Lognormal and Normal Distributions

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Questions and Answers

What type of distribution do continuously compounded stock returns follow in the binomial model?

  • Lognormal distribution (correct)
  • Normal distribution
  • Uniform distribution
  • Binomial distribution

Which characteristic is true of the lognormal distribution?

  • It is skewed to the right. (correct)
  • It can take negative values.
  • It is symmetric around its mean.
  • It has a uniform shape.

What happens to sums of binomial random variables as the sample size increases?

  • They approach uniform distribution.
  • They remain binomially distributed.
  • They approach normality. (correct)
  • They diverge with no specific distribution.

What is the expected value of an exponentiated normal random variable, $E(e^x)$, if $x ∼ N(m, v^2)$?

<p>$e^{m + 2v}$ (B)</p> Signup and view all the answers

What aspect of exponentiation is highlighted as being asymmetric?

<p>Positive changes yield larger increases than negative changes cause decreases. (B)</p> Signup and view all the answers

In the context of lognormal distributions, what does it mean when it is stated that the distribution is 'bounded below by zero'?

<p>The values always start from zero. (D)</p> Signup and view all the answers

Which of the following parameters would most likely lead to a lognormal distribution that resembles the normal distribution?

<p>High mean and low variance. (A)</p> Signup and view all the answers

What is the lognormal density function primarily dependent on?

<p>Both the mean and variance of the underlying normal variable. (A)</p> Signup and view all the answers

What is the expected value of S2 after 2 years if the continuously compounded return is 20% and the volatility is 0.4243?

<p>$122.14 (B)</p> Signup and view all the answers

How is the median stock price calculated when the volatility is 60%?

<p>$85.21 (A)</p> Signup and view all the answers

What concept is illustrated by the stock price moving one standard deviation up over 2 years?

<p>$100e(0.1− 2 0.3 )×2+σ (B)</p> Signup and view all the answers

What is the median stock price based on given calculations with a volatility of 30%?

<p>$111.63 (C)</p> Signup and view all the answers

If Z = -1, what is the implication for the stock return under a normal distribution?

<p>The stock return is equal to the mean minus one standard deviation. (B)</p> Signup and view all the answers

What does a one standard deviation move down equal when volatility is 0.30?

<p>$73.03 (A)</p> Signup and view all the answers

What does the expected value and median price being different indicate about the stock price distribution?

<p>The distribution is skewed. (D)</p> Signup and view all the answers

In the provided stock price calculations, what is a key characteristic of lognormally distributed data?

<p>Data is positively skewed. (B)</p> Signup and view all the answers

What is the primary assumption commonly made in option pricing regarding asset prices?

<p>Asset prices are assumed to be lognormally distributed. (B)</p> Signup and view all the answers

Which of the following terms is necessary to fully describe the normal distribution?

<p>Mean and standard deviation (A)</p> Signup and view all the answers

Which function is used to describe the probability of a random variable in a normal distribution?

<p>Density function (D)</p> Signup and view all the answers

What characteristic does the standard normal density have?

<p>Mean of 0 and standard deviation of 1. (A)</p> Signup and view all the answers

What is the formula for the normal density function?

<p>$\phi(x; \mu, \sigma) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2}}$ (C)</p> Signup and view all the answers

Why is the lognormal assumption based on stock prices important to understand?

<p>It simplifies complex models for easier calculations. (A)</p> Signup and view all the answers

What conclusion can be drawn about stock price data in relation to lognormality?

<p>Stock prices show some consistency with lognormality but are not exactly lognormal. (C)</p> Signup and view all the answers

What does the equation $\phi(x; \mu, \sigma) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ represent?

<p>The probability density function of the normal distribution. (B)</p> Signup and view all the answers

What does the expression $E(St)$ represent in this context?

<p>The expected value of the stock price at time $t$ (A)</p> Signup and view all the answers

Which term is subtracted to maintain the interpretability of the parameter $ u$ as the continuously compounded expected capital gain?

<p>$ rac{1}{2} u^2$ (B)</p> Signup and view all the answers

According to the model, what will happen if $ u$ is large in terms of the stock's returns?

<p>The stock will lose money more than half of the time. (A)</p> Signup and view all the answers

What does the term $ rac{1}{2} u^2$ represent in the equation for expected stock price?

<p>The expected variability in stock returns (C)</p> Signup and view all the answers

In the expression $E(St) = S_0 e^{( u - eta - rac{1}{2} u^2) t}$, which variable corresponds to the initial stock price?

<p>$S_0$ (D)</p> Signup and view all the answers

What conclusion can be drawn about the median stock price compared to the mean stock price from the model?

<p>The median stock price is usually below the mean stock price. (D)</p> Signup and view all the answers

What is the definition of a lognormally distributed random variable?

<p>A variable for which the natural logarithm is normally distributed. (D)</p> Signup and view all the answers

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?

<p>The stock could exhibit both significant gains and losses. (A)</p> Signup and view all the answers

What does the equation $St = S0e^{R(0, t)}$ represent?

<p>The future stock price based on past returns. (D)</p> Signup and view all the answers

What does $ u - eta$ in the equation denote?

<p>The expected capital gain in continuous terms (C)</p> Signup and view all the answers

What does exponentiation do to continuously compounded returns?

<p>Transforms them into lognormally distributed stock prices. (B)</p> Signup and view all the answers

Which of the following statements is true regarding lognormal distributions?

<p>The product of lognormal random variables is lognormal. (A)</p> Signup and view all the answers

If $x_1$ and $x_2$ are normally distributed, what can be said about $y_1 = e^{x_1}$ and $y_2 = e^{x_2}$?

<p>The product $y_1 × y_2$ is lognormally distributed. (B)</p> Signup and view all the answers

Why can't a lognormal stock price be negative?

<p>Because it is based on exponential functions. (B)</p> Signup and view all the answers

What is the significance of the central limit theorem in relation to lognormal distributions?

<p>It provides evidence that continuously compounded returns are normally distributed. (A)</p> Signup and view all the answers

Which equation describes the continuously compounded return from an initial stock price $S_0$ to a future stock price $S_t$?

<p>$R(0, t) = ln(S_t / S_0)$ (C)</p> Signup and view all the answers

What does the standard deviation indicate in a normal distribution?

<p>How spread out the values are (D)</p> Signup and view all the answers

Which of the following represents a normal distribution with mean 0 and a variance of 1?

<p>z ∼ N (0, 1) (C)</p> Signup and view all the answers

How does increasing the standard deviation affect the normal distribution?

<p>Spreads out the distribution (A)</p> Signup and view all the answers

In the formula φ(μ + a; μ, σ), what do the symbols represent?

<p>A value deviating from the mean (D)</p> Signup and view all the answers

What is the probability of drawing a specific value from a normal distribution?

<p>It is zero (B)</p> Signup and view all the answers

What is the relationship between the means of two normal distributions with equal variances?

<p>They can be any value (A)</p> Signup and view all the answers

What does the notation φ(x) denote in a standard normal distribution?

<p>The probability density function (C)</p> Signup and view all the answers

What does it mean for a normal distribution to be symmetric around the mean?

<p>The probabilities are the same on both sides of the mean (B)</p> Signup and view all the answers

Flashcards

Lognormally distributed stock

A stock price whose logarithm is normally distributed.

Expected value of S2

The average stock price after two years, calculated using the continuously compounded return and volatility.

Median stock price (2 years)

The stock price at which half the possible stock prices after two years fall below and half above.

One standard deviation move up (stock price)

The stock price after two years, which is one standard deviation above the expected price.

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One standard deviation move down (stock price)

The stock price after two years, which is one standard deviation below the expected price.

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Continuously compounded return

The return on an investment compounded continuously over time. It is the continuously compounded return over the specified period.

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Stock volatility

The degree of variation or fluctuation in the stock price over a period of time, expressed as a percentage.

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2-year volatility

The volatility of stock prices over a period of two years, used in the lognormal pricing model.

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Expected Stock Price (St)

The expected future stock price at time t, calculated from the initial stock price (S0) and the continuously compounded expected rate of appreciation.

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Continuously Compounded Expected Rate of Appreciation

(α − δ) Represents the expected rate at which the stock's value grows over time, considering both expected returns and dividends.

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Lognormal Stock Price Model

A model where the logarithm of stock prices follows a normal distribution which dictates that stock prices are always positive.

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Median Stock Price

The stock price value where 50% of future stock prices will be higher, and 50% will be lower.

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α − δ − 2σ²

The crucial component within the equation used to calculate the expected future stock price (St), which determines the continuous growth rate expected along a statistical distribution.

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Volatility (σ)

A measure of the price fluctuations of a stock, how much stock price varies around its expected value.

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Mean vs. Median Stock Price

In a lognormal distribution, the median stock price is always lower than the mean, because more than half of the time stock prices will be lower than their average.

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Stock Loss Probability

In a lognormally distributed stock, the probability of a stock losing value (St < S0) is greater than 50% if the volatility is relatively high, highlighting the risk in stock investments.

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Lognormal Distribution

A probability distribution of a random variable whose logarithm is normally distributed.

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Normal Distribution

A probability distribution that is symmetrical around the mean, with most values clustered near the mean.

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Parameters of Normal Distribution

The mean (μ) and standard deviation (σ) fully describe the normal distribution.

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Standard Normal Density

A normal distribution with a mean of 0 and a standard deviation of 1.

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Asset Price Lognormality

An assumption in option pricing that asset prices follow a lognormal distribution.

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Two Parameter Distribution

A probability distribution that requires only two values (parameters) to fully describe itself.

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Normal Density Function

Mathematical formula that describes the probability distribution of a normally distributed random variable.

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Standard Deviation (σ)

A measure of the dispersion of a dataset around the mean; a larger standard deviation implies wider spread.

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Standard Normal Distribution

A normal distribution with a mean of 0 and a standard deviation of 1.

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Normal Density

The probability density function of a normal distribution. It describes the relative likelihood of different values.

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Normal Distribution Equation in Excel

NormDist(x, μ, σ, False) is the function for calculating the normal density.

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Normal Distribution Variance

Affected by the standard deviation—a larger standard deviation indicates a more spread-out distribution. This impacts how probabilities cluster around the mean

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Normal Distribution Mean

The center of the normal distribution. A value around which the data clusters.

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x ~ N(μ, σ^2)

Notation indicating that a random variable x follows a normal distribution with mean μ and variance σ^2.

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Probability of a specific value in a continuous distribution

Zero. Continuous distributions deal with ranges, not single points.

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z ~ N(0, 1)

Notation for a random variable 'z' following the standard normal distribution.

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Lognormal Distribution

A distribution where the natural logarithm of a variable is normally distributed.

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Continuously Compounded Return

Return calculated continuously over time, often used in stock price models.

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Lognormal Stock Price

A stock price whose continuously compounded returns follow a normal distribution.

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Normal Distribution is Preserved

Addition of normal variables results in another normal variable. This concept is crucial mathematically.

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Lognormality is Preserved

Multiplication of lognormal variables results in a new lognormal one.

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Stock Price (Lognormal)

The stock price is lognormal if the continuously compounded returns are normally distributed, meaning the stock price can't be negative.

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Central Limit Theorem

A theorem stating that the sum of many independent random variables (like stock prices) tends towards a normal distribution.

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Binomial Model

A model generating a stock price distribution that resembles lognormal distribution, a particular instance of the Central Limit Theorem.

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Lognormal Distribution

A probability distribution of a random variable whose logarithm follows a normal distribution.

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Continuously Compounded Return

A return calculated by compounding interest continuously, often used in financial models.

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Stock Price Distribution

The distribution used to model future stock prices in investment analysis.

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Lognormal Density Function

A mathematical function describing the probability density of a lognormal distribution.

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E(ex)

The expected value of "e" to the power of x, when x itself is normally distributed.

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Lognormal Stock Price Model

An option pricing model in which the price is assumed to be lognormally distributed.

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Mean of a Lognormal Variable

The average of a lognormally-distributed random variable (greater than the exponential of the underlying normal's mean).

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Asymmetric Exponentiation

Positive random draws lead to bigger increases than negative ones have decreases.

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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√2Ï€) * 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√2Ï€) * 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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