When computing an average financial return over time (rather than cross-sectionally), the geometric mean is generally preferred to the arithmetic mean because it accounts for the geometric growth caused by compounding.
This is especially important when using simple returns of the type $r_t = p_t/p_{t-1} -1$, rather than log returns of the type $r_t = \ln(p_t/p_{t-1})$. Simple returns are typically preferred by practitioners because they are intuitive, but their mathematical properties are not as nice.
The volatility, which is commonly computed in finance, is defined as the arithmetic standard deviation of returns. This means that it should only applied to quantities for which the arithmetic mean is appropriate (i.e., log returns). When trying to compute volatilities from simple returns, shouldn't one instead use the geometric standard deviation
$$\sigma_g = \exp\left( \frac{\sum_{t=1}^T \left( \ln \frac{A_i}{\mu_g} \right)^2}{T} \right),$$
(where $\mu_g$ is the geometric mean), as the geometric mean is the preferred mean for simple returns over time?
Why is this never used and how could it be used in finance practice? Why might it still be okay to use the standard (arithmetic) standard deviation on simple returns?
