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Five Years of Phase Space Dynamics of the Standard & Poor’s 500

Veniamin Smirnov
Veniamin Smirnov
 y 
Dimitri Volchenkov
Dimitri Volchenkov
   | 28 jun 2019
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Article Category: Papers dedicated to the memory of Valetnin Afraymovich (1945-2018)
Publicado en línea: 28 jun 2019
Páginas: 209 - 222
Recibido: 30 oct 2018
Aceptado: 02 ene 2019
DOI: https://doi.org/10.2478/AMNS.2019.1.00019
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© 2019 Smirnov, V., Volchenkov, D., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

The smoothed phase space trajectories (individual stock prices at market close, one observation per day). a.) The Amazon company stocks (from 12/18/2013 to 01/10/2014); b.) The Apple company stocks (from 02/21/2018 to 03/14/2018). The regions of negative return are colored in red; the regions of positive returns are green colored.
The smoothed phase space trajectories (individual stock prices at market close, one observation per day). a.) The Amazon company stocks (from 12/18/2013 to 01/10/2014); b.) The Apple company stocks (from 02/21/2018 to 03/14/2018). The regions of negative return are colored in red; the regions of positive returns are green colored.

Fig. 2

a.) The state diagram for tossing a unfair coin, in which each state (’heads’ or ’tails’) repeats itself with the probability 0 ≤ p ≤ 1. b.) The information components (D)$(\mathscr{D})$(p), the past-future mutual information (excess entropy [14]); (U)$(\mathscr{U})$ (p), the conditional mutual information available at the present state of the chain and relevant to the future states; (E)$(\mathscr{E})$ (p), the ephemeral information existing only in the present state of the chain, being neither a consequence of the past, nor of consequence for the future.
a.) The state diagram for tossing a unfair coin, in which each state (’heads’ or ’tails’) repeats itself with the probability 0 ≤ p ≤ 1. b.) The information components (D)$(\mathscr{D})$(p), the past-future mutual information (excess entropy [14]); (U)$(\mathscr{U})$ (p), the conditional mutual information available at the present state of the chain and relevant to the future states; (E)$(\mathscr{E})$ (p), the ephemeral information existing only in the present state of the chain, being neither a consequence of the past, nor of consequence for the future.

Fig. 3

The daily snapshots of S&P 500 stocks in phase space during a.) a stock market crush; b.) a stock market rally; c.) & e.) and d.) & f.) a market tumbling phenomenon. The regions of phase space characterized by the positive/ negative values of return are colored by ’green’ and ’red’, respectively).
The daily snapshots of S&P 500 stocks in phase space during a.) a stock market crush; b.) a stock market rally; c.) & e.) and d.) & f.) a market tumbling phenomenon. The regions of phase space characterized by the positive/ negative values of return are colored by ’green’ and ’red’, respectively).

Fig. 4

a.) The color - coded histogram indicating the number of visits of the phase space cells throughout the entire period of observations; b.) The color - coded histogram representing the information content of phase space cells; c.) The predictability of states based on 1-day precursors; d.) The predictability of states based on 24-day precursors.
a.) The color - coded histogram indicating the number of visits of the phase space cells throughout the entire period of observations; b.) The color - coded histogram representing the information content of phase space cells; c.) The predictability of states based on 1-day precursors; d.) The predictability of states based on 24-day precursors.

Fig. 5

T–days mutual information of S&P 500.
T–days mutual information of S&P 500.

Fig. 6

The shares of transition information predictable from the historic time series ((D)$(\mathscr{D})$), from the present observation ((U)$(\mathscr{U})$), and ephemeral information (E)$(\mathscr{E})$ vs. Shannon’s entropy H measuring uncertainty of daily return for every studied company. The lines represent the polynomial trends for the empirically observed relations.
The shares of transition information predictable from the historic time series ((D)$(\mathscr{D})$), from the present observation ((U)$(\mathscr{U})$), and ephemeral information (E)$(\mathscr{E})$ vs. Shannon’s entropy H measuring uncertainty of daily return for every studied company. The lines represent the polynomial trends for the empirically observed relations.

Fig. 7

Uncertainty of daily stock price measured by Shannon’s entropy H along with the information components, D,U$\mathscr{D},\,\mathscr{U}$and E,$\mathscr{E},$for 468 major companies from the S&P 500.
Uncertainty of daily stock price measured by Shannon’s entropy H along with the information components, D,U$\mathscr{D},\,\mathscr{U}$and E,$\mathscr{E},$for 468 major companies from the S&P 500.

Fig. 8

A "unfair coin" of the S& P 500 stock market. a.) Unpredictable (E)$(\mathscr{E})$vs. predictable information (D+U)$(\mathscr{D}\,+\,\mathscr{U})$in unfair coin tossing Fig. (2. a); b.) Predictable information available from the present state alone (U)$(\mathscr{U})$vs. predictable information available from the past series (D)$(\mathscr{D})$in unfair coin tossing Fig. (2. a); c.) Unpredictable (E)$(\mathscr{E})$vs. predictable information (D+U)$(\mathscr{D}\,+\,\mathscr{U})$in the S& P 500; d.) Predictable information available from the present state alone (U)$(\mathscr{U})$vs. predictable information available from the past series (D)$(\mathscr{D})$in the S& P 500.
A "unfair coin" of the S& P 500 stock market. a.) Unpredictable (E)$(\mathscr{E})$vs. predictable information (D+U)$(\mathscr{D}\,+\,\mathscr{U})$in unfair coin tossing Fig. (2. a); b.) Predictable information available from the present state alone (U)$(\mathscr{U})$vs. predictable information available from the past series (D)$(\mathscr{D})$in unfair coin tossing Fig. (2. a); c.) Unpredictable (E)$(\mathscr{E})$vs. predictable information (D+U)$(\mathscr{D}\,+\,\mathscr{U})$in the S& P 500; d.) Predictable information available from the present state alone (U)$(\mathscr{U})$vs. predictable information available from the past series (D)$(\mathscr{D})$in the S& P 500.
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