the exercise price.
None of the options are correct.
The exercise price is negatively correlated with the call option price.
References
Multiple Choice
Difficulty: 2 Intermediate
Other things equal, the price of a stock call option is positively correlated with which of the following factors?
The stock price, only
The time to expiration, only
The stock volatility, only
The exercise price, only
The stock price, time to expiration, and stock volatility
The exercise price is negatively correlated with the call option price.
References
Multiple Choice
Difficulty: 2 Intermediate
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Other things equal, the price of a stock call option is negatively correlated with which of the following factors?
The stock price, only
The time to expiration, only
The stock volatility, only
The exercise price, only
The stock price, time to expiration, and stock volatility
The exercise price is negatively correlated with the call option price.
References
Multiple Choice
Difficulty: 2 Intermediate
Other things equal, the price of a stock put option is positively correlated with the following factors
except
:
the stock price.
the time to expiration.
the stock volatility.
the exercise price.
None of the options are correct.
The put option price is negatively correlated with the stock price.
References
Multiple Choice
Difficulty: 2 Intermediate
Other things equal, the price of a stock put option is positively correlated with which of the following factors?
The stock price, only
The time to expiration, only
The stock volatility, only
The exercise price, only
The time to expiration, stock volatility, and exercise price
The put option price is negatively correlated with the stock price.
References
Multiple Choice
Difficulty: 2 Intermediate
Other things equal, the price of a stock put option is negatively correlated with which of the following factors?
The stock price, only
The time to expiration, only
The stock volatility, only
The exercise price, only
The time to expiration, stock volatility, and exercise price
The exercise price is negatively correlated with the stock price.
References
Multiple Choice
Difficulty: 2 Intermediate
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The price of a stock put option is _________ correlated with the stock price and _________ correlated with the strike price.
positively; positively
negatively; positively
negatively; negatively
positively; negatively
not; not
The lower the stock price, the more valuable the put option. The higher the striking price, the more valuable the put option.
References
Multiple Choice
Difficulty: 2 Intermediate
The price of a stock call option is _________ correlated with the stock price and _________ correlated with the strike price.
positively; positively
negatively; positively
negatively; negatively
positively; negatively
not; not
The higher the stock price, the more valuable the call option. The lower the striking price, the more valuable the call option.
References
Multiple Choice
Difficulty: 2 Intermediate
All the inputs in the Black-Scholes option pricing model are directly observable
except
:
the price of the underlying security.
the risk-free rate of interest.
the time to expiration.
the variance of returns of the underlying asset return.
None of the options are correct.
The variance of the returns of the underlying asset is not directly observable, but must be estimated from historical data, from scenario analysis, or from the prices of other options.
References
Multiple Choice
Difficulty: 2 Intermediate
Which of the inputs in the Black-Scholes option pricing model are directly observable?
The price of the underlying security
The risk-free rate of interest
The time to expiration
The variance of returns of the underlying asset return
The price of the underlying security, risk-free rate of interest, and time to expiration
The variance of the returns of the underlying asset is not directly observable, but must be estimated from historical data, from scenario analysis, or from the prices of other options.
References
Multiple Choice
Difficulty: 2 Intermediate
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Delta is defined as:
the change in the value of an option for a dollar change in the price of the underlying asset.
the change in the value of the underlying asset for a dollar change in the call price.
the percentage change in the value of an option for a 1% change in the value of the underlying asset.
the change in the volatility of the underlying stock price.
None of the options are correct.
An option's hedge ratio (delta) is the change in the price of an option for a $1 increase in the stock price.
References
Multiple Choice
Difficulty: 2 Intermediate
A hedge ratio of 0.70 implies that a hedged portfolio should consist of:
long 0.70 calls for each short stock.
short 0.70 calls for each long stock.
long 0.70 shares for each short call.
long 0.70 shares for each long call.
None of the options are correct.
The hedge ratio is the slope of the option value as a function of the stock value. A slope of 0.70 means that as the stock increases in value by $1, the option increases by approximately $0.70. Thus, for every call written, 0.70 shares
of stock would be needed to hedge the investor's portfolio.
References
Multiple Choice
Difficulty: 2 Intermediate
A hedge ratio of 0.85 implies that a hedged portfolio should consist of:
long 0.85 calls for each short stock.
short 0.85 calls for each long stock.
long 0.85 shares for each short call.
long 0.85 shares for each long call.
None of the options are correct.
The hedge ratio is the slope of the option value as a function of the stock value. A slope of 0.85 means that as the stock increases in value by $1, the option increases by approximately $0.85. Thus, for every call written, 0.85
shares of stock would be needed to hedge the investor's portfolio.
References
Multiple Choice
Difficulty: 2 Intermediate
A hedge ratio for a call option is _________, and a hedge ratio for a put option is _________.
negative; positive
negative; negative
positive; negative
positive; positive
zero; zero
Call option hedge ratios must be positive and less than 1.0, and put option ratios must be negative, with a smaller absolute value than 1.0.
References
Multiple Choice
Difficulty: 2 Intermediate
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A hedge ratio for a call is always:
equal to one.
greater than one.
between zero and one.
between negative one and zero.
of no restricted value.
Call option hedge ratios must be positive and less than 1.0, and put option ratios must be negative, with a smaller absolute value than 1.0.
References
Multiple Choice
Difficulty: 2 Intermediate
A hedge ratio for a put is always:
equal to one.
greater than one.
between zero and one.
between negative one and zero.
of no restricted value.
Call option hedge ratios must be positive and less than 1.0, and put option ratios must be negative, with a smaller absolute value than 1.0.
References
Multiple Choice
Difficulty: 2 Intermediate
The dollar change in the value of a stock call option is always:
lower than the dollar change in the value of the stock.
higher than the dollar change in the value of the stock.
negatively correlated with the change in the value of the stock.
higher than the dollar change in the value of the stock and negatively correlated with the change in the value of the stock.
lower than the dollar change in the value of the stock and negatively correlated with the change in the value of the stock.
The slope of the call option valuation function is less than one.
References
Multiple Choice
Difficulty: 2 Intermediate
The percentage change in the stock call-option price divided by the percentage change in the stock price is called:
the elasticity of the option.
the delta of the option.
the theta of the option.
the gamma of the option.
None of the options are correct.
Option price elasticity measures the percent change in the option price as a function of the percent change in the stock price.
References
Multiple Choice
Difficulty: 2 Intermediate
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The elasticity of an option is:
the volatility level for the stock that the option price implies.
the continued updating of the hedge ratio as time passes.
the percentage change in the stock call-option price divided by the percentage change in the stock price.
the sensitivity of the delta to the stock price.
None of the options are correct.
Option price elasticity measures the percent change in the option price as a function of the percent change in the stock price.
References
Multiple Choice
Difficulty: 2 Intermediate
The elasticity of a stock call option is always:
greater than one.
smaller than one.
negative.
infinite.
None of the options are correct.
Option prices are much more volatile than stock prices, as option premiums are much lower than stock prices.
References
Multiple Choice
Difficulty: 2 Intermediate
The elasticity of a stock put option is always:
positive.
smaller than one.
negative.
infinite.
None of the options are correct.
As put options become more valuable as stock prices decline, the elasticity of a put option must be negative.
References
Multiple Choice
Difficulty: 2 Intermediate
The gamma of an option is:
the volatility level for the stock that the option price implies.
the continued updating of the hedge ratio as time passes.
the percentage change in the stock call-option price divided by the percentage change in the stock price.
the sensitivity of the delta to the stock price.
None of the options are correct.
The gamma of an option is the sensitivity of the delta to the stock price.
References
Multiple Choice
Difficulty: 2 Intermediate
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Delta neutral:
is the volatility level for the stock that the option price implies.
is the continued updating of the hedge ratio as time passes.
is the percentage change in the stock call-option price divided by the percentage change in the stock price.
means the portfolio has no tendency to change value as the underlying portfolio value changes.
None of the options are correct.
Delta neutral means the portfolio has no tendency to change value as the underlying portfolio value changes.
References
Multiple Choice
Difficulty: 2 Intermediate
Dynamic hedging is:
the volatility level for the stock that the option price implies.
the continued updating of the hedge ratio as time passes.
the percentage change in the stock call-option price divided by the percentage change in the stock price.
the sensitivity of the delta to the stock price.
None of the options are correct.
Dynamic hedging is the continued updating of the hedge ratio as time passes.
References
Multiple Choice
Difficulty: 2 Intermediate
Volatility risk is:
the volatility level for the stock that the option price implies.
the risk incurred from unpredictable changes in volatility.
the percentage change in the stock call-option price divided by the percentage change in the stock price.
the sensitivity of the delta to the stock price.
None of the options are correct.
Volatility risk is the risk incurred from unpredictable changes in volatility.
References
Multiple Choice
Difficulty: 2 Intermediate
Portfolio A consists of 150 shares of stock and 300 calls on that stock. Portfolio B consists of 575 shares of stock. The call delta is 0.7. Which portfolio has a higher dollar exposure to a change in stock price?
Portfolio B
Portfolio A
The two portfolios have the same exposure.
Portfolio A if the stock price increases and portfolio B if it decreases
Portfolio B if the stock price increases and portfolio A if it decreases
300 calls × 0.7 + 150 shares = 210 shares + 150 shares = 360 shares;
575 shares > 360 Shares
References
Multiple Choice
Difficulty: 3 Challenge
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Portfolio A consists of 500 shares of stock and 500 calls on that stock. Portfolio B consists of 800 shares of stock. The call delta is 0.6. Which portfolio has a higher dollar exposure to a change in stock price?
Portfolio B
Portfolio A
The two portfolios have the same exposure.
Portfolio A if the stock price increases and portfolio B if it decreases
Portfolio B if the stock price increases and portfolio A if it decreases
500 calls × 0.6 + 500 shares = 300 shares + 500 shares = 800 shares
References
Multiple Choice
Difficulty: 3 Challenge
Portfolio A consists of 400 shares of stock and 400 calls on that stock. Portfolio B consists of 500 shares of stock. The call delta is 0.5. Which portfolio has a higher dollar exposure to a change in stock price?
Portfolio B
Portfolio A
The two portfolios have the same exposure.
Portfolio A if the stock price increases and portfolio B if it decreases
Portfolio B if the stock price increases and portfolio A if it decreases
400 calls × 0.5 + 400 shares = 200 shares + 400 shares = 600 shares;
600 shares > 500 shares.
References
Multiple Choice
Difficulty: 3 Challenge
Portfolio A consists of 600 shares of stock and 300 calls on that stock. Portfolio B consists of 685 shares of stock. The call delta is 0.3. Which portfolio has a higher dollar exposure to a change in stock price?
Portfolio B
Portfolio A
The two portfolios have the same exposure.
Portfolio A if the stock price increases, and portfolio B if it decreases
Portfolio B if the stock price increases, and portfolio A if it decreases
300 calls × 0.3 + 600 shares = 90 shares + 600 shares = 690 shares;
690 shares > 685 shares.
References
Multiple Choice
Difficulty: 3 Challenge
A portfolio consists of 100 shares of stock and 1500 calls on that stock. If the hedge ratio for the call is 0.7, what would be the dollar change in the value of the portfolio in response to a $1 decline in the stock price?
+$700
+$500
−
$1,150
−
$520
None of the options are correct.
−
$100 + (
−
$1,500 × 0.7) =
−
$1,150
References
Multiple Choice
Difficulty: 3 Challenge
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45.
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A portfolio consists of 800 shares of stock and 100 calls on that stock. If the hedge ratio for the call is 0.5, what would be the dollar change in the value of the portfolio in response to a $1 decline in the stock price?
+$700
−
$850
−
$580
−
$520
None of the options are correct.
−
$800 + (
−
$100 × 0.5) =
−
$850
References
Multiple Choice
Difficulty: 3 Challenge
A portfolio consists of 225 shares of stock and 300 calls on that stock. If the hedge ratio for the call is 0.4, what would be the dollar change in the value of the portfolio in response to a $1 decline in the stock price?
−
$345
+$500
−
$580
−
$520
None of the options are correct.
−
$225 + (
−
$300 × 0.4) =
−
$345
References
Multiple Choice
Difficulty: 3 Challenge
A portfolio consists of 400 shares of stock and 200 calls on that stock. If the hedge ratio for the call is 0.6, what would be the dollar change in the value of the portfolio in response to a $1 decline in the stock price?
+$700
+$500
−
$580
−
$520
None of the options are correct.
−
$400 + (
−
$200 × 0.6) =
−
$520
References
Multiple Choice
Difficulty: 3 Challenge
If the hedge ratio for a stock call is 0.30, the hedge ratio for a put with the same expiration date and exercise price as the call would be:
0.70.
0.30.
−
0.70.
−
0.30.
−
0.17.
Call hedge ratio =
N
(
d
1);
Put hedge ratio =
N
(
d
1)
−
1 = 0.3
−
1.0 =
−
0.7.
References
Multiple Choice
Difficulty: 3 Challenge
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If the hedge ratio for a stock call is 0.50, the hedge ratio for a put with the same expiration date and exercise price as the call would be:
0.30.
0.50.
−
0.60.
−
0.50.
−
0.17.
Call hedge ratio =
N
(
d
1);
Put hedge ratio =
N
(
d
1)
−
1 = 0.5
−
1.0 =
−
0.5.
References
Multiple Choice
Difficulty: 3 Challenge
If the hedge ratio for a stock call is 0.60, the hedge ratio for a put with the same expiration date and exercise price as the call would be:
0.60.
0.40.
−
0.60.
−
0.40.
−
0.17.
Call hedge ratio =
N
(
d
1);
Put hedge ratio =
N
(
d
1)
−
1 = 0.6
−
1.0 =
−
0.4.
References
Multiple Choice
Difficulty: 3 Challenge
If the hedge ratio for a stock call is 0.70, the hedge ratio for a put with the same expiration date and exercise price as the call would be:
0.70.
0.30.
−
0.70.
−
0.30.
−
0.17.
Call hedge ratio =
N
(
d
1);
Put hedge ratio =
N
(
d
1)
−
1 = 0.7
−
1.0 =
−
0.3.
References
Multiple Choice
Difficulty: 3 Challenge
A put option is currently selling for $6 with an exercise price of $50. If the hedge ratio for the put is 0.30, and the stock is currently selling for $46, what is the elasticity of the put?
2.76
2.30
−
7.67
−
2.76
−
2.30
%
P
Stock
= ($47 − $46) ÷ $46 = 0.021739
%
P
Option
= ($5.70 − $6.00) ÷ $6.00 = −0.05
Elasticity
Option
= −0.05 ÷ 0.021739 = −2.30
References
Multiple Choice
Difficulty: 3 Challenge
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A put option on the S&P 500 Index will best protect a portfolio:
of 100 shares of IBM stock.
of 50 bonds.
that corresponds to the S&P 500.
of 50 shares of AT&T and 50 shares of Xerox stocks.
that replicates the Dow.
The S&P 500 Index is more like a portfolio that corresponds to the S&P 500 and thus is more protective of such a portfolio than of any of the other assets.
References
Multiple Choice
Difficulty: 1 Basic
Higher dividend-payout policies have a _________ impact on the value of the call and a _________ impact on the value of the put compared to lower dividend-payout policies.
negative; negative
positive; positive
positive; negative
negative; positive
zero; zero
Dividends lower the expected stock price, and thus lower the current call option value and increase the current put option value.
References
Multiple Choice
Difficulty: 2 Intermediate
Lower dividend-payout policies have a _________ impact on the value of the call and a _________ impact on the value of the put compared to higher dividend-payout policies.
negative; negative
positive; positive
positive; negative
negative; positive
zero; zero
Dividends lower the expected stock price, and thus lower the current call option value and increase the current put option value.
References
Multiple Choice
Difficulty: 2 Intermediate
A $1 decrease in a call option's exercise price would result in a(n) _________ in the call option's value of _________ one dollar.
increase; more than
decrease; more than
decrease; less than
increase; less than
increase; exactly
Option prices are less than stock prices, thus changes in stock prices (market or exercise) are greater (in absolute terms) than are changes in prices of options.
References
Multiple Choice
Difficulty: 2 Intermediate
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Which one of the following variables influences the value of call options?
1. Level of interest rates
2. Time to expiration of the option
3. Dividend yield of underlying stock
4. Stock price volatility
1 and 4 only
2 and 3 only
1, 2, and 4 only
1, 2, 3, and 4
1, 2, and 3 only
All of the variables affect call option prices.
References
Multiple Choice
Difficulty: 2 Intermediate
Which one of the following variables influences the value of put options?
1. Level of interest rates
2. Time to expiration of the option
3. Dividend yield of underlying stock
4. Stock price volatility
1 and 4 only
2 and 3 only
1, 2, and 4 only
1, 2, 3, and 4
1, 2, and 3 only
All of the variables affect put option prices.
References
Multiple Choice
Difficulty: 2 Intermediate
An American call-option buyer on a nondividend-paying stock will:
always exercise the call as soon as it is in the money.
only exercise the call when the stock price exceeds the previous high.
never exercise the call early.
buy an offsetting put whenever the stock price drops below the strike price.
None of the options are correct.
An American call-option buyer will not exercise early if the stock does not pay dividends; exercising forfeits the time value. Rather, the option buyer will sell the option to collect both the intrinsic value and the time value.
References
Multiple Choice
Difficulty: 2 Intermediate
Relative to European puts, otherwise identical American put options:
are less valuable.
are more valuable.
are equal in value.
will always be exercised earlier.
None of the options are correct.
It is valuable to exercise a put option early if the stock drops below a threshold price; thus American puts should sell for more than European puts.
References
Multiple Choice
Difficulty: 2 Intermediate
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Use the two-state put-option value in this problem.
S
O
= $100
;
X
= $120; the two possibilities for
S
T
are $150 and $80. The range of
P
across the two states is _________, and the hedge ratio is _________.
$0 and $40;
−
4 ÷ 7
$0 and $50; +4 ÷ 7
$0 and $40; +4 ÷ 7
$0 and $50;
−
4 ÷ 7
$20 and $40; +1 ÷ 2
When
S
T
= $150;
P
= $0; when
S
T
= $80:
P
= $40;
Hedge Ratio = ($0 − $40) ÷ ($150 − $80) = −4 ÷ 7
References
Multiple Choice
Difficulty: 3 Challenge
Empirical tests of the Black-Scholes option pricing model:
show that the model generates values fairly close to the prices at which options trade, only.
show that the model tends to overvalue deep in-the-money calls and undervalue deep out-of-the-money calls, only.
indicate that the mispricing that does occur is due to the possible early exercise of American options on dividend-paying stocks, only.
show that the model generates values fairly close to the prices at which options trade and indicate that the mispricing that does occur is due to the possible early exercise of American options on dividend-paying
stocks.
All of the options are correct.
Studies have shown that the model tends to undervalue deep in-the-money calls and to overvalue deep out-of-the-money calls.
References
Multiple Choice
Difficulty: 3 Challenge
Options sellers who are delta-hedging would most likely:
sell when markets are falling.
buy when markets are rising.
sell when markets are falling and buy when markets are rising.
sell whether markets are falling or rising.
buy whether markets are falling or rising.
Options sellers who are delta-hedging would most likely sell when markets are falling and buy when markets are rising.
References
Multiple Choice
Difficulty: 2 Intermediate
An American-style call option with six months to maturity has a strike price of $35. The underlying stock now sells for $44. The call premium is $14. What is the intrinsic value of the call?
$14
$9
$0
$23
None of the options are correct.
$44
−
$35 = $9.
References
Multiple Choice
Difficulty: 1 Basic
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An American-style call option with six months to maturity has a strike price of $35. The underlying stock now sells for $43. The call premium is $14. What is the time value of the call?
$8
$14
$0
$6
Cannot be determined without more information
$14
−
($43
−
$35) = $6.
References
Multiple Choice
Difficulty: 2 Intermediate
The hedge ratio of an at-the-money call option on KO is 0.2. The hedge ratio of an at-the-money put option is
−
0.3. What is the hedge ratio of an at-the-money straddle position on KO?
0.1
−
0.3
−
0.1
0.2
None of the options are correct.
Hedge straddle = 0.2
−
0.3 =
−
0.1
References
Multiple Choice
Difficulty: 3 Challenge
An American-style call option with six months to maturity has a strike price of $35. The underlying stock now sells for $43. The call premium is $12. If the company unexpectedly announces it will pay its first-ever dividend three
months from today, you would expect that:
the call price would increase.
the call price would decrease.
the call price would not change.
the put price would decrease.
the put price would not change.
As an approximation, subtract the present value of the dividend from the stock price and recompute the Black-Scholes value with this adjusted stock price. Since the stock price is lower, the option value will be lower.
References
Multiple Choice
Difficulty: 2 Intermediate
The intrinsic value of an out-of-the-money call option is equal to:
the call premium.
zero.
the stock price minus the exercise price.
the striking price.
None of the options are correct.
The fact that the owner of the option can buy the stock at a price greater than the market price gives the contract an intrinsic value of zero, and the holder will not exercise.
References
Multiple Choice
Difficulty: 1 Basic
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Since deltas change as stock values change, portfolio hedge ratios must be constantly updated in active markets. This process is referred to as:
portfolio insurance.
rebalancing.
option elasticity.
gamma hedging.
dynamic hedging.
Dynamic hedgers will convert equity into cash in market declines to adjust for changes in option deltas.
References
Multiple Choice
Difficulty: 2 Intermediate
In volatile markets, dynamic hedging may be difficult to implement because:
prices move too quickly for effective rebalancing.
as volatility increases, historical deltas are too low.
price quotes may be delayed so that correct hedge ratios cannot be computed.
volatile markets may cause trading halts.
All of the options are correct.
All of the options correctly describe the problems associated with dynamic hedging in volatile markets.
References
Multiple Choice
Difficulty: 1 Basic
Rubinstein (1994) observed that the performance of the Black-Scholes model had deteriorated in recent years, and he attributed this to:
investor fears of another market crash.
higher-than-normal dividend payouts.
early exercise of American call options.
decreases in transaction costs.
None of the options are correct.
Options on the same stock with the same strike price should have the same implied volatility, but they exhibit progressively different implied volatilities. Rubinstein believes this is due to fear of another market crash.
References
Multiple Choice
Difficulty: 2 Intermediate
The time value of a call option is
1. the difference between the option's price and the value it would have if it were expiring immediately.
2. the same as the present value of the option's expected future cash flows.
3. the difference between the option's price and its expected future value.
4. different from the usual time value of money concept.
1
1 and 2
2 and 3
2
1 and 4
The time value of an option is described by 1 and is different from the time value of money concept frequently used in finance.
References
Multiple Choice
Difficulty: 1 Basic
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The time value of a put option is
1. the difference between the option's price and the value it would have if it were expiring immediately.
2. the same as the present value of the option's expected future cash flows.
3. the difference between the option's price and its expected future value.
4. different from the usual time value of money concept.
1
1 and 2
2 and 3
2
1 and 4
The time value of an option is described by 1 and is different from the time value of money concept frequently used in finance.
References
Multiple Choice
Difficulty: 1 Basic
The intrinsic value of an at-the-money call option is equal to:
the call premium.
zero.
the stock price plus the exercise price.
the striking price.
None of the options are correct.
The fact that the owner of the option can buy the stock at a price equal to the market price gives the contract an intrinsic value of zero.
References
Multiple Choice
Difficulty: 1 Basic
As the underlying stock's price increased, the call option valuation function's slope approaches:
zero.
one.
two times the value of the stock.
one-half the value of the stock.
infinity.
As the stock price increases the value of the call option increases in price one for one with the stock price. The option is very likely to be exercised.
References
Multiple Choice
Difficulty: 2 Intermediate
The intrinsic value of an in-of-the-money call option is equal to:
the call premium.
zero.
the stock price minus the exercise price.
the striking price.
None of the options are correct.
The fact that the owner of the option can buy the stock at a price less than the market price gives the contract a positive intrinsic value.
References
Multiple Choice
Difficulty: 1 Basic
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The Black-Scholes formula assumes that
1. the risk-free interest rate is constant over the life of the option.
2. the stock price volatility is constant over the life of the option.
3. the expected rate of return on the stock is constant over the life of the option.
4. there will be no sudden extreme jumps in stock prices.
1 and 2
1 and 3
2 and 3
1, 2, and 4
1, 2, 3, and 4
The risk-free rate and stock price volatility are assumed to be constant, but the option value does not depend on the expected rate of return on the stock. The model also assumes that stock prices will not jump markedly.
References
Multiple Choice
Difficulty: 3 Challenge
The intrinsic value of an in-the-money put option is equal to:
the stock price minus the exercise price.
the put premium.
zero.
the exercise price minus the stock price.
None of the options are correct.
The intrinsic value of an in-the-money put option contract is the strike price less the stock price, since the holder can buy the stock at the market price and sell it for the strike.
References
Multiple Choice
Difficulty: 2 Intermediate
The hedge ratio of an option is also called the option's:
alpha.
beta.
sigma.
delta.
rho.
The two terms mean the same thing.
References
Multiple Choice
Difficulty: 1 Basic
The intrinsic value of an at-the-money put option is equal to:
the stock price minus the exercise price.
the put premium.
zero.
the exercise price minus the stock price.
None of the options are correct.
The intrinsic value of an at-the-money put option contract is zero.
References
Multiple Choice
Difficulty: 2 Intermediate
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An American-style call option with six months to maturity has a strike price of $44. The underlying stock now sells for $50. The call premium is $14. What is the intrinsic value of the call?
$12
$10
$6
$23
None of the options are correct.
$50
−
$44 = $6.
References
Multiple Choice
Difficulty: 1 Basic
An American-style call option with six months to maturity has a strike price of $44. The underlying stock now sells for $54. The call premium is $14. What is the time value of the call?
$8
$12
$6
$4
Cannot be determined without more information
$14
−
($54
−
$44) = $4.
References
Multiple Choice
Difficulty: 2 Intermediate
An American-style call option with six months to maturity has a strike price of $42. The underlying stock now sells for $50. The call premium is $14. If the company unexpectedly announces it will pay its first-ever dividend four
months from today, you would expect that:
the call price would increase.
the call price would decrease.
the call price would not change.
the put price would decrease.
the put price would not change.
As an approximation, subtract the present value of the dividend from the stock price and recompute the Black-Scholes value with this adjusted stock price. Since the stock price is lower, the option value will be lower.
References
Multiple Choice
Difficulty: 2 Intermediate
The intrinsic value of an out-of-the-money put option is equal to:
the stock price minus the exercise price.
the put premium.
zero.
the exercise price minus the stock price.
None of the options are correct.
The intrinsic value of an out-of-the-money put option contract is zero.
References
Multiple Choice
Difficulty: 2 Intermediate
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Vega is defined as:
the change in the value of an option for a dollar change in the price of the underlying asset.
the change in the value of the underlying asset for a dollar change in the call price.
the percentage change in the value of an option for a 1% change in the value of the underlying asset.
the change in the volatility of the underlying stock price.
the sensitivity of an option's price to changes in volatility.
An option's hedge ratio (delta) is the change in the price of an option for $1 increase in the stock price.
Vega
is defined as the sensitivity of an option's price to changes in volatility.
References
Multiple Choice
Difficulty: 2 Intermediate
The price of a stock is currently $50. Over the course of the next year, the price is anticipated to rise to $55 or decline to $46. If both outcomes are equally likely and the risk-free interest rate is 3%, what is the price of a one year
call option with an exercise price of $50 using the binomial model?
$2.43
$2.50
$2.66
$2.70
None of the options are correct.
C
= (($55
−
$50) × 0.5 + 0 × 0.5)) ÷ 1.03 = $2.43
References
Multiple Choice
Difficulty: 3 Challenge
The price of a stock is currently $37. Over the course of the next year, the price is anticipated to rise to $40 or decline to $36. If the upside has a 65% probability of occurring and the risk-free interest rate is 3%, what is the price of a
one year call option with an exercise price of $35 using the binomial model?
$1.43
$1.50
$1.26
$2.70
None of the options are correct.
C
= (($37
−
$35) × 0.65 + 0 × 0.35) ÷ 1.03 = $1.26
References
Multiple Choice
Difficulty: 3 Challenge
The price of a stock is currently $38. Over the course of the next year, the price is anticipated to rise to $41 or decline to $36. If the upside has a 65% probability of occurring and the risk-free interest rate is 3%, what is the price of a
six month call option with an exercise price of $35 using the binomial model?
$3.89
$4.18
$4.25
$4.70
None of the options are correct.
C
= (($41 − $35) × 0.65 + ($36 − $35) × 0.35) ÷ 1.03
0.5
= $4.18
References
Multiple Choice
Difficulty: 3 Challenge
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A call option was purchased last month for $2.50 with an exercise price of $45. The option is exercised when the market price of the stock is $48. What is the net profit or loss to the investor?
$3.00
$2.50
$0.50
$0.60
None of the options are correct.
Profit = $48
−
$45
−
$2.50 = $0.50
References
Multiple Choice
Difficulty: 2 Intermediate
A put option was purchased two months ago for $3.70 with an exercise price of $50. The option is exercised when the market price of the stock is $48. What is the net profit or loss to the investor?
$3.00
$2.70
−
$1.70
−
$2.00
None of the options are correct.
Profit = $50
−
$48
−
$3.70 =
−
$1.70 (loss)
References
Multiple Choice
Difficulty: 2 Intermediate
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Related Questions
Answer the following questions including the one on the image provided
2. What is the present values of Team's Offer using a 14 percent discount rate? (Round answer to 2 decimal places, e.g. 15.25.)
3.What is the present values of Counteroffer using a 14 percent discount rate? (Round answer to 2 decimal places, e.g. 15.25.)
4. Which of the three contacts has the highest present value?
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A broker wants to sell a customer an investment costing $100 with an expected payoff in one year of $108.6. The customer indicates
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percent interest to help pay for the investment.
a. What is the customer's expected return if she borrows the money? (Round your answer to 1 decimal place.)
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Use the table below to calculate the premium of the call. This is a 3-year option; the present value must account for
T 3. Use (1+r) for discounting.
4. Use the table below to calculate the premium of the call.
This is a 3-year option; the present value must account for T=3.
Use (1+r) for discounting.
SO
r
$50
6.00%
u
2
0.5
3
F3
$59.55
37.3%
Strike
$80.00
CALL with strike of $80.00
Outcome
Ups
Downs
Probability
1
3
0
5.20%
probability
100.00%
Cumulative Underlier
price
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Revenue
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Exercise?
from sale
purchase
Payoff
Yes
$400.00
$80.00
$320.00
2
2
1
26.20%
94.80%
$100.00
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$100.00
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$20.00
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43.98%
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interest to help pay for the investment
a. What is the customer's expected return if she borrows the money?
Customer's expected return
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parts a through c.
B' (+) =0.03B - 600, B(0) = 40,000
a) Find the solution of the initial value problem and explain why B is an increasing solution.
B(t) =
Why is B an increasing function?
O A. The function is increasing because it is an exponential function with a positive coefficient and a negative exponent.
O B. The function is increasing because it is an exponential function with a positive coefficient and a positive exponent.
O C. The function is increasing because it is an exponential function with a positive exponent.
O D. The function is increasing because it is an exponential function with a positive coefficient.
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The…
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Group of answer choices
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son.1
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Fisher Equation: Nominal, Real, and Inflation
=
10%. Expected inflation is E(T) =
Q1) You lend money to a business colleague at a rate of i
3.5%. What is your real rate of return r? Use the exact Fisher Equation.
=
Q2) How much would you charge a colleague on a loan of $100 if you want your real rate or
return to be r = 6% and the expected inflation is E(л) = 4%. E.g., what is the nominal rate i?
Use the exact Fisher Equation.
Q3) A corporate bond pays investors both a fixed and variable rate of return. The fixed rate is
constitutes the real return which is r = 4.5%. The variable rate is indexed to the CPI and is to
compensate investors for inflation. If the expected inflation in the next year is E(T) = 2.5%,
what is the nominal rate of return i the company will have to pay?
Enter your answer in the table below in the cells colored in yellow.
You must show all work to receive credit.
Q1) r=
|Q2) i=
||Q3) i=
Answer
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your assumption and write it on the submitted file. [example: sa
interest rate or years are not mentioned. you assume any value, do your problem, write your assumptions/answer on
work paper.
Question 1
You have $2,000 to invest, the MARR would be at least what you need pay for borrowing $2000
True
OFalse
MacBook Pro
F6
F4
FS
F2
F3
%23
24
%
2
3
4
5
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the math of interest. please indicate if you are unsure or totally sure about the answer
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not use ai please
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Q)You are given the future value of an annuity, A, the monthly payment, R, and the annual interest rate, r. Find the number of monthly payments, n. Round your answer to the nearest whole number if necessary.A = $4000; R = $70; r = 7%
Solve it correctly not use excel
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uestion 1: Solve the following TVM problems using Excel formulas. You MUST use Excel formulas (FV or PV) to receive credit.
ou can assume that all payments are made at the beginning of the period and use "1" for the "type" argument in the formula.
A. Suppose you invest
11,400 today. What is the future value of the investment in
29 years, if interest at
7% is compounded annually?
B. Suppose you invest
$ 11,400 today. What is the future value of the investment in
29 years, if interest at
7% is compounded quarterly?
C. Suppose you invest
$
570 monthly. What is the future value of the investment in
29
29
years, if interest at
+
5% is compounded monthly?
5
6
7
8
19
20
21
22
23
24
25
26
27
28
29
Question 1
Question 2
+
Ready Accessibility: Investigate
MAR
17
A
国
W
X
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please help me find how to get that answer for this question, please help NO EXCEL
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Plz solve as soon as possible
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Mr. Saso is planning to take a loan of $5 million to be repaid over 10 years. He is contemplating different strategies and decides to take a 10-year $5 million loan at a fixed rate of 4.5%. What has Mr. Saso strategy achieved?
Question 13 options:
The strategy eliminates credit risk but not repricing risk
The strategy leaves Mr. Saso exposed to both credit risk and repricing risk
The strategy eliminates both credit risk and repricing risk
The strategy eliminates repricing risk but not credit risk
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please give me answer step by step
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Hello
Can you show how to calculate the compensating balance requirement?
In my solution guide i have this:
76/953 = 7.975%
My question:
Why do we devide 76/953 and not say 1076/953 to calulate the effective annual cost?
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None
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