ARCH Model

ARCH Model (Autoregressive Conditional Heteroskedasticity)

The ARCH(1) model, introduced by Robert Engle (1982), captures volatility clustering by modeling the conditional variance as a function of past squared returns:

h[t] = ω + α · r[t-1]²

where:

• h[t] = conditional variance at time t
• ω = long-run variance base (omega > 0)
• α = ARCH coefficient (0 < α < 1)
• r[t-1] = return at t-1
• h[0] = variance of the return series (sample variance)

The return at each step is: r[t] = ε[t] · √h[t] where ε[t] ~ N(0,1).

Task

Implement:

• arch_variance(returns, omega, alpha) → list of ARCH(1) conditional variances
• arch_simulate(n, omega, alpha, seed) → simulate n ARCH(1) returns using a seeded RNG