January 2, 2020

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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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 =

√

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

The standard normal probabilities are:

Calculating the price of the call option yields:

3. d1 =

.37 ×

√

100 / 365

= .0280

√

The standard normal probabilities are:

Calculating the price of the call option yields:

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4. d1 =

ln(63 / 60 )+(.04 - . 02 +. 432/ 2)× 45 / 365

. 43 ×

√

45 / 365

= .4150

√

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

√

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

√

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

√

The standard normal probabilities are:

Calculating the price of the put option yields:

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8. d1 =

ln(67 / 80 )+(.03 +. 302/ 2 )× 60 / 365

.30 ×

√

60 / 365

= –1.3566

√

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 = -

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

=

Call =

ΔSu(1 + r − u )+ Cu

1+r

=

12. Up price = $74(1.2) = $88.80

Down price = $74(.8) = $59.20

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Delta =

Cu - Cd

Su - Sd

=

Call =

ΔSu(1 + r − u )+ Cu

1+r

=

13. Up price = $58(1.13) = $65.54

Delta =

Cu - Cd

Su - Sd

=

Call =

ΔSu(1 + r − u )+ Cu

1+r

=

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 =

√

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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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 =

These standard normal probabilities are given:

The number of put option contracts is then calculated as

Number of 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 =

In turn, the new ESO delta is

For the put option, we obtain this value for d1

d1 =

and this put option delta

The new number of contracts required is:

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Number of 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

=

Deltad =

Cu - Cd

Su - Sd

=

So the call value after an up move will be:

Callu =

ΔSu(1 + r − u )+ Cuu

1+r

=

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

=

So, the value of call today is:

Call =

ΔS(1 + r − u )+ Cu

1+r

=

21. The stock price in one period will be:

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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

=

Deltad =

Cu - Cd

Su - Sd

=

= 0

So the call value after an up move will be:

Callu =

ΔSu(1 + r − u )+ Cuu

1+r

=

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

=

= .4005

So, the value of call today is:

Call =

ΔS(1 + r − u )+ Cu

1+r

=

Using put-call parity:

P + S0 = C + K / (1 + r)

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Spreadsheet Answers

CFA Exam Review by Kaplan Schweser

1. c

d1 =

√

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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3. b

4. b

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