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15. You get to keep the premium in all cases. For 10 contracts and a $1 premium, thatвЂ™ s

$1,000. If the stock price is $20 or $30, the options expire worthless, so your net profit is

$1,000. If the stock price is $10, you lose $10 per share on each of 1,000 shares, or

$10,000 in all. You still have the premium, so your net loss is $9,000.

B-50 CORRADO AND JORDAN

16. The contract costs $2,000. At maturity, an in-the-money SPX option is worth 100 times

the difference between the S&P index and the strike, or $5,000 in this case You net profit

is $3,000.

17. The call is selling for less than its intrinsic value; an arbitrage opportunity exists. Buy the

call for $10, exercise the call by paying $35 in return for a share of stock, and sell the

stock for $50. You've made a riskless $5 profit.

18. 42 contracts were traded, 25 calls and 17 puts; this represents options on 4,200 shares of

Milson stock.

19. The calls are in the money. The intrinsic value of the calls is $4.

20. The puts are out of the money. The intrinsic value of the puts is $0.

21. The March call and the October put are mispriced. The call is mispriced because it is

selling for less than its intrinsic value. The arbitrage is to buy the call for $3.50, exercise it

and pay $55 for a share of stock, and sell the stock for $59 for a riskless profit of $0.50.

The October put is mispriced because it sells for less than the July put. To take advantage

of this, sell the July put for $3.63 and buy the October put for $3.25, for a cash inflow of

$0.38. The exposure of the short position is completely covered by the long position in the

October put, with a positive cash inflow today.

22. The covered put would represent writing put options on the stock. This strategy is

analogous to a covered call because the upside potential of the underlying position, (which

in the case of a short sale would be a decline in the stock price), is capped in exchange for

the receipt of the option premium for certain.

The protective call would represent the purchase of call options as a form of

insurance for the short sale position. If the stock price rises, then losses incurred on the

short sale are offset, or insured, by gains on the call options; however, if the stock price

falls, which represents a profit to the short seller, then only the purchase price of the

option is lost.

FUNDAMENTALS OF INVESTMENTS B-51

Chapter 15

Option Valuation

Answers to Questions and Problems

Core Questions

1. The six factors are the stock price, the strike price, the time to expiration, the risk-free

interest rate, the stock price volatility, and the dividend yield.

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 option to increase in value. For example, imagine an out-of-the-

money option that is about to expire. Because the option is essentially worthless,

increasing the time to expiration obviously would increase its value.

3. An increase in volatility acts to increase both put and call values because greater volatility

increases the possibility of favorable in-the-money payoffs.

4. An increase in dividend yields 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 company that sells all its assets, pays off its debts, and then pays out

the remaining cash in a final, liquidating dividend. The stock price would fall to zero,

which is great for put holders, but not so great for call holders.

5. Interest rate increases are good for calls and bad for puts. The reason is that if a call is

exercised in the future, we have to pay a fixed amount at that time. The higher is the

interest rate, the lower is the present value of that fixed amount. The reverse is true for

puts in that we receive a fixed amount.

6. Rearranging the put-call parity condition to solve for P, the put price, and plugging in the

other numbers get us:

P = C в€’ S + Ke в€’ rT

= $10 в€’ $85 + $80e в€’ .06Г—.25

= $3.81

B-52 CORRADO AND JORDAN

7. Rearranging the put-call parity condition to solve for S, the stock price, and plugging in

the other numbers get us:

S = C в€’ P + Ke в€’ rT

= $10 в€’ $8 + $80e в€’ .04Г—.25

= $81.20

8. Using the option calculator with the following inputs:

S = current stock price = $100,

K = option strike price = $70,

r = risk-free rate = .05,

Пѓ = stock volatility = .30, and

T = time to expiration = 30 days

results in a call option price of $30.29.

9. Using the option calculator with the following inputs:

S = current stock price = $20,

K = option strike price = $22,

r = risk-free rate =.04,

Пѓ = stock volatility = .50,

T = time to expiration = 60 days, and

y = dividend yield = .02

results in a call option price of $.90.

10. Using the option calculator with the following inputs:

S = current stock price = $60,

K = option strike price = $65,

r = risk-free rate =.05,

Пѓ = stock volatility = .25, and

T = time to expiration = 180 days

results in a put option price of $6.24.

FUNDAMENTALS OF INVESTMENTS B-53

11. The call is worth more. To see this, we can rearrange the put-call parity condition as

follows:

C в€’ P = S в€’ Ke в€’ rT

If the options are at the money, S = K, so the right-hand side of this expression is equal to

the strike minus the present value of the strike price. This is necessarily positive.

Intuitively, if both options are at the money, the call option offers a much bigger potential

payoff, so itвЂ™ worth more.

s

12. Looking at the previous answer, if the call and put have the same price (i.e., C - P = 0), it

must be that the stock price is equal to the present value of the strike price, so the put is in

the money.

13. Looking at Question 7 above, a stock can be replicated by a long call (to capture the

upside gains), a short put (to reflect the downside losses), and a T-bill (to capture the

time-value componentвЂ“the вЂњwaitвЂќ factor).

14. An optionвЂ™ delta tells us the (approximate) dollar change in the optionвЂ™ value that will

s s

result from a change in the stock price. If a call sells for $2.00 with a delta of .60, a $1

stock price increase will add $.60 to option price, increasing it to $2.60.

15. The delta relates dollar changes in the stock to dollar changes in the option. The eta

relates percentage changes. So, the stock price rises by 2 percent ($100 to $102), an eta of

12 implies that the option price will rise by 24 percent.

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