Define cdf and pdf as:
Then moments of x are:
Source: Modeling and Generating Daily Changes in Market Variables Using A Multivariate Mixture of Normal Distributions. My special thanks to Dr. Jin Wang for his correction to the skew formula.
Mixture of Normal, Formulas for Skew and Kurtosis
Mixture of normal distributions is a simple and intuitive way to create distributions with skew and kurtosis by mixing or adding two or more normal distributions. These are well known formulas, but for some reason every time I need to use them I have trouble locating them. So here they are for everyone's reference.
Define cdf and pdf as:
=\sum_{i=1}^{n}{p_i}\Phi%20(\frac{x-\mu_i}{\sigma_i})%20\\%20f(x)=\sum_{i=1}^{n}{p_i}\phi%20(x;\mu_i,\sigma_i^2)%20\\%20\phi%20(x;\mu_i,\sigma_i^2)=\frac{1}{\sqrt{2\pi}\sigma_i}e^{-\frac{(x-\mu_i)^2}{2\sigma_i^2}}%20\\%200\leq%20p_i%20\leq%201,%20\sum_{i=1}^n{p_i}=1 "\dpi{150} \bg_black \dpi{150} \bg_black \\ F(x)=\sum_{i=1}^{n}{p_i}\Phi (\frac{x-\mu_i}{\sigma_i}) \\ f(x)=\sum_{i=1}^{n}{p_i}\phi (x;\mu_i,\sigma_i^2) \\ \phi (x;\mu_i,\sigma_i^2)=\frac{1}{\sqrt{2\pi}\sigma_i}e^{-\frac{(x-\mu_i)^2}{2\sigma_i^2}} \\ 0\leq p_i \leq 1, \sum_{i=1}^n{p_i}=1")
Then moments of x are:
-\mu^2%20\\%20skew%20=%20\frac{1}{\sigma^3}\sum_{i=1}^{n}{p_i}(\mu_i-\mu)[3\sigma_i^2+(\mu_i-\mu)^2]%20\\%20kurtosis%20=%20\frac{1}{\sigma^4}\sum_{i=1}^{n}{p_i}[3\sigma_i^4+6(\mu_i-\mu)^2\sigma_i^2+(\mu_i-\mu)^4] "\dpi{150} \bg_black \\ \mu=\sum_{i=1}^{n}{p_i}\mu_i \\ \sigma^2 = \sum_{i=1}^{n}{p_i}(\sigma_i^2+\mu_i^2)-\mu^2 \\ skew = \frac{1}{\sigma^3}\sum_{i=1}^{n}{p_i}(\mu_i-\mu)[3\sigma_i^2+(\mu_i-\mu)^2] \\ kurtosis = \frac{1}{\sigma^4}\sum_{i=1}^{n}{p_i}[3\sigma_i^4+6(\mu_i-\mu)^2\sigma_i^2+(\mu_i-\mu)^4]")
Source: Modeling and Generating Daily Changes in Market Variables Using A Multivariate Mixture of Normal Distributions. My special thanks to Dr. Jin Wang for his correction to the skew formula.
Define cdf and pdf as:
Then moments of x are:
Source: Modeling and Generating Daily Changes in Market Variables Using A Multivariate Mixture of Normal Distributions. My special thanks to Dr. Jin Wang for his correction to the skew formula.
Subscribe to:
Post Comments (Atom)
Weekly market report
Wall st delivered a mixed bag of news with VIX, VNKY, and VSTOXX and their underlying markets almost unchanged. VXD - volatility index based...
-
As I am sure all of you know Russia has began a full scale war against my home country Ukraine. Please make no mistake - Putin's goal ...
-
Many investors are looking at VIX and VSTOXX indexes as a leading indicators of volatility in equity markets, however many are confused by t... -
Deutsche Bank Currency Volatility Index was developed to provide an implied volatility benchmark for major currency markets. The index is d...
No comments:
Post a Comment