TY  - JOUR
A1  - Ritschel, Stefan
A1  - Cherstvy, Andrey G.
A1  - Metzler, Ralf
T1  - Universality of delay-time averages for financial time series
BT  - analytical results, computer simulations, and analysis of historical stock-market prices
JF  - Journal of physics. Complexity
N2  - We analyze historical data of stock-market prices for multiple financial indices using the concept of delay-time averaging for the financial time series (FTS). The region of validity of our recent theoretical predictions [Cherstvy A G et al 2017 New J. Phys. 19 063045] for the standard and delayed time-averaged mean-squared 'displacements' (TAMSDs) of the historical FTS is extended to all lag times. As the first novel element, we perform extensive computer simulations of the stochastic differential equation describing geometric Brownian motion (GBM) which demonstrate a quantitative agreement with the analytical long-term price-evolution predictions in terms of the delayed TAMSD (for all stock-market indices in crisis-free times). Secondly, we present a robust procedure of determination of the model parameters of GBM via fitting the features of the price-evolution dynamics in the FTS for stocks and cryptocurrencies. The employed concept of single-trajectory-based time averaging can serve as a predictive tool (proxy) for a mathematically based assessment and rationalization of probabilistic trends in the evolution of stock-market prices.
KW  - econophysics
KW  - geometric Brownian motion
KW  - time-series analysis
Y1  - 2021
U6  - https://doi.org/10.1088/2632-072X/ac2220
SN  - 2632-072X
VL  - 2
IS  - 4
PB  - IOP Publ. Ltd.
CY  - Bristol
ER  - 
TY  - JOUR
A1  - Stojkoski, Viktor
A1  - Sandev, Trifce
A1  - Basnarkov, Lasko
A1  - Kocarev, Ljupco
A1  - Metzler, Ralf
T1  - Generalised geometric Brownian motion
BT  - theory and applications to option pricing
JF  - Entropy
N2  - Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics, due to irregularities found when comparing its properties with empirical distributions. As a solution, we investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) in order to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate that the performance of a kernel ultimately depends on the maturity of the option and its moneyness.
KW  - geometric Brownian motion
KW  - Fokker– Planck equation
KW  - Black– Scholes model
KW  - option pricing
Y1  - 2020
U6  - https://doi.org/10.3390/e22121432
SN  - 1099-4300
VL  - 22
IS  - 12
PB  - MDPI
CY  - Basel
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
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  - GEN
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
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.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 347 
KW  - diffusion
KW  - financial time series
KW  - geometric Brownian motion
KW  - time averaging
Y1  - 2017
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-400541
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  -