Abstract
The dynamics of complex systems, from financial markets to the brain, can be monitored in terms of multiple time series of activity of the constituent units, such as stocks or neurons, respectively. While the main focus of time series analysis is on the magnitude of temporal increments, a significant piece of information is encoded into the binary projection (i.e. the sign) of such increments. In this paper we provide further evidence of this by showing strong nonlinear relations between binary and non-binary properties of financial time series. These relations are a novel quantification of the fact that extreme price increments occur more often when most stocks move in the same direction. We then introduce an information-theoretic approach to the analysis of the binary signature of single and multiple time series. Through the definition of maximum-entropy ensembles of binary matrices and their mapping to spin models in statistical physics, we quantify the information encoded into the simplest binary properties of real time series and identify the most informative property given a set of measurements. Our formalism is able to accurately replicate, and mathematically characterize, the observed binary/non-binary relations. We also obtain a phase diagram allowing us to identify, based only on the instantaneous aggregate return of a set of multiple time series, a regime where the so-called ‘market mode’ has an optimal interpretation in terms of collective (endogenous) effects, a regime where it is parsimoniously explained by pure noise, and a regime where it can be regarded as a combination of endogenous and exogenous factors. Our approach allows us to connect spin models, simple stochastic processes, and ensembles of time series inferred from partial information.
Highlights
In large systems, the observed dynamics or activity of each unit can be represented by a discrete time series providing a sequence of measurements of the state of that unit
Other studies have documented that the binary projections of various financial [16] and neural [17] time series exhibit non-trivial dynamical features that resemble those of the original data
Having established that the binary projections of real time series contain non-trivial information, in the rest of the paper we introduce a theory of binary time series aimed, among other things, at reproducing the observed nonlinear relationships showed in figures 2 and 3
Summary
The observed dynamics or activity of each unit can be represented by a discrete time series providing a sequence of measurements of the state of that unit. We first provide robust empirical evidence of novel relationships between binary and non-binary properties of real financial time series To this end, we use the daily closing prices of all stocks of three markets (S&P500, FTSE100 and NIKKEI225) over the period 2001–2011. These empirical relations quantify in a novel way the strong correlations existing between the increments of individual stocks and the overall level of synchronization among all stocks in the market Building on this evidence, we introduce a formalism to analytically characterize random ensembles of single and multiple time series with desired constraints. In the case of interest here, we introduce ensembles of maximum-entropy binary matrices that represent projections of single and multiple binary time series, subject to a set of desired constraints defined as simple empirical measurements. ‘Weighted’ (left) versus ‘binary’ (right) time series of log-returns of the Apple stock over a period of 50 days starting from 7 May 2011
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
