Volatility, in the context of pure math (not necessarily finance), refers to the coefficient of dW term. Therefore,
* it implies a measure,
* it implies a process, a stoch process
Therefore, if a vol number is 5%, it is, conceptually and physically, different from a stdev of 0.05.
* Stdev measures the dispersion of a static population, or a snapshot as I like to say. Again, think of the histogram.
* variance parameter (vol^2) of BM shows diffusion speed.
* if we quadruple the variance param (doubling the vol) value, then the terminal snapshot’s stdev will double.
At any time, there’s an instantaneous vol value, like 5%. This could last a brief interval before it increase or decreases. Vol value changes, as specified in most financial models, but it changes slowly — quasi-constant… (see other blog posts)
There is also a Black-Scholes vol. See other posts.