Return Volatility: Log Returns, Simple Returns, and Geometric Dispersion
Summary
The document asks whether volatility computed from simple returns should use a geometric analogue of standard deviation because geometric averaging accounts for compounding. One response says that volatility in options practice is generally estimated from log returns, formed as the natural logarithm of the ending price divided by the beginning price; the standard deviation of those returns is used as the volatility estimate.
Another response supplies a geometric standard deviation formula that takes the square root of the average squared log distance from the geometric mean, then exponentiates it. It suggests that simple returns are often retained partly to avoid assuming lognormal returns, since financial return distributions may be skewed and heavy-tailed. The discussion is short and does not establish a general rule for choosing a dispersion measure. In particular, geometric dispersion and the standard deviation of arithmetic simple returns describe different quantities, so the choice depends on the return representation and modeling purpose; the thread does not develop a practical comparison or validation method.
Key ideas
- The geometric mean is often used to summarize compounded returns over time.
- Options volatility is commonly estimated as the standard deviation of log returns.
- A geometric standard deviation can be constructed from log distances to the geometric mean.
- Simple returns may be used without assuming lognormality, though the thread gives only a brief rationale.
- The measures describe different quantities and should be selected for the intended return representation.
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Full text
# Should the geometric standard deviation be used to compute the volatility of financial returns?
# 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?
## Answer by AlRacoon (score 3)
https://quant.stackexchange.com/a/37564
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.
## Answer by Nipper (score 1)
https://quant.stackexchange.com/a/57358
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).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.