TY  - JOUR
A1  - Cherstvy, Andrey G.
A1  - Vinod, Deepak
A1  - Aghion, Erez
A1  - Chechkin, Aleksei V.
A1  - Metzler, Ralf
T1  - Time averaging, ageing and delay analysis of financial time series
JF  - New journal of physics
N2  - 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.
KW  - time averaging
KW  - diffusion
KW  - geometric Brownian motion
KW  - financial time series
Y1  - 2017
U6  - https://doi.org/10.1088/1367-2630/aa7199
SN  - 1367-2630
VL  - 19
SP  - 1
EP  - 11
PB  - IOP
CY  - London
ER  - 
TY  - JOUR
A1  - Cherstvy, Andrey G.
A1  - Vinod, Deepak
A1  - Aghion, Erez
A1  - Chechkin, Aleksei V.
A1  - Metzler, Ralf
T1  - Time averaging, ageing and delay analysis of financial time series
JF  - New journal of physics : the open-access journal for physics
N2  - 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.
KW  - time averaging
KW  - diffusion
KW  - geometric Brownian motion
KW  - financial time series
Y1  - 2017
U6  - https://doi.org/10.1088/1367-2630/aa7199
SN  - 1367-2630
VL  - 19
SP  - 135
EP  - 147
PB  - IOP Publ. Ltd.
CY  - Bristol
ER  - 
TY  - JOUR
A1  - Cherstvy, Andrey G.
A1  - Vinod, Deepak
A1  - Aghion, Erez
A1  - Sokolov, Igor M.
A1  - Metzler, Ralf
T1  - Scaled geometric Brownian motion features sub- or superexponential ensemble-averaged, but linear time-averaged mean-squared displacements
JF  - Physical review : E, Statistical, nonlinear and soft matter physics
N2  - Various mathematical Black-Scholes-Merton-like models of option pricing employ the paradigmatic stochastic process of geometric Brownian motion (GBM). The innate property of such models and of real stock-market prices is the roughly exponential growth of prices with time [on average, in crisis-free times]. We here explore the ensemble- and time averages of a multiplicative-noise stochastic process with power-law-like time-dependent volatility, sigma(t) similar to t(alpha), named scaled GBM (SGBM). For SGBM, the mean-squared displacement (MSD) computed for an ensemble of statistically equivalent trajectories can grow faster than exponentially in time, while the time-averaged MSD (TAMSD)-based on a sliding-window averaging along a single trajectory-is always linear at short lag times Delta. The proportionality factor between these the two averages of the time series is Delta/T at short lag times, where T is the trajectory length, similarly to GBM. This discrepancy of the scaling relations and pronounced nonequivalence of the MSD and TAMSD at Delta/T << 1 is a manifestation of weak ergodicity breaking for standard GBM and for SGBM with s (t)-modulation, the main focus of our analysis. The analytical predictions for the MSD and mean TAMSD for SGBM are in quantitative agreement with the results of stochastic computer simulations.
Y1  - 2021
U6  - https://doi.org/10.1103/PhysRevE.103.062127
SN  - 2470-0045
SN  - 2470-0053
VL  - 103
IS  - 6
PB  - American Physical Society
CY  - College Park
ER  -