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Questions and Answers
What type of distribution do continuously compounded stock returns follow in the binomial model?
What type of distribution do continuously compounded stock returns follow in the binomial model?
Which characteristic is true of the lognormal distribution?
Which characteristic is true of the lognormal distribution?
What happens to sums of binomial random variables as the sample size increases?
What happens to sums of binomial random variables as the sample size increases?
What is the expected value of an exponentiated normal random variable, $E(e^x)$, if $x ∼ N(m, v^2)$?
What is the expected value of an exponentiated normal random variable, $E(e^x)$, if $x ∼ N(m, v^2)$?
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What aspect of exponentiation is highlighted as being asymmetric?
What aspect of exponentiation is highlighted as being asymmetric?
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In the context of lognormal distributions, what does it mean when it is stated that the distribution is 'bounded below by zero'?
In the context of lognormal distributions, what does it mean when it is stated that the distribution is 'bounded below by zero'?
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Which of the following parameters would most likely lead to a lognormal distribution that resembles the normal distribution?
Which of the following parameters would most likely lead to a lognormal distribution that resembles the normal distribution?
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What is the lognormal density function primarily dependent on?
What is the lognormal density function primarily dependent on?
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What is the expected value of S2 after 2 years if the continuously compounded return is 20% and the volatility is 0.4243?
What is the expected value of S2 after 2 years if the continuously compounded return is 20% and the volatility is 0.4243?
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How is the median stock price calculated when the volatility is 60%?
How is the median stock price calculated when the volatility is 60%?
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What concept is illustrated by the stock price moving one standard deviation up over 2 years?
What concept is illustrated by the stock price moving one standard deviation up over 2 years?
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What is the median stock price based on given calculations with a volatility of 30%?
What is the median stock price based on given calculations with a volatility of 30%?
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If Z = -1, what is the implication for the stock return under a normal distribution?
If Z = -1, what is the implication for the stock return under a normal distribution?
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What does a one standard deviation move down equal when volatility is 0.30?
What does a one standard deviation move down equal when volatility is 0.30?
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What does the expected value and median price being different indicate about the stock price distribution?
What does the expected value and median price being different indicate about the stock price distribution?
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In the provided stock price calculations, what is a key characteristic of lognormally distributed data?
In the provided stock price calculations, what is a key characteristic of lognormally distributed data?
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What is the primary assumption commonly made in option pricing regarding asset prices?
What is the primary assumption commonly made in option pricing regarding asset prices?
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Which of the following terms is necessary to fully describe the normal distribution?
Which of the following terms is necessary to fully describe the normal distribution?
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Which function is used to describe the probability of a random variable in a normal distribution?
Which function is used to describe the probability of a random variable in a normal distribution?
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What characteristic does the standard normal density have?
What characteristic does the standard normal density have?
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What is the formula for the normal density function?
What is the formula for the normal density function?
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Why is the lognormal assumption based on stock prices important to understand?
Why is the lognormal assumption based on stock prices important to understand?
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What conclusion can be drawn about stock price data in relation to lognormality?
What conclusion can be drawn about stock price data in relation to lognormality?
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What does the equation $\phi(x; \mu, \sigma) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ represent?
What does the equation $\phi(x; \mu, \sigma) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ represent?
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What does the expression $E(St)$ represent in this context?
What does the expression $E(St)$ represent in this context?
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Which term is subtracted to maintain the interpretability of the parameter $
u$ as the continuously compounded expected capital gain?
Which term is subtracted to maintain the interpretability of the parameter $ u$ as the continuously compounded expected capital gain?
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According to the model, what will happen if $
u$ is large in terms of the stock's returns?
According to the model, what will happen if $ u$ is large in terms of the stock's returns?
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What does the term $ rac{1}{2}
u^2$ represent in the equation for expected stock price?
What does the term $ rac{1}{2} u^2$ represent in the equation for expected stock price?
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In the expression $E(St) = S_0 e^{(
u - eta - rac{1}{2}
u^2) t}$, which variable corresponds to the initial stock price?
In the expression $E(St) = S_0 e^{( u - eta - rac{1}{2} u^2) t}$, which variable corresponds to the initial stock price?
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What conclusion can be drawn about the median stock price compared to the mean stock price from the model?
What conclusion can be drawn about the median stock price compared to the mean stock price from the model?
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What is the definition of a lognormally distributed random variable?
What is the definition of a lognormally distributed random variable?
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If the expected rate of return on a stock is $
u=10%$ and the volatility is $
u=30%$, what effect does this combination have on the potential outcomes?
If the expected rate of return on a stock is $ u=10%$ and the volatility is $ u=30%$, what effect does this combination have on the potential outcomes?
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What does the equation $St = S0e^{R(0, t)}$ represent?
What does the equation $St = S0e^{R(0, t)}$ represent?
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What does $
u - eta$ in the equation denote?
What does $ u - eta$ in the equation denote?
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What does exponentiation do to continuously compounded returns?
What does exponentiation do to continuously compounded returns?
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Which of the following statements is true regarding lognormal distributions?
Which of the following statements is true regarding lognormal distributions?
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If $x_1$ and $x_2$ are normally distributed, what can be said about $y_1 = e^{x_1}$ and $y_2 = e^{x_2}$?
If $x_1$ and $x_2$ are normally distributed, what can be said about $y_1 = e^{x_1}$ and $y_2 = e^{x_2}$?
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Why can't a lognormal stock price be negative?
Why can't a lognormal stock price be negative?
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What is the significance of the central limit theorem in relation to lognormal distributions?
What is the significance of the central limit theorem in relation to lognormal distributions?
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Which equation describes the continuously compounded return from an initial stock price $S_0$ to a future stock price $S_t$?
Which equation describes the continuously compounded return from an initial stock price $S_0$ to a future stock price $S_t$?
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What does the standard deviation indicate in a normal distribution?
What does the standard deviation indicate in a normal distribution?
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Which of the following represents a normal distribution with mean 0 and a variance of 1?
Which of the following represents a normal distribution with mean 0 and a variance of 1?
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How does increasing the standard deviation affect the normal distribution?
How does increasing the standard deviation affect the normal distribution?
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In the formula φ(μ + a; μ, σ), what do the symbols represent?
In the formula φ(μ + a; μ, σ), what do the symbols represent?
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What is the probability of drawing a specific value from a normal distribution?
What is the probability of drawing a specific value from a normal distribution?
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What is the relationship between the means of two normal distributions with equal variances?
What is the relationship between the means of two normal distributions with equal variances?
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What does the notation φ(x) denote in a standard normal distribution?
What does the notation φ(x) denote in a standard normal distribution?
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What does it mean for a normal distribution to be symmetric around the mean?
What does it mean for a normal distribution to be symmetric around the mean?
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Study Notes
The Lognormal Distribution
- The lognormal distribution is frequently used in option pricing to model asset prices.
- It's based on the normal distribution.
- Stock prices are not precisely lognormal, but the assumption is useful for pricing models.
The Normal Distribution
- A random variable, x, follows a normal distribution if its probability is described by the normal density function.
- Formula: φ(χ; μ, σ) = 1/(σ√2π) * e^(-(x-μ)² / (2σ²))
- Characterized by the mean (μ) and standard deviation (σ).
- The normal distribution is symmetric around the mean.
- The normal density with μ = 0 and σ = 1 is called the standard normal density. Represented as N(z).
Lognormal Distribution
- A random variable, y, is lognormal if ln(y) is normally distributed.
- It's always positive.
- Useful for modeling asset prices because they can't be negative.
- The formula for the lognormal density function is g(y; m, v) = 1/(yv√2π) * e^(-(ln(y)-m)² / (2v²)). Where m is the mean of ln(y) and v is its standard deviation.
Calculations and Properties
- Probabilities of values within a range of x can be found using the cumulative normal distribution function, N(a). This is denoted as N(a).
- The cumulative normal distribution function (N(a)) is the area under the curve to the left of a.
- Used in calculating important financial metrics, such as calculating the probability that a random number is less than a given number(a).
Relationship Between Normal and Lognormal
- If x is a normally distributed random variable, then y=ex is lognormally distributed.
- The sum of independent lognormal variables is not lognormal.
- The product of independent lognormal variables is lognormal.
Central Limit Theorem
- The normal distribution arises naturally when multiple independent random variables are added together.
- The sum of independent random variables tend towards a normal distribution.
Lognormal Model of Stock Prices
- Stock prices are often modeled as lognormal.
- Continuously compounded returns are frequently assumed to be normally distributed.
- This assumption leads to a lognormal distribution for the stock price.
Continuous Compounded Returns
- Continuous compounding calculations return a result that shows prices as lognormally distributed
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Description
This quiz focuses on the lognormal and normal distributions, essential concepts in statistics and finance. It covers the properties, formulas, and applications of these distributions in modeling asset prices and option pricing. Test your understanding of these important statistical tools!
