  A central problem of Quantitative Finance is that of formulating a
probabilistic model of the time evolution of asset prices allowing reliable
predictions on their future volatility. As in several natural phenomena, the
predictions of such a model must be compared with the data of a single process
realization in our records. In order to give statistical significance to such a
comparison, assumptions of stationarity for some quantities extracted from the
single historical time series, like the distribution of the returns over a
given time interval, cannot be avoided. Such assumptions entail the risk of
masking or misrepresenting non-stationarities of the underlying process, and of
giving an incorrect account of its correlations. Here we overcome this
difficulty by showing that five years of daily Euro/US-Dollar trading records
in the about three hours following the New York market opening, provide a rich
enough ensemble of histories. The statistics of this ensemble allows to propose
and test an adequate model of the stochastic process driving the exchange rate.
This turns out to be a non-Markovian, self-similar process with non-stationary
returns. The empirical ensemble correlators are in agreement with the
predictions of this model, which is constructed on the basis of the
time-inhomogeneous, anomalous scaling obeyed by the return distribution.
