Question 1

A stock price is 20, 22, 19, 21, 24, and 24 on six successive Fridays. Which of the following is closest to the volatility per annum estimated from this data?
A) 50%
B) 60%
C) 70%
D) 80%

Accepted Answer

The price relative for the first week is 22/20 or 1.1.The natural log of the price relative is ln(1.1)or 0.09531.Similarly the ln of the price relatives for the other weeks are -0.1466,0.1001,0.1335,and 0.The standard deviation of 0.09531,-0.1466,0.1001,0.1335,and 0 is 0.1138.The volatility per week is therefore 11.38%.This corresponds to a volatility per year of 0.1138 multiplied by the square root of 52 or about 82%.The answer is therefore D.

Question 2

Which of the following is a definition of volatility?
A) The standard deviation of the return, measured with continuous compounding, in one year
B) The variance of the return, measured with continuous compounding, in one year
C) The standard deviation of the stock price in one year
D) The variance of the stock price in one year

Accepted Answer

Volatility is a statistical measure of the dispersion of returns for a given security or market index, and standard deviation is the most common measure of volatility. The option A defines volatility as the standard deviation of the return, measured with continuous compounding, in one year, which is an accurate definition of volatility. Option B defines variance, not volatility. Option C and D define the standard deviation and variance of the stock price, respectively, but volatility is concerned with the variability of returns, not stock price.

Question 3

A stock provides an expected return of 10% per year and has a volatility of 20% per year. What is the continuously compounded expected return in one year?
A) 6%
B) 8%
C) 10%
D) 12%

Accepted Answer

The continuously compounded expected return can be calculated using the formula:

ln(1+expected return) = ln(1+annual return) - (volatility^2/2)

ln(1+0.10) = ln(1+annual return) - (0.20^2/2)

0.0953 = ln(1+annual return) - 0.02

ln(1+annual return) = 0.1153

1+annual return = e^0.1153

Annual return = 1.1226 - 1 = 0.1226 or 12.26%

Therefore, the continuously compounded expected return in one year is 12.26%. However, since this is not an option in the answer choices, the closest option is B, which is 8%.

Question 4

The risk-free rate is 5% and the expected return on a non-dividend-paying stock is 12%. Which of the following is a way of valuing a derivative?
A) Assume that the expected growth rate for the stock price is 17% and discount the expected payoff at 12%
B) Assuming that the expected growth rate for the stock price is 5% and discounting the expected payoff at 12%
C) Assuming that the expected growth rate for the stock price is 5% and discounting the expected payoff at 5%
D) Assuming that the expected growth rate for the stock price is 12% and discounting the expected payoff at 5%

Accepted Answer

Risk-neutral valuation shows that a derivative can be correctly valued by assuming that the stock grows at the risk-free rate and discounting the expected payoff at the risk-free rate.It follows that C is the correct answer.

Question 5

Which of the following is true when there are dividends?
A) It is never optimal to exercise a call option on the stock early
B) It can be optimal to exercise a call option at any time
C) It is only ever optimal to exercise a call option immediately after an ex-dividend date
D) None of the above

Accepted Answer

The optimal time to exercise a call option depends on various factors, including the stock price, the strike price, the time remaining until expiration, and the volatility of the stock. Dividends can also affect the value of a stock and its options. Therefore, there is no one-size-fits-all answer to when it is optimal to exercise a call option, and none of the options presented in the question are always true.  It depends on the individual circumstances.

Question 6

A stock price is $100. Volatility is estimated to be 20% per year. What is an estimate of the standard deviation of the change in the stock price in one week?
A) $0.38
B) $2.77
C) $3.02
D) $0.76

Accepted Answer

The standard deviation of the stock price change over a given time period can be estimated using the formula for annual volatility: $ \sigma_{\text{annual}} = \text{Volatility} \times \text{Price} $. For a week, this needs to be adjusted by the square root of the time fraction (1 week / 52 weeks = 1/52), because volatility scales with the square root of time. Thus, the weekly standard deviation is $ 0.20 \times 100 \times \sqrt{1/52} \approx \$2.77 $.

Question 7

Which of the following is measured by the VIX index?
A) Implied volatilities for stock options trading on the CBOE
B) Historical volatilities for stock options trading on CBOE
C) Implied volatilities for options trading on the S&P 500 index
D) Historical volatilities for options trading on the S&P 500 index

Accepted Answer

The VIX index measures the implied volatilities for options trading on the S&P 500 index. It is often referred to as the "fear index" because it reflects market expectations for future volatility, particularly during times of greater uncertainty and risk aversion.

Question 8

Which of the following is a way of extending the Black-Scholes-Merton formula to value a European call option on a stock paying a single dividend?
A) Reduce the maturity of the option so that it equals the time of the dividend
B) Subtract the dividend from the stock price
C) Add the dividend to the stock price
D) Subtract the present value of the dividend from the stock price

Accepted Answer

The correct way to extend the Black-Scholes-Merton formula to value a European call option on a stock paying a single dividend is to subtract the present value of the dividend from the stock price. This adjustment is needed because when the stock pays a dividend during the life of the option, the stock price will decrease by the amount of the dividend, which negatively affects the value of the call option. By subtracting the present value of the dividend from the stock price, the impact of the dividend is effectively removed from the calculation and the Black-Scholes-Merton formula can be used to calculate the option value.

Question 9

What does N(x) denote?
A) The area under a normal distribution from zero to x
B) The area under a normal distribution up to x
C) The area under a normal distribution beyond x
D) The area under the normal distribution between -x and x

Accepted Answer

N(x) denotes the area under a normal distribution up to x.

Question 10

The original Black-Scholes and Merton papers on stock option pricing were published in which year?
A) 1983
B) 1984
C) 1974
D) 1973

Accepted Answer

The correct answer is 1973.By coincidence this is also the year that organized trading in call options started.Put option trading started a few years later.

Question 11

What was the original Black-Scholes-Merton model designed to value?
A) A European option on a stock providing no dividends
B) A European or American option on a stock providing no dividends
C) A European option on any stock
D) A European or American option on any stock

Accepted Answer

The original Black-Scholes-Merton model was designed to value a European option on a stock providing no dividends.

Question 12

When the non-dividend paying stock price is $20, the strike price is $20, the risk-free rate is 5%, the volatility is 20% and the time to maturity is 3 months, which of the following is the price of a European put option on the stock?
A) 19.7N(-0.1)-20N(-0.2)
B) 20N(-0.1)-20N(-0.2)
C) 19.7N(-0.2)-20N(-0.1)
D) 20N(-0.2)-20N(-0.1)

Accepted Answer

The formula for the option price is
$\begin{array} { l } 
K e ^ { - r T } N \left( - d _ { 2 } \right) - S _ { 0 } N \left( - d _ { 1 } \right) \
d _ { 1 } = \frac { \ln \left( S _ { 0 } / K \right) + \left( r + \sigma ^ { 2 } / 2 \right) T } { \sigma \sqrt { T } } \quad \text { and } \quad d _ { 2 } = d _ { 1 } - \sigma \sqrt { T }
\end{array}$
In this case $S _ { 0 } = K = 20 , r = 0.06 , \sigma = 0.2$, and $T = 0.25$ so that $K e ^ { - r T } = 20 \mathrm { e } ^ { - 0.06 \times 0.25 } = 19.7$. Also $d _ { 1 } = [ \ln ( 1 ) + ( 0.06 + 0.04 / 2 ) \times 0.25 ] / ( 0.2 \times 0.5 ) = 0.2$ and $d _ { 2 } = 0.2 - 0.2 \times 0.5 = 0.1$
$\mathrm { A }$ is therefore the correct answer.

Question 13

What is the number of trading days in a year usually assumed for equities?
A) 365
B) 252
C) 262
D) 272

Accepted Answer

The number of trading days in a year for equities is usually assumed to be 252. This is because weekends and public holidays are not counted as trading days. Therefore, there are 52 weeks in a year, minus an average of 8 public holidays, which gives 252 trading days.

Question 14

When the Black-Scholes-Merton and binomial tree models are used to value an option on a non-dividend-paying stock, which of the following is true?
A) The binomial tree price converges to a price slightly above the Black-Scholes-Merton price as the number of time steps is increased
B) The binomial tree price converges to a price slightly below the Black-Scholes-Merton price as the number of time steps is increased
C) Either A or B can be true
D) The binomial tree price converges to the Black-Scholes-Merton price as the number of time steps is increased

Accepted Answer

As the number of time steps is increased, the binomial tree model becomes more and more granular and approaches a continuous-time model similar to the Black-Scholes-Merton model. Therefore, the binomial tree price will converge to the Black-Scholes-Merton price as the number of time steps is increased.

Question 15

When there are two dividends on a stock, Black's approximation sets the value of an American call option equal to which of the following?
A) The value of a European option maturing just before the first dividend
B) The value of a European option maturing just before the second (final) dividend
C) The greater of the values in A and B
D) The greater of the value in B and the value assuming no early exercise

Accepted Answer

For Black's approximation we calculate a)the value of the option assuming no early exercise and b)the value of the option assuming that the exercise decision is made immediately before the final ex-dividend date.The value of the option is set equal to the greater of these two values.

Question 16

The volatility of a stock is 18% per year. What is the volatility per month?
A) 1.5%
B) 3.0%
C) 5.2%
D) None of the above

Accepted Answer

To find the volatility per month, we need to convert the annual volatility to monthly volatility. We can do this by dividing the annual volatility by the square root of 12 (since there are 12 months in a year). So, monthly volatility = 18% / sqrt(12) = 5.2%. Therefore, the answer is C) 5.2%.

Question 17

Which of the following is NOT true?
A) Risk-neutral valuation assumes that investors are risk neutral
B) Options can be valued based on the assumption that investors are risk neutral
C) In risk-neutral valuation the expected return on all investment assets is set equal to the risk-free rate
D) In risk-neutral valuation the risk-free rate is used to discount expected cash flows

Accepted Answer

The answer of Which of the following is NOT true?
A)...

Question 18

An investor has earned 2%, 12% and -10% on equity investments in successive years (annually compounded). This is equivalent to earning which of the following annually compounded rates for the three year period.
A) 1.33%
B) 1.23%
C) 1.13%
D) 0.93%

Accepted Answer

The answer of An investor has earned 2%, 12% and...

Question 19

When the non-dividend paying stock price is $20, the strike price is $20, the risk-free rate is 6%, the volatility is 20% and the time to maturity is 3 months, which of the following is the price of a European call option on the stock?
A) 20N(0.1)-19.7N(0.2)
B) 20N(0.2)-19.7N(0.1)
C) 19.7N(0.2)-20N(0.1)
D) 19.7N(0.1)-20N(0.2)

Accepted Answer

Using the Black-Scholes formula, we have:
d1 = [ln(20/20) + (0.06 + 0.2^2/2)(0.25)] / (0.2 * sqrt(0.25)) = 0.2
d2 = d1 - 0.2 * sqrt(0.25) = 0.1
N(d1) = 0.5793, and N(d2) = 0.5398
Therefore, the price of a European call option is:
C = 20N(d1) - 20e^(-0.06*0.25)*N(d2) = 20(0.5793) - 20e^(-0.06*0.25)*(0.5398) = 1.21
The closest option is B: 20N(0.2) - 19.7N(0.1), where N(0.1) = 0.5398 and N(0.2) = 0.5793.

Question 20

Which of the following is assumed by the Black-Scholes-Merton model?
A) The return from the stock in a short period of time is lognormal
B) The stock price at a future time is lognormal
C) The stock price at a future time is normal
D) None of the above

Accepted Answer

Black-Scholes-Merton assumes that the return from a stock in a short period of time is normally distributed.This means that the stock price at a future time is lognormally distributed.

Quiz 13: Valuing Stock Options: the Bsm Model

A stock price is 20, 22, 19, 21, 24, and 24 on six successive Fridays. Which of the following is closest to the volatility per annum estimated from this data?

Which of the following is a definition of volatility?

A stock provides an expected return of 10% per year and has a volatility of 20% per year. What is the continuously compounded expected return in one year?

The risk-free rate is 5% and the expected return on a non-dividend-paying stock is 12%. Which of the following is a way of valuing a derivative?

Which of the following is true when there are dividends?

A stock price is $100. Volatility is estimated to be 20% per year. What is an estimate of the standard deviation of the change in the stock price in one week?

Which of the following is measured by the VIX index?

Which of the following is a way of extending the Black-Scholes-Merton formula to value a European call option on a stock paying a single dividend?

What does N(x) denote?

The original Black-Scholes and Merton papers on stock option pricing were published in which year?

What was the original Black-Scholes-Merton model designed to value?

When the non-dividend paying stock price is $20, the strike price is $20, the risk-free rate is 5%, the volatility is 20% and the time to maturity is 3 months, which of the following is the price of a European put option on the stock?

What is the number of trading days in a year usually assumed for equities?

When the Black-Scholes-Merton and binomial tree models are used to value an option on a non-dividend-paying stock, which of the following is true?

When there are two dividends on a stock, Black's approximation sets the value of an American call option equal to which of the following?

The volatility of a stock is 18% per year. What is the volatility per month?

Which of the following is NOT true?

An investor has earned 2%, 12% and -10% on equity investments in successive years (annually compounded) . This is equivalent to earning which of the following annually compounded rates for the three year period.

When the non-dividend paying stock price is $20, the strike price is $20, the risk-free rate is 6%, the volatility is 20% and the time to maturity is 3 months, which of the following is the price of a European call option on the stock?

Which of the following is assumed by the Black-Scholes-Merton model?