Question 1

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 refers to the degree of variation of a stock's price over time. The standard deviation of the return, measured with continuous compounding in one year, is a common way to quantify volatility in financial markets. The variance of the return (choice B) and the variance and standard deviation of the stock price (choices C and D) are related but are not as commonly used in practice.

Question 2

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 formula for continuously compounded expected return is:
Expected return = ln(1+R)
where R is the expected return in decimal form.
Using the given information, we can solve for R:
R = 0.1 - (0.2^2 / 2) = 0.06
Now we can plug in R to get the continuously compounded expected return:
Expected return = ln(1+0.06) = 0.057 = 5.7%
Therefore, the best choice is B, 8%.

Question 3

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 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 4

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 5

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 6

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)Risk-neutral...

Question 7

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 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 adjust 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 is because a dividend payment reduces the value of the underlying stock, and therefore the option price, since the holder of the option is not entitled to receive the dividend. Subtracting the present value of the dividend from the stock price effectively adjusts the stock price downward to account for the expected reduction in value due to the dividend.

Question 9

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, as stock markets are generally closed on weekends and holidays. This is why the number 252 is often referred to as the "trading days" or "business days" in a year.

Question 10

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 of one-month options trading on the S&P 500 index.

Question 11

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 -10%...

Question 12

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 decision to exercise a call option depends on various factors, such as the stock price, option strike price, time to expiration, and interest rates, among others. Dividends can affect the stock price and option pricing, but they do not dictate the optimal exercise strategy. In some cases, it may be optimal to exercise a call option early to capture the dividend, while in other cases, it may be better to wait until closer to expiration. Therefore, none of the given choices are always true, and the optimal exercise decision needs to be evaluated on a case-by-case basis.

Question 13

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 approaches the continuous nature of the Black-Scholes-Merton model. Therefore, the binomial tree price converges to the Black-Scholes-Merton price as the number of time steps is increased.

Question 14

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 15

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

We can use the Black-Scholes formula to calculate the price of the European call option:

C = S*N(d1) - Ke^(-rt)*N(d2)

where:
S = stock price = $20
K = strike price = $20
r = risk-free rate = 6% = 0.06
t = time to maturity = 3/12 = 0.25 (since time is in months)
σ = volatility = 20% = 0.20

We need to calculate d1 and d2 as follows:

d1 = [ln(S/K) + (r + σ^2/2)*t]/(σ*sqrt(t))
    = [ln(20/20) + (0.06 + 0.20^2/2)*0.25]/(0.20*sqrt(0.25))
    = 0.20

d2 = d1 - σ*sqrt(t)
    = 0.20 - 0.20*sqrt(0.25)
    = 0.10

Using these values, we can calculate the option price:

C = 20*N(0.2) - 20*e^(-0.06*0.25)*N(0.1)
  = 20*0.57926 - 20*0.96930
  = $0.21

Therefore, the correct choice is B, 20N(0.2)-19.7N(0.1).

Question 16

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

To estimate the standard deviation of the change in the stock price in one week, we need to first estimate the standard deviation of the change in the stock price in one day, since there are 5 trading days in one week. We can use the formula:
daily standard deviation = annual volatility / square root of trading days per year
daily standard deviation = 20% / square root of 252 (assuming 252 trading days in a year)
daily standard deviation = 1.26%
Now, to estimate the standard deviation of the change in the stock price in one week, we simply multiply the daily standard deviation by the square root of the number of trading days in one week:
weekly standard deviation = daily standard deviation x square root of number of trading days in one week
weekly standard deviation = 1.26% x square root of 5
weekly standard deviation = 1.26% x 2.236
weekly standard deviation = 2.81%
Finally, we convert this percentage to a dollar amount by multiplying by the stock price:
weekly standard deviation = 2.81% x $100
weekly standard deviation = $2.81
Therefore, the estimate of the standard deviation of the change in the stock price in one week is $2.81, which is closest to answer choice B ($2.77).

Question 17

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.

Question 18

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 cumulative distribution function of a normal distribution, which represents the area under the curve up to the point x.

Question 19

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 20

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%.

Quiz 13: Valuing Stock Options: The BSM Model

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?

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 put option on the stock?

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

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?

Which of the following is NOT true?

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

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

Which of the following is measured by the VIX index?

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.

Which of the following is true when there are dividends?

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?

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 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 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 assumed by the Black-Scholes-Merton model?

What does N(x) denote?

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?

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