6. Volatility clustering: why calm and storm arrive in runs

Level 5: Time series: why the past does not repeat itself the same way

Artificially generate a series of daily returns by drawing each value completely independently from one and the same distribution with a fixed standard deviation.

It is the kind of i.i.d. process from module 1. Days with a large return and days with a small return will come out spread more or less at random over time, with no clustering pattern.

The real market looks nothing like that.

Calm and storm arrive in runs

What is observed in real markets is markedly different from that simulation. Days of large movements tend to cluster in runs, followed by runs of quiet days, instead of being spread out uniformly.

This phenomenon is called volatility clustering. It is one of the most robust and best documented empirical regularities of financial series.

Where to look in order to see it

The correct way to measure this phenomenon was anticipated in module 3, and it is not the one a beginner would try first:

• Do not look at the autocorrelation of the returns themselves. That tends to be small, consistent with the scant directional predictability of module 4.
• Do look at the autocorrelation of a measure of the size of the movement: the squared return, or equivalently the absolute value of the return.

If that autocorrelation is positive and sizable at several consecutive lags — not only at lag 1, but sustained over several days or even weeks — it is a statistical signature worth reading carefully.

What that signature says is that the size of today's movements is related to the size of recent movements, regardless of sign.

How far the persistence reaches

It is common to observe in indices such as the S&P 500 that this autocorrelation of volatility, measured on the squared returns, is still detectable even at lags of 20 or 30 days.

That is far more persistent than any autocorrelation of the raw returns, which typically disappears into the noise within a few days if it appears at all.

The arithmetic of two very different autumns

A concrete historical example helps to fix the scale of the phenomenon. Compare two periods:

• During the autumn of 2008, in the central weeks of the financial crisis, the S&P 500 strung together sessions with daily swings of 5%, 7%, even more than 10% on particular days, for several weeks in a row.
• In sharp contrast, during much of 2017 the index went months with daily swings below 0.5%, an unusually prolonged period of calm even by historical standards.

Now suppose that daily returns were really independent with a fixed variance, the simplified assumption that is sometimes used when first introducing the concept of the normal distribution in a basic statistics course.

Under that assumption, observing such long runs of consecutive extreme days, or of consecutive quiet days, would be statistically almost impossible.

The fact that these runs occur recurrently throughout market history is precisely the evidence that variance is not constant over time, but instead goes through regimes. It directly contradicts one of the three stationarity conditions seen in module 2.

What it implies for measuring risk

This phenomenon has a very direct practical consequence for risk management, and it is worth stating it plainly.

Recent volatility contains genuine predictive information about the volatility of the coming days. It is something that the simple absence of autocorrelation in the returns might lead one to dismiss mistakenly.

It follows that it is more informative to estimate the short-term risk of a position using a window of recent data, rather than a long historical average mixing calm and turbulent regimes.

There is, moreover, a whole family of statistical models designed specifically to capture this persistence, known generically as GARCH models and their variants.

Without going into their mathematical formulation, the conceptual idea is that tomorrow's expected variance is modeled as a combination of two ingredients:

• The recent observed variance, which supplies the short-term persistence.
• A long-term variance level to which the series tends to return gradually, which supplies the tendency to normalize over time.

The problem with volatility in a single number

There is a consequence that is commonly overlooked.

Using a single historical standard deviation computed over many years, as is usually done when presenting an asset's volatility with a single number, averages across very different regimes and may misrepresent the risk of the present moment in either direction:

• If the market is currently going through a period of unusual calm, that long historical number will overstate the real short-term risk.
• If it is going through a period of recent turbulence, it will understate it just when having it well measured matters most.

There is an underlying reason here, beyond the simple updating of data. It is the reason why risk metrics computed on recent rolling windows — the subject of the last module of this level — are preferable to a fixed static number computed once and for all over the entire available history.

The leverage effect: it is not even symmetric

An additional nuance enriches the picture: the so-called asymmetry or leverage effect.

Volatility tends to spike somewhat more after negative returns than after positive returns of comparable size.

It is a pattern consistent with sharp falls generating more uncertainty and defensive reaction — hurried selling, hedging of positions, revision of expectations — than sharp rises of equivalent size.

This asymmetry does not change the module's central conclusion, but it does qualify it: volatility clustering is not perfectly symmetric between rises and falls.

It is a relevant detail for any risk model aspiring to capture the real behavior of the market with somewhat more fidelity than a simple average of magnitudes.