Lognormal Distribution

Modeling variables that cannot be negative, like stock prices and asset values.

The Stock Price Distribution

The Lognormal Distribution is essential in finance because it models variables that are always positive, like stock prices or asset values. If a variable's logarithm is normally distributed, then the variable itself has a lognormal distribution.

This is a perfect fit for modeling stock returns. If we assume that the continuously compounded returns of a stock are normally distributed, then the future price of that stock will be lognormally distributed. This elegantly solves the problem of prices going below zero, which a normal distribution would allow.

The Formula

The probability density function (PDF) is given by:

f(x) = \frac{1}{x\sigma\sqrt{2\pi}} \exp\left(-\frac{(\ln(x) - \mu)^2}{2\sigma^2}\right)

$x$ is the variable (e.g., stock price), must be $> 0$ .
$\mu$ (mu) is the mean of the underlying normal distribution (location parameter).
$\sigma$ (sigma) is the standard deviation of the underlying normal distribution (scale parameter).

Interactive Lognormal Distribution

Adjust the location (μ) and scale (σ) parameters of the underlying normal distribution to see how the shape of the lognormal curve changes.

Location (μ): 0.00

Scale (σ): 0.50