# 波动性

## 数学定义

${\displaystyle \sigma _{T}=\sigma {\sqrt {T}}.\,}$

${\displaystyle \sigma ={\frac {\sigma _{SD}}{\sqrt {P}}}.\,}$

${\displaystyle \sigma ={\frac {0.01}{\sqrt {\frac {1}{252}}}}=0.1587.}$

${\displaystyle \sigma _{\text{month}}=0.1587{\sqrt {\frac {1}{12}}}=0.0458.}$

The formula used above to convert returns or波动性measures from one time period to another assume a particular underlying model or process. These formulas are accurate extrapolations of a random walk, or Wiener process, whose steps have finite variance. However, more generally, for natural stochastic processes, the precise relationship between volatility measures for different time periods is more complicated. Some use the Lévy stability exponent α to extrapolate natural processes:

${\displaystyle \sigma _{T}=T^{1/\alpha }\sigma .\,}$

If α = 2 you get the Wiener process scaling relation, but some people believe α < 2 for financial activities such as stocks, indexes and so on. This was discovered by Benoît Mandelbrot, who looked at cotton prices and found that they followed a Lévy alpha-stable distribution with α = 1.7. (See New Scientist, 19 April 1997.) Mandelbrot's conclusion is, however, not accepted by mainstream financial 计量经济学家。