In this paper we propose a new model for volatility fluctuations in financial time series. This model relies on a nonstationary Gaussian process that exhibits aging behavior. It turns out that its properties, over any finite time interval, are very close to continuous cascade models. These latter models are indeed well known to reproduce faithfully the main stylized facts of financial time series. However, it involves a large-scale parameter (the so-called "integral scale" where the cascade is initiated) that is hard to interpret in finance. Moreover, the empirical value of the integral scale is in general deeply correlated to the overall length of the sample. This feature is precisely predicted by our model, which, as illustrated by various examples from daily stock index data, quantitatively reproduces the empirical observations.