It has been known for a long time, since Mandelbrot’s papers in the 1960s that returns are heavytailed, and the research on nonGaussian heavy tailed distributions is intense. I was quite confused regarding what model to use for volatility jumps for weeks. I knew that kurtosis was not infinite for volatility jumps, and I had been considering the issue of whether to model jumps by the truncated Levy distributions or the simpler Student t distributions. Today I found some guidance from empirical tail indices and kurtosis for volatility jump distributions which is very clean. A simple linear relation between these exist in empirical data (after removal of outliers). This result implies that neither truncated Levy nor Student t models are quite right and there exist some parametric distribution whose kurtosis is approximately linear in tail index. Perhaps the distributions considered from the MittagLeffler family may produce the right distribution.
summary(fit)
Call:
lm(formula = y ~ x, data = D)
Residuals:
Min 1Q Median 3Q Max
17.5020 0.2785 0.0259 0.2899 18.6118
Coefficients:
Estimate Std. Error t value Pr(>t)
(Intercept) 18.61180 0.05102 364.8 <2e16 ***
x 13.42259 0.01582 848.3 <2e16 ***
—
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.684 on 1873 degrees of freedom
(24 observations deleted due to missingness)
Multiple Rsquared: 0.9974, Adjusted Rsquared: 0.9974
Fstatistic: 7.197e+05 on 1 and 1873 DF, pvalue: < 2.2e16
DISTRIBUTION WITH TAIL INDEX 0<alpha<1 WITH KURTOSIS GROWING WITH alpha

9:52 PM (38 minutes ago)


Rather than picking from known parametric distributions to match volatility jump distributions, we can consider distributions that solve a fractional diffusion equation.
For example, the fundamental solution of the fractional equation
(d/dt)^alpha = Laplacian
has fourth moment 24 t^{2*alpha)/Gamma(2*alpha+1) as described in this paper:
This is actually not only a simple way of matching empirical law or regularity I discovered today between tail indices of volatility jumps and empirical kurtosis but is probably a step toward a deeper understanding of the laws for volatility.
Preview attachment estimatedtailindexversuskurtosis.png
Preview attachment fphy0300011.pdf
Leave a Reply