Abstract
We introduce three strategies for the analysis of financial time series based on time averaged observables. These comprise the time averaged mean squared displacement (MSD) as well as the ageing and delay time methods for varying fractions of the financial time series. We explore these concepts via statistical analysis of historic time series for several Dow Jones Industrial indices for the period from the 1960s to 2015. Remarkably, we discover a simple universal law for the delay time averaged MSD. The observed features of the financial time series dynamics agree well with our analytical results for the time averaged measurables for geometric Brownian motion, underlying the famed Black–Scholes–Merton model. The concepts we promote here are shown to be useful for financial data analysis and enable one to unveil new universal features of stock market dynamics.
Highlights
In 1900, Bachelier pioneered the concept that prices on financial markets are stochastic and may follow the laws of Brownian motion [1, 2]
As we demonstrate from statistical analyses of real financial time series, these approaches are highly useful and reveal universal features of the market dynamics, which may be relevant for the further development of financial market models
The above dependence (6) of the time averaged mean squared displacement (MSD) on the trace length T reflects the phenomenon of ageing, a characteristic property of non-stationary stochastic processes [55, 58]
Summary
Andrey G Cherstvy, Deepak Vinod, Erez Aghion, Aleksei V Chechkin and Ralf Metzler. These comprise the time averaged mean squared displacement (MSD) as well as the this work must maintain attribution to the ageing and delay time methods for varying fractions of the financial time series. We explore these author(s) and the title of concepts via statistical analysis of historic time series for several Dow Jones Industrial indices for the the work, journal citation and DOI. The observed features of the financial time series dynamics agree well with our analytical results for the time averaged measurables for geometric Brownian motion, underlying the famed Black–Scholes–Merton model. The concepts we promote here are shown to be useful for financial data analysis and enable one to unveil new universal features of stock market dynamics
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