Volatility Clustering

Have you ever wondered in your sleep what is Volatility Clustering? How do we model Volatility? Why GARCH models are used for volatility?

So you can today have sound sleep because this is what we’re covering today! 🙂

Volatility clustering and fat tails are most common features of financial data. 

To illustrate the point, image below reports monthly returns for SP500 and NASDAQ from 1990 till 2018. First thing to note here is that returns are centered to zero. The second thing to note is that there are lot of spikes, so there are extreme positives and extreme negatives. There is an interesting regularity anyway: when the market is nervous, then the next period is going to be nervous. When the market is calm, then the next period is going to be calm. This characteristic is present also along the entire sample span, so it is independent of the point at which it happens.

To summarize, while it is difficult to forecast the mean of the distribution of return, it is somehow easier to forecast the volatility of returns, because when the market is nervous, the next period is going to be nervous. When the market is calm, the next period is going to be calm. This is true also if we change the index. The next graph reports the same picture for the NASDAQ. The period and the frequency of the observations are the same. This market has is a higher level of volatility because the market is smaller (there are fewer people working in this market) so it takes longer for the information to spread.

However, the mean of the distribution of returns is still equal to zero, and there are still a lot of spikes in the graph, both positive and negative. Again, when the market is nervous, the next period is going to be nervous. When the market is calm, the next period is going to be calm. This property is called volatility clustering: the variance cluster in periods.

You can appreciate that even with different frequencies, daily, and different periods, the main characteristics of financial markets are still there:

• The mean is equal to zero
• there are a lot of positive and negative spikes
• when the market is calm, then it’s going to be followed by a period where the market is calm.

From the data we have described, we can build the distribution of returns. The first approach we can follow is to assume that the data is normally distributed. In this case, you calculate the mean and the variance of the distribution of returns, and you substitute these values in the formula for the normal distribution. Both the images below show that there are much more observations on the tails than if data are normally distributed. Moreover, both NASDAQ and SP500 returns are negatively skewed.

As we decrease the frequency of returns (SP500 & NASDAQ) from daily to monthly, we will observe that distributional looks closer to normal distribution. This is called Aggregational Gaussianity.

Another piece of evidence suggests that the mean is constant and equal to zero: returns are not autocorrelated, so you can’t use past observations of returns to try and guess future returns. But note that absolute values of returns and squared returns are positively autocorrelated, and non-zero correlation persists even at higher lags. That means we can use squared returns to forecast squared future returns.

To summarize, we can say that in financial markets, periods of high volatility tend to be followed by periods of high volatility and, conversely, periods of low volatility tend to be followed by periods of low volatility. This means that we can try and predict the volatility of the market, which is of interest here as it is a measure of risk.

The linear models cannot explain this volatility feature, and hence these are captured by the so-called autoregressive conditional heteroscedastic models (ARCH models) and by the Generalised Autoregressive Conditional Heteroscedastic (GARCH) models. The word conditional heteroscedasticity means that variance is not constant and depends on observations. Meaning, in periods of high volatility, variance is high and in periods of low volatility, it is low. In general, GARCH (1,1) model will be sufficient to capture the volatility clustering in the data.