A GARCH(1,1) fit on daily returns gives omega = 0.000002, alpha = 0.08 and beta = 0.90. Compute the long-run daily variance, the annualised long-run volatility at 252 trading days, and the half-life of a shock to the conditional variance.

A GARCH(1,1) fit on daily returns gives omega = 0.000002, alpha = 0.08 and beta = 0.90. Compute the long-run daily variance, the annualised long-run volatility at 252 trading days, and the half-life of a shock to the conditional variance.

Approach: Take unconditional expectations of the variance recursion to get the stationary level, then read the decay rate of deviations from it.

0.0001. Covariance stationarity of a GARCH(1,1) needs alpha + beta < 1. Taking unconditional expectations of sigma_t^2 = omega + alpha*r_{t-1}^2 + beta*sigma_{t-1}^2, and using E[r^2] = E[sigma^2], gives the long-run variance omega/(1 - alpha - beta) = 0.000002/0.02 = 0.0001. That is a daily volatility of 0.01, and 0.01*sqrt(252) = 0.1587. Annualised that is close to 16%. Deviations of the conditional variance from that level decay by the persistence factor alpha + beta, equal to 0.98, each day, so the half-life is log(0.5)/log(0.98), which is 34.3 days. High persistence is why one large move keeps raising forecast variance for weeks, and why volatility clustering is visible out to many lags. It also means the long-run level itself is poorly identified from a few years of data, since alpha + beta near 1 leaves few effective independent variance cycles in the sample.

Follow-up: The fit returns alpha + beta = 1.001. What model does that correspond to, and what forecast property does it have?

Key concepts: GARCH, unconditional variance, persistence, volatility clustering.