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CHAPTER 5: OPTION PRICING MODELS: THE BLACK-SCHOLES-MERTON MODEL
MULTIPLE CHOICE TEST QUESTIONS
The following information is given about options on the stock of a certain company.
S0 = 23 X = 20
rc = 0.09 T = 0.5
2 = 0.15
No dividends are expected.
Use this information to answer questions 1 through 8.
1. What value does the Black-Scholes-Merton model predict for the call? (Due to differences in rounding your
calculations may be slightly different. “none of the above” should be selected only if your answer is
different by more than 10 cents.)
a. 5.35
b. 1.10
c. 4.73
d. 6.50
e. none of the above
2. Suppose you feel that the call is overpriced. What strategy should you use to exploit the apparent
misvaluation? (Due to differences in rounding your calculations may be slightly different. “none of the
above” should be selected only if your answer is different by more than 10 shares.)
a. buy 791 shares, sell 1,000 calls
b. buy 705 shares, sell 1,000 calls
c. sell short 791 shares, buy 1,000 calls
d. sell short 705 shares, buy 1,000 calls
e. none of the above
3. The price of a put on the stock is: (Due to differences in rounding your calculations may be slightly
different. “none of the above” should be selected only if your answer is different by more than 10 cents.)
a. 0.85
b. 8.64
c. 2.35
d. 4.88
e. none of the above
4. To construct a riskless hedge, the number of puts per 100 shares purchased is: (Due to differences in
rounding your calculations may be slightly different. “none of the above” should be selected only if your
answer is different by more than 0.01.)
a. 0.7580
b. 0.2420
c. –0.2480
d. –0.6628
e. none of the above
5. The call’s vega is: (Due to differences in rounding your calculations may be slightly different. “none of the
above” should be selected only if your answer is different by more than 0.05.)
a. –3.02
b. 0.046
c. –0.792
d. 4.67
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e. none of the above
6. If the actual call price is 3.79, the implied standard deviation is
a. 0.25
b. greater than 0.25
c. less than 0.25
d. infinite
e. none of the above
7. If we now assume that the stock pays a dividend at a known constant rate of 3.5 percent, what stock price
should we use in the model? (Due to differences in rounding your calculations may be slightly different.
“none of the above” should be selected only if your answer is different by more than 10 cents.)
a. 22.60
b. 19.65
c. 23.00
d. 21.99
e. none of the above
8. If we now assume that the stock pays a single dividend of 2.25 in three months, what stock price should we
use in the model? (Due to differences in rounding your calculations may be slightly different. “none of the
above” should be selected only if your answer is different by more than 10 cents.)
a. 17.75
b. 20.75
c. 20.00
d. 20.80
e. none of the above
9. If the simple return on a Treasury bill is 8.5 percent, the risk-free rate in the Black-Scholes-Merton model is
a. 8.77 percent
b. 8.93 percent
c. 8.55 percent
d. 8.20 percent
e. none of the above
10. Which of the following variables in the Black-Scholes-Merton option pricing model is the most difficult to
obtain?
a. the volatility
b. the risk-free rate
c. the stock price
d. the time to expiration
e. the exercise price
11. The binomial price will theoretically equal the Black-Scholes-Merton price under which of the following
conditions?
a. when the number of time periods is large
b. when the option is at–the-money
c. when the option is in-the-money
d. when the option is out-of-the-money
e. none of the above
12. If the stock price is 44, the exercise price is 40, the put price is 1.54, and the Black-Scholes-Merton price
using 0.28 as the volatility is 1.11, the implied volatility will be
a. higher than 0.28
b. lower than 0.28
c. 0.28
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d. lower than the risk-free rate
e. none of the above
13. Which of the following statements about the Black-Scholes-Merton model is not true?
a. decreasing the volatility lowers the call price
b. the expected stock price plays a role in the model
c. the risk-free rate is continuously compounded
d. the model is consistent with put-call parity
e. none of the above
14. Which of the following characteristics of the Black-Scholes-Merton model is not correct?
a. it is a discrete time model
b. it is the limit of the binomial model
c. it is a continuous time model
d. it gives the price of a European option
e. none of the above
15. Which of the following assumptions of the Black-Scholes-Merton model is not correct?
a. the stock volatility is constant
b. the stock return follows a normal distribution
c. there are no transaction costs
d. there are no taxes
e. none of the above
16. Which of the following statements about the delta is not true?
a. it ranges from zero to one
b. it converges to zero or one at expiration
c. it is given by N(d1) in the Black-Scholes-Merton model
d. it changes slowly near expiration if the option is at-the-money
e. none of the above
17. Which of the following “Greeks” is not a measure of the option’s sensitivity to a change in one of its input
values?
a. delta
b. gamma
c. rho
d. theta
e. sigma
18. Which of the following statements is true about the relationship between the option price and the risk-free
rate?
a. a call price is nearly linear with respect to the risk-free rate
b. a call price is highly sensitive to the risk-free rate
c. the risk-free rate affects a call but not a put
d. the risk-free rate does not affect a call price
e. none of the above
19. The relationship between the volatility and the time to expiration is called the
a. volatility smile
b. volatility skew
c. term structure of volatility
d. theta
e. none of the above
20. What is the reason for executing a gamma hedge?
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a. the volatility can change
b. the stock price can make a large move
c. the stock price moves are too small for a delta hedge to work
d. there is no true risk-free rate
e. none of the above
21. Which of the following statements about the volatility is not true?
a. the implied volatility often differs across options with different exercise prices
b. the implied volatility equals the historical volatility if the option is correctly priced
c. the implied volatility is determined by trial and error
d. the implied volatility is nearly linearly related to the option price
e. none of the above
22. The relationship between the option price and the exercise price is called
a. the gamma
b. the vega
c. the omega
d. the zeta
e. none of the above
23. What happens when the volatility is zero in the Black-Scholes-Merton model?
a. the option price converges to either zero or the lower bound
b. the option price converges to the intrinsic value
c. the option automatically expires out of the money
d. the gamma and delta converge
e. none of the above
24. Which of the following is not correct about a call’s gamma?
a. it is the same as a put’s gamma
b. it is large when the call is at-the-money
c. it can be viewed as a measure of the risk of the delta
d. it is a source of risk that can be hedged only by using another option
e. none of the above
25. Which of the following statements is incorrect about the historical volatility?
a. if used in the Black-Scholes-Merton model, it gives the current market price
b. it is based on the volatility of the log return on the stock
c. it requires a sample of recent returns
d. it should be converted to an annualized volatility
e. none of the above
26. A hedge portfolio is established and maintained by constantly adjusting the relative proportions of stock
and options, a process referred to as
a. actively managing
b. continuous reconciliation
c. marking to market
d. dynamic trading
e. none of the above
27. The standard normal random variable used in the calculation of cumulative normal probabilities within the
Black-Scholes-Merton option pricing model is
a. the lognormal distribution
b. the d1 and d2 statistic
c. the z statistic
d. the f distribution
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e. none of the above
28. The pattern of volatility across exercise prices is often called
a. the price-fluctuation graph
b. the volatility smile
c. the term structure of implied volatility
d. the skew
e. none of the above
29. The Black-Scholes-Merton model for European puts, obtained by applying put-call parity to the Black-
Scholes-Merton model for European calls, is customarily expressed by which of the following:
a.
P=Xe −rcTN(−d2)−S0N(−d1)
b.
P=X(1+r)−TN(−d2)−S0N(−d1)
c.
P=X(1+r)−TN(−d1)−S0N(−d2)
d.
P=Xe −rcTN(−d1)−S0N(−d2)
e. none of the above
30. The implied volatility is obtained by finding the standard deviation that, when used in the Black-Scholes-
Merton model, makes the
a. model price expire at zero
b. model price equal the market price of the option
c. model price such that it exceeds currently traded market option values
d. model price equal the intrinsic value of the underlying stock
e. none of the above
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CHAPTER 5: OPTION PRICING MODELS: THE BLACK-SCHOLES-MERTON MODEL
TRUE/FALSE TEST QUESTIONS
T F 1. The Black-Scholes-Merton model is the discrete time limit to the binomial model.
T F 2. In the Black-Scholes-Merton model, stock prices are assumed to behave randomly.
T F 3. The values of N(d1) and N(d2) are called risk neutral probabilities.
T F 4. An option’s gamma represents the risk of the delta changing.
T F 5. The binomial model always gives the same option price as the Black-Scholes-Merton
model.
T F 6. The Black-Scholes-Merton model assumes that the volatility does not change throughout
the option’s life.
T F 7. One of the variables that influences the price of the option is the expected return on the
stock.
T F 8. The option’s rate of time value decay is represented by its theta.
T F 9. The option’s delta is approximately the change in the option price for a change in the
stock price.
T F 10. Since dividends could trigger an early exercise of an American call, the Black-Scholes-
Merton dividend adjustment will provide the correct price of an American call.
T F 11. The Black-Scholes-Merton option price is relatively insensitive to changes in the risk–
free rate.
T F 12. One of the inputs to the Black-Scholes-Merton model is the volatility over a recent time
period.
T F 13. When the risk–free rate is zero, the Black–Scholes formula converges to the intrinsic
value.
T F 14. In order to compute the implied volatility, one must force the option to be correctly
priced by the model.
T F 15. The Black-Scholes-Merton model combined with put-call parity give the theoretical price
of an American put option.
T F 16. Vega captures the combined effects of time decay and volatility.
T F 17. The volatility smile is the relationship between implied volatility and historical volatility.
T F 18. The Black-Scholes-Merton formula requires cumulative probabilities from the lognormal
distribution.
T F 19. In the term structure of volatility, the forward volatility is the expected future volatility.
T F 20. A riskless hedge requires more shares of stock than call options.
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T F 21. An approximate implied volatility for an at-the-money call can be solved directly.
T F 22. The implied volatilities of a call and a put with the same terms should be the same.
T F 23. The historical volatility is the same value as the implied volatility.
T F 24. The option’s sensitivity to an interest rate change is called rho.
T F 25. The time to expiration of an option is based on a 360-day year.
T F 26. The Black-Scholes-Merton model can be used with currency options by replacing the
dividend yield with the foreign interest rate.
T F 27. The Black-Scholes-Merton model is the best model for valuing all types of options.
T F 28. The Black-Scholes-Merton model assumes the underlying instrument movement is
lognormally distributed.
T F 29. The Black-Scholes-Merton model assumes that the underlying company never goes
bankrupt.
T F 30. The level of liquidity of the underlying instrument will influence the behavior of related
options.
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