We propose a local likelihood estimation for the log-transformed ARCH(1) model in the financial field. Our nonparametric estimator is constructed within the likelihood framework for non-Gaussian observations: It is different from standard kernel regression smoothing, where the innovations are assumed to be normally distributed. We derive consistency and asymptotic normality for our estimators and conclude from simulation and real data analysis that the local likelihood estimator has better predictive potential than classical local regression.