Volatility clustering

Calm follows calm and storms follow storms. Build a GARCH market and see it happen.

Long-run volatility20% a year

Both markets share this average level.

Reaction to news, α0.100

How much yesterday's surprise raises today's volatility.

Memory, β0.880

How much of yesterday's volatility carries over.

Persistence0.980

Half-life34 d

ω3.2e-6

Excess kurtosis, clustered3.9

Excess kurtosis, constant0.10 for a bell curve

Days beyond 4σ, clustered12

Days beyond 4σ, constant0Bell curve expects 0.06

Clustered volatility (GARCH)

Constant volatility, same random shocks

Volatility the market is running at, a year

Both markets average 20% volatility and use identical random shocks. Only one remembers. With persistence of 0.980, a shock to volatility fades by half in about 34 days, so its volatility swung between 10% and 68%. Mixing calm and stormy days produces fat tails even though every single day is drawn from a bell curve.

The shape of the days

Clustered: daily returns in standard deviations

Constant: daily returns in standard deviations

The dashed curve is the bell curve. The clustered market is taller in the middle and longer in the tails: more boring days, and more extreme ones.

Direction has no memory, size does

Autocorrelation asks whether today looks like the day before, two days before, and so on up to 20 days. Returns themselves stay inside the band in both markets: you can't predict tomorrow's direction. Squared returns, a measure of size, are a different story.

Clustered

Returns

Squared returns

20 of 20 squared-return lags sit outside the shaded band that pure chance would stay in.

Constant

Returns

Squared returns

0 of 20 squared-return lags sit outside the shaded band that pure chance would stay in.

In the real S&P 500

Size of each monthly move since 2005, up or down.

Big months arrive in bunches. The correlation between one month's squared return and the next is 0.31, well outside the ±0.12 that chance would give.

Try this

Set α to 0. The clustered market becomes the constant one. What happens to kurtosis?
Push memory β up until persistence is near 0.99. How long do storms last now?
Raise α and lower β with the same persistence. Are the storms sharper or longer?

The math

\sigma_t^{2} = \omega + \alpha\, r_{t-1}^{2} + \beta\, \sigma_{t-1}^{2}, \qquad \bar\sigma^{2} = \frac{\omega}{1 - \alpha - \beta}

$\sigma_t^{2}$: today's variance
$r_{t-1}$: yesterday's return, the surprise
$\alpha$: reaction to news
$\beta$: memory of past volatility
$\bar\sigma^{2}$: long-run variance it returns to

Ask the Loop about this Next: Tail risk and black swans