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

Question

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.

Answer 1

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 $\sigma$.

Otherwise I am not sure what your notation is for a stock which pays dividends, so can't comment on that.

Look up any introductory book on financial derivatives or stochastic calculus for financial applications and this should cover the issues in more detail.

I hope that helps.

Answer 2

You can see e^(rt) =St/S0, which is continuous and can generate r when t=1 time lag r= In(St/S0).