Financial time series is often assumed to be stationary and has a normal distribution in the literature. Both assumptions are however unrealistic. This paper proposes a new methodology with a focus on volatility estimation that is able to account for nonstationarity and heavy tails simultaneously. In particular, a local exponential smoothing (LES) approach is developed, in which weak estimates with different memory parameters are aggregated in a locally adaptive way. The procedure is fully automatic and the parameters are tuned by a new propagation approach. The extensive and practically oriented numerical results confirm the desired properties of the constructed estimate: it performs stable in a nearly time homogeneous situation and is sensitive to structural shifts. Our main theoretical oracle result claims that the aggregated estimate performs as good as the best estimate in the considered family. The results are stated under realistic and unrestrictive assumptions on the model.