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# Problems on call and put options.

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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.

#### Solution Preview

The answers can be seen in the attached file:

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.

Put call parity : c + Present Value of X = p + S
where c is the value of call, X is the strike price, p is the value of put and S is the stock price
If we increase the volatility,keeping everything else the same, the value of call would go up
Present value of X and S remain the same.
Therefore, to maintain equality sign, the value of put would go up.

Numerical example:
Let
Stock Price= S= \$60.00
Exercise price = X= \$55.00
Time to expiration = T-t= 3 months
Risk free rate = r = 10.00%
standard deviation = volatility= s = 25.00%

Then the value of call= \$7.07
Value of put= \$0.72

If we now increase the value of volatility to 50%, keeping all else constant
Then the value of call= \$9.39
Value of put= \$3.03

Thus both the value of call and put increase when volatility increases
(Note: the values of calls and puts have been calculated using option calculator)

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.

When the stocks do not pay any dividends
Value of both call and put increase with an increase in volatility (s), time to expiration (T)
Value of call increases and put decreases with an increase in stock price (S)
Value of call decreases and put increases with an increase in strike price (X)

I
Put T X s Price of Option
A 0.5 50 0.2 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

Since price of put options are the same, and T and X are the same, and B has a higher volatility (which increases put value) stock A must be written on a stock with lower price (which increases put value)
In stock A , a lower stock price compensates for a lower volatility, leading to put option prices being the same.

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