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ARCH Models

Historical Volatility

  • Historical Volatility (HV) measures the variability of an asset's returns over a specific period in the past.
  • It reflects the uncertainty or risk associated with price movements during a given historical period.
Situation Interpretation
High HV \(\implies\) Large price swings, higher level of risk.
Low HV \(\implies\) Smaller price fluctuations and lower risk; characteristic of stable assets (e.g., government bonds).
Changes in HV \(\implies\) Sudden spikes or drops \(\implies\) market uncertainty, upcoming events, or a shift in investor sentiment.

Limitations

  • Backward-looking: Only reflects past behavior.
  • Sensitivity to Time Period: Results vary significantly based on the window (e.g., 20 days vs. 100 days).
  • Outliers and Noise: Extreme events can skew the measure, making it less representative of typical asset behavior.
  • Ignores Structural Changes: Assumes the future will be identical to the past, which may not hold during regime changes or crises.

Volatility Models

Let \(F_{t-1}\) be the information set at time \(t-1\).
- \(E(y_{t}| F_{t-1}) = \mu_{t}\) (Conditional Mean)
- \(V(y_{t}| F_{t-1}) = \sigma_{t}^2\) (Conditional Variance)

If \(y_{t}\) follows an \(ARMA(p,q)\) model:
$\(y_{t} = \mu_{t} + e_{t}\)$
Where the mean equation is:
$\(E(y_{t}| F_{t-1}) = \mu_{t} = \phi_{0} + \sum_{i=1}^{p} \phi_{i} y_{t-i} + \sum_{j=1}^{q} \theta_{j} e_{t-j}\)$
And the volatility is defined as: \(\sigma_{t} = + \sqrt{ \sigma_{t}^{2} }\).

Since volatility evolves over time, we must model \(\sigma_{t}^2\) as well.
- Stochastic Volatility (SV) Models: Use a stochastic equation to describe \(\sigma_{t}^{2}\).
- ARCH/GARCH Models: Use an exact function to govern the evolution of \(\sigma_{t}^{2}\).

Model Building Steps

  1. Specify mean equation: Test for serial dependence or build a standard time series model.
  2. Check for ARCH effects: Use the residuals from the mean equation to check for changing variance tendencies.
  3. Specify Volatility Model: If ARCH effects are statistically significant, perform joint estimation of the mean and volatility equations.
  4. Diagnostic Checks: Validate the model fit and residual properties.

ARCH Model

The Autoregressive Conditional Heteroscedasticity (ARCH) model is used for time series with varying volatility, where current volatility is dependent on previous shocks (\(e_{t-1}, e_{t-2}, \dots\)).

\(ARCH(m)\) Model Specification:
- \(e_{t} = \sigma_{t}\epsilon_{t}\)
- \(\sigma_{t}^{2} = \alpha_{0} + \alpha_{1}e^{2}_{t-1} + \dots + \alpha_{m}e^{2}_{t-m}\)
- \(\epsilon_{t} \sim iid(0, 1)\)1
- Constraints: \(\alpha_{0} > 0\) and \(\alpha_{i} \geq 0\) for \(i > 0\) to ensure positive variance.

\(\epsilon_{t}\) typically follows:
- Standard Normal distribution
- Standardized T-distribution (to capture fat tails)
- Generalized Error Distribution (GED)

Note

Leverage effects hold under the ARCH framework, though standard ARCH models treat positive and negative shocks symmetrically.

ARCH(1) Model

The simplest form, where volatility depends only on the immediately preceding shock:
- Mean: \(E(e_{t})=0\)
- Unconditional Variance: \(Var(e_{t}) = \dfrac{\alpha_{0}}{1-\alpha_{1}}\)
- Condition: \(0 \leq \alpha_1 < 1\) for the variance to be finite and positive.

Limitations of ARCH Models

  • Symmetry: The model depends on the square of previous shocks, meaning positive and negative shocks of the same magnitude have the same effect on volatility.
  • Over-prediction: It often over-predicts volatility because it responds slowly to large, isolated "one-off" shocks.
  • Parameter Explosion: The model becomes complex and computationally expensive as the order \(m\) increases to capture long-term persistence.

  1. By mentioning it this way, we are saying that the white noise is distribution-agnostic. ↩