Lesson 5 of 25 · 7 min

Volatility clustering, described and measured

Tomorrow's direction is close to unforecastable; the size of tomorrow's move is not, and the asymmetry has a number.

Track 3 lesson 19 named volatility clustering as one of the places the pricing model is known to be wrong, and Track 4 lesson 3 used it to justify talking about regimes. Both treated it as a description. It is measurable, and the measurement is a specific pair of correlations that point in opposite directions.

Take the thirty-session series from lesson 4 and compute the correlation between each return and the one immediately after it. The answer is −0.0745, which is close enough to zero to call nothing. Direction does not persist: a down session tells you essentially nothing about the sign of the next.

Now throw away the signs and do the same computation on the absolute returns. The answer is +0.4289. Magnitude persists strongly. A large move, in either direction, is followed by another large move, in either direction, far more often than chance allows. That pair of numbers — roughly zero for the return, roughly a half for its absolute value — is what clustering is.

A second measurement of the same thing is concentration. Add up the squared returns across all thirty sessions and ask where they came from. The five largest sessions account for 55.66 percent of the total, and the three largest account for 38.93 percent. Volatility over a month is not spread across the month; it is delivered on a handful of days. The average absolute return in the quiet half is 0.70 percent against 2.33 percent in the turbulent half, a ratio of 3.32.

A haircut is a forecast of magnitude, not of direction. Clustering says magnitude is the one of the two that is actually forecastable — so the lender's model is aimed at the only part of the problem that has any signal in it.

Key terms

Autocorrelation at lag one: The correlation between a series and the same series shifted forward by one observation. Near zero for returns, strongly positive for their absolute values.
Absolute return: A return with its sign removed. The quantity that persists from session to session, and the quantity a discount is sized against.
Variance concentration: The share of a window's total squared return contributed by its few largest sessions. High concentration is clustering seen from a different angle.

Worked example

Clustering, measured two ways on one series

The thirty-session illustrative series from lesson 4. All figures computed from that series.

Correlation of each return with the next: −0.0745
Correlation of each absolute return with the next: +0.4289
Share of total squared return from the top five sessions: 55.66%
Share from the top three sessions: 38.93%
Average absolute return, quiet half: 0.70%
Average absolute return, turbulent half: 2.33%

Two numbers from the same thirty observations: one says the future is unknowable, the other says part of it is not. Both are true because they are about different things.

Check yourself

A series shows a lag-one autocorrelation of −0.07 in its returns and +0.43 in its absolute returns. What can and cannot be forecast from yesterday?

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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.