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## PETER CLARK’S SUBORDINATED PROCESS FOR STOCKS

Trading time being random is due to Peter Clark (1971) Peter-Clark-SubordinatedProcess1971. I had not noticed that he had precise formulae for the kurtosis effects on price changes in terms of statistics of the random time directing process.  So this is an interesting exercise:  we know that the renewal times of volatility shocks in the equities is fractional Poisson so an exercise is to update Clark’s analysis for this case including some analysis of kurtosis.

If the directing process is $T(t)$ with $z=\Delta T(t)$ with mean $\alpha$, then the kurtosis of the price process $\Delta X(T(t))$ is

$k_{\Delta X(T(t))} = \frac{3 (\alpha^2 + Var(z))}{\alpha^2}$

according to Clark’s Corollary 4.1.

Instead of looking at the kurtosis of price changes directly, probably the best thing to do is to consider the model

$r_t = e^{h_t/2} \epsilon_t$

and consider Clark’s formula as it applies to $h_t$, because we know that the volatility shock distribution is regular.  This is a synthesis of approaches to describing the kurtosis of price changes.  Rather than ‘stochastic volatility is the answer’ or ‘random times is the answer’ we have a synthesis because it turns out that the volatility series has fractional Poisson features and excess kurtosis.