Question 1

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

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 2

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

Accepted Answer

The number of trading days in a year for equities is usually assumed to be 252. This is because the stock market is closed on weekends and holidays, and there are also a few scheduled days off throughout the year. Therefore, the number of actual trading days is less than the total number of days in a year (365). The most commonly used number for trading days is 252, which assumes that there are no trading days on weekends or holidays throughout the year.

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 call option on the stock

Accepted Answer

Using the Black-Scholes formula, we can calculate the price of the European call option as:
C = S*N(d1) - Ke^(-rt)*N(d2)
Where:
S = stock price = $20
K = strike price = $20
r = risk-free rate = 6% = 0.06
σ = volatility = 20% = 0.2
T = time to maturity = 0.25 (3 months expressed in years)
d1 = (ln(S/K) + (r + σ^2/2)*T) / (σ*sqrt(T)) = 0.1
d2 = d1 - σ*sqrt(T) = -0.1
N(d1) = 0.5398
N(d2) = 0.4602
Plugging in the values, we get:
C = $20*0.5398 - $20*e^(-0.06*0.25)*0.4602
C = $11.32
So the price of the European call option is $11.32, which is equivalent to the formula in option B: 20N(0.2)-19.7N(0.1).

Question 4

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?

Accepted Answer

As the number of time steps increases, the binomial tree becomes more and more similar to the continuous-time 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 5

What does N(x) denote?

Accepted Answer

N(x) denotes the cumulative distribution function of a normal distribution, which gives the probability of getting a value less than or equal to x in a normal distribution. Therefore, the area under a normal distribution up to x is the correct interpretation of N(x).

Question 6

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?

Accepted Answer

We can use the formula for continuously compounded expected return, which is e^(mean+variance/2)-1. Plugging in the given values, we get e^(0.1-0.2^2/2)-1 = 0.077 or 7.7%. Therefore, the closest answer choice is B) 8%.

Question 7

Which of the following is a definition of volatility

Accepted Answer

Volatility refers to the degree of variation of a stock's price over time. It is commonly measured by calculating the standard deviation of the stock's return over a specific period. Option A defines volatility as the standard deviation of the return, measured with continuous compounding, in one year, which is a commonly accepted definition of volatility. Option B also refers to the variance of the return, which is related to volatility, but is not as commonly used as the standard deviation in finance. Option C and D refer to the standard deviation and variance of the stock price, respectively, which are related to volatility but do not capture the specific relationship between the price of a stock and its expected return.

Question 8

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?

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 9

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

Accepted Answer

The original Black-Scholes and Merton papers on stock option pricing were published in 1973.

Question 10

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?

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 11

Which of the following is NOT true?

Accepted Answer

Risk-neutral valuation does not assume that investors are risk neutral; it is a mathematical technique used to price derivatives by assuming that all investors are indifferent to risk, allowing the use of the risk-free rate for discounting.

Question 12

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?

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 13

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

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 14

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.

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

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

Accepted Answer

The Black-Scholes-Merton model assumes that the stock price at a future time is lognormal, and this assumption is central to the model's ability to calculate option prices. The model assumes that the return from the stock in a short period of time is normally distributed, but not lognormal. Therefore, option prices cannot be accurately calculated using the assumption in answer choice A. Answer choice C is incorrect because the model assumes lognormality, not normality.

Question 17

Which of the following is true when there are dividends

Accepted Answer

The optimal time to exercise a call option with dividends depends on various factors such as time to expiration, interest rates, and the dividend amount. It may be optimal to exercise the call option before the ex-dividend date to capture the dividend or after the ex-dividend date to avoid the dividend effect on the stock price. Therefore, the decision to exercise a call option with dividends is not always straightforward, and none of the given options represents the complete truth.

Question 18

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?

Accepted Answer

To account for a single dividend, we subtract the present value of the dividend from the stock price. This is because the dividend reduces the value of the stock, and we need to adjust our calculations accordingly. The other options are not correct: reducing the maturity of the option does not account for the dividend, subtracting the dividend from the stock price is too simplistic (as it does not consider the time value of money or the interest rate), and adding the dividend to the stock price would overestimate the value of the option.

Question 19

Which of the following is measured by the VIX index

Accepted Answer

The VIX index measures the implied volatilities of one-month options trading on the S&P 500 index.

Question 20

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

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.

Quiz 13: Valuing Stock Options: the Bsm Model