Deck 17: The Option Greeks

Question 1

The current price of a call is $5 and the stock is trading at $50. The delta and gamma of the call are 0.5 and 0.05, respectively. If the stock price increases to $50.50, then approximate the new call price using the information given:
(a) $5.24375
(b) $5.25
(c) $5.25625
(d) $5.2625

Answer 1

C.

Question 2

Which of the following statements is true? Consider options written on a non-dividend-paying stock.
(a) All American options have negative theta and positive gamma.
(b) American calls have negative theta and positive gamma, but American puts may sometimes have positive theta and negative gamma.
(c) All American options have negative theta and all American calls have positive gamma, but American puts may sometimes have negative gamma.
(d) All American options have positive gamma and American calls have negative theta, but American puts may sometimes have positive theta.

Answer 2

A.

Question 3

You hold a portfolio of a long position in a call and a long position in a put, both for the same strike and maturity. Which of the following statements is true?
(a) When the stock price increases, the delta of the portfolio increases if the call is in-the-money.
(b) When the stock price increases, the delta of the portfolio decreases if the call is out-of-the-money.
(c) When the stock price increases, the delta of the portfolio increases whether or not the call is in-the-money.
(d) There is not enough information to answer this question.

Answer 3

C.

Question 4

The delta of a call option is 0.6. The current price of the call is $5 and that of a put at the same strike is $4, and the stock is at $100. What is the approximate price of the put if the stock price increases to $100.50?
(a) $3.60
(b) $3.80
(c) $4.20
(d) $4.50

Question 5

The current stock price is $50, and a put is priced at $3.59201. If the stock rises to $50.10, the price of the put will be $3.549152 and if the stock drops to $49.90, the price of the put will be $3.635261. What is the approximate gamma of the put?

A) $0.4$
B) $0.04$
C) $0.004$
D) Cannot be calculated from the given data.

Question 6

You hold a portfolio of a long position in a call and a short position in a put, both for the same strike and maturity, both written on a non-dividend paying stock. Which of the following statements is most correct?
(a) The delta of the portfolio increases when the stock price increases.
(b) The delta of the portfolio stays the same when the stock price increases.
(c) The delta of the portfolio decreases when the stock price increases.
(d) The delta of the portfolio may increase or decrease when the stock price increases.

Question 7

The delta of a call option is 0.6. The current price of the call is $5 and the stock is at $100. What is the approximate price of the call if the stock price increases to $100.50?
(a) $4.70
(b) $5.30
(c) $5.60
(d) $8.00

Question 8

Which of the following statements is true? Consider options written on a non-dividend-paying stock.
(a) All European and American options have negative theta.
(b) European options and American calls have negative theta, but American puts may have positive theta.
(c) European options have negative theta, but all American options may have positive or negative theta.
(d) Theta may be positive or negative for both American and European options.

Question 9

Which of the following statements is true? Consider options written on a non-dividend-paying stock.
(a) All European and American options have positive gamma.
(b) European options and American calls have positive gamma, but American puts may have negative gamma.
(c) European options have positive gamma, but American options may have positive or negative gamma.
(d) Gamma may be positive or negative for both American and European options.

Question 10

You are short a put on $A B C$ stock and have delta-hedged yourself using the stock. The delta of the put is $- 0.71$ and the gamma of the put is $+ 0.04$ . If the price of the underlying registers an increase of $0.50, to maintain your delta-hedge, you must

A) Sell 0.04 units of the stock.
B) Buy 0.04 units of the stock.
C) Sell 0.02 units of the stock.
D) Buy 0.02 units of the stock.
E) None of the above.

Question 11

The delta of an option measures, approximately,
(a) The dollar change in option value for a $1 change in the price of the underlying.
(b) The percentage change in option value for a 1% change in the price of the underlying.
(c) The risk-neutral probability that the option finishes in-the-money.
(d) The reaction of the option to a sudden jump in the price of the underlying.

Question 12

Which of the following statements is valid for European options?
(a) The gamma of a call is positive and that of a put is negative.
(b) The gamma of a call is negative and that of a put is negative.
(c) The gamma of a call is negative and that of a put is positive.
(d) The gamma of a call is positive and that of a put is positive.

Question 13

Which of the following statements is false?
(a) The change in value of a portfolio of puts and calls on the same stock is approximated by the sum of the product of each option's delta with the change in the stock price.
(b) The change in value of a portfolio of puts and calls on the same stock is approximated by the product of the sum of each option's delta with the change in the stock price.
(c) The change in value of a portfolio of puts and calls on the same stock is approximated by the sum of the product of each option's delta with the change in the stock price plus the sum of the cross-correlated changes of each option with the other options for the same change in the stock price.
(d) The change in value of a portfolio of puts and calls on the same stock is approximated by the sum of the product of each option's delta with the change in the stock price plus half the sum of the product of option gammas with squared changes in the stock price.

Question 14

The gamma of an option is
(a) The dollar change in the option delta for a $1 change in the price of the underlying.
(b) The percentage change in option delta for a 1% change in the price of the underlying.
(c) The dollar change in the option price for a sudden unit jump change in the price of the underlying.
(d) The percentage change in the option price for a sudden unit jump change in the price of the underlying.

Question 15

The current stock price is $50, and a put is priced at $3.59201. If the stock rises to $50.10, the price of the put will be $3.549152 and if the stock drops to $49.90, the price of the put will be $3.635261. What is the best estimate of the delta of the put?

A) $- 0.42858$
B) $- 0.43251$
C) $- 0.43055$
D) Cannot be calculated from the given data.

Question 16

You hold a straddle on a stock that you bought a month ago and that still has two months to expiry. Assume the options are European. An unexpected increase of $1 in the price of the stock
(a) Will increase the value of your straddle.
(b) Will decrease the value of your straddle.
(c) Will have no effect on the value of your straddle.
(d) Can increase, decrease, or leave unchanged the value of the straddle.

Question 17

You hold a straddle on a stock that you bought a month ago and that still has two months to expiry. Assume the options are European. An unexpected increase of $1 in the price of the stock
(a) Changes the value of your straddle by more than $1.
(b) Changes the value of your straddle by at most $1.
(c) Leaves the value of your straddle unchanged.
(d) Any of the above can happen.

Question 18

The current stock price is $50, and a put is priced at $3.59201. If the stock rises to $50.10, the price of the put will be $3.549152 and if the stock drops to $49.90, the price of the put will be $3.635261. If the stock jumps to $52, what is your best estimate of the new price of the put using the put's delta and gamma?
(a) 2.8095
(b) 2.7309
(c) 2.6523
(d) 4.5317

Question 19

A stock is currently trading at $50. A three-month at-the-money European put option on the stock costs 2.178. The delta of the put is $- 0.4355$ and the gamma of the put is 0.063. Given these values, if the stock price decreases by $5.00, then the best estimate for the new value of the put is

A) $4.35$ .
B) $5.14$ .
C) $0.79$ .
D) $3.57$ .
E) Zero.

Question 20

A stock is trading at $80. You hold a delta-hedged portfolio in which you are short a call and long $\Delta$ units of the stock. The delta of the call is 0.65 and the gamma of the call is 0.06. If the stock registers an unexpected price decrease of $4, the value of your delta-hedged portfolio will

A) not change.
B) decrease by approximately $0.12.
C) increase by approximately $0.48.
D) decrease by approximately $0.48.
E) increase by approximately $0.12.

Question 21

Which of the following statements is true? Consider options written on a non-dividend-paying stock.
(a) All European options have negative theta and positive gamma.
(b) European calls have negative theta and positive gamma, but European puts may sometimes have positive theta and negative gamma.
(c) All European options have negative theta and all European calls have positive gamma, but European puts may sometimes have negative gamma.
(d) All European options have positive gamma and European calls have negative theta, but European puts may sometimes have positive theta.

Question 22

For options that are at-the-money, which of the following statements is typically valid as maturity nears?
(a) Gamma increases as does theta.
(b) Gamma increases and theta decreases.
(c) Gamma decreases and theta increases.
(d) Gamma and theta both decrease.

Question 23

A stock is trading at $24. A three-month European put option with a strike of $35 costs $10.855 and has a theta of 1.735. The passage of one trading day ( $\approx 0.004$ years) causes the value of the put to, approximately,

A) Fall by $0.004$ .
B) Fall by $0.007$ .
C) Fall by $0.043$ .
D) Rise by $0.007$ .

Question 24

The vega of a ________ is highest when it is_________.
(a) call; deep in-the-money.
(b) put; deep out-of-the-money.
(c) call or put; far away-from-the-money.
(d) None of the above.

Question 25

A stock is trading at $20. A one-month put with a strike of 19 is valued at $0.360 and has a rho of $- 0.50$ . A decrease of 10 basis points in the interest rate changes the put value by

A) $- 0.005$ .
B) $+ 0.005$ .
C) $- 0.05$
D) $+ 0.0005$ .

Question 26

Gamma is a risk measure that is related to the volatility (particularly jump-risk) of the underlying stock. Which of the following is most valid?
(a) When you buy vanilla call options it is useful to hedge away gamma to minimize jump risk.
(b) When you sell vanilla put options it is useful to hedge away gamma to minimize jump risk.
(c) Both (a) and (b).
(d) Neither (a) nor (b).

Question 27

You expect a sizable jump in the stock price but are not sure of the direction in which the stock will go. Which of the following alternatives would be best?
(a) Buy call options.
(b) Buy put options.
(c) Sell a straddle.
(d) Buy a call and delta hedge it.

Question 28

The gamma of a put is typically highest when
(a) The stock price is much lower than the strike price.
(b) The stock price is in the region of the strike price.
(c) The stock price is much higher than the strike price.
(d) The information is insufficient to answer this question.

Question 29

The absolute value of theta is highest for options
(a) That are deep out-of-the-money.
(b) That are near-the-money.
(c) That are deep in-the-money.
(d) There is not enough information to determine the answer.

Question 30

A stock is trading at $132. A one-month call with a strike of $125 costs $9.773 and has a theta of $- 30.53$ . The passage of one trading day ( $\approx 0.004$ years) will cause the call value to, approximately,

A) Rise by $0.04$ .
B) Fall by $0.003$
C) Fall by $0.122$ .
D) Fall by $0.04$

Question 31

Consider options written on a non-dividend-paying stock. Deep in-the-money put options may
(a) Have negative gamma and positive theta if they are European in style.
(b) Have negative gamma and negative theta if they are American in style.
(c) Have positive gamma and positive theta if they are European in style.
(d) Have positive gamma and positive theta if they are American in style.

Question 32

A stock is trading at $20. A one-month call with a strike of 19 is valued at $1.439 and has a rho of 1.09. An increase in interest rates of 10 basis points changes the call value to, approximately,
(a) 1.440
(b) 1.450
(c) 1.548
(d) 2.529

Question 33

Theta is always negative except possibly for certain
(a) European calls.
(b) European puts.
(c) American calls.
(d) American puts.

Question 34

You hold a portfolio of a long position in a call and a long position in a put, both for the same strike and maturity. Which of the following statements is true?
(a) When the stock price increases, the rho of the portfolio increases if and only if the call is in-the-money.
(b) When the stock price increases, the rho of the portfolio decreases if and only if the call is out-of-the-money.
(c) When the stock price increases, the rho of the portfolio increases whether or not the call is in-the-money.
(d) When the stock price increases, the rho of the portfolio may increase, decrease or stay constant depending on volatility and depth-in-the-money.

Question 35

The price of a European call option is $5 at an implied volatility of 0.25. The vega of the call is 20. If the implied volatility increases to 0.26, what is the new value of a European put option with the same strike and maturity as the call that is currently priced at $6?
(a) $0.8
(b) $11.2
(c) $5.8
(d) $6.2

Question 36

Which of the following is NOT valid about the time decay of European put and call options with the same strike and maturity?
(a) The time decay is measured in dollars per unit time.
(b) The call decays at the same rate as the put.
(c) The call decays faster than the put.
(d) The difference between the time decay of decay of the call and put depends on the moneyness of the options.