38. The two possible stock prices and the corresponding put values are:
uS 0 = 120 Pu = 0
dS 0 = 80 Pd = 20
The hedge ratio is
0 0
0 20 1
120 80 2
u d
P P
HuS dS
––
= = =-
– –
Form a riskless portfolio by buying one share of stock and buying two puts. The cost of the
The payoff for the riskless portfolio equals $120:
Riskless
Portfolio S = 80 S = 120
Therefore, find the value of the put by solving
According to put-call parity P + S = C + PV(X)
Our estimates of option value satisfy this relationship:
39. If we assume that the only possible exercise date is just prior to the ex-dividend date, then
the relevant parameters for the Black-Scholes formula are:
S 0 = 60
If instead, one commits to foregoing early exercise, then we reduce the stock price by the
present value of the dividends. Therefore, we use the following parameters:
40. True. The call option has an elasticity greater than 1.0. Therefore, the call’s percentage rate
of return is greater than that of the underlying stock. Hence the GM call responds more
41. False. The elasticity of a call option is higher the more out of the money is the option.
(Even though the delta of the call is lower, the value of the call is also lower. The
42. As the stock price increases, conversion becomes increasingly more assured. The hedge
43. Goldman Sachs believes that the market assessment of volatility is too high. It should sell
options because the analysis suggests the options are overpriced with respect to true volatility.
44. If the stock market index increases 1%, the 1 million shares of stock on which the options
are written would be expected to increase by
The options would increase by:
45. S = 100; current value of portfolio
a. Using the Black-Scholes formula, we find that
Therefore, total funds to be managed equals $110.27 million: $100 million portfolio
Therefore, sell off 25.78% of the equity portfolio, placing the remaining funds in T-bills.
b. At the new portfolio value of 97, the put delta is
This means that you must reduce the delta of the portfolio by
You should sell an additional 2.01% of the equity position and use the proceeds to
buy T-bills. Since the stock price is now at only 97% of its original value, you need to
sell
46. a. Because you believe the calls are underpriced (selling at an implied volatility that is too
low), you will buy calls.
47. The calls are cheap (implied σ = 0.30) and the puts are expensive (implied
48. a. To calculate the hedge ratio, suppose that the market index increases by 1%. Then the
stock portfolio would be expected to increase by:
Given the option delta of 0.8, the option portfolio would increase by
JP Morgan Chase’s liability from writing these options would increase by the same
amount. The market index portfolio would increase in value by 1%. Therefore, JP
b. The delta of a put option is
Therefore, for every 1% the market increases, the index will rise by 10 points and the
value of the put option contract will change by
49. a,b,c
Subperiods Δt=T/n u=exp(σ√ Δt)d=exp(-σ√ Δt)
50. Since the spread between u and d reflects the volatility of the rate of return and u and d
depend on that volatility, next year’s rate of return should increase accordingly by the σ√ Δt
51. P = C – S0 + PV(X). When at-the-money, X= S0. PV(X) will always be less than S0 and due to
52. Using the risk-neutral shortcut, we must first calculate the risk-neutral probability p.
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❑
❑
❑
❑
❑❑
which is exactly equal to the two-state approach.
53.
a. If the stock price rises, the payoff will be zero as profit from a put is only made when the
stock price is less than the exercise price.
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❑
❑
❑
❑
❑❑
d. This is exactly equal to the two-stage approach.
CFA PROBLEMS
1. Statement a: The hedge ratio (determining the number of futures contracts to sell) ought to
be adjusted by the beta of the equity portfolio, which is 1.20. The correct hedge ratio would
be
$100 million β 4,000 β 4,000 1.2 4,800
$100 250 ´ = ´ = ´ =
´
Statement b: The portfolio will be hedged and should therefore earn the risk-free rate, not
2. a. The value of the call option decreases if underlying stock price volatility decreases.
The less volatile the underlying stock price, the less the chance of extreme price
The value of the call option is expected to increase if the time to expiration of the
b. i. When European options are out of the money, investors are essentially saying that
they are willing to pay a premium for the right, but not the obligation, to buy or sell
the underlying asset. The out-of-the-money option has no intrinsic value, but, since
ii. With American options, investors have the right, but not the obligation, to exercise
the option prior to expiration, even if they exercise for noneconomic reasons. This
3. a. American options should cost more (have a higher premium). American options give
the investor greater flexibility than European options since the investor can choose
c. i. An increase in short-term interest rate PV(exercise price) is lower, and call
value increases.
4. a. The two possible values of the index in the first period are:
The possible values of the index in the second period are:
uuS0 = (1.20)2 × 50 = 72
b. The call values in the second period are:
To compute Cu, first compute the hedge ratio:
0 0
12 0 1
72 48 2
uu ud
C C
HuuS udS
––
= = =
– –
Form a riskless portfolio by buying one share of stock and writing two calls.
The cost of the portfolio is: S – 2Cu = $60 – 2Cu
The payoff for the riskless portfolio equals $48:
Riskless
Therefore, find the value of the call by solving:
(continued on next page)
To compute C, compute the hedge ratio:
0 0
7.358 0 0.3679
60 40
u d
C C
HuS dS
––
= = =
– –
Form a riskless portfolio by buying 0.3679 of a share and writing one call.
The cost of the portfolio is 0.3679S – C = $18.395 – C
The payoff for the riskless portfolio equals $14.716:
Riskless Portfolio S = 40 S = 60
Therefore, find the value of the call by solving:
c. The put values in the second period are:
Puu = 0
To compute Pu, first compute the hedge ratio:
0 0
0 12 1
72 48 2
uu ud
P P
HuuS udS
––
= = =-
– –
Form a riskless portfolio by buying one share of stock and buying two puts.
The cost of the portfolio is: S + 2Pu = $60 + 2Pu
The payoff for the riskless portfolio equals $72:
Riskless
Portfolio S = 48 S = 72
Buy 1 share 48 72
Buy 2 puts 24 0
Total 72 72
Therefore, find the value of the put by solving:
(continued on next page)
To compute Pd, compute the hedge ratio:
0 0
12 28 1.0
48 32
du dd
P P
HduS ddS
––
= = =-
– –
Form a riskless portfolio by buying one share and buying one put.
The payoff for the riskless portfolio equals $60:
Riskless
Portfolio S = 32 S = 48
Therefore, find the value of the put by solving
To compute P, compute the hedge ratio:
0 0
3.962 16.604 0.6321
60 40
u d
P P
HuS dS
––
= = =-
– –
Form a riskless portfolio by buying 0.6321 of a share and buying one put.
The payoff for the riskless portfolio equals $41.888:
Riskless Portfolio S = 40 S = 60
Therefore, find the value of the put by solving:
2
$60
1.06
This is the value of the call calculated in part (b) above.
5. a.(i) Index increases to 1,193. The combined portfolio will suffer a loss. The written calls
expire in the money; the protective put purchased expires worthless. Let’s analyze the
The net cost of the portfolio when the option positions are established is:
(ii) Index remains at 1,136. Both options expire out of the money. The portfolio will thus be
(iii) Index declines to 1,080. The calls expire worthless. The portfolio will be worth $1,130,
b. (i) Index increases to 1,193. The delta of the call approaches 1.0 as the stock goes deep into
(iii) Index declines to 1,080. The call is out of the money as expiration approaches. Delta
c. The call sells at an implied volatility (22.00%) that is less than recent historical volatility