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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
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)
Given a value of S and a risk-free interest rate, if C increases because of an increase in
volatility, so must P in order to keep the parity equation in balance.
Numerical example:
Suppose you have a put with exercise price 100, and that the stock price can take on
one of three values: 90, 100, 110. The payoff to the put for each stock price is:
Now suppose the stock price can take on one of three alternate values also centered
around 100, but with less volatility: 95, 100, 105. The payoff to the put for each stock
price is:
5. a. (1) Put A must be written on the lower-priced stock. Otherwise, given the lower
volatility of stock A, put A would sell for less than put B.
Chapter 16 - Option Valuation
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.
6. H = Cu － Cd
uS0 － dS0
uS0 – dS0 = 120 – 90 = 30
X Cu - Cd Hedge Ratio
a. 120 0 – 0 0/30=0.000
7. We first calculate d1= ln(S0/X) + (r － + 2/2)T
T
, and then find N(d1), which is the
Black Scholes hedge ratio for the call. We can observe from the following that when
the stock price increases, N(d1) increases as well.
X
50
r
3%
8. a. When ST = $130, then P = 0.
When ST = $80, then P = $30.
Chapter 16 - Option Valuation
b.
Riskless Portfolio
ST = 80
ST = 130
3 shares
240
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
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
12. The best estimate for the change in price of the option is: Change in asset price × delta
Chapter 16 - Option Valuation
13. The number of call options necessary to delta hedge is
51,750 75,000
0.69 =
options or 750
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.
16. d1 = ln(S0/X) + (r － + 2/2)T
T
= ln(50/50) + (.03 － + .52/2) × .5
.5 ×
.5
= 0.2192
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:
C + P = S0e−T N(d1) − Xe–rT N(d2) + Xe–rT [1 − N(d2)] − S0e−T [1 − N(d1)]
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.
22. Holding beta constant, the stock with high firm-specific risk has higher total volatility.
Therefore, the option on the stock with a lot of firm-specific risk is worth more.
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
26. The put option’s implied volatility has increased. If this were not the case, then the put
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,
29. The hedge ratio of the straddle is the sum of the hedge ratios for the two options:
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)
Chapter 16 - Option Valuation
b. The spreadsheet below shows the standard deviation has increased to: .3568
INPUTS OUTPUTS FORMULA FOR OUTPUT IN COLUMN E
Exercise price 105 B/S call value 9.0000 B5*EXP(-B7*B3)*E4 - B6*EXP(-B4*B3)*E5
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
Risk-free rate (annual) 0.05 N(d1) 0.4927 NORMSDIST(E2)
Stock Price 100 N(d2) 0.3997 NORMSDIST(E3)
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
Risk-free rate (annual) 0.05 N(d1) 0.5917 NORMSDIST(E2)
e. The decrease in stock price decreases the value of the call. In order for the
option price to remain at $8, implied volatility increases to .3566.
INPUTS OUTPUTS FORMULA FOR OUTPUT IN COLUMN E
Stock Price 98 N(d2) 0.3819 NORMSDIST(E3)
31. A put is more in the money, and has a hedge ratio closer to –1, when its exercise price
is higher:
32. a.
Position
ST < X
ST > X
Stock
ST + 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
c. The cost of the stock-plus-put portfolio is (S0 + P). The cost of the call-plus-
zero portfolio is: [C + PV(X + D)]. Therefore:
33. a. The delta of the collar is calculated as follows:
Delta
Stock
1.0
b. If S becomes very large, then the delta of the collar approaches zero. Both
N(d1) terms approach 1 so that the delta for the short call position approaches –
Chapter 16 - Option Valuation
c. As S approaches zero, the delta of the collar also approaches zero. Both N(d1)
terms approach 0 so that the delta for the short call position approaches zero and
34. a. Choice A: Calls have higher elasticity than shares. For equal dollar
investments, the capital gain potential for calls is higher than for stocks.
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
possible call values are: Cu = $20 and Cd = $0.
36. 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.
Chapter 16 - Option Valuation
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)
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 $110 exercise price). Therefore, the hedge ratio at
this point is:
Thus, the following portfolio will be worth $121 at option expiration regardless of the
ultimate stock price:
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
duS0 = $104.50) or rise to a value of Pdd = $19.75 (since at this point, the stock price is
ddS0 = $90.25). Therefore, the hedge ratio at this point is −1.0, which reflects the fact that
the put will necessarily expire in the money if the stock price falls to $95 in the first
period:
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
Chapter 16 - Option Valuation
The portfolio must have a current market value equal to the present value of $110:
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
= $1.746 (at this point, the stock price is uS0 = $110). Therefore, the hedge ratio at this
point is:
Thus, the following portfolio will be worth $60.53 at option expiration regardless of the
ultimate stock price:
Riskless portfolio
dS0 = $95
uS0 = $110
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
#4 that C = $4.434. Put-call parity requires that:
Except for minor rounding error, put-call parity is satisfied.
38.
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
39.
Chapter 16 - Option Valuation
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
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
want to hold (1 – .5) = .5 shares of stock, which costs $50, and place the
remaining funds ($52.38) in bills, earning 5% interest.
Stock price
S = 90
S = 110
Half share
45
55
Bills
55
55
Total
100
110
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.
40. u = exp(
t
); d = exp(–
t
)
a. 1 period of one year
41. u = 1.5 = exp(
t
) = exp(
1
) =
42. 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).
43. We first calculate the risk neutral probability that the stock price will increase:
p = 1 + rf － d
u － d = 1 + .1 － .8
1.2 － .8 = .75
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:
uE(CF) = .6667 0 + .3333 $5.5 = $1.8333
a. Pu = E(CF)
1 + rf = $1.8333
1.05 = $1.7460
CFA 1
Answer:
Chapter 16 - Option Valuation
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
payout $54, less the short option price of $27.20, while each put will lose the
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.
ii. The option price will increase.
Chapter 16 - Option Valuation
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:
$72
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
Chapter 16 - Option Valuation
Therefore, the Cu = 60 + =
The two-period binomial tree for the option values is as follows:
$12
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
payoffs: ST + (+ rf) = Payoff
72 + =
Chapter 16 - Option Valuation
The two-period binomial tree for the option values is as follows:
$0
c. The put-call parity relationship is:
C – P = S0 – PV(X)
Substituting the values for this problem:
