Modeling the Non-Markovian, Non-stationary Scaling Dynamics of Financial Markets

Abstract

3 bovina@pd.infn.it 4 camana@pd.infn.it 5 stella@pd.infn.it † Summary. 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. †

Scaling symmetry, renormalization, and time series modeling: The case of financial assets dynamics

“…This fine-graining, reverse renormalization group strategy for the description of market dynamics has been already exemplified in previous contributions [23,25,26], especially dealing with high frequency processes [35][36][37]. Unlike in cases for which a single time series is available, in Refs.…”

“…Unlike in cases for which a single time series is available, in Refs. [35][36][37] we focused on a particular, fixed window of the daily evolution of an asset, and extracted from the available records an ensemble of histories which have been assumed to be independent realizations of the same stochastic process. The manifest time inhomogeneous character of this process and its limited duration in time significantly simplify a modeling approach based on the above fine-graining strategy.…”

“…(7): X t = a t Y t . In principle, these rescaling factors convey a time inhomogeneity to the increments X t 's, a property which has been exploited in the modeling of ensembles of histories [35][36][37]. However, a proper randomization of the time argument of the a t 's restores the stationarity of the X t 's, making them suitable for describing single time series whose statistical properties are thought to be independent of time [10].…”

“…The background ideas exposed in Refs. [23,34] already stimulated some contributions along various lines [23][24][25][26][35][36][37]. In particular, in [25] a model with nonstationary increments and lacking a volatility feedback mechanism has been discussed in detail, pointing out its potential interest and missing features.…”

Aftershock Prediction for High-Frequency Financial Markets’ Dynamics

“…These limits reflect a lack of adequate modeling for the dynamics of financial indexes, especially in regimes like those covered by the Omori law. In recent contributions [3,4], some of the present authors have proposed a model for the dynamics at high frequency of exchange rates or stock market indexes, which takes into account most of the relevant stylized facts. Among them, the martingale character of index evolution, the manifest non-stationarity of volatility detected in well defined daily windows of trading activity, the anomalous scaling properties of the aggregate return probability density function (PDF) in the same windows, and the strong time autocorrelation of the elementary absolute return.…”