The lognormal distribution

In short

• Y is lognormal if $\ln Y$ is normally distributed. Equivalently, if X is normal, then $Y = e^X$ is lognormal.
• A lognormal variable is bounded below by zero and skewed to the right (it has a long right tail).
• That shape fits asset prices, which cannot fall below zero. Returns are usually modelled as (approximately) normal.
• Its two parameters are the mean $\mu$ and variance $\sigma^2$ of the associated normal distribution (of $\ln Y$), not of Y itself.
• The mean of Y is $e^{\mu + 0.5\sigma^2}$, which is above $e^{\mu}$: more volatility pushes the mean up.