I am self-studying for an actuarial exam, Models for Financial Economics.
It's stated as a given in my manual that $\sigma$ is the volatility of the stock, $\sqrt{\text{Var}(\ln(S_t/S_0))}$ and that the volatility of the continuously compounded return on a stock is given by $\sqrt{\text{Var}(\ln(S))}$.
Clearly $\sqrt{\text{Var(X)}}$ is volatility, but where does the $\ln(S_t/S_0)$ and $\ln(S)$ come from, respectively?
Also stated without proof is that if a stock pays continuous dividends, $\sqrt{\text{Var}(\ln(S_t/S_0))} = \sqrt{\text{Var}\big(F_{0, t}^p(S)\big)}$. i.e. the volatility of the stock is the same as the volatility of the prepaid forward on the stock.
I was hoping someone could provide reference to a derivation, or an explanation.