Question 1

What is the probability that a number drawn from the standard normal distribution will NOT be between -1 and 1?
A) 0.22
B) 0.32
C) 0.42
D) 0.52

Accepted Answer

The area under the standard normal distribution curve between -1 and 1 is approximately 0.68, which represents the probability of a number falling in that range. Therefore, the probability of a number not being in that range is 1 - 0.68 = 0.32.

Question 2

In a lognormal model of stock price movement,describe the mean and variance of the continuously compounded returns.

Accepted Answer

The return is normally distributed while both the return and variance grow proportionally with time.

Question 3

Why do we assume a lognormal distribution in option pricing? Ask the class to explain the pluses and minuses to this assumption.Once the downfalls are established,probe students to find out if a better alternative exists.

Accepted Answer

We assume a lognormal distribution in option pricing because it is a common and convenient way to model the distribution of stock prices. The lognormal distribution is often used because it reflects the fact that stock prices cannot be negative and tend to have positive skewness. Additionally, the lognormal distribution is easy to work with mathematically and has been found to provide a good approximation for the distribution of stock prices in many cases.

However, there are some downsides to assuming a lognormal distribution in option pricing. One major drawback is that the lognormal distribution assumes that stock prices can increase without bound, which may not always be the case in reality. Additionally, the lognormal distribution may not accurately capture extreme events or "fat tails" in the distribution of stock prices, leading to potential underestimation of the risk of extreme price movements.

To address these downsides, students can be prompted to consider alternative distributions such as the normal distribution, which assumes that stock prices follow a symmetric bell-shaped curve, or the more flexible and heavy-tailed distributions such as the Student's t-distribution or the generalized hyperbolic distribution. Each of these alternatives has its own advantages and disadvantages, and the choice of distribution should be based on the specific characteristics of the underlying asset and the goals of the option pricing model. By exploring these alternatives, students can gain a deeper understanding of the assumptions and limitations of option pricing models and the importance of choosing the most appropriate distribution for the specific context.

Question 4

A stock is valued at $28.00.The annual expected return is 9.0% and the standard deviation of annualized returns is 19.0%.If the stock is lognormally distributed,what is the expected median stock price after 4 years?
A) $28.00
B) $32.33
C) $40.13
D) $54.60

Accepted Answer

Using the formula: 
Expected Median Stock Price = initial stock price * e^( (mu - 0.5*sigma^2)*time)
where mu is the expected return and sigma is the standard deviation, we have:
Expected Median Stock Price = 28 * e^( (0.09 - 0.5*(0.19)^2)*4)
Expected Median Stock Price = $32.33. Therefore, the best choice is B.

Question 5

Given a mean of -4.3 and a standard deviation of 26,what is the equivalent draw from a normal distribution for a standard normal sample variable of 0.67?
A) -13.12
B) 03.12
C) 13.12
D) 23.12

Accepted Answer

To convert a standard normal variable (Z) to a normal variable with given mean (μ) and standard deviation (σ), we use the formula:

X = Zσ + μ

Substituting the given values, we get:

X = 0.67 x 26 + (-4.3) = 13.12

Therefore, the equivalent draw from a normal distribution for a standard normal sample variable of 0.67 is 13.12. The closest choice is C) 13.12.

Question 6

A stock is valued at $55.00.The annual expected return is 12.0% and the standard deviation of annualized returns is 22.0%.If the stock is lognormally distributed,what is the price of the stock given a one standard deviation move up after 3 years?
A) $64.41
B) $74.41
C) $84.41
D) $94.41

Accepted Answer

The stock price after a one standard deviation move up can be calculated using the formula for the future price of a lognormally distributed stock: $ S_t = S_0 \times e^{(\mu + \sigma) \times t} $. Here, $ S_0 = 55 $, $ \mu = 0.12 $, $ \sigma = 0.22 $, and $ t = 3 $. Plugging in these values gives $ S_t = 55 \times e^{(0.12 + 0.22) \times 3} \approx 84.41 $.

Question 7

What is the area under the standard normal distribution curve and is less than 0.654?
A) 0.5115
B) 0.6215
C) 0.7434
D) 0.8283

Accepted Answer

The answer of What is the area under the standard...

Question 8

Given a mean of 45 and a standard deviation of 32 from a normally distributed sample,what is the probability of an observation being between 35 and 75?
A) 0.35
B) 0.45
C) 0.55
D) 0.65

Accepted Answer

The answer of Given a mean of 45 and a...

Question 9

Given a mean of -7.8 and a standard deviation of 16 from a normally distributed sample,what is the probability of an observation being below 12.0?
A) 0.51
B) 0.61
C) 0.71
D) 0.81

Accepted Answer

First, we need to standardize the value using the formula:

z = (x - μ) / σ

where: 
x = 12 (the observation we want to find the probability for) 
μ = -7.8 (mean) 
σ = 16 (standard deviation)

So,

z = (12 - (-7.8)) / 16 
z = 1.1175

Next, we look up the probability of z being less than 1.1175 in the standard normal distribution table. We can also use a calculator or software to find this probability, which is approximately 0.8708.

However, we are interested in the probability of the observation being below 12, which is the same as the probability of z being less than 1.1175.

So, the answer is D) 0.81, which is the probability of a standard normal distribution having a value less than 1.1175.

Question 10

Given a mean of 4.5 and a standard deviation of 12 from a sample of variables,what is the equivalent draw from a standard normal distribution for 6.0?
A) 0.065
B) 0.075
C) 0.095
D) 0.125

Accepted Answer

The answer of Given a mean of 4.5 and a...

Question 11

A stock is valued at $55.00.The annual expected return is 12.0% and the standard deviation of annualized returns is 22.0%.If the stock is lognormally distributed,what is the expected price after 3 years?
A) $78.83
B) $88.83
C) $98.83
D) $108.83

Accepted Answer

The expected price after 3 years is $78.83. This is calculated using the formula:Expected price = Current price * e^(expected return * time + 0.5 * standard deviation^2 * time^2)Expected price = $55.00 * e^(0.12 * 3 + 0.5 * 0.22^2 * 3^2)Expected price = $55.00 * e^(0.36 + 0.726)Expected price = $55.00 * e^1.086Expected price = $78.83

Question 12

A stock is valued at $28.00.The annual expected return is 9.0% and the standard deviation of annualized returns is 19.0%.If the stock is lognormally distributed,what is the expected price after 4 years?
A) $28.00
B) $32.33
C) $40.13
D) $54.60

Accepted Answer

We can use the formula for lognormal distribution to find the expected price after 4 years, which is:

Expected Price = Current Price * e^[(μ - σ^2/2) * t + σ * sqrt(t) * Z]

where:
- μ is the expected return
- σ is the standard deviation
- t is the number of years
- Z is a standard normal random variable

Plugging in the values given, we get:

Expected Price = $28.00 * e^[(0.09 - 0.19^2/2) * 4 + 0.19 * sqrt(4) * Z]
Expected Price = $28.00 * e^[-0.5016 + 0.38Z]

Using a calculator or a normal distribution table, we can find that the expected value of e^(0.38Z) is 1.4397. Therefore:

Expected Price = $28.00 * 1.4397
Expected Price = $40.13

So the best choice is C.

Question 13

What is the probability that a number drawn from the standard normal distribution will be between -0.60 and 0.45?
A) 0.40
B) 0.50
C) 0.60
D) 0.70

Accepted Answer

The probability that a number drawn from the standard normal distribution will be between -0.60 and 0.45 is 0.40. This can be found using a standard normal distribution table or a calculator.

Question 14

A stock is valued at $28.00.The annual expected return is 9.0% and the standard deviation of annualized returns is 19.0%.If the stock is lognormally distributed,what is the price of the stock given a one standard deviation move up after 4 years?
A) $28.00
B) $32.33
C) $40.13
D) $54.60

Accepted Answer

The answer of A stock is valued at $28.00.The annual...

Question 15

Why might normally distributed returns appear non-normal?

Accepted Answer

The answer of Why might normally distributed returns appear non-normal?...

Question 16

For a stock price that was initially $55.00,what is the price after 4 years if the continuously compounded returns for these 4 years are 4.5%,6.2%,8.9%,-3.2%?
A) $64.80
B) $74.80
C) $84.80
D) $94.80

Accepted Answer

The price after 4 years can be calculated using the formula for continuously compounded returns: $P = P_0 e^{(r_1+r_2+r_3+r_4)}$, where $P_0$ is the initial price and $r_1, r_2, r_3, r_4$ are the annual return rates. Plugging in the values: $P = 55 \times e^{(0.045+0.062+0.089-0.032)} = 55 \times e^{0.164} \approx 64.80$.

Question 17

Give a very brief definition of conditional expected stock price.

Accepted Answer

The answer of Give a very brief definition of conditional...

Question 18

A stock is valued at $55.00.The annual expected return is 12.0% and the standard deviation of annualized returns is 22.0%.If the stock is lognormally distributed,what is the expected median stock price after 3 years?
A) $57.67
B) $67.67
C) $77.67
D) $87.67

Accepted Answer

The median of a lognormal distribution is given by the initial stock price multiplied by the exponential of the expected return minus half the variance, all multiplied by the time period. Here, it is $55 * exp((0.12 - 0.5 * 0.22^2) * 3) = $57.67.

Question 19

What assumption is made in the Black-Scholes model concerning volatility?

Accepted Answer

The answer of What assumption is made in the Black-Scholes...

Question 20

How are partial expectation prices converted to conditional expectation prices?

Accepted Answer

The answer of How are partial expectation prices converted to...

Quiz 18: The Lognormal Distribution

What is the probability that a number drawn from the standard normal distribution will NOT be between -1 and 1?

In a lognormal model of stock price movement,describe the mean and variance of the continuously compounded returns.

Why do we assume a lognormal distribution in option pricing? Ask the class to explain the pluses and minuses to this assumption.Once the downfalls are established,probe students to find out if a better alternative exists.

A stock is valued at $28.00.The annual expected return is 9.0% and the standard deviation of annualized returns is 19.0%.If the stock is lognormally distributed,what is the expected median stock price after 4 years?

Given a mean of -4.3 and a standard deviation of 26,what is the equivalent draw from a normal distribution for a standard normal sample variable of 0.67?

A stock is valued at $55.00.The annual expected return is 12.0% and the standard deviation of annualized returns is 22.0%.If the stock is lognormally distributed,what is the price of the stock given a one standard deviation move up after 3 years?

What is the area under the standard normal distribution curve and is less than 0.654?

Given a mean of 45 and a standard deviation of 32 from a normally distributed sample,what is the probability of an observation being between 35 and 75?

Given a mean of -7.8 and a standard deviation of 16 from a normally distributed sample,what is the probability of an observation being below 12.0?

Given a mean of 4.5 and a standard deviation of 12 from a sample of variables,what is the equivalent draw from a standard normal distribution for 6.0?

A stock is valued at $55.00.The annual expected return is 12.0% and the standard deviation of annualized returns is 22.0%.If the stock is lognormally distributed,what is the expected price after 3 years?

A stock is valued at $28.00.The annual expected return is 9.0% and the standard deviation of annualized returns is 19.0%.If the stock is lognormally distributed,what is the expected price after 4 years?

What is the probability that a number drawn from the standard normal distribution will be between -0.60 and 0.45?

A stock is valued at $28.00.The annual expected return is 9.0% and the standard deviation of annualized returns is 19.0%.If the stock is lognormally distributed,what is the price of the stock given a one standard deviation move up after 4 years?

Why might normally distributed returns appear non-normal?

For a stock price that was initially $55.00,what is the price after 4 years if the continuously compounded returns for these 4 years are 4.5%,6.2%,8.9%,-3.2%?

Give a very brief definition of conditional expected stock price.

A stock is valued at $55.00.The annual expected return is 12.0% and the standard deviation of annualized returns is 22.0%.If the stock is lognormally distributed,what is the expected median stock price after 3 years?

What assumption is made in the Black-Scholes model concerning volatility?

How are partial expectation prices converted to conditional expectation prices?