Basics about the scaling property of volatility

Question

It is a usual practice to calculate realized volatility $\sigma$ using the square root of the usual variance estimator $\hat{{\sigma}²}$. This is done using the stock log returns (practitioners sometimes BS variance). It is well known that the volatility scales as square root of time $\sigma_T = \sqrt{T} \cdot \sigma_1$. This is a trivial result when you model the stock dynamics as exponential brownian motion.

My questions are now the following, would any scaling property hold if you calculate the volatility as square root of variance of stock prices, as after all one can calculate the variance of a exponential brownian motion.

Answer 1

I don't know what you mean by "any scaling" rule. For the square-root of time I can say that it only needs uncorrelated returns.

Assume that the return from time point $1$ to $T$ is called $r_{1,T}$ and that it is given as $r_{1,T} = r_1 + r_2 + \cdots + r_T = \sum_{t=1}^T r_t$ where $r_t, t=1,\ldots,T$ are the one-period (e.g. one day) returns. The condition (the sum) holds excactly true for log-returns and approximately for simple returns.

Then we can calculate the variance $$ VAR(r_{1,T}) = VAR(\sum_{t=1}^T r_t) = \sum_{t=1}^T VAR(r_t), $$ where the covariance terms vanish as we assume that the returns are uncorrelated. Otherwise we would have more covariance terms $COV(r_i,r_j)$ for $i \neq j$.

If we furthermore assume that $VAR(r_t) = \sigma^2$ for $t=1,\ldots,T$ (this is stationary variance) then we get $$ VAR(r_{1,T}) = \sum_{t=1}^T \sigma^2 = T \sigma^2. $$ and by taking the square-root we get the square-root of time scaling for volatility. Note that we have used log-returns and that we have assume uncorrelated returns and stationary variance - nothing more.