1.4 lognormal distribution Flashcards
The lognormal distribution
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.
The lognormal distribution is completely described by which two parameters?
which formulas can we derive from this
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)
- The continuously compounded return for each time period can be calculated as follows:
- Then, the continuously compounded return to the investment horizon:
what does it all imply?
- rt,t+1 = ln ((St+1)/St)
- r0,T = ln(ST/S0)
this implies: ST = S0*e^(r0,T)
Assuming the period returns are independently and identically distributed yields the following:
E(r0,T) = μT
σ^2(r0,T) = σ^2T
Volatility
standard deviation of the continuously compounded returns on the underlying asset
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.
B
have outcomes that cannot be negative.
The Student’s t-distribution, or t-distribution for short
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
what is the effect of the t-distribution having longer, fatter tails compared to the normal distribution?
extreme values are more likely than the normal distribution, producing a more conservative downside risk estimate
The t-distribution only has one parameter, known as the?
degrees of freedom (df)
degrees of freedom (df)
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
t-ratio formula
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))
what happens to the t-distribution when the degrees of freedom increase?
he t-distribution approaches the standard normal distribution
a t-distribution with fewer degrees of freedom has fatter tails
The chi-square distribution and the F-distribution
both asymmetric distributions
The normal distribution is closely related to the chi-square distribution, which is related to the F-distribution
Chi-square distribution
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
what happens to the chi-square distribution when the degrees of freedom increase?
the shape of the distribution becomes similar to a bell curve
F-distribution
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
f-distribution formula
F = (χ^2of1 / m) / (χ^2of2 / n)
what happens to the f-distribution when the numerator and denominator degrees of freedom increase?
its shape approaches a bell shape
explain if the negative value is allowed and the Hypothesis Tests of Returns for the following:
Student’s t distribution
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
explain if the negative value is allowed and the Hypothesis Tests of Returns for the following:
chi-square distribution
negative value: not allowed
Hypothesis Tests of Returns:
Test of variance of a normally distributed population
explain if the negative value is allowed and the Hypothesis Tests of Returns for the following:
f-distribution
negative value: not allowed
Hypothesis Tests of Returns:
Test of equality of variances of normally distributed populations from two independent random samples
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.
A
It has shorter (i.e., thinner) tails than the normal distribution.
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
C
The F-distribution is expressed as the sum of two chi-squared distributions
Monte Carlo simulation
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
The following steps provide a general overview of the Monte Carlo simulation:
- Specify the quantities of interest in terms of underlying variables.
- Split the time horizon into subperiods.
- Specify the distributional assumptions for the underlying risk factors.
- Use a computer program to draw K random values of each risk factor.
- Calculate the underlying variables based on the random values drawn.
- 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.
Historical simulation (or back simulation)
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
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.
C
test the sensitivity of a model to changes in assumptions—for example, on distributions of key variables.
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.
C
its inability to independently specify cause-and-effect relationships.
