Chapter 16 – Option Valuation
CHAPTER 16
OPTION VALUATION
1. Intrinsic value = S0 – X = $55 – $50 = $5.00
Time value = C – Intrinsic value = $6.50 – $5.00 = $1.50
3. Using put-call parity: Put = C – S0 + PV(X) + PV(Dividends)
$2.50 = $3.00 – S0 + $75/(1 + .08) + 0 S0 = $69.94
4. Put values also increase as the volatility of the underlying stock increases. We see this
from the parity relationship as follows:
C = P + S0 – PV(X) – PV(Dividends)
Chapter 16 – Option Valuation
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consent of McGraw-Hill Education.
c. (2) Call B. Despite the higher price of stock B, call B is cheaper than call A.
This can be explained by a lower time to maturity.
d. (2) Call B. This would explain its higher price.
e. (3) Not enough information. The call with the lower exercise price sells for
more than the call with the higher exercise price. The values given are
consistent with either stock having higher volatility.
6. H = Cu － Cd
uS0 － dS0
uS0 – dS0 = 120 – 90 = 30
X Cu – Cd Hedge Ratio
8. a. When ST = $130, then P = 0.
When ST = $80, then P = $30.
b.
Riskless Portfolio
ST = 80
ST = 130
3 shares
240
390
5 puts
150
0
Total
390
390
9. The hedge ratio for the call is: H = Cu － Cd
uS0 － dS0 = 20 － 0
130 － 80 = .4
Riskless Portfolio
S =80
S = 130
2 shares
160
260
Short 5 calls
0
-100
Total
160
160
10. a. 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
11. a. 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.
12. The best estimate for the change in price of the option is: Change in asset price × delta
= (–$6) (– .65) = $3.90. The option price is estimated to increase by $3.90.
51,750 75,000
14. The number of calls needed to create a delta-neutral hedge is inversely proportional to
15. 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.
17. P = Xe–rT [1− N(d2)] − S0 e–T [1 − N(d1)] = $6.60
18. Use the Black-Scholes spreadsheet and change the input for each of the followings:
a. Time to expiration = 3 months = .25 year C falls to $5.14
19. A straddle is a call and a put. The Black-Scholes value is:
Chapter 16 – Option Valuation
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consent of McGraw-Hill Education.
E6 + E7 = B5*EXP(−B7*B3)*(2*E4 − 1) + B6*EXP(−B4*B3)*(1 − 2*E5)
20. The call price will decrease by less than $1. The change in the call price would be $1
21. Holding firm-specific risk constant, higher beta implies higher total stock volatility.
Therefore, the value of the put option increases as beta increases.
23. The call option with a high exercise price has a lower hedge ratio. Both d1 and N(d1)
24. The call option is more sensitive to changes in interest rates. The option elasticity
25. The call option’s implied volatility has increased. If this were not the case, then the call
price would have fallen.
27. As the stock price becomes infinitely large, the hedge ratio of the call option [N(d1)]
28. The hedge ratio of a put option with a very small exercise price is zero. As X decreases,
30. a. The spreadsheet appears as follows:
Chapter 16 – Option Valuation
INPUTS OUTPUTS FORMULA FOR OUTPUT IN COLUMN E
Standard deviation (annual) 0.3213 d1 0.0089 (LN(B5/B6)+(B4-B7+.5*B2^2)*B3)/(B2*SQRT(B3))
Maturity (in years) 0.5 d2 –0.2183 E2-B2*SQRT(B3)
Risk–free rate (annual) 0.05 N(d1) 0.5036 NORMSDIST(E2)
Stock Price 100 N(d2) 0.4136 NORMSDIST(E3)
Exercise price 105 B/S call value 8.0000 B5*EXP(-B7*B3)*E4 – B6*EXP(-B4*B3)*E5
Dividend yield (annual) 0 B/S put value 10.4075 B6*EXP(-B4*B3)*(1-E5) – B5*EXP(–B7*B3)*(1-E4)
The implied standard deviation is .3213.
b. The spreadsheet below shows the standard deviation has increased to: .3568
INPUTS OUTPUTS FORMULA FOR OUTPUT IN COLUMN E
Standard deviation (annual) 0.3568 d1 0.0318 (LN(B5/B6)+(B4-B7+.5*B2^2)*B3)/(B2*SQRT(B3))
Maturity (in years) 0.5 d2 –0.2204 E2-B2*SQRT(B3)
Risk–free rate (annual) 0.05 N(d1) 0.5127 NORMSDIST(E2)
Stock Price 100 N(d2) 0.4128 NORMSDIST(E3)
Exercise price 105 B/S call value 9.0000 B5*EXP(-B7*B3)*E4 – B6*EXP(-B4*B3)*E5
Dividend yield (annual) 0 B/S put value 11.4075 B6*EXP(-B4*B3)*(1–E5) – B5*EXP(-B7*B3)*(1–E4)
Implied volatility has increased because the value of an option increases with greater volatility.
c. Implied volatility increases to .4087 when maturity decreases to four months.
The shorter maturity decreases the value of the option; therefore, in order for the
option price to remain unchanged at $8, implied volatility must increase.
INPUTS OUTPUTS FORMULA FOR OUTPUT IN COLUMN E
Standard deviation (annual) 0.4087 d1 –0.0182 (LN(B5/B6)+(B4-B7+.5*B2^2)*B3)/(B2*SQRT(B3))
Maturity (in years) 0.3333 d2 -0.2541 E2–B2*SQRT(B3)
Risk–free rate (annual) 0.05 N(d1) 0.4927 NORMSDIST(E2)
Stock Price 100 N(d2) 0.3997 NORMSDIST(E3)
Exercise price 105 B/S call value 8.0000 B5*EXP(–B7*B3)*E4 – B6*EXP(-B4*B3)*E5
Dividend yield (annual) 0 B/S put value 11.2645 B6*EXP(–B4*B3)*(1–E5) – B5*EXP(–B7*B3)*(1–E4)
d. Implied volatility decreases to .2406 when exercise price decreases to $100.
The decrease in exercise price increases the value of the call, so that in order for
the option price to remain at $8, implied volatility decreases.
INPUTS OUTPUTS FORMULA FOR OUTPUT IN COLUMN E
Standard deviation (annual) 0.2406 d1 0.2320 (LN(B5/B6)+(B4-B7+.5*B2^2)*B3)/(B2*SQRT(B3))
Maturity (in years) 0.5 d2 0.0619 E2-B2*SQRT(B3)
Risk–free rate (annual) 0.05 N(d1) 0.5917 NORMSDIST(E2)
Stock Price 100 N(d2) 0.5247 NORMSDIST(E3)
Exercise price 100 B/S call value 8.0010 B5*EXP(–B7*B3)*E4 – B6*EXP(–B4*B3)*E5
Dividend yield (annual) 0 B/S put value 5.5320 B6*EXP(–B4*B3)*(1–E5) – B5*EXP(–B7*B3)*(1–E4)
Chapter 16 – Option Valuation
INPUTS OUTPUTS FORMULA FOR OUTPUT IN COLUMN E
Standard deviation (annual) 0.3566 d1 -0.0484 (LN(B5/B6)+(B4-B7+.5*B2^2)*B3)/(B2*SQRT(B3))
Maturity (in years) 0.5 d2 –0.3005 E2-B2*SQRT(B3)
Risk–free rate (annual) 0.05 N(d1) 0.4807 NORMSDIST(E2)
Stock Price 98 N(d2) 0.3819 NORMSDIST(E3)
Exercise price 105 B/S call value 8.0014 B5*EXP(–B7*B3)*E4 – B6*EXP(–B4*B3)*E5
Dividend yield (annual) 0 B/S put value 12.4089 B6*EXP(–B4*B3)*(1–E5) – B5*EXP(–B7*B3)*(1-E4)
31. A put is more in the money, and has a hedge ratio closer to –1, when its exercise price
is higher:
Put
X
Delta
A
10
– .1
B
20
– .5
C
30
– .9
32. a.
Position
ST < X
ST > X
Stock
ST + D
ST + D
Put
X –ST
0
Total
X + D
ST + D
b. The total value for each of the two strategies is the same, regardless of the stock
price (ST).
Position
ST < X
ST > X
Call
0
ST – X
Zeroes
X + D
X + D
Total
X + D
ST + D
c. The cost of the stock-plus-put portfolio is (S0 + P). The cost of the call-plus-
33. a. The delta of the collar is calculated as follows:
Delta
Stock
1.0
Short call
–N(d1) = – .35
Long put
N(d1) – 1 = – .40
Total
.25
Chapter 16 – Option Valuation
If the stock price increases by $1, the value of the collar increases by $ .25. The
stock will be worth $1 more, the loss on the short put is $ .40, and the call
written is a liability that increases by $ .35.
b. If S becomes very large, then the delta of the collar approaches zero. Both
c. As S approaches zero, the delta of the collar also approaches zero. Both N(d1)
34. a. Choice A: Calls have higher elasticity than shares. For equal dollar
35. Step 1: Calculate the option values at expiration. The two possible stock prices are: S+ =
$120 and S– = $80. Therefore, since the exercise price is $100, the corresponding two
36. Step 1: Calculate the option values at expiration. The two possible stock prices are: S+
Chapter 16 – Option Valuation
Step 3: Form a riskless portfolio made up of one share of stock and two written calls.
37.
a. 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)
3
50.104$121$
udSuuS
00
−
−
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
Buy 3 puts at price Pu
16.50
0.00
Total
$121.00
$121.00
The portfolio must have a current market value equal to the present value of $121:
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
25.90$50.104$
ddSduS
00
−
−
Chapter 16 – Option Valuation
Thus, the following portfolio will be worth $110 at option expiration regardless of the
ultimate stock price:
Riskless portfolio
ddS0 = $90.25
duS0 = $104.50
Buy 1 share at price dS0 = $95
$90.25
$104.50
Buy 1 put at price Pd
19.75
5.50
Total
$110.00
$110.00
The portfolio must have a current market value equal to the present value of $110:
95 + Pd = $110/1.05 = $104.762 Pd = $9.762
Finally, we solve for P using the values of Pu and Pd . From its initial value, the put can rise
to a value of Pd = $9.762 (at this point, the stock price is dS0 = $95) or fall to a value of Pu
95$110$
dSuS
00
−
−
Thus, the following portfolio will be worth $60.53 at option expiration regardless of the
ultimate stock price:
Riskless portfolio
dS0 = $95
uS0 = $110
Buy 0.5344 share at price S = $100
$50.768
$58.784
Buy 1 put at price P
9.762
1.746
Total
$60.530
$60.530
The portfolio must have a market value equal to the present value of $60.53:
b. Finally, we check put-call parity. Recall from Example 15.1 and Concept Check
S0 = 100 (current value of portfolio)
Chapter 16 – Option Valuation
a. The put delta is: N(d1) – 1 = 0.7422 – 1 = – .2578
39. a.
Stock price
110
90
Put payoff
0
10
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
c. The goal is a portfolio with the same exposure to the stock as the
hypothetical protective put portfolio. Since the put’s hedge ratio is – .5, we
S = 110
Chapter 16 – Option Valuation
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written
consent of McGraw-Hill Education.
c. 12 subperiods, each 1 month
u = exp(
12/1
) = 1.1224; d = exp(–
12/1
) = .8909
41. u = 1.5 = exp(
t
) = exp(
1
) =
t
42. Given S0 = X when the put and the call are at-the-money, the relationship of put-call
43. We first calculate the risk neutral probability that the stock price will increase:
44. We first calculate the risk neutral probability that the stock price will increase:
p = 1 + rf － 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 Pu and Pd:
Chapter 16 – Option Valuation
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written
consent of McGraw-Hill Education.
c. P = E(CF)
1 + rf = $4.4180
1.05 = $4.208
It matches the value we found in problem 37.
CFA 1
Answer:
a. i. The combined portfolio will earn a return. The written call will expire in the
money. The protective put purchased will expire worthless. Each short call will
b. i. The delta of the call will approach 1.0 as the stock goes deep into the money,
while expiration of the call approaches and exercise becomes essentially certain.
The put delta will approach zero.
c. The call sells at an implied volatility (20.00%) that is less than recent historical
CFA 2
Answer:
a. i. The option price will decline.
Chapter 16 – Option Valuation
b. i. Besides Weber’s belief that the implied volatility may differ from the market,
the Black Scholes model assumes the volatility, risk-free rate, and dividend
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:
u = 1 + percentage increase in a period if the stock price rises = 1.20
Chapter 16 – Option Valuation
We use a portfolio combining the underlying stock and bond to replicate the
payoffs: ST + (+ rf) = Payoff
72 + =
Solve the equations for and B = 0.5, B = –22.6415
The two-period binomial tree for the option values is as follows:
$0
$7.3585
$0
$0
$12
$4.5123
60 + = 7.3585
b. The value of a call option at expiration is: Max(0, X – S)
Puu = Max (0, $60 – $72) = $0
We use a portfolio combining the underlying stock and bond to replicate the
Chapter 16 – Option Valuation
48 + =
Solve the equations for and B = –0.5, B = 33.9623
The two-period binomial tree for the option values is as follows:
$12
$3.9623
$28
$16.6038
$0
$7.9108
60 + = 3.9623
c. The put-call parity relationship is:
C – P = S0 – PV(X)
Substituting the values for this problem:
d. Despite a small rounding error, the put-call parity relationship stands.