Financial data give an opportunity to uncover the non-stationarity which may be hidden in many single time-series. Five years of daily Euro/ Dollar trading records in the about three hours following the New York opening session are shown to give an accurate ensemble representation of the self-similar , non-Markovian stochastic process with nonstationary increments recently conjectured to generally underlie financial assets dynamics PNAS%DIFDELCMD < {\bf %%% 104 , 19741 (2007)%DIFDELCMD < ]%%% . Introducing novel quantitative tools in the analysis of non-Markovian time-series we show that empirical non-linear correlators are in remarkable agreement with model predictions based only on the anomalous scaling form of the logarithmic return distribution.
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 %DIFDELCMD < {\bf %%% %DIFDELCMD < ]%%% 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.
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Financial data give an opportunity to uncover the non-stationarity which may be hidden in many single time-series. Five
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
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Dollar
US-Dollar
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opening session are shown to give an accurate ensemble representation of the
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,
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, non-Markovian stochastic process with nonstationary increments recently conjectured to generally underlie financial assets dynamics
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PNAS
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104
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, 19741 (2007)
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. Introducing novel quantitative tools in the analysis of non-Markovian time-series we show that empirical non-linear
process with non-stationary returns. The empirical ensemble
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remarkable agreement with model predictions based only on the anomalous scaling form of the logarithmic
agreement with the predictions of this model, which is constructed on the basis of the time-inhomogeneous, anomalous scaling obeyed by the