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AdvancedQuantitative MethodsPython

Run this module

cd "Quantitative Methods - GARCH"
python "garch.py"

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GARCH Volatility Models

GARCH (Generalized Autoregressive Conditional Heteroskedasticity) captures volatility clustering — high-volatility days tend to follow high-volatility days. Used for risk forecasting, option pricing, and VaR.

Functions

Function Description
ewma_volatility(returns, lambda_) RiskMetrics EWMA conditional volatility
fit_garch(returns) MLE estimation of GARCH(1,1) parameters
garch_forecast(fit, last_return, horizon) Multi-step variance forecast
garch_log_likelihood(params, returns) Gaussian negative log-likelihood

Model

GARCH(1,1):

sigma_t^2 = omega + alpha * r_{t-1}^2 + beta * sigma_{t-1}^2
  • alpha: ARCH term — reaction to recent shocks.
  • beta: GARCH term — persistence of past variance.
  • alpha + beta: persistence (must be < 1 for stationarity).
  • omega / (1 - alpha - beta): unconditional variance.

Example

from garch import fit_garch, garch_forecast

fit = fit_garch(returns)
print(fit['alpha'], fit['beta'], fit['persistence'])

vol_5d = garch_forecast(fit, returns[-1], horizon=5)

Practical Notes

  • Most equity GARCH fits show alpha ~ 0.05-0.15, beta ~ 0.80-0.92.
  • Persistence near 1 → integrated GARCH (IGARCH) — shocks have permanent effects.
  • For thicker tails, use Student-t innovations (extension).
  • EWMA is GARCH(1,1) with omega=0 and fixed alpha+beta=1.

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