Modeling the non-Markovian, non-stationary scaling dynamics of financial markets

TL;DR

This paper develops a non-Markovian, non-stationary stochastic model for financial asset prices, specifically the Euro/US-Dollar exchange rate, capturing complex scaling and correlation properties observed in real trading data.

Contribution

It introduces a novel model accounting for non-stationarity and non-Markovian dynamics in financial markets, validated with extensive empirical data from currency trading.

Findings

Model accurately reproduces empirical correlators

Returns exhibit non-stationary, self-similar scaling

Empirical data supports non-Markovian process assumptions

Abstract

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.