Chapter 13 - Properties of Options (FRM Part 1 - Book 3)
Chapter 13 - Properties of Options
Chapter 13 - Properties of Options
1. Which of the following increases the value of a call option?
A. Increase in current stock price
B. Increase in strike price
C. Decrease in time to expiration
D. Increase in dividend payout
As the current stock price increases, the call option becomes more in-the-money, thereby increasing its value.
2. How does an increase in strike price affect the value of a call option?
A. Increases the call value
B. Decreases the call value
C. Has no effect
D. Makes the option risk-free
As the strike price increases, the call option becomes further out-of-the-money, reducing its value.
3. Which factor does not directly affect the value of an option?
A. Current stock price
B. Strike price
C. Company's brand value
D. Volatility of stock price
Brand value does not directly affect the mathematical pricing model of an option, unlike the other listed factors.
4. What effect does an increase in volatility (σ) have on an option's value?
A. Increases value of both call and put options
B. Increases call value but decreases put value
C. Decreases the option value
D. No impact
Higher volatility increases the potential range of stock movements, thereby increasing the value of both call and put options.
5. Which of the following would likely decrease the value of a put option?
A. Increase in strike price
B. Increase in time to expiration
C. Increase in volatility
D. Increase in current stock price
When the current stock price increases, a put option becomes less valuable as it's more out-of-the-money.
6. What is the general impact of increased time to expiration on the value of American-style options?
A. Increases the option value
B. Decreases the option value
C. No effect on option value
D. Makes the option risk-free
More time to expiration increases the chance of the option ending in-the-money, increasing its value for American-style options.
7. Why might a one-month European call option be worth more than a three-month call on the same stock with a dividend expected in two months?
A. Because short-term options always have more time value
B. Due to lower premium on short-term options
C. The stock price and the longer-term option value may fall when the dividend is paid
D. Dividends increase call value
Since the dividend will reduce the stock price in two months, it can make the longer-term call less valuable than the one-month call.
8. What is the impact of an increase in the risk-free interest rate on the value of a call option?
A. It decreases call option value
B. It increases call option value
C. It has no effect
D. It increases the strike price
A higher risk-free rate increases the present value of expected gains, thereby increasing the value of the call option.
9. How does an increase in dividend affect the value of a call option?
A. Increases the call value
B. Has no impact
C. Doubles the intrinsic value
D. Decreases the call value
Since call holders do not receive dividends, the expected drop in stock price when a dividend is paid decreases the value of the call.
10. Why is volatility considered one of the most important factors in option pricing?
A. Because it increases the probability of the option expiring in-the-money
B. Because it decreases the intrinsic value
C. Because it guarantees profit
D. Because it limits losses
Higher volatility increases the chance of a favorable price movement, raising the option’s value due to its asymmetric payoff.
11. What is the upper bound for the value of a European call option on a non-dividend paying stock?
A. Strike price of the option
B. Current stock price (S₀)
C. Risk-free rate times stock price
D. Intrinsic value
A European call gives the right to buy the stock, so it cannot be worth more than the stock itself. Hence, c ≤ S₀.
12. Why can't a European call option's value exceed the price of the underlying stock?
A. Because it would be unfair to the buyer
B. Because interest rate parity would be violated
C. Because arbitrage opportunities would exist
D. Because options have lower margins
If an option were worth more than the stock, traders could short the option, buy the stock, and pocket a riskless arbitrage profit.
13. What is the upper bound for the value of a European put option?
A. Present value of the strike price
B. The strike price itself
C. The current price of the stock
D. It has no upper bound
Since the European put cannot be exercised early, its maximum value is the present value of the amount you'd receive at maturity.
14. What is the upper bound of an American put option?
A. Present value of the stock
B. Intrinsic value
C. Present value of the strike price
D. Strike price (X)
The American put can be exercised any time, so its value can go up to the full strike price, i.e., P ≤ X.
15. For an American call option on a non-dividend paying stock, what is the correct upper bound?
A. Present value of strike price
B. Strike price plus premium
C. Current stock price (S₀)
D. Sum of intrinsic and time value
Just like European calls, American calls on non-dividend paying stocks cannot exceed the stock price: C ≤ S₀.
16. What is the standard put-call parity relationship for European options on non-dividend paying stocks?
A. c + PV(X) = S + p
B. c + X = S + p
C. c + p = S + X
D. c + p = S − PV(X)
The classic put-call parity for European options is: c + PV(X) = S + p. This ensures no-arbitrage between a fiduciary call and a protective put.
17. Which of the following portfolios replicates the payoff of a long stock position using European options?
A. Long call + Long put + Risk-free bond
B. Long call − Long put + PV(X)
C. Short call + Long put + Risk-free bond
D. Long put − Long call + S
To replicate a stock (S), use: S = c − p + PV(X). This involves a long call, a short put, and a long bond.
18. How is put-call parity affected when the underlying stock pays dividends?
A. No change; the formula remains the same
B. Replace X with X + dividends
C. Adjust the stock price for the present value of expected dividends
D. Add dividends to the right-hand side
For dividend-paying stocks, the stock value in the parity equation is adjusted: c + PV(X) = S − PV(Div) + p.
19. What is the synthetic equivalent of a European put option according to put-call parity?
A. p = c − S + PV(X)
B. p = S + c + PV(X)
C. p = S − c + PV(X)
D. p = S + c − PV(X)
Rearranging the parity formula gives: p = c − S + PV(X), showing how to synthetically create a put.
20. Expressing put-call parity in terms of the forward price, what is the relationship?
A. c + X = p + F₀
B. c − p = X + F₀
C. c + p = F₀ − X
D. c − p = S − PV(X) = F₀ − X
Put-call parity can be rewritten using the forward price: c − p = S − PV(X), and since F₀ = S − PV(dividends), it implies c − p = F₀ − X.
21. Why is it never optimal to exercise an American call option early on a non-dividend-paying stock?
A. Early exercise allows for greater profit potential
B. The investor can earn more interest by exercising early
C. Exercising early forgoes interest that could be earned by holding the option
D. It increases the stock price to its intrinsic value immediately
It is never optimal to exercise an American call early on a non-dividend-paying stock because exercising early means forgoing the interest that could be earned by holding the option. This interest is preferable to the early exercise value.
22. What is the primary difference between an American and a European option?
A. American options can be exercised early, while European options cannot
B. American options have a lower price bound than European options
C. European options are traded on the exchange, while American options are not
D. American options cannot be exercised at expiration
The primary difference between American and European options is that American options can be exercised before the expiration date, while European options can only be exercised at expiration.
23. Which of the following is a reason for not exercising an American call option early on a non-dividend-paying stock?
A. The stock price is expected to drop significantly before expiration
B. The value of the option would increase after early exercise
C. Exercising the option would lead to no profit
D. The investor can earn interest by holding the option instead of exercising it
The reason for not exercising an American call option early is that the investor can earn interest by holding the option, rather than forgoing the interest by exercising the option early.
24. According to the lower pricing bound for a European call option, which of the following is true?
A. c ≥ max(S0 − X, 0)
B. c ≥ max(S0 − X, 1)
C. c ≥ max(S0 − PV(X), 0)
D. c ≥ min(S0 − PV(X), 0)
The lower pricing bound for a European call option is c ≥ max(S0 − PV(X), 0), where c is the price of the call, S0 is the stock price, and PV(X) is the present value of the exercise price.
25. What is the relationship between the prices of an American call option and its European counterpart?
A. The price of an American call is always less than the price of a European call
B. The price of an American call is always greater than or equal to the price of a European call
C. The price of an American call is always equal to the price of a European call
D. The price of an American call is always less than the price of a European call due to early exercise feature
The price of an American call option is always greater than or equal to the price of a European call option, since an American call option can be exercised early, but it is optimal to wait until expiration unless there are dividends.
26. When is it optimal to exercise an American put option on a non-dividend-paying stock?
A. When the stock price is very high
B. When the stock price is close to the exercise price
C. When the option is out-of-the-money
D. When the option is sufficiently in-the-money
An American put option is optimally exercised early when it is sufficiently in-the-money. This occurs when the intrinsic value of the put is high, and exercising early allows the investor to invest the payoff to earn interest.
27. What is the lower pricing bound for an American put option on a non-dividend-paying stock?
A. P ≥ max(X − S0, 0)
B. P ≥ max(S0 − X, 0)
C. P ≥ max(PV(X) − S0, 0)
D. P ≥ PV(X) − S0
The lower pricing bound for an American put option on a non-dividend-paying stock is P ≥ max(X − S0, 0), where X is the exercise price and S0 is the stock price. This ensures that the option price cannot fall below the intrinsic value of the put.
28. Why is it never optimal to exercise an American call option early on a non-dividend-paying stock?
A. The stock price will decrease significantly in the future
B. The investor would forgo interest that could be earned by holding the option
C. The early exercise reduces the total payoff of the option
D. The exercise value is always higher than the option's future value
It is never optimal to exercise an American call option early because the investor would forgo the interest that could be earned by holding the option. The value of the option is generally greater when held until expiration.
29. When is it more beneficial to exercise an American put option early on a non-dividend-paying stock?
A. When the stock price is high and volatility is low
B. When the put is out-of-the-money
C. When the stock price is low and the option is deep in-the-money
D. When the option is expiring soon and has no intrinsic value
Exercising an American put option early is more beneficial when the stock price is low and the option is deep in-the-money. This allows the investor to lock in a high payoff and earn interest by investing the exercise value.
30. In the extreme case when S0 approaches zero, what is the relationship between exercising an American put option and the present value of the exercise price?
A. Exercising the put early results in less value than waiting
B. Exercising the put immediately is irrelevant as the stock price is high
C. The future value of the exercised cash value, PV(X), is always worth more than later exercise
D. There is no difference between early and late exercise when S0 is close to zero
When S0 is close to zero, the future value of the exercised cash value (PV(X)) is always worth more than a later exercise. In this case, exercising the American put early is optimal.
31. What is the relationship between the American call and put options according to the inequality for American options?
A. S0 − X ≤ C − P ≤ S0 − PV(X)
B. S0 − X ≤ C − P ≤ S0 + PV(X)
C. C − P ≤ S0 − X
D. C − P ≥ S0 − X
The inequality for American options is: S0 − X ≤ C − P ≤ S0 − PV(X). This places upper and lower bounds on the difference between the American call and put options.
32. How does the payment of a dividend affect the pricing of a call option?
A. It increases the value of the call option
B. It decreases the value of the call option
C. It has no effect on the call option value
D. It makes the call option more volatile
The payment of a dividend decreases the value of a call option because the stock price typically drops by the amount of the dividend, lowering the potential payoff from the call.
33. What is the lower pricing bound for a European call option in the presence of a dividend payment?
A. c ≥ S0 − PV(X)
B. c ≥ S0 − D − PV(X)
C. c ≥ max(S0 − D − PV(X), 0)
D. c ≥ max(S0 − D − X, 0)
The payment of a dividend reduces the lower pricing bound for a call option, and the new bound is: c ≥ S0 − D − PV(X), where D is the dividend and PV(X) is the present value of the exercise price.
34. How does the payment of a dividend affect the pricing of a put option?
A. It increases the value of the put option
B. It decreases the value of the put option
C. It has no effect on the put option value
D. It makes the put option more volatile
The payment of a dividend increases the value of a put option because it generally causes the stock price to fall, making the put more valuable.
35. What is the lower pricing bound for a European put option in the presence of a dividend payment?
A. p ≥ max(PV(X) − S0, 0)
B. p ≥ max(PV(X) − S0 − D, 0)
C. p ≥ D + PV(X) − S0
D. p ≥ max(S0 − D − PV(X), 0)
The payment of a dividend increases the lower pricing bound for a put option, which is given by: p ≥ D + PV(X) − S0. This reflects the impact of dividends on increasing the value of the put.
36. Under what condition might an American call option be optimally exercised early due to dividends?
A. When the dividend exceeds the amount of interest forgone from early exercise
B. When the stock price is high enough to make early exercise optimal
C. When the dividend is paid after the option expires
D. When the dividend is small enough to not impact the option price
American call options might be optimally exercised early when the dividend exceeds the interest forgone by early exercise. In such cases, exercise should occur just before the ex-dividend date.
37. What is the modified put-call parity equation when dividends are considered?
A. p + S0 = c + PV(X)
B. p + S0 = c + D + PV(X)
C. p + S0 = c + D
D. p + S0 − D = c + PV(X)
The modified put-call parity equation when dividends are considered is: p + S0 = c + D + PV(X). This adjustment reflects the impact of dividends on the pricing of options.
38. How is the relationship between American call and put options modified to account for dividends?
A. S0 − X ≤ C − P ≤ S0 − D
B. S0 − X − D ≤ C − P ≤ S0 − PV(X)
C. S0 − X − D ≤ C − P ≤ S0 − PV(X)
D. S0 − X ≤ C − P ≤ S0 − PV(X) − D
The modified inequality for the relationship between American call and put options with dividends is: S0 − X − D ≤ C − P ≤ S0 − PV(X). This accounts for the dividend payout reducing the value of the call and affecting the difference between the options.
39. Which of the following factors influence the value of an option?
A. Current value of the underlying asset, strike price, time to expiration, volatility, risk-free rate, and dividends
B. Time to expiration, only
C. Only the current stock price and volatility
D. Current value of the underlying asset, strike price, and time to expiration only
Six factors influence the value of an option: current value of the underlying asset, the strike price, the time to expiration, volatility, the risk-free rate, and dividends. With the exception of time to expiration, these factors impact both European and American-style options in the same way.
40. What is the maximum value of a call option?
A. The present value of the strike price
B. The strike price
C. The underlying security's current value
D. The present value of the underlying security's future price
Call options cannot be worth more than the underlying security itself. Therefore, the maximum value of a call option is the current value of the underlying stock.
41. Which of the following is the minimum value of a European call option?
A. The strike price minus the current stock price
B. The difference between the current stock price and the present value of the strike price
C. The difference between the present value of the strike price and the current stock price
D. The present value of the stock price
European call options cannot be worth less than the difference between the current stock price and the present value of the strike price.
42. What is the minimum value of a European put option?
A. The difference between the present value of the strike price and the current stock price
B. The strike price minus the current stock price
C. The present value of the strike price
D. The present value of the stock price
European put options cannot be worth less than the difference between the present value of the strike price and the current stock price.
43. Which of the following is the correct expression for put-call parity for European-style options?
A. p + PV(X) = c + S0
B. p + S0 = c + PV(F)
C. p + S0 = c + PV(X)
D. p + S0 = c + S0
Put-call parity is a no-arbitrage relationship for European-style options. It states that the value of a portfolio consisting of a call option and a zero-coupon bond with a face value equal to the strike price must be equal to a portfolio consisting of the corresponding put option and the stock. The correct expression is: p + S0 = c + PV(X).
44. Which of the following expressions represents put-call parity in terms of forward prices?
A. p + S0 = c + PV(X)
B. p + PV(F) = c + PV(X)
C. p + S0 = c + S0
D. p + S0 = c + F
Put-call parity can be expressed in terms of forward prices as: p + PV(F) = c + PV(X). This relationship allows for the comparison between the value of a put and a call using forward prices instead of spot prices.
45. It is never optimal to exercise an American call option on non-dividend-paying stock before expiration. This is because:
A. The investor would forgo the interest on the strike price.
B. The call option would be worthless if exercised early.
C. The value of the stock is always lower than the strike price.
D. There is no benefit to exercising a call option early on non-dividend-paying stocks.
It is never optimal to exercise an American call option on non-dividend-paying stock before expiration because the investor would forgo the interest that could have been earned on the cash used for early exercise, making it better to hold the option until expiration.
46. When are American put options on non-dividend-paying stocks optimally exercised before expiration?
A. Only when the stock price reaches its highest value
B. Always, because the value of the option increases with time
C. When the put option is sufficiently in-the-money
D. When the option reaches its expiration date
American put options on non-dividend-paying stocks can be optimally exercised before expiration if the put is sufficiently in-the-money. This is because the immediate payoff from exercising the option may exceed the value of waiting for expiration.
