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Chapter 16
Option Valuation
Concept Questions
1. The six factors are the stock price, the strike price, the time to expiration, the risk-free interest rate,
2. Increasing the time to expiration increases the value of an option. The reason is that the option gives
the holder the right to buy or sell. The longer the holder has that right, the more time there is for the
3. An increase in volatility acts to increase both put and call values because greater volatility increases
4. An increase in dividend yield reduces call values and increases put values. The reason is that, all else
the same, dividend payments decrease stock prices. To give an extreme example, consider a
5. Interest rate increases are good for calls and bad for puts. The reason is that if a call is exercised in
6. The time value of both a call option and a put option is the difference between the price of the option
7. An option’s delta tells us the (approximate) dollar change in the option’s value that will result from a
8. Vesting refers to the date at which an option can be exercised. For example, if the option has a 4 year
9. There are two possible benefits. First, awarding employee stock options may better align the
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Education.
10. The fact that employee stock options are not tradeable decreases its value relative to a tradeable
Solutions to Questions and Problems
NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple
steps. Due to space and readability constraints, when these intermediate steps are included in this
solutions manual, rounding may appear to have occurred. However, the final answer for each problem is
found without rounding during any step in the problem.
Core Questions
1. d1 =
ln(84/80 )+(.04 +. 422/ 2)× 135 / 365
. 42 ×
√
135 / 365
= .3766
d2 = .3766 – .42
√
135 / 365
= .1212
The standard normal probabilities are:
Calculating the price of the call option yields:
2. d1 =
ln(81 / 90)+(.03 +. 502/ 2)× 60 / 365
.50 ×
√
60 / 365
= –.3940
60 / 365
The standard normal probabilities are:
Calculating the price of the call option yields:
3. d1 =
ln(73 / 75 )+(. 05 +. 372/ 2)× 100 / 365
.37 ×
√
100 / 365
= .0280
100 / 365
The standard normal probabilities are:
Calculating the price of the call option yields:
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Education.
4. d1 =
ln(63 / 60 )+(.04 - . 02 +. 432/ 2)× 45 / 365
. 43 ×
√
45 / 365
= .4150
45 / 365
The standard normal probabilities are:
Calculating the price of the call option yields:
5. d1 =
ln(44 / 40 )+(. 041 - .025 +. 452/ 2)× 65 / 365
. 45 ×
√
65 / 365
= .6119
65 / 365
The standard normal probabilities are:
Calculating the price of the call option yields:
6. d1 =
ln(68 / 70 )+(.06 +. 412/ 2 )× 45 / 365
. 41 ×
√
45 / 365
= -0.0780
45 / 365
The standard normal probabilities are:
Calculating the price of the put option yields:
7. d1 =
ln(42 / 35 )+(. 05+. 472/ 2)× 140 / 365
. 47 ×
√
140 / 365
= .8378
140 / 365
The standard normal probabilities are:
Calculating the price of the put option yields:
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Education.
8. d1 =
ln(67 / 80 )+(.03 +. 302/ 2 )× 60 / 365
.30 ×
√
60 / 365
= –1.3566
60 / 365
The standard normal probabilities are:
Calculating the price of the put option yields:
9. Number of option contracts = -
Portfolio beta × Portfolio value
Option delta × Option contract value
Number of option contracts = -
1 . 07 × $300,000,000
.62 × 2030 × $100
= -2,550 (contracts to write)
10. You can either buy put options or sell call options. In either case, gains or losses on your stock
Number of option contracts = -
Portfolio beta × Portfolio value
Option delta × Option contract value
11. Up price = $45(1.15) = $51.75
Delta =
Cu - Cd
Su - Sd
=
$1. 75 - 0
$51. 75 - 39. 15
= .1389
Call =
ΔSu(1 + r − u )+ Cu
1+r
=
(.1389 )($45 )(1+. 025−1. 15)+$1 .75
1+. 025
= $0.95
12. Up price = $74(1.2) = $88.80
Down price = $74(.8) = $59.20
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Education.
Delta =
Cu - Cd
Su - Sd
=
$13 .80 - 0
$88 .80 - 59 .20
= .4662
Call =
ΔSu(1 + r − u )+ Cu
1+r
=
(. 4662)( $74 )(1+. 042−1. 20)+$13 . 80
1+.042
= $8.01
13. Up price = $58(1.13) = $65.54
Delta =
Cu - Cd
Su - Sd
=
$10 .54 - 0
$65 .54 - 51 . 04
= .7269
Call =
ΔSu(1 + r − u )+ Cu
1+r
=
(.7269 )($58 )(1+. 03−1 .13 )+$10. 54
1+. 03
= $6.14
Using put-call parity:
P + S0 = C + K / (1 + r)
Intermediate Questions
14. K = 0, so C = S = $70
15. = 0, so d1 and d2 go to +8, so N(d1) and N(d2) go to 1.
17. d1 =
ln(20 . 72/23 .15 )+(. 043 +. 292/2)× 3. 5
.29 ×
√
3. 5
= .3443
d2 = .3443 – .29
√
3. 5
= –.1983
These standard normal probabilities are given:
Calculating the price of the employee stock options yields:
18. This is a hedging problem in which you wish to hedge one option position with another. Your
employee stock option (ESO) position represents 10,000 shares, and you need to know how many
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Education.
Using values from the previous answer, the ESO delta is
ESO (Call) Delta = N(d1) = .6347
For the put option, we get this value for d1
d1 =
ln(20 . 72 / 22 .50 )+(. 043 +.292/ 2)×. 25
.29 ×
√
.25
= –.4217
These standard normal probabilities are given:
The number of put option contracts is then calculated as
Number of option contracts = –
ESO delta ×10,000
Put option delta × 100
= –
.6347 × 10,000
−.6634 × 100
Performing the calculation yields 95.67, or about 96, put option contracts.
19. After the volatility shift, we need to recalculate deltas for both options. The new value of d1 for
the ESO is:
d1 =
ln(20 . 72/23.15 )+(. 043 +. 452/2)× 3 .5
. 45 ×
√
3 .5
= 0.4680
In turn, the new ESO delta is
For the put option, we obtain this value for d1
d1 =
ln(20 . 72/22.50 )+(. 043 +. 452/2)×. 25
. 45 ×
√
. 25
= –.2060
and this put option delta
The new number of contracts required is:
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Education.
Number of option contracts = -
.6801×10 ,000
−. 5816×100
Which yields 116.93, or about 117, put option contracts.
20. The stock price in one period will be:
In two periods, the stock price will be:
Suu = $60(1.15)(1.15) = $79.35
The call value for each node is:
Value of call Suu = Max($79.35 – 60, 0) = $19.35
The delta of the up and down moves will be:
Deltau =
Cu - Cd
Su - Sd
=
$19 .35 - 0 . 03
$79 .35 - 60 .03
= 1.000
Deltad =
Cu - Cd
Su - Sd
=
$0 .03 - 0
$60 .03 - 45 . 41
= .0021
So the call value after an up move will be:
Callu =
ΔSu(1 + r − u )+ Cuu
1+r
=
(1. 00 )($69 )(1+. 032−1. 15 )+$19 . 35
1+. 032
= $10.86
The value of a call with a first down move is $0 since it will always be worthless. The delta today is:
Delta =
Cu - Cd
Su - Sd
=
$10 .86 - 0. 02
$69 .00 - 52 .20
= .6455
So, the value of call today is:
Call =
ΔS(1 + r − u )+ Cu
1+r
=
(.6455 )($60)(1+. 032−1 .15 )+$10. 86
1+.032
= $6.10
21. The stock price in one period will be:
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Education.
In two periods, the stock price will be:
The call value for each node is:
The delta of the up and down moves will be:
Deltau =
Cu - Cd
Su - Sd
=
$8 .73 - 0
$48 .73 - 35 .11
= .6408
Deltad =
Cu - Cd
Su - Sd
=
$0 - 0
$35 .11 - 25 .29
= 0
So the call value after an up move will be:
Callu =
ΔSu(1 + r − u )+ Cuu
1+r
=
(.6408 )($41 .30 )(1+.03−1 .18 )+$8. 73
1+. 03
= $4.63
The value of a call with a first down move is $0 since it will always be worthless. The delta today is:
Delta =
Cu - Cd
Su - Sd
=
$4 . 63 - 0
$41. 30-29 . 75
= .4005
So, the value of call today is:
Call =
ΔS(1 + r − u )+ Cu
1+r
=
(. 4005)( $35 )(1+. 03−1. 18)+$4 .63
1+. 03
= $2.45
Using put-call parity:
P + S0 = C + K / (1 + r)
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Education.
Spreadsheet Answers
CFA Exam Review by Kaplan Schweser
1. c
ln(100 / 100)+(. 07 +. 202/ 2)× 1
.20 ×
1
The standard normal probabilities are:
Calculating the price of the call option yields:
2. a
Put-call parity states: S + Vp = Vc + Xe-rt
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Education.
3. b
4. b
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Education.
