GARCH (1,1) and Volatility Clustering

Financial markets exhibit an important property: volatility clustering and reversion toward a long-run average.

In other words, large movements tend to be followed by large movements, and calm periods tend to be followed by calm periods. However, over time volatility tends to move back toward a long-term average level.

To capture this behavior, the GARCH(1,1) model is commonly used.

In a previous post we discussed Moving Average (MA) and Exponentially Weighted Moving Average (EWMA) methods. Both attempt to estimate the current volatility of an asset’s returns using historical data.

However:

• MA cannot capture volatility clustering,
• EWMA cannot capture mean reversion.

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What GARCH Means

GARCH stands for:

Generalized Autoregressive Conditional Heteroskedasticity

The name sounds complicated, but the idea is quite simple.

Word	Meaning
Generalized	an extension of an earlier model (ARCH)
Autoregressive	volatility depends on its own past values
Conditional	depends on past information
Heteroskedasticity	variance is not constant

In simple terms:

today’s volatility depends on the long-run volatility, yesterday’s volatility, and yesterday’s shock.

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The Core Formula

The model updates the variance (the square of volatility) using the equation

$$\sigma_t^2 = \omega + \alpha u_{t-1}^2 + \beta \sigma_{t-1}^2$$where:

Symbol	Meaning
$u_{t-1}$ut−1​	yesterday’s return
$u_{t-1}^2$ut−12​	the size of yesterday’s shock
$\sigma_{t-1}^2$σt−12​	yesterday’s variance
$\omega$ω	base variance level
$\alpha$α	reaction to new shocks
$\beta$β	volatility persistence

Here$$\omega = V_L (1-\alpha-\beta)$$

where$$V_L \approx \text{the long-run average variance of returns}$$

Unlike EWMA, GARCH includes a long-run average variance, which can be written as$$V_L = \frac{\omega}{1-\alpha-\beta}$$

The corresponding long-run volatility is$$\sigma_L = \sqrt{V_L}$$

This represents the volatility level toward which the model tends to move over time.

However, it is important to understand that volatility will not stay close to this level every day. The long-run variance simply acts like gravity, slowly pulling volatility toward equilibrium over time.

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Why Volatility Clustering Appears

Looking again at the GARCH equation$$\sigma_t^2 = \omega + \alpha u_{t-1}^2 + \beta \sigma_{t-1}^2$$

If volatility is high today, the next period will still contain the term

$$\beta \sigma_{t-1}^2$$

Because β is usually large (0.85–0.95), volatility declines only gradually. This persistence creates volatility clustering.

Mathematically, the weight applied to past shocks becomes$$\alpha \beta^{i}$$

which means the influence of shocks decays exponentially through time.

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Parameter Interpretation

For daily financial data we often observe parameter values in the following ranges:

Parameter	Typical Range
α	0.05 – 0.12
β	0.85 – 0.94

For the model to be stable we require$$\alpha + \beta < 1$$α+β<1

If this sum is very close to one, the impact of shocks disappears very slowly. In real financial markets the value is often between 0.95 and 0.99.

Note also that if$$\omega = 0$$ω=0

then GARCH reduces to EWMA.

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Why It Is Called GARCH (1,1)

The general notation is GARCH(p,q).

Where:

• p = number of past shocks included
• q = number of past variance estimates included

Thus GARCH(1,1) means the model uses:

• the most recent squared return
• the most recent variance estimate

Despite its simplicity, GARCH(1,1) is the most widely used volatility model in finance.

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Example

The chart below is constructed using historical data for Tesla (TSLA):

Excel model: GARCH(1,1)

There is one interesting detail that explains why GARCH sometimes moves further away from equilibrium than EWMA, even though intuitively we might expect the opposite.

Intuitively we think:

EWMA → inertia
GARCH → inertia + gravity

Therefore we might expect GARCH to stay closer to equilibrium.

However, the graph shows something different.

The reason is that EWMA reacts more slowly to shocks.

Let’s compare the formulas.

EWMA:$$\sigma_t^2 = \lambda\sigma_{t-1}^2 + (1-\lambda)u_{t-1}^2$$σt2​=λσt−12​+(1−λ)ut−12​

GARCH:$$\sigma_t^2 = \omega + \alpha u_{t-1}^2 + \beta\sigma_{t-1}^2$$σt2​=ω+αut−12​+βσt−12​

Now look at the weights.

RiskMetrics EWMA uses$$\lambda = 0.94$$λ=0.94

which means the weight on the shock is$$1-\lambda = 0.06$$1−λ=0.06

Typical GARCH parameters (Hull example) are$$\alpha = 0.13,\quad \beta = 0.86$$α=0.13,β=0.86

Thus the shock weight in EWMA is about 6%, while in GARCH it is about 13%.

In other words, GARCH reacts roughly twice as strongly to shocks, which is why its path can move further away from equilibrium.

At the same time, ω is usually very small, meaning the gravitational pull toward the long-run average is quite weak in the short run.

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Adapted from:

John C. Hull — Options, Futures and Other Derivatives

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