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## STOCHASTIC VOLATILITY PROCESS AND EXCESS KURTOSIS

This is standard material but since we are considering the issue of kurtosis, I thought to include this computation.  For lognormal $X \sim N(\alpha,\beta^2)$ one has $E(X) = \exp(\mu + \sigma^2/2)$.  So the moments

$E[\sigma_t^k] = \exp[ k\alpha + k^2 \beta^2 /2]$
$Var(r_t) = E((r_t - \mu)^2) = E[\sigma^2\epsilon_t^2] = \exp(2\alpha + 2\beta^2)$

And kurtosis is

$E[(r_t - \mu)^4]/E((r_t-\mu)^2)^2 = 3 \exp(4\beta^2)$

In other words here kurtosis increases with variance of $\log(\sigma_t)$.  The kurtosis of returns is produced by any stochastic volatility model.  Our concern is at the moment kurtosis of volatility jumps themselves.

In particular, this argument can be used to show that if $\log r_t^2 = \log\sigma_t^2 + \log\epsilon_t^2 \sim N(\alpha,\beta^2)$ then because $\log(r_t^4) \sim N(2\alpha, 4\beta^2)$ that

$kurtosis = E(r_t^4)/E(r_t^2)^2 = \exp(2\alpha + 2\beta^2)/\exp(2\alpha + \beta^2) = \exp(\beta^2)$

In particular the variance of volatility measures kurtosis of returns.