How GARCH Produces Volatility Clustering After a Shock

Volatility clustering means large absolute returns tend to occur near other large absolute returns, while quiet days tend to arrive together. A GARCH model captures this pattern by increasing its next variance forecast after a large surprise and carrying part of the previous variance forecast forward.

After an unexpected announcement, uncertainty may stay elevated while markets absorb new information and assess its consequences. The next move can be large in either direction. If daily returns go from

+4\%

-3\%

+2.5\%

, their signs alternate, while their magnitudes all indicate an unusually volatile period. A GARCH model summarises that recent history with a single conditional variance.

Follow a shock through the variance recursion

Let

\varepsilon_t=r_t-\mu_t

be the surprise in return

r_t

after accounting for its conditional mean. Write

\varepsilon_t=\sqrt{h_t}z_t

, where the innovation

z_t

has conditional mean zero and variance one. GARCH(1,1) uses

h_{t+1}=\omega+\alpha\varepsilon_t^2+\beta h_t.

Here $\omega>0$ is the baseline term, $\alpha\ge0$ controls the response to the last squared surprise, and $\beta\ge0$ controls how much previous conditional variance carries forward. Since $E[\varepsilon_t^2]=E[h_t]$ , taking expectations in the recursion gives the finite long-run variance $\bar h=\omega/(1-\alpha-\beta)$ when $\alpha+\beta<1$ and the process is in its stationary regime.

After observing a large shock, let $h_{t+1}$ be today's forecast for tomorrow. Because future expected squared surprises equal the corresponding variance forecasts, the expected deviation from $\bar h$ decays by the factor $\alpha+\beta$ at each additional step. For persistence between zero and one, the number of periods for the expected deviation to halve is $\log(0.5)/\log(\alpha+\beta)$ . This half-life refers to an expected variance forecast following a shock. On a realised path, another surprise may interrupt the decay.

For a multi-step forecast made after observing the shock at time

t

, the next variance

h_{t+1}

is known under the fitted model. The next squared surprise has conditional expectation

h_{t+1}

, because the standardised innovation has conditional second moment one. Substituting that expectation gives the next forecast

\omega+(\alpha+\beta)h_{t+1}

. Subtract the long-run level

\omega/(1-\alpha-\beta)

from both sides to see that the gap shrinks geometrically when

\alpha+\beta<1

. The half-life concerns this forecasted gap. Any individual future shock can push variance back up.

Measure returns in percentage points. Take

\omega=0.05

\alpha=0.10

\beta=0.85

and start with

h_t=1

. A

-4

percentage point surprise gives

h_{t+1}=0.05+0.10(16)+0.85(1)=2.50

, or about

1.58\%

volatility. A

+4

percentage point surprise has the same effect in this model. The expected variance excess is

1.50

above the long-run value of

1

. After ten further steps, its expected excess is

1.50(0.95)^{10}\approx0.90

. The expected half-life is about 13.5 steps.

Plot the variance forecast after a shock

import numpy as np

omega, alpha, beta = 0.05, 0.10, 0.85
long_run = omega / (1 - alpha - beta)
tomorrow = omega + alpha * (-4.0)**2 + beta * long_run
days_after_tomorrow = np.arange(21)
expected_variance = long_run + (tomorrow - long_run) * (
    alpha + beta
)**days_after_tomorrow
print(np.sqrt(expected_variance[[0, 10, 20]]))

This computes volatility from the expected future variance. It illustrates the volatility implied by variance forecasts, while the expected square root of future variance is a different quantity.

How GARCH Produces Volatility Clustering After a Shock

Follow a shock through the variance recursion

Plot the variance forecast after a shock

Explore how quickly a GARCH shock decays

References and next steps