Validation of Volatility Clustering

Models of return volatility such as EWMA and GARCH aim to explain volatility clustering. In real markets, calm periods tend to be followed by calm periods, while turbulent periods are followed by turbulent periods. The better a model explains this clustering, the higher its quality.

For example, the quality of a GARCH model can be assessed conceptually as follows: if the ratio of squared returns to variance (u²/σ², i.e., z²) does not exhibit autocorrelation, then volatility has been correctly estimated. If this ratio still shows autocorrelation, it remains clustered, which means that clustering persists outside the model — in other words, the model has failed to fully capture the dynamics of volatility.

It is also important to understand why we use squared values. The average return is close to zero, so a direct comparison is not meaningful. By squaring returns, we measure the magnitude of movements, which provides the correct basis for comparison with volatility.

In a practical example, this can be calculated as follows: in Excel, where we already have columns for returns and GARCH volatility, we add a column for z = u/σ, and next to it 2 columns for z² (shifted by one period). We then calculate the correlation between these two columns.

As we can see in our example, the correlation is close to zero, which indicates that the model parameters are well specified and the volatility estimate adequately reflects historical data.

From the chart, we observe that variance follows the pattern of squared returns, but reacts more smoothly to shocks and is less extreme — which is exactly the objective of the model.

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Excel — Validation of Volatility Clustering

Adapted from:

Options, Futures & Other Derivatives, John C. Hull