Function fitGarch

Maximum likelihood by BFGS in an unconstrained space where every point is a valid, covariance-stationary model:

ω = e^{x1}·var(r)   P = σ(x2) = α + γ/2 + β   β = P·σ(x3)
q = P − β = α + γ/2   γ = 2q·tanh(x4)   α = q − γ/2

so ω > 0, β ≥ 0, α ≥ 0, α + γ ≥ 0 and P < 1 hold by construction. One start per persistence level (the best α / γ at that level), plus the warm start; the highest likelihood wins. The GARCH(1,1) likelihood is multimodal on short or jumpy series, and a single start stops in whichever mode is nearest.