波动性

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波动性(又稱“波幅”),指金融资产在一定时间段的变化性。通常以一年内涨落的标准方差来测量。 光的波动性指两列光波在空中相遇时发生叠加,在某些区域总加强,某些区域减弱,相间的条纹或者彩色条纹的现象。 金融市场中,投资的波动性与其风险有着密切的联系。

数学定义[编辑]

年化波动率\sigma定义为对象资产的年回报率的对数值的标准差。

一般的时间长度为T(年)的一段时间内的波动率\sigma_T则定义为:

\sigma_T = \sigma \sqrt{T}.\,

因此,如果一种资产(如股票)在长度为P的一段时间内,日均的对数回报率的标准差为\sigma_{SD},那么年化波动性为

\sigma = \frac {\sigma_{SD}}{\sqrt{P}} .\,

在美国,通常认为P = \frac {1}{252}(因为美国每年有252个交易日)。因此如果某资产的\sigma_{SD}=0.01,那么就可以算出年化波动率为

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

当然也可依上式算出月波动率等:T = \frac {1}{12}时,

\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:

\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。

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