1.4 lognormal distribution

Question 1

The lognormal distribution

Answer 1

also widely used in financial modeling, including by the Black-Scholes-Merton model

If a random variable Y follows a lognormal distribution, then ln(Y) is normally distributed

The lognormal distribution is bounded below by 0 and is skewed to the right

–> This makes it a good choice for asset prices

If a stock’s continuously compounded return is normally distributed, then the future stock price is lognormally distributed. In fact, the future stock price may be lognormally distributed even if the returns are not normally distributed.

Question 2

The lognormal distribution is completely described by which two parameters?

which formulas can we derive from this

Answer 2

the mean (μ) and variance (σ2) of ln(Y)

μY = e^(μ + 0.5*σ^2)

σ^2Y = (e^(2μ + σ^2))*(eσ^2 − 1)

Question 3

1. The continuously compounded return for each time period can be calculated as follows:
2. Then, the continuously compounded return to the investment horizon:

what does it all imply?

Answer 3

1. rt,t+1 = ln ((St+1)/St)
2. r0,T = ln(ST/S0)

this implies: ST = S0*e^(r0,T)

Question 4

Assuming the period returns are independently and identically distributed yields the following:

Answer 4

E(r0,T) = μT

σ^2(r0,T) = σ^2T

Question 5

Volatility

Answer 5

standard deviation of the continuously compounded returns on the underlying asset

Question 6

In contrast to normal distributions, lognormal distributions most likely:

A
are skewed to the left.

B
have outcomes that cannot be negative.

C
are more suitable for describing asset returns than asset prices.

Answer 6

B
have outcomes that cannot be negative.

Question 7

The Student’s t-distribution, or t-distribution for short

Answer 7

frequently used in statistical modeling and hypothesis testing

similar to the normal distribution because they both form symmetric bell curves

–> However, the t-distribution has longer, fatter tails

Question 8

what is the effect of the t-distribution having longer, fatter tails compared to the normal distribution?

Answer 8

extreme values are more likely than the normal distribution, producing a more conservative downside risk estimate

Question 9

The t-distribution only has one parameter, known as the?

Answer 9

degrees of freedom (df)

Question 10

degrees of freedom (df)

Answer 10

can be interpreted as the number of independent variables used in defining sample statistics

–>The larger the sample size, the larger the degrees of freedom because there are more independent variations

Question 11

t-ratio formula

Answer 11

Assume X¯ and s represent the sample mean and sample standard deviation, respectively.

The ratio below follows a t-distribution with a mean of 0 and n−1 degrees of freedom

t = (X¯ - μ) / (s/√n))

Question 12

what happens to the t-distribution when the degrees of freedom increase?

Answer 12

he t-distribution approaches the standard normal distribution

a t-distribution with fewer degrees of freedom has fatter tails

Question 13

The chi-square distribution and the F-distribution

Answer 13

both asymmetric distributions

The normal distribution is closely related to the chi-square distribution, which is related to the F-distribution

Question 14

Chi-square distribution

Answer 14

Defined as the sum of the squares of k independent normal random variables

χ^2ofk = Z^2#1+Z^2#2+…+Z^2#k

χ^2ofk follows a chi-square distribution with k
degrees of freedom

–> cannot be negative

the chi-square distribution is asymmetric

Question 15

what happens to the chi-square distribution when the degrees of freedom increase?

Answer 15

the shape of the distribution becomes similar to a bell curve

Question 16

F-distribution

Answer 16

Defined as a ratio of two chi-square random variables

It has two values of degrees of freedom:

–> the numerator degree of freedom

–> the denominator degree of freedom

the F-distribution cannot be negative

Question 17

f-distribution formula

Answer 17

F = (χ^2of1 / m) / (χ^2of2 / n)

Question 18

what happens to the f-distribution when the numerator and denominator degrees of freedom increase?

Answer 18

its shape approaches a bell shape

Question 19

explain if the negative value is allowed and the Hypothesis Tests of Returns for the following:

Student’s t distribution

Answer 19

negative value: allowed

Hypothesis Tests of Returns:

• Test of a single population mean
• Test of differences between two population means
• Test of mean difference between paired populations
• Test of population correlation coefficient

Question 20

explain if the negative value is allowed and the Hypothesis Tests of Returns for the following:

chi-square distribution

Answer 20

negative value: not allowed

Hypothesis Tests of Returns:

Test of variance of a normally distributed population

Question 21

explain if the negative value is allowed and the Hypothesis Tests of Returns for the following:

f-distribution

Answer 21

negative value: not allowed

Hypothesis Tests of Returns:

Test of equality of variances of normally distributed populations from two independent random samples

Question 22

Which one of the following statements about Student’s t-distribution is most likely false?

A
It has shorter (i.e., thinner) tails than the normal distribution.

B
It is symmetrically distributed around its mean value, like the normal distribution.

C
As its degrees of freedom increase, Student’s t-distribution approaches the normal distribution.

Answer 22

A
It has shorter (i.e., thinner) tails than the normal distribution.

Question 23

Which of the following statements is least likely accurate concerning both the chi-square and F
-distributions?

A
Both distributions are asymmetric

B
Neither distribution allows for negative values

C
The F-distribution is expressed as the sum of two chi-squared distributions

Answer 23

C
The F-distribution is expressed as the sum of two chi-squared distributions

Question 24

Monte Carlo simulation

Answer 24

uses computers to generate many random samples from a specified probability distribution

It can be used for many purposes, such as estimating Value-at-Risk (VaR) or valuing securities with embedded options

Question 25

The following steps provide a general overview of the Monte Carlo simulation:

Answer 25

1. Specify the quantities of interest in terms of underlying variables.
2. Split the time horizon into subperiods.
3. Specify the distributional assumptions for the underlying risk factors.
4. Use a computer program to draw K random values of each risk factor.
5. Calculate the underlying variables based on the random values drawn.
6. Compute the quantities of interest.

Repeat steps 4 to 6 for N trials.

–> The Monte Carlo estimate is the mean quantity of interest over the N trials.

Question 26

Historical simulation (or back simulation)

Answer 26

an alternative to Monte Carlo simulation

It assumes the past applies to the future

However, unlike the Monte Carlo method, historical simulation does not let itself to “what if” analyses because it only reflects the tendencies that appear in the data

Question 27

A Monte Carlo simulation can most likely be used to:

A
directly provide precise valuations of call options.

B
simulate a process from historical records of returns.

C
test the sensitivity of a model to changes in assumptions—for example, on distributions of key variables.

Answer 27

C
test the sensitivity of a model to changes in assumptions—for example, on distributions of key variables.

Question 28

A limitation of Monte Carlo simulation is:

A
its failure to do “what if” analysis.

B
that it requires historical records of returns.

C
its inability to independently specify cause-and-effect relationships.

Answer 28

C
its inability to independently specify cause-and-effect relationships.