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L47 ARCH Models

Historical Volatility

  • Historical Volatility (HV) measures variability of an asset's returns over a specific period in the past
  • Reflects 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, stable assets (government bonds)
Changes in HV \(\implies\) Sudden spikes or drops \(\implies\) market uncertainty, upcoming events, shift in investor sentiment

Limitations

  • Backward-looking
    • only reflects past behavior
  • Sensitivity to Time Period
    • (20 days vs 100 days)
  • Outliers and Noise
    • makes it less representative of typical asset behavior
  • Ignores Structural changes
    • assumes that future will be identical to past
    • may not hold during regime changes or crises

Practical Examples

  • Forecasting Future Asset prices
  • market sentiment and behavioral finance

Volatility Models

\(F_{t_{1}}\) is the information set

  • \(E(y_{t}| F_{t-1}) = \mu_{t}\)
  • \(V(y_{t}| F_{t-1}) = \sigma_{t}\)

Let \(y_{t}\) follow an \(ARMA(p,q)\) model

\[ y_{t} = \mu_{t} + e_{t} \]

with,

\[ E(y_{t}| F_{t-1}) = \mu_{t} = \phi_{0} + \sum_{i=1}^{p} \phi_{i} y_{t-i} + \sum_{j=1}^{q} \theta_{j} e_{t-j} \]

and

\[ V(y_{t}| F_{t-1}) = \sigma_{t}^{2} \]

Volatility: \(\sigma_{t} = + \sqrt{ \sigma_{t}^{2} }\)

Since it evolves over time, we should model \(\sigma_{t}\) too.

  • Classical Heteroscedastic model

    • Stochastic equation to describe \(\sigma_{t}^{2}\) → SV Models
    • Use an exact function to govern evolution of \(\sigma_{t}^{2}\)
    • ARCH or GARCH

  • Model building

    1. Specify mean equation by testing for serial dependence. Or build a time series model
    2. Use residuals to check for ARCH effects (changing variance tendencies)
    3. Specify a volatility model if ARCH effects statistically significant. Perform joint estimation of the mean and volatility equations
    4. Perform diagnostic checks

ARCH Model

  • Model TS with varying volatility

  • \(e_{t} \sim e_{t-1}, e_{t-2}\dots\)

  • \(ARCH(m) \text{ model}\)

    • \(e_{t} = \sigma_{t}\epsilon_{t}\)
    • \(\sigma_{t}^{2} = \alpha_{0} + \alpha_{1}e^{2}_{t-1}+ .. + \alpha_{m}e^{2}_{t-m}\)
    • \(e_{t} \sim iid(0, \sigma^{2})\)1, \(\alpha_{0}\gt 0\) and \(\alpha_{i} \geq 0\) for \(i>0\)
  • \(\alpha_{i}\) satisfy regularity condition so that \(e_{t}\) is finite
  • \(\epsilon_{t}\) follows
    • standard normal
    • standardized T distribution
    • generalized error distribution (GED)
  • Leverage effects hold under ARCH framework

ARCH(1) Model

  • Mean, \(E(e_{t})=0\)
  • Variance, \(Var(e_{t}) = \dfrac{\alpha_{0}}{1-\alpha_{1}} \geq 0\) where \(0 \leq \alpha \leq 1\)

Limitations

  • Model depends on square of the previous shocks (positive and negative shocks have same effect)
  • Often over-predicts volatility
    • as it responds slowly to large isolated shocks
  • Parameter explosion
    • model becomes complex and computationally expensive with \(q\)

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