Lesson 7 of 25 · 6 min

The volatility of volatility

The input is more variable than most of the things it measures, and it cannot fall below zero while it can rise without limit.

Volatility is treated as a parameter — a level that describes an asset. It is better understood as a series in its own right, with its own mean and its own dispersion, and its dispersion is large.

Measure it on the thirty-session series from lesson 4 by splitting it into six non-overlapping five-session blocks and annualising each. The readings are 11.31, 12.52, 14.52, 39.21, 52.39 and 34.90 percent. Their mean is 27.475 percent and their standard deviation is 17.124 percent, so the dispersion is 62.33 percent of the level. The highest reading is 4.63 times the lowest.

Compare that with the thing being measured. A broad equity holding might have an annualised volatility in the twenties. Volatility's own volatility, on this sample, is above sixty percent. The parameter moves around more than the price it describes, which is a strange property for a parameter to have and explains why a single figure is such a poor summary.

The distribution is also asymmetric, for a reason that requires no empirical work. Volatility cannot be negative, so the worst a reading can do is approach zero — a fall of at most a hundred percent. Upward there is no bound, and readings that quadruple in a fortnight are ordinary. So the distribution of a volatility figure is stretched to the right, and its mean sits above its median.

This propagates straight into a collateral discount, because the model in lesson 19 is linear in volatility. A 4.63-fold range in the input is a 4.63-fold range in the volatility-driven component of the haircut. A borrower whose advance rate is computed from a short window is holding a number that inherits all of this dispersion, and inherits the right-hand stretch too: the surprises are larger in the direction that reduces lending value.

Key terms

Volatility of volatility: The dispersion of a volatility series, usually expressed as its standard deviation divided by its mean. On the sample here it is 62 percent.
Non-overlapping window: A window that shares no observations with the next one. Overlapping windows understate how much an estimate moves, because consecutive readings share most of their data.
Right-skewed distribution: A distribution with a long upper tail and a bounded lower one. Volatility has this shape because it cannot go below zero and has no ceiling.

Worked example

Six readings from one month

The thirty-session illustrative series split into six non-overlapping five-session blocks, each annualised at 252 sessions.

Blocks one to three: 11.31%, 12.52%, 14.52%
Blocks four to six: 39.21%, 52.39%, 34.90%
Mean of the six readings: 27.475%
Standard deviation of the six readings: 17.124%
Dispersion relative to the level: 62.33%
Highest reading against lowest: 4.63 times

The input to every model in the second half of this track ranged over a factor of four and a half inside a single month.

Check yourself

Six consecutive five-session readings average 27.475 percent with a standard deviation of 17.124 percent. Why does that make a single quoted volatility figure a weak summary?

Saved in this browser only. No account, nothing sent anywhere.

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.