Variance of random walk, mean reverting and trending series

Question

I am reading Ernie Chan's blog post "Mean reversion, momentum, and volatility term structure". It says that

To be precise, if $z$ is the log price, then volatility, sampled at intervals of $\tau$, is $$\text{Volatility}(\tau)=\sqrt{\text{Var}(z_t-z_{t-\tau})}$$ where $\text{Var}$ means taking the variance over many sample times.

In general, we can write $$\text{Var}(z_t-z_{t-\tau})\propto \tau^{2H}$$ where H is called the "Hurst exponent", and it is equal to 0.5 for a true geometric random walk, but will be less than 0.5 for mean reverting prices, and greater than 0.5 for trending prices.

It looks correct intuitively, as for a trending price, the variance increases faster as the sample interval gets larger, and for a mean reverting price, the variance doesn't change that much accordingly.

Can anyone give some explanation (or proof) on that (and why it is equal to 0.5 for a true geometric random walk)? Thanks!

Answer 1

A geometric random walk looks something like $S_t=S_0\exp((\mu-\sigma^2/2)t+\sigma W_t)$, where $W_t$ is standard brownian motion. Thus $\log(S_t)$ has fluctuations $\sigma W_t$, and it is a fact that $W_{t}-W_{t-\tau}$ has standard deviation $\sqrt{\tau}$.

For a simpler explanation, see here. Effectively, $\log(S_t)\approx \log(S_{t-\tau})+e_{t-\tau}$, where $e_t$ is normally distributed with mean 0 and standard deviation $\sqrt{\tau}$.