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TL;DR

A random variable X is log-normally distributed if \( \ln X \sim N(\mu, \sigma^2) \). Its mean is \( e^{\mu + \sigma^2/2} \) and its median is \( e^\mu \). It is always positive and right-skewed.

By Valenke Exam Prep Team·Last updated 2026-06-03

Log-Normal Distribution

A random variable X is log-normally distributed if \( \ln X \sim N(\mu, \sigma^2) \). Its mean is \( e^{\mu + \sigma^2/2} \) and its median is \( e^\mu \). It is always positive and right-skewed.

Why it matters for interviews

Stock prices under GBM are log-normally distributed. Understanding log-normal properties (mean > median, multiplicative structure) is essential for Black-Scholes and for correctly computing expected returns.

Definition and Mathematical Foundation

A random variable X is log-normally distributed if \( \ln X \sim N(\mu, \sigma^2) \). Its mean is \( e^{\mu + \sigma^2/2} \) and its median is \( e^\mu \). It is always positive and right-skewed.

Application in Quantitative Finance

Stock prices under GBM are log-normally distributed. Understanding log-normal properties (mean > median, multiplicative structure) is essential for Black-Scholes and for correctly computing expected returns.

Related Concepts

Brownian Motion (Wiener Process)

Related Terms

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Frequently Asked Questions

Why is the mean of a log-normal greater than the median?▼
The log-normal is right-skewed: \( E[X] = e^{\mu + \sigma^2/2} > e^\mu = \text{median} \). The \( \sigma^2/2 \) correction (Ito's correction) inflates the mean due to Jensen's inequality on the convex exponential function.
How do you compute the expected value of a log-normal?▼
\( E[e^Y] = e^{E[Y] + \text{Var}(Y)/2} \) for \( Y \sim N(\mu, \sigma^2) \). This follows from the MGF of the normal distribution evaluated at t=1.
What is the product of log-normals?▼
The product of independent log-normals is log-normal: \( \ln(XY) = \ln X + \ln Y \sim N(\mu_1+\mu_2, \sigma_1^2+\sigma_2^2) \). This multiplicative closure makes log-normals natural for modeling compounded returns.