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Journals
Applied Mathematics and Nonlinear Sciences
Volume 4 (2019): Issue 1 (January 2019)
Open Access
Five Years of Phase Space Dynamics of the Standard & Poor’s 500
Veniamin Smirnov
Veniamin Smirnov
and
Dimitri Volchenkov
Dimitri Volchenkov
| Jun 28, 2019
Applied Mathematics and Nonlinear Sciences
Volume 4 (2019): Issue 1 (January 2019)
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Article Category:
Papers dedicated to the memory of Valetnin Afraymovich (1945-2018)
Published Online:
Jun 28, 2019
Page range:
209 - 222
Received:
Oct 30, 2018
Accepted:
Jan 02, 2019
DOI:
https://doi.org/10.2478/AMNS.2019.1.00019
Keywords
Discrete-time Markov processes
,
Measures of information
,
entropy
,
Mathematical finance
© 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.
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.
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).
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.
Fig. 5
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.
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.
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.
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