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P15-16. Explain the following paradox. A put option is a highly volatile security. If the underlying stock has a positive beta, then a put option on that stock will have a negative beta (because the put and the stock move in opposite directions).
According to the CAPM, an asset with a negative beta, such as the put option, has an expected return below the risk-free rate. How can an equilibrium exist in which a highly risky security such as a put option offers an expected return below a much safer security such as a Treasury bill?

P15-17. A particular stock sells for $27. A call option on this stock is available, with a strike price of $28 and an expiration date in four months. If the risk-free rate equals 6 percent and the standard deviation of the stock's return is 40 percent, what is the price of the call option? Next, recalculate your answer assuming that the market price of the stock is $28. How much does the option price change in dollar terms? How much does it change in percentage terms?

P15-18. Temex Foods stock currently sells for $48. A call option on this stock is available, with a strike price of $45 and an expiration date six months in the future. The standard deviation of the stock's return is 45 percent, and the risk-free interest rate is 4 percent. Calculate the value of the call option. Next, use the put-call parity to determine the value of a Temex put option that also has a $45 strike price and six months until expiration.

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Answers to questions on put, call options

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Textbook: Essentials of Investments. Chapter 16 (1, 2, 5, 6, 7 & 12). Problems on call and put options.

We showed in the text that the value of a call option increases with the volatility of the stock. Is this also true of put option values? In each of the following questions, you are asked to compare two options with parameters as given. What is the hedge ratio of the put? Verify that the put-call parity relationship is satisfied by your answers. Use the Black-Scholes formula to find the value of a call option on the following stock. 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.

1.We showed in the text that the value of a call option increases with the volatility of the stock. Is this also true of put option values? Use the put-call parity relationship as well as a numerical example to prove your answer.

2.In each of the following questions, you are asked to compare two options with parameters as given. The risk-free interest rate for all cases should be assumed to be 6%. Assume the stocks on which these options are written pay no dividends.

I.

Put T X s Price of Option
0.5 50 0.20 10
B 0.5 50 0.25 10

Which put option is written on the stock with the lower price?
(1) A
(2) B
(3) Not enough information

II.
Put T X s Price of Option
A 0.5 50 0.2 10
B 0.5 50 0.2 12
Which put option must be written on the stock with the lower price?
a. A
b.B
c. Not enough information

III.
Call S X s Price of Option
A 50 50 0.20 12
B 55 50 0.20 10
Which call option must have the lower time to expiration?
a. A
b. B
c. Not enough information

IV.
Call T X S Price of Option
A 0.5 50 55 10
B 0.5 50 55 12
Which call option is written on the stock with higher volatility?
a. A
b. B
c. Not enough information

Call T X S Price of Option
A 0.5 50 55 10
B 0.5 55 55 7
Which call option is written on the stock with higher volatility?
a. A
b. B
c.Not enough information

5. We will derive a two-state put option value in this problem. Data: S0 = 100; X = 110; 1 + r = 1.10. The two possibilities for ST are 130 and 80.
1. 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?
2. 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?
3. 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 expiration = 6 months
Standard deviation = 50% per year
Exercise price = $50
Stock price = $50
Interest rate = 10%

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.

Please see attached for problem. Thanks.

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