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Bodieâˆ’Kaneâˆ’Marcus: V. Derivative Markets 15. Option Valuation Â© The McGrawâˆ’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

559

15 Option Valuation

e. Price of

Call T X S Option

A 0.5 50 55 10

B 0.5 55 55 7

Which call option is written on the stock with higher volatility?

(1) A

(2) B

(3) Not enough information

3. Reconsider the determination of the hedge ratio in the two-state model, where we

showed that one-half share of stock would hedge one option. What is the hedge ratio at

each of the following exercise prices: $115, $100, $75, $50, $25, and $10? What do you

conclude about the hedge ratio as the option becomes progressively more in the money?

4. Show that Black-Scholes call option hedge ratios also increase as the stock price

increases. Consider a one-year option with exercise price $50 on a stock with annual

standard deviation 20%. The T-bill rate is 8% per year. Find N(d1) for stock prices $45,

$50, and $55.

5. We will derive a two-state put option value in this problem. Data: S0 100; X 110;

1 r 1.1. The two possibilities for ST are 130 and 80.

a. Show that the range of S is 50 while that of P is 30 across the two states. What is the

hedge ratio of the put?

b. Form a portfolio of three shares of stock and five puts. What is the (nonrandom)

payoff to this portfolio? What is the present value of the portfolio?

c. Given that the stock currently is selling at 100, show that the value of the put must

be 10.91.

6. Calculate the value of a call option on the stock in problem 5 with an exercise price of

110. Verify that the put-call parity relationship is satisfied by your answers to problems

5 and 6. (Do not use continuous compounding to calculate the present value of X in this

example, because the interest rate is quoted as an effective annual yield.)

7. Use the Black-Scholes formula to find the value of a call option on the following stock:

Time to maturity 6 months

Standard deviation 50% per year

Exercise price $50

Stock price $50

Interest rate 10%

8. Find the Black-Scholes value of a put option on the stock in the previous problem with

the same exercise price and maturity as the call option.

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9. What would be the Excel formula in Figure 15.4 for the Black-Scholes value of a

straddle position?

10. Recalculate the value of the option in problem 7, successively substituting one of the

changes below while keeping the other parameters as in problem 7:

a. Time to maturity 3 months

b. Standard deviation 25% per year

c. Exercise price $55

d. Stock price $55

e. Interest rate 15%

Bodieâˆ’Kaneâˆ’Marcus: V. Derivative Markets 15. Option Valuation Â© The McGrawâˆ’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

560 Part FIVE Derivative Markets

Consider each scenario independently. Confirm that the option value changes in

accordance with the prediction of Table 15.1.

11. Would you expect a $1 increase in a call optionâ€™s exercise price to lead to a decrease

in the optionâ€™s value of more or less than $1?

12. All else being equal, is a put option on a high beta stock worth more than one on a low

beta stock? The firms have identical firm-specific risk.

13. All else being equal, is a call option on a stock with a lot of firm-specific risk worth

more than one on a stock with little firm-specific risk? The betas of the stocks are equal.

14. All else being equal, will a call option with a high exercise price have a higher or lower

hedge ratio than one with a low exercise price?

15. Should the rate of return of a call option on a long-term Treasury bond be more or less

sensitive to changes in interest rates than the rate of return of the underlying bond?

16. If the stock price falls and the call price rises, then what has happened to the call

optionâ€™s implied volatility?

17. If the time to maturity falls and the put price rises, then what has happened to the put

optionâ€™s implied volatility?

18. According to the Black-Scholes formula, what will be the value of the hedge ratio of a

call option as the stock price becomes infinitely large? Explain briefly.

19. According to the Black-Scholes formula, what will be the value of the hedge ratio of

a put option for a very small exercise price?

20. The hedge ratio of an at-the-money call option on IBM is 0.4. The hedge ratio of an

at-the-money put option is 0.6. What is the hedge ratio of an at-the-money straddle

position on IBM?

21. These three put options all are written on the same stock. One has a delta of 0.9, one

a delta of 0.5, and one a delta of 0.1. Assign deltas to the three puts by filling in the

table below.

Put X Delta

A 10

B 20

C 30

22. In this problem, we derive the put-call parity relationship for European options on

stocks that pay dividends before option expiration. For simplicity, assume that the stock

makes one dividend payment of $D per share at the expiration date of the option.

a. What is the value of the stock-plus-put position on the expiration date of the option?

b. Now consider a portfolio comprising a call option and a zero-coupon bond with the

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same maturity date as the option and with face value (X D). What is the value of

this portfolio on the option expiration date? You should find that its value equals that

of the stock-plus-put portfolio, regardless of the stock price.

c. What is the cost of establishing the two portfolios in parts (a) and (b)? Equate the

cost of these portfolios, and you will derive the put-call parity relationship,

Equation 15.3.

23. A collar is established by buying a share of stock for $50, buying a six-month put option

with exercise price $45, and writing a six-month call option with exercise price $55.

Based on the volatility of the stock, you calculate that for an exercise price of $45 and

maturity of six months, N(d1) .60, whereas for the exercise price of $55, N(d1) .35.

Bodieâˆ’Kaneâˆ’Marcus: V. Derivative Markets 15. Option Valuation Â© The McGrawâˆ’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

561

15 Option Valuation

a. What will be the gain or loss on the collar if the stock price increases by $1?

b. What happens to the delta of the portfolio if the stock price becomes very large?

Very small?

24. You are very bullish (optimistic) on stock EFG, much more so than the rest of the

market. In each question, choose the portfolio strategy that will give you the biggest

dollar profit if your bullish forecast turns out to be correct. Explain your answer.

a. Choice A: $100,000 invested in calls with X 50.

Choice B: $100,000 invested in EFG stock.

b. Choice A: 10 call options contracts (for 100 shares each), with X 50.

Choice B: 1,000 shares of EFG stock.

25. Imagine you are a provider of portfolio insurance. You are establishing a four-year

program. The portfolio you manage is currently worth $100 million, and you promise to

provide a minimum return of 0%. The equity portfolio has a standard deviation of 25%

per year, and T-bills pay 5% per year. Assume for simplicity that the portfolio pays no

dividends (or that all dividends are reinvested).

a. What fraction of the portfolio should be placed in bills? What fraction in equity?

b. What should the manager do if the stock portfolio falls by 3% on the first day of

trading?

26. You would like to be holding a protective put position on the stock of XYZ Co. to lock

in a guaranteed minimum value of $100 at year-end. XYZ currently sells for $100. Over

the next year, the stock price will either increase by 10% or decrease by 10%. The T-bill

rate is 5%. Unfortunately, no put options are traded on XYZ Co.

a. Suppose the desired put option were traded. How much would it cost to purchase?

b. What would have been the cost of the protective put portfolio?

c. What portfolio position in stock and T-bills will ensure you a payoff equal to the

payoff that would be provided by a protective put with X $100? Show that the

payoff to this portfolio and the cost of establishing the portfolio matches that of the

desired protective put.

27. You are attempting to value a call option with an exercise price of $100 and one year

to expiration. The underlying stock pays no dividends, its current price is $100, and

you believe it has a 50% chance of increasing to $120 and a 50% chance of decreasing

to $80. The risk-free rate of interest is 10%. Calculate the call optionâ€™s value using the

two-state stock price model.

28. Consider an increase in the volatility of the stock in problem 27. Suppose that if the

stock increases in price, it will increase to $130, and that if it falls, it will fall to $70.

Show that the value of the call option is now higher than the value derived in

problem 27.

29. Return to Example 15.1. Use the binomial model to value a one-year European put

option with exercise price $110 on the stock in that example. Does your solution for

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the put price satisfy put-call parity?

Bodieâˆ’Kaneâˆ’Marcus: V. Derivative Markets 15. Option Valuation Â© The McGrawâˆ’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

562 Part FIVE Derivative Markets

WEBMA STER

Option Value and Greeks

Go to http://www.thegumpinvestor.com/options/home.asp. This site offers extensive in-

formation on options. From the quote tab, find the option quotes for both puts and

calls for Dell Computer (DELL). Select the item that shows options within nine months

to expiration with strike prices that are close to the underlying stock price (near the

money). After examining the data, answer the following questions.

1. Does the Black-Scholes model predict the option prices perfectly?

2. What is the largest error noted in your screen?

3. What do the delta and theta of an option indicate?

4. Are the estimates of implied volatility similar for all of the options?

SOLUTIONS TO 1. Yes. Consider the same scenarios as for the call.

>

Concept Stock price $10 $20 $30 $40 $50

CHECKS Put payoff 20 10 0 0 0

Stock price 20 25 30 35 40

Put payoff 10 5 0 0 0

The low volatility scenario yields a lower expected payoff.

If This Variable Increases ... The Value of a Put Option

2.

S Decreases

X Increases

Increases

T Increases/Uncertain*

rf Decreases

Dividend payouts Increases

*For American puts, increase in time to expiration must increase value. One can always

choose to exercise early if this is optimal; the longer expiration date simply expands the

range of alternatives open to the option holder, thereby making the option more valuable.

For a European put, where early exercise is not allowed, longer time to expiration can

have an indeterminate effect. Longer maturity increases volatility value since the final

stock price is more uncertain, but it reduces the present value of the exercise price that

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will be received if the put is exercised. The net effect on put value is ambiguous.

3. Because the option now is underpriced, we want to reverse our previous strategy.

Cash Flow in 1 Year for Each

Possible Stock Price

Initial

Cash Flow S $50 S $200

Buy 2 options $ 48 $ 0 $ 150

Short-sell 1 share 100 50 200

Lend $52 at 8% interest rate 52 56.16 56.16

Total $ 0 $ 6.16 $ 6.16

Bodieâˆ’Kaneâˆ’Marcus: V. Derivative Markets 15. Option Valuation Â© The McGrawâˆ’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

563

15 Option Valuation

4. a. C+ C $6.984 0 $6.984

b. S S $110 $95 $15

c. 6.984/15 .4656

Value in Next Period as

d.

Function of Stock Price

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