# What is the difference between log volatility and simple volatility in a GBM? [closed]

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...

• Hi @PlatinumMaths, could you please be a bit more specific in your question? Thanks a lot! Jan 12 at 8:04
• 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 be produce less forecast error. I wonder why? Log volatility $\sigma = \frac{1}{N}\sum_{i=1}^N(logS_{ti}- logS_{ti-1})$ Jan 12 at 19:19
• Also, I'm not able to find this in any book.. Pehaps it's to do because the GBM is lognormal Jan 12 at 19:24

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!!!]