Let's assume we have close prices of one stock of period one year. As far as I understand historical volatility is standard deviation of log return, however I do not understand what this actually mean.
Let's assume last traded price was $120 and HV is 0.29.
What does this really mean? That we can assume next price will be within the range of 29,58 (which is 29 percent of 120) above or bellow $120?
No. Volatility by itself is a meaningless number.
Volatility becomes relevant when used in a context, such as when comparing between assets, or to identify how much the stability of a financial instrument has changed over time. In these two examples, dividing volatility by the mean rate of return gives a more useful, dimensionless number known as coefficient of variation.
Historical volatility may also serve as input parameter for simulation of an asset price, provided that the analyst already has in mind a probabilistic model. The example you outline is akin to estimating a confidence interval. Your example reflects an assumption that the asset's behavior is modeled with a probability density function which is symmetrical with respect to the latest price.
There are other uses of historical volatility, but the ones above are among the most typical applications of that concept.
