# What's the explanation for the formula for the volatility of a stock / volatility of the continuously compounded return of a stock?

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.

The standard starting point with modelling a stock price process is to use the Black-Scholes model for the stock price. This simply asserts that the changes in the stock price are described by the following stochastic differential equation (SDE) $$\dfrac{\textrm{d}S}{S} = \mu\:\textrm{d}t + \sigma\:\textrm{d}W_t$$ where $W_t$ is a standard Brownian motion (which is a random variable), and typically $\mu$ and $\sigma$ are taken to be positive constants, (although this assumption can be dropped, the final result is more complicated).
A standard argument to solve this is to then apply Ito's Lemma to the process $\log(S)$ by considering $\textrm{d}\big(\log(S)\big)$. Although to formally appreciate the subtle details requires a course in stochastic calculus. The final result though is that the solution to the above SDE is $$S = S_0\exp\left(\left(\mu - \dfrac{\sigma^2}{2}\right)t + \sigma W_t\right)\:.$$ Notice that again $S$ is still a random variable (technically a log-normal random variable). Then if we consider $\log\left(\dfrac{S}{S_0}\right)$ we see that $$\log\left(\dfrac{S}{S_0}\right) = \left(\mu - \dfrac{\sigma^2}{2}\right)t + \sigma W_t$$ which is a normally distributed random variable with mean $\left(\mu - \dfrac{\sigma^2}{2}\right)t$ and standard deviation $\sigma t^{\frac{1}{2}}$. What is then typically done is a time scale is picked such that $t\to1$ and so the standard deviation of (aka the noise) evaluates to $\sigma$.