Log-normal and logistic distributions, and the moments of key distributions

In short

• Y is log-normal if ln Y is normal. Y is then always positive and skewed to the right, a natural fit for asset prices.
• Log returns $\ln(P_1/P_0)$ add across periods and never imply a negative price. Simple returns do not add, and a normal model of them allows returns below −100%.
• Projected log price: $\ln S_T = \ln S_0 + (\mu - \sigma^2/2)T$, with log-return standard deviation $\sigma\sqrt{T}$.
• If $X \sim N(\mu, \sigma^2)$, then $Y = e^X$ has mean $e^{\mu + \sigma^2/2}$ and variance $(e^{\sigma^2} - 1)e^{2\mu + \sigma^2}$.
• The logistic distribution has mean μ (its location) and variance $s^2\pi^2/3$ (s is the scale). It is symmetric with excess kurtosis 1.2, so its tails are heavier than the normal's.