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## WHAT IS THE RIGHT MODEL OF VOLATILITY JUMP DISTRIBUTION?

It has been known for a long time, since Mandelbrot’s papers in the 1960s that returns are heavy-tailed, and the research on non-Gaussian 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 Mittag-Leffler 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 <2e-16 ***
x 13.42259 0.01582 848.3 <2e-16 ***

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 R-squared: 0.9974, Adjusted R-squared: 0.9974
F-statistic: 7.197e+05 on 1 and 1873 DF, p-value: < 2.2e-16

## DISTRIBUTION WITH TAIL INDEX 0<alpha<1 WITH KURTOSIS GROWING WITH alpha

### Zulfikar Ahmed<wile.e.coyote.006@gmail.com>

9:52 PM (38 minutes ago)

 to David, bcc: aimee
Sir,

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.

3 Attachments

Preview attachment estimated-tail-index-versus-kurtosis.png

Preview attachment fphy-03-00011.pdf