Chapter 16 – Option Valuation
1.
a. Choice A: Calls have higher elasticity than shares. For equal dollar
b. Choice B: Calls have hedge ratios less than 1.0. For equal numbers of shares
2. Step 1: Calculate the option values at expiration. The two possible stock prices are: S+ =
Step 2: Calculate the hedge ratio: (Cu – Cd)/(uS0 – dS0) = (20 – 0)/(120 – 80) = .5
3. Step 1: Calculate the option values at expiration. The two possible stock prices are: S+
= $130 and S– = $70. Therefore, since the exercise price is $100, the corresponding two
possible call values are: Cu = $30 and Cd = $0.
4. We start by finding the value of Pu . From this point, the put can fall to an expiration-date
value of Puu = $0 (since at this point the stock price is uuS0 = $121) or rise to a final value
of Pud = $5.50 (since at this point the stock price is udS0 = $104.50, which is less than the
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distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 16 – Option Valuation
Thus, the following portfolio will be worth $121 at option expiration regardless of the
ultimate stock price:
Riskless portfolio udS0 = $104.50 uuS0 = $121
Buy 1 share at price uS0 = $110 $104.50 $121.00
Next we find the value of Pd . From this point (at which dS0 = $95), the put can fall to an
expiration-date value of Pdu = $5.50 (since at this point the stock price is
duS0 = $104.50) or rise to a value of Pdd = $19.75 (since at this point, the stock price is
Chapter 16 – Option Valuation
Buy 0.5344 share at price S = $100 $50.768 $58.784
Buy 1 put at price P 9.762 1.746
5.
S0 = 100 (current value of portfolio)
X = 100 (floor promised to clients, 0% return)
a. The put delta is: N(d1) – 1 = 0.7422 – 1 = –.2578
b. At the new portfolio value, the put delta becomes –.2779, so that the amount
6.
a.
Stock price 110 90
b. The cost of the protective put portfolio is the cost of one share plus the cost of
one put: $100 + $2.38 = $102.38
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distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 16 – Option Valuation
c. The goal is a portfolio with the same exposure to the stock as the hypothetical
7. u = exp(
t
); d = exp(–
t
)
a. 1 period of one year
1
1
b. 4 subperiods, each 3 months
4/1
4/1
c. 12 subperiods, each 1 month
12/1
12/1
8. u = 1.5 = exp(
t
) = exp(
1
)
9. Given S0= X when the put and the call are at-the-money, the relationship of put-call
parity, P = C – S0 + PV(X)can be written as: P = C – S0 + PV(S0).
S0
10. We first calculate the risk neutral probability that the stock price will increase:
16-4
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distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 16 – Option Valuation
p =
1+ r f － d
u －d
=
1 + .1 －.8
1.2 －.8
= .75
Then use the probability to find the expected cash flow at expiration, and discount it by
the risk free rate:
E(CF) = .75 $20 + .25 0 = $15
C =
E(CF)
1 + r f
=
$ 15
1.1
=$13.64
It matches the value we found in problem 35.
11. We first calculate the risk neutral probability that the stock price will increase:
p =
1 + r f － d
u －d
=
1 + .05 －. 95
1.1 －.95
= .6667
Then use the probability to find the expected cash flows at expiration, and discount it
by the risk free rate to find Puand Pd:
uE(CF) = .6667 0 + .3333 $5.5 = $1.8333
Pu =
E(CF)
1 + r f
=
$1 .8333
1.05
=$1.7460
dE(CF) = .6667 $5.5 + .3333 $19.75 = $10.25
Pd =
E(CF)
1 + r f
=
$ 10.25
1.05
=$9.7619
Finally, compute the expected cash flow in 6 month, and discount the E(CF) by the 6
month risk free rate:
E(CF) = .6667 1.7460 + .3333 $9.7619 = $4.4180
P =
E(CF)
1 + r f
=
$4. 4180
1.05
=$4.208
It matches the value we found in problem 37.
CFA 1
Answer:
a. i. The combined portfolio will suffer a loss. The written calls will expire in the
money. The protective put purchased will expire worthless. Each short call will
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distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 16 – Option Valuation
ii. Both options expire out of the money, and are thus not exercised. The
iii. The calls will expire worthless and the portfolio will retain the short call
option price. The put option will be exercised and the proceeds used to offset
the purchase price of the put and the decline in value of the portfolio.
b. i. The delta of the call will approach 1.0 as the stock goes deep into the money,
ii. Both options expire out of the money. Delta of each will approach zero as
iii. The call is out of the money as expiration approaches. Delta approaches
c. The call sells at an implied volatility (20.00%) that is less than recent historical
volatility (21.00%); the put sells at an implied volatility (22.00%) that is greater
CFA 2
Answer:
a. i. The option price will decline.
ii. The option price will increase.
b. i. Besides Weber’s belief thatthe implied volatility may differ from the market,
ii. An American option may be exercised early, and the value of early exercise
CFA 3
Answer:
a. Over two periods, the stock price must follow one of four patterns: up-up,
up-down, down-down, or down-up.
The binomial parameters are:
The two-period binomial tree is as follows:
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distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 16 – Option Valuation
The calculations for the values shown above are as follows:
uS0 = $50 × 1.20 = $60
a. The value of a call option at expiration is: Max(0, S – X)
Cuu = Max (0, $72 – $60) = $12
We use a portfolio combining the underlying stock and bond to replicate the
payoffs: ST rf)Payoff
Solve the equations for and B = 0.5, B = –22.6415
Therefore, the Cu = 60
Solve the equations for and B = 0, B = 0
Therefore, the Cd = 40
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distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 16 – Option Valuation
607.3585
400
b. The value of a call option at expiration is: Max(0, X – S)
We use a portfolio combining the underlying stock and bond to replicate the
payoffs: ST rf)Payoff
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distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.
Chapter 16 – Option Valuation
603.9623
40
Solve the equations for and B = –0.6321, B = 39.5158
Therefore, the C = 50
c. The put-call parity relationship is:
16-9
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distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in
whole or part.