GARCH(1,1) Model

GARCH(1,1) Model

The GARCH(1,1) (Generalized ARCH) model by Bollerslev (1986) extends ARCH by also incorporating lagged conditional variances:

h[t] = ω + α · r[t-1]² + β · h[t-1]

where:

• h[t] = conditional variance at time t
• ω = base constant (omega > 0)
• α = ARCH coefficient (sensitivity to recent shocks)
• β = GARCH coefficient (persistence of variance)
• Stationarity requires α + β < 1

The initial variance h[0] is set to the sample variance of the returns.

GARCH(1,1) is the workhorse volatility model in finance — it captures both volatility clustering and mean reversion in variance.

Task

Implement garch_variance(returns, omega, alpha, beta_g) that returns the list of conditional variances.