Log Returns, Volatility Drag, and GBM Growth
Summary
This note explains why volatility measured from simple returns can imply a different growth picture from volatility measured in log returns. Its central concept is volatility drag: alternating gains and losses do not cancel when wealth compounds, even if the arithmetic average return is zero. The example compares equally likely gains of 50% and losses of 50%, showing that the resulting price path declines over time despite the zero arithmetic mean. It also describes the usual relationship under normal-return assumptions: the log growth rate is lower than the arithmetic growth rate by half the variance.
The note says log returns are commonly used to reason about compounded wealth, while simple and log volatility are often close in practical cases. It does not provide a derivation or empirical forecast comparison, and its displayed definitions of volatility are not standard variance or standard-deviation formulas. The discussion is therefore a qualitative explanation of return conventions and compounding effects, not a procedure for estimating volatility or forecasting prices.
Key ideas
- Simple returns and log returns describe growth using different conventions.
- Compounding means equal-sized gains and losses do not offset in wealth terms.
- Under normal-return assumptions, log growth is reduced relative to arithmetic growth by half the variance.
- Simple and log volatility can be close when return changes are modest.
- The displayed volatility formulas should not be treated as standard estimators.
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# What is the difference between log volatility and simple volatility in a GBM?
# What is the difference between log volatility and simple volatility in a GBM?
What difference do they make? Why do many people seem to find more accurate simulations with log volatility?
standard volatility in GBM is defined as $\sigma = \frac{1}{N}\sum_{i=1}^N(x_i-\mu)$ where $x_i$ is the rate of return and $\mu$ is the mean of the rate of return, also called the drift. Log volatitily on the other hand is defined as follows, I read someone using it in a research paper and it proved to produce less forecast error. I wonder why? Log volatility $\sigma = \frac{1}{N}\sum_{i=1}^N(logS_{ti}- logS_{ti-1})$ where $(logS_{ti}$ Is the price of the asset at time $t$.
Also, i'm not able to find this in any book...
## Answer by demully (score 1, accepted)
https://quant.stackexchange.com/a/60480
In its simplest form, the difference is "variance drag", ie how volatility itself affects your mu above.
Imagine a series of random returns of +50% versus -50% with a 50% probability of each. Evidently this will have a mu of zero, and a sigma of 0.5.
But if prices halve and appreciate by 50% with equal probability, they will clearly decline over time. True "Mu" is not really zero. Estimates of linear volatility are thus biased.
On all the conventional assumptions of normality, the difference between the two mu's will be half the (linear/conventional) variance. So if linear mu and sigma are say 0% and 10% respectively, then you should expect the asset to depreciate by 50bps a year. Equally, if you always held a fixed amount of this asset (and did not let your holding in it compound, appreciate or depreciate), then this zero-return asset generate positive returns of the same 50bps.
If you want to understand this effect more intuitively, imagine if it doubled or went to zero with equal chance (instead of +/-50%). What then would be your expectations? Higher vol (eventually) guarantees gambler's ruin.
So market participants often just default to looking at everything in log terms (as per your textbook) simply to get around the distortive effects of sticking with the standard linear approach, taught to students in traditional statistics courses.
In most practical cases, linear and log vol will be almost identical to each other. The implication is more normally (no pun intended) the differing impact of this volatility on linear (arithmetic) versus log (geometric) averages [and resulting wealth outcomes!!!]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.