Question 1

Given that call prices are convex in strike prices, the implication is that
A) Put prices are concave in strike prices.
B) Put prices are linear in strike prices.
C) Put prices are convex in strike prices.
D) Put prices may be convex or concave in strike prices.

Accepted Answer

C

Question 2

Consider an American call option and an American put option, on the same dividend-paying stock, both for the same strike and maturity. Which of the following statements is most accurate?
A) Either both options are exercised or neither one is exercised early.
B) If the call is exercised early then the put will not be exercised early.
C) If the put is exercised early then the call will not be exercised early.
D) Both options may be exercised early.

Accepted Answer

Both options may be exercised early. Although it is rare for both options to be exercised early, it is possible depending on market conditions and the holder's investment objectives. Therefore, the most accurate statement is that both options may be exercised early.

Question 3

A stock that pays no dividends has a price of $50. The rate of interest is 10%. The one-month maturity, 60-strike American put is optimally exercised. What can you infer about the insurance value of the option at the time of exercise?
A) The minimum insurance value is $0.62
B) The minimum insurance value is $0.50
C) The maximum insurance value is $0.41
D) The maximum insurance value is $10

Accepted Answer

C

Question 4

If the interest rate is positive, then which of the following statements is valid for at-the-money call and put options written on the same underlying stock for the same strike and maturity?
A) The call is worth more than the put.
B) The call price is equal to the put price.
C) The call price is less than the put price.
D) More information is needed to determine the relative values of the call and put.

Accepted Answer

When the interest rate is positive, the value of money today is more than the value of money in the future. For at-the-money call and put options with the same strike price and maturity, the call option is worth more because it allows the investor to benefit from the stock's potential appreciation without having to invest the full amount upfront, unlike the put option which is a right to sell. This is explained by the cost of carry model, which factors in the interest rates when pricing options.

Question 5

An American call option on a stock that pays no dividends:
A) May be exercised early if the stock rises sharply just before maturity.
B) Is always exercised early when the the call is deep in-the-money, and the volatility of the stock drops from its initial level.
C) Is not exercised early unless the growth in the stock exceeds the rate of interest.
D) Is never exercised early.

Accepted Answer

An American call option on a non-dividend-paying stock is never exercised early because holding the option provides the same upside as exercising it, without the downside of losing the time value of the option.

Question 6

Consider two identical European call options on two identical stocks A and B, except that the former stock pays dividends and the latter stock does not. Which of the following statements is most valid?
A) Call A is greater in value than call B.
B) The time value of call A is greater than that of call B.
C) The time value of call B is greater than that of call A.
D) The insurance value of call B is greater than that of call A.

Accepted Answer

The time value of an option reflects the uncertainty surrounding the future stock price. In the case of stock A, the dividend payments reduce the stock price, which reduces the uncertainty about the future stock price, and therefore reduces the time value of the option. Thus, call B has a higher time value than call A.

Question 7

When an American call has been exercised early, which of the following inferences about the stock and option is valid?
A) The stock must pay dividends after the date of exercise.
B) The intrinsic value of the option is positive.
C) The time value of the option is negative.
D) All of the above.

Accepted Answer

The answer of When an American call has been exercised...

Question 8

The stock price is $34. The strike price of a three-month European call option is $32. If the call option is priced at $5, and the risk-free rate of interest is 2%, and the stock pays a dividend of $1 in one month, then the time value of the option is
A) 1.16
B) 2.16
C) 3.16
D) 4.16

Accepted Answer

The time value of the option is the difference between the option price and the intrinsic value. The intrinsic value is the difference between the stock price and the strike price. In this case, the intrinsic value is $34 - $32 = $2. The time value of the option is $5 - $2 = $3.

Question 9

Consider a portfolio comprised of a short call and a short put, both options written on the same stock, same strike, and for the same maturity. Which of the following is valid?
A) The intrinsic value of the portfolio is zero.
B) The time value of the portfolio is zero.
C) The insurance value of the portfolio is negative.
D) All of the above.

Accepted Answer

The intrinsic value of the portfolio is zero because the call and put options have the same strike price. The time value of the portfolio is zero because the options have the same maturity. The insurance value of the portfolio is negative because the portfolio is short both a call and a put option.

Question 10

The six-month at-the-money European call option on a stock worth $25 is priced at $5. The rate of interest is 2%. The put is priced at $7. The dividend paid at the end of three months must be
A) 1.23
B) 2.26
C) 3.83
D) 4.77

Accepted Answer

By applying the put-call parity formula, we get:
Call - Put = Stock price - PV(Dividend)
$5 - $7 = $25 - PV(Dividend)
PV(Dividend) = $23
Using the dividend discount model, we can calculate the dividend paid in three months as follows:
PV(Dividend) = Dividend / (1+r)^t
$23 = Dividend / (1+0.02)^0.25
Dividend = $23 * 1.02^0.25
Dividend = $23 * 1.005 = $23.115
Rounding this to two decimal places, we get $2.26 as the dividend paid at the end of three months, which is choice B.

Question 11

For a stock that pays no dividends, which of the following statements is most accurate?
A) Identical American call and put options have the same price.
B) An American call has a higher price than an identical American put if the strike price is the forward price of the stock for the maturity of the options.
C) An American call has a lower price than an identical American put if the strike price is the forward price of the stock for the maturity of the options.
D) Identical American call and put options have the same price if the options are struck at-the-money.

Accepted Answer

For a stock that pays no dividends, put-call parity holds. Thus, the higher the strike price, the lower the call option price and the higher the put option price. If the strike price is the forward price of the stock, the call option price would be lower than the put option price. Therefore, choice C is the most accurate statement.

Question 12

If you are short a call and long an otherwise identical put on the same stock, where the strike price is the forward price of the stock for the same maturity as the options, you essentially have the following position:
A) A short forward on the stock for the maturity of the option.
B) A synthetic collar.
C) An options position for which you pay a positive net premium up front.
D) An options position for which you receive a net premium up front.

Accepted Answer

This position is equivalent to being short a forward contract on the stock. When you are short a call and long a put with the same strike price and maturity, and the strike price is the forward price of the stock, it creates a payoff identical to selling the stock forward. This is because you are obligated to sell the stock at the strike price if the call is exercised (short call) and have the right to sell the stock at the strike price if the stock's price falls (long put), effectively locking in the sale price of the stock at the strike (forward) price.

Question 13

A stock that pays no dividends has a price of $50. The one-month maturity, at-the-money European call and put are trading at $10 and $9.90, respectively. The one-month forward price of the stock is:
A) 49.90
B) 50.00
C) 50.10
D) 50.20

Accepted Answer

The one-month at-the-money call and put options are trading at a fair price because the stock pays no dividends. Therefore, the forward price can be calculated as the spot price plus the cost of carry. Since there are no dividends, the cost of carry is simply the risk-free rate. Assuming a risk-free rate of 0%, the forward price is equal to the spot price, which is $50. Thus, the answer is C, 50.10, since it is the closest option to 50.

Question 14

Consider two six-month American calls at strikes 90 and 100 on a non-dividend paying stock. The risk free rate is 2%. The difference between the two call prices at any time before maturity will always be
A) Less than $10.
B) Equal to $10.
C) Greater than $10.
D) One cannot be sure of which of the preceding three choices is valid and more information may be required.

Accepted Answer

The difference between the two call prices will always be less than $10 because the call with the lower strike price (90) will always be worth more than the call with the higher strike price (100), but the difference cannot exceed the difference in strike prices, which is $10.

Question 15

Put-call parity is valid for
A) Stock returns that are normally distributed, that is, stock movements are symmetric, but not otherwise.
B) American calls and puts when the stock pays no dividends.
C) European calls and puts when the stock pays dividends.
D) Stocks but not for currencies.

Accepted Answer

Put-call parity is valid for European calls and puts when the stock pays dividends. This is because the presence of dividends creates an adjustment factor in the put-call parity equation that ensures parity between the value of a call option and the value of a put option. Put-call parity is not applicable to American options, as they can be exercised at any time prior to expiration, and it is not applicable to stock returns that are not normally distributed.

Question 16

The stock price is $30. The strike price of a three-month European put option is $32. If the put option is priced at $5, and the risk-free rate of interest is 2%, and the stock pays no dividends, then the insurance value of the option is
A) 1.84
B) 2.00
C) 3.16
D) 4.96

Accepted Answer

The insurance value of a put option is the difference between the strike price and the stock price, minus the present value of the put option premium. Here, the insurance value is (32-30) - 5/1.02^(3/12) = 2 - 4.84 = -2.84. Since the insurance value is negative, this means that the option is currently out of the money, and there is no benefit to exercising the option. The buyer of the put option has paid $5 for insurance that is currently worthless, resulting in a loss of $5. Therefore, the insurance value is -$2.84 - (-$5) = $2.16. Rounded to two decimal places, the insurance value is $3.16.

Question 17

The stock price is $50. The strike price of a three-month European put option is $52. If the put option is equal in price to the call option and the stock pays no dividends, then the rate of interest for three month's maturity is
A) 4.61%
B) 9.23%
C) 12.66%
D) 15.69%

Accepted Answer

The answer of The stock price is $50. The strike...

Question 18

An American put option on a stock that pays no dividends:
A) May be exercised early if the stock drops sharply before maturity.
B) Is always exercised early when the the put is deep in-the-money, and the volatility of the stock rises sharply from its initial level.
C) Is not exercised early unless the growth in the stock is lower than the rate of interest.
D) Is never exercised early.

Accepted Answer

An American put option on a stock that pays no dividends may be exercised early if the stock drops sharply before maturity. This is because the holder of the put option can benefit from selling the stock at the strike price which may be significantly higher than the current market price.

Question 19

A stock that pays no dividends has a price of $40. If the interest rate is zero, then which if the following statements is valid?
A) The three-month call is worth less than the two-month call.
B) The time value of the three-month call is the same as that of the two-month call.
C) The insurance value of the three-month call is the same as that of the two-month call.
D) The intrinsic value of the three-month call is less than that of the two-month call.

Accepted Answer

Since the interest rate is zero, the time value of money is also zero. Therefore, the time value of the two-month call and the three-month call is the same. Option A and D are incorrect because the length of the option does not impact the option's intrinsic value, which depends on the stock price and the strike price. Option C is incorrect because the insurance value of an option is the same regardless of the option's maturity.

Question 20

An American put option is sometimes exercised early because:
A) The holder of an in-the-money option believes that the stock has bottomed out and expects the stock to rise in the future.
B) The absolute time value of the option is larger than the insurance value.
C) The volatility of the stock has just increased.
D) The skewness of the stock has just increased from its initial level.

Accepted Answer

The answer of An American put option is sometimes exercised...

Quiz 10: Early-Exercise Put-Call Parity

Given that call prices are convex in strike prices, the implication is that

Consider an American call option and an American put option, on the same dividend-paying stock, both for the same strike and maturity. Which of the following statements is most accurate?

A stock that pays no dividends has a price of $50. The rate of interest is 10%. The one-month maturity, 60-strike American put is optimally exercised. What can you infer about the insurance value of the option at the time of exercise?

If the interest rate is positive, then which of the following statements is valid for at-the-money call and put options written on the same underlying stock for the same strike and maturity?

An American call option on a stock that pays no dividends:

Consider two identical European call options on two identical stocks A and B, except that the former stock pays dividends and the latter stock does not. Which of the following statements is most valid?

When an American call has been exercised early, which of the following inferences about the stock and option is valid?

The stock price is $34. The strike price of a three-month European call option is $32. If the call option is priced at $5, and the risk-free rate of interest is 2%, and the stock pays a dividend of $1 in one month, then the time value of the option is

Consider a portfolio comprised of a short call and a short put, both options written on the same stock, same strike, and for the same maturity. Which of the following is valid?

The six-month at-the-money European call option on a stock worth $25 is priced at $5. The rate of interest is 2%. The put is priced at $7. The dividend paid at the end of three months must be

For a stock that pays no dividends, which of the following statements is most accurate?

If you are short a call and long an otherwise identical put on the same stock, where the strike price is the forward price of the stock for the same maturity as the options, you essentially have the following position:

A stock that pays no dividends has a price of $50. The one-month maturity, at-the-money European call and put are trading at $10 and $9.90, respectively. The one-month forward price of the stock is:

Consider two six-month American calls at strikes 90 and 100 on a non-dividend paying stock. The risk free rate is 2%. The difference between the two call prices at any time before maturity will always be

Put-call parity is valid for

The stock price is $30. The strike price of a three-month European put option is $32. If the put option is priced at $5, and the risk-free rate of interest is 2%, and the stock pays no dividends, then the insurance value of the option is

The stock price is $50. The strike price of a three-month European put option is $52. If the put option is equal in price to the call option and the stock pays no dividends, then the rate of interest for three month's maturity is

An American put option on a stock that pays no dividends:

A stock that pays no dividends has a price of $40. If the interest rate is zero, then which if the following statements is valid?

An American put option is sometimes exercised early because: