# Should the geometric standard deviation be used to compute the volatility of financial returns?

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?

## 2 Answers

Volatility, at least in the options world, is typically not calculated using arithmetic returns. The returns are generated by taking the natural log of ending price/beginning price. The standard deviation of this return series is considered the volatility of the underlying.

• In the options world yes, but in finance outside of academia in general, there is a preference for simple (non-log) returns of the type $r_t = \frac{p_t}{p_{t-1}}-1$ because they appear to be more intuitive to some people (I still prefer log, but a lot of people don't). My question is whether, when using such simple returns, it would be more appropriate to compute their geometric standard deviation as a dispersion measure, since the geometric mean is the preferred measure of location for simple returns. Jan 5 '18 at 10:28

The correct formula for the geometric standard deviation is the following:

$$\sigma_g = exp\left(\sqrt{\frac{\sum_{t=1}^{T}\bigg(ln\frac{A_t}{\mu_g}\bigg)^2}{T}}\right)$$

For what I understand outside academia papers in general non log returns are used not because they appear to be more intuitive but because doing so one does not assumes that returns follow a log normal distribution (it is demonstrated that returns distributions often show fat tails and skewness).