# Detecting stochastic volatility

I have a time series extracted from a financial time series (so my series of prices is described by an arithmetic model $X(t)+Y(t)+Z(t)$, my series is $Z(t)$). I'm trying to model $Z(t)$ by a Levy process. However, it might be the case that $Z(t)=\int_0^t \omega_{s-} dV_s$, where $\omega_{t}$ is a stochastic volatility and $V(t)$ - a Levy subordinator.   We can see that:

• My fit seems to be reasonable, $Z(t)$ has slightly fatter tails than the model.
• Increments of $Z(t)$ are uncorrelated.
• Squared increments of $Z(t)$ are slightly correlated.

My question: How can I "detect" stochastic volatility? I tried to look at ACF functions of the increments of $Z(t)$ (and squared increments), but I'm not sure what it would tell me. I'm looking for easy tests that could give me some intuition, not a definite answer.

## 1 Answer

I'm not a time series expert but one idea occurs to me: look at the distribution of the increments if Z(t). If the w are stochastic , that distribution should have fat tails relative to the distribution that is generating the Levy process.

• This sounds like a good idea. However, the qq-plot indicates that the tails are only slightly fatter. Which might be a sign of "very slightly stochastic" (however we interpret it) volatility. – Paula Apr 19 '17 at 11:33