Using Increment Distributions and Squared Returns to Explore Stochastic Volatility

Summary

The document asks how to explore whether a time series component modeled as a Lévy process may instead have stochastic volatility. The described observations are uncorrelated increments, slight correlation in squared increments, and tails somewhat heavier than the fitted Lévy model. These patterns motivate diagnostics, but they do not by themselves identify a stochastic volatility process.

The response proposes comparing the empirical increment distribution with the distribution implied by the underlying Lévy process: random volatility can create heavier-tailed increments. The question also raises autocorrelation of increments and their squares as exploratory checks. The material offers intuition rather than a formal hypothesis test, calibration procedure, or account of alternative causes such as jumps, model misspecification, or finite-sample effects. Any apparent excess tail weight or dependence should therefore be treated as suggestive evidence, not proof.

Key ideas

• Uncorrelated increments can coexist with dependence in squared increments.
• Stochastic volatility may make increments heavier-tailed than the driving Lévy process would imply.
• Comparing empirical increment distributions with a fitted model can provide an exploratory diagnostic.
• Autocorrelation in squared increments is suggestive but does not establish stochastic volatility.
• The proposed checks are informal and do not rule out jumps or model misspecification.

Tags

# Detecting stochastic volatility


# 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.

## Answer by dm63 (score 3)

https://quant.stackexchange.com/a/33742

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.