This paper studies the weak convergence of renorming volatilities in a family of GARCH(1,1) models from a functional point of view. After suitable renormalization, it is shown that the limiting distribution is a geometric Brownian motion when the associated top Lyapunov exponent γ > 0 and is an exponential functional of the maximum process of a Brownian motion when γ = 0. This indicates that the volatility of the GARCH(1,1)-type model has a completely different random structure according to the sign of γ. The obtained results further strengthen our understanding of volatilities in GARCH-type models. Simulation studies are carried out to assess our findings.