Volatility Futures & Options: 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:
$\dpi{150} \bg_black \dpi{150} \bg_black \\ F(x)=\sum_{i=1}^{n}{p_i}\Phi (\frac{x-\mu_i}{\sigma_i}) \\ f(x)=\sum_{i=1}^{n}{p_i}\phi (x;\mu_i,\sigma_i^2) \\ \phi (x;\mu_i,\sigma_i^2)=\frac{1}{\sqrt{2\pi}\sigma_i}e^{-\frac{(x-\mu_i)^2}{2\sigma_i^2}} \\ 0\leq p_i \leq 1, \sum_{i=1}^n{p_i}=1$

Then moments of x are:
$\dpi{150} \bg_black \\ \mu=\sum_{i=1}^{n}{p_i}\mu_i \\ \sigma^2 = \sum_{i=1}^{n}{p_i}(\sigma_i^2+\mu_i^2)-\mu^2 \\ skew = \frac{1}{\sigma^3}\sum_{i=1}^{n}{p_i}(\mu_i-\mu)[3\sigma_i^2+(\mu_i-\mu)^2] \\ kurtosis = \frac{1}{\sigma^4}\sum_{i=1}^{n}{p_i}[3\sigma_i^4+6(\mu_i-\mu)^2\sigma_i^2+(\mu_i-\mu)^4]$

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.

Volatility Futures & Options

Mixture of Normal, Formulas for Skew and Kurtosis

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