Wandering Drunkards Walk after Fibonacci Rabbits: How the Presence of Shared Market Opinions Modifies the Outcome of Uncertainty
<p>Different possible combinations after six trades, ‘a’ marking any price change, ‘H’ marking no change.</p> "> Figure 2
<p>Different possible combinations posting a six-letter chain.</p> "> Figure 3
<p>Different ways to see the same rise in price from a bottom to a top as depicted in ‘a’, with ‘b’ presenting an l-composition (l standing for left) of ‘a’, and with ‘c’ presenting an r-composition (r standing for right) of ‘a’. On both ‘b’ and ‘c’, each horizontal line is an ‘H’.</p> "> Figure 4
<p>Grouping different ‘tiles’ of an eight-letter chain composed with ‘1’s’ (one tick up or down) and ‘0’s’ (no change). (<b>A</b>) shows the set of tiles for moves going up, and (<b>B</b>) shows the set of tiles for moves going down.</p> "> Figure 5
<p>Probability structure of Fibonacci <span class="html-italic">n</span>-letter chains, or classes, probability levels on the y-axis.</p> "> Figure 6
<p>Showing how the tile sets T{9} T{8} T{7} are related. Black squares indicate no price changes (‘H’ or ‘0’), and all other squares indicate a price change of one unit of price (‘1’).</p> ">
Abstract
:1. Introduction
2. Methods
3. Discussion
3.1. Bachelier and Methods of Vote Counting
3.2. Pathwise Is Not Path Dependency
3.3. The Price Approach
3.4. Alphabet and Words Describing Market Moves
- (i)
- The first price is noted a;
- (ii)
- Any ai superior to its predecessor would become Ui;
- (iii)
- Any ai inferior to its predecessor would become Di;
- (iv)
- Any H would remain a H.
3.5. Self-Similarities within the Compositions of Market Price Sequences
3.6. Consequence of Transitivity
- (i)
- The way market participants adjust price is said to be motivated by a change in the news flow that affects the matching of value (let us say intrinsic value at least) with price, which are supposed to be equal as most of the time as price reflects value, once more in the context of a set of shared market opinions;
- (ii)
- The dynamic of the voting process reinforces the presence of self-similarity during the progression of market price sequences, irrespective of their dimension [29];
- (iii)
- The reaction to these new informative elements should be self-similar, at least in the long run, as what causes the information to permeate into the market price is identical in nature: price does not reflect value anymore (or the opinions and beliefs that most of the market participants have of value);
- (iv)
- Therefore, price should follow similar paths within the self-similar multidimensional structure of tile classes that describe all possible courses of a price trajectory, all the different ways to approach an expected new ‘correct expected price’ or a new target price. This is also true when considering percolation models, described by logistic (Verhulst) functions;
- (v)
- Disregarding how they group in different sets, market participants are, once more, most of the time acting in a similar way when comparing actual prices with their expectations considering the news flow, and this process is mostly a continuous and stable one, here, without even necessitating any demanding qualification regarding the rationality of market participants [3]. It is the process which matters and which is self-similar, even considering the so-called ‘irrational behaviours’ of market participants. Identical configurations of voter expectations generate similar courses among the various tiling classes. In any case, path dependency is a necessary attribute of complex systems [30];
- (vi)
- Hence, the fact that the different dimensions of the tiling structure are related to each other by Fibonacci ratios, be it in absolute terms or in average terms, implies that the effects of price adjustments within market moves motivated by similar causes and are potentially well-described by a self-similar structure will tend to be homeomorphic to this self-similar structure itself;
- (vii)
- Hence, market moves tend to be related to each other by the same ratios as the one found between the different dimensions of the structure;
- (viii)
- Additionally, the greater the price trajectories—the longer they are, the more this will be true, as this tendency will show up even more consistently in larger dimensions. As the process at work is consistent, this structural self-similarity implies fractal characteristics on price trajectories, and the latter brings in an additional constraining factor, multidimensional coherence, which weighs on the way smaller price trajectories are built. Market phases tend to be proportionate to one another, not only between correctly selected increases and decreases but also between moves in similar directions;
- (ix)
- The presence of these ratios is not a necessity. It is rather, and exactly, a structural tendency.
- (i)
- What value, price and information are one to another, and in particular, the fact they are forming a complex system: price defines value as much as it measures it and is dependent on the information as much as it is part of it;
- (ii)
- That market participants can be largely gathered in more or less consistent groups in the function of their opinions (due to the access to uniformised media, the access to ‘efficient’ information, and the similarity of the valuing tools they use);
- (iii)
- That the structure of each tiling is forgiving in statistical terms, as the very particular profile of the probability structure of Fibonacci n-letter chains, as depicted in Figure 5, allows for approximations;
- (iv)
- That the reading of market price histories can be performed using a left or a right composition, which therefore allows greater leeway in the adequation of the actual move and its ‘structured’ profile.
4. Conclusions
Funding
Data Availability Statement
Conflicts of Interest
References
- Benhabib, J.; Liu, X.; Wang, P. Sentiments, financial markets, and macroeconomic fluctuations. J. Financ. Econ. 2016, 120, 420–443. [Google Scholar] [CrossRef]
- Lisciandra, C.; Korbmacher, J. Multiple models, one explanation. J. Econ. Methodol. 2021, 28, 186–206. [Google Scholar] [CrossRef]
- Gu, C.; Chen, D.; Stan, R. Investor sentiment and the market reaction to macroeconomic news. J. Futures Mark. 2021, 41, 1412–1426. [Google Scholar] [CrossRef]
- Albeverio, S.; Schachermayer, W.; Talagrand, M.; Bernard, P. (Eds.) Introduction: Bachelier’s Thesis from 1900. In Lectures on Probability Theory and Statistics, Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany, 2003; pp. 111–126. [Google Scholar] [CrossRef]
- Boldyrev, I.; Ushakov, A. Adjusting the model to adjust the world: Constructive mechanisms in postwar general equilibrium theory. J. Econ. Methodol. 2016, 23, 38–56. [Google Scholar] [CrossRef]
- Callado, A.A.C.; Leitão, C.R.S. Dynamics of Stock Prices and Market Efficiency. Int. Bus. Res. 2018, 11, 29. [Google Scholar] [CrossRef]
- Douady, S.; Couder, Y. Phyllotaxis as a Dynamical Self Organizing Process Part I: The Spiral Modes Resulting from Time-Periodic Iterations. J. Theor. Biol. 1996, 178, 255–273. [Google Scholar] [CrossRef]
- Douady, S.; Couder, Y. Phyllotaxis as a physical self-organized growth process. Phys. Rev. Lett. 1992, 68, 2098–2101. [Google Scholar] [CrossRef] [PubMed]
- Okabe, T. Evolutionary origins of Fibonacci phyllotaxis in land plants. Heliyon 2024, 10, e27812. [Google Scholar] [CrossRef]
- Rozin, B. Towards solving the mystery of spiral phyllotaxis. Prog. Biophys. Mol. Biol. 2023, 182, 8–14. [Google Scholar] [CrossRef]
- Allahyari Soeini, R.; Niroomand, A.; Kheyrmand Parizi, A. Using fibonacci numbers to forecast the stock market. Int. J. Manag. Sci. Eng. Manag. 2012, 7, 268–279. [Google Scholar] [CrossRef]
- Duan, H.; Xiao, X.; Yang, J.; Zeng, B. Elliott wave theory and the Fibonacci sequence-gray model and their application in Chinese stock market. J. Intell. Fuzzy Syst. 2018, 34, 1813–1825. [Google Scholar] [CrossRef]
- Gurrib, I.; Nourani, M.; Bhaskaran, R.K. Energy crypto currencies and leading U.S. energy stock prices: Are Fibonacci retracements profitable? Financ. Innov. 2022, 8, 8. [Google Scholar] [CrossRef]
- Tsinaslanidis, P.; Guijarro, F.; Voukelatos, N. Automatic identification and evaluation of Fibonacci retracements: Empirical evidence from three equity markets. Expert Syst. Appl. 2022, 187, 115893. [Google Scholar] [CrossRef]
- Bachelier, L. Théorie de la spéculation. Ann. Sci. École Norm. Sup. 1900, 17, 21–86. [Google Scholar] [CrossRef]
- Lim, C.; Zhang, W. Social opinion dynamics is not chaotic. Int. J. Mod. Phys. B 2016, 30, 1541006. [Google Scholar] [CrossRef]
- Vilela, A.L.M.; Wang, C.; Nelson, K.P.; Stanley, H.E. Majority-vote model for financial markets. Phys. A Stat. Mech. Its Appl. 2019, 515, 762–770. [Google Scholar] [CrossRef]
- Taqqu, M.S. Bachelier and his times: A conversation with Bernard Bru. Financ. Stochast 2001, 5, 3–32. [Google Scholar] [CrossRef]
- Poincaré, H. Sur le problème des trois corps. Bull. Astron. 1891, 8, 12–24. [Google Scholar] [CrossRef]
- Stavroglou, S.K.; Pantelous, A.A.; Stanley, H.E.; Zuev, K.M. Hidden interactions in financial markets. Proc. Natl. Acad. Sci. USA 2019, 116, 10646–10651. [Google Scholar] [CrossRef]
- Ureña, R.; Kou, G.; Dong, Y.; Chiclana, F.; Herrera-Viedma, E. A review on trust propagation and opinion dynamics in social networks and group decision making frameworks. Inf. Sci. 2019, 478, 461–475. [Google Scholar] [CrossRef]
- Connes, A. Géométrie non Commutative; InterEditions: Paris, France, 1990. [Google Scholar]
- Maloumian, N. Unaccounted forms of complexity: A path away from the efficient market hypothesis paradigm. Soc. Sci. Humanit. Open 2022, 5, 100244. [Google Scholar] [CrossRef]
- Fibonacci, L.; Sigler, L. Fibonacci’s Liber Abaci: A Translation into Modern English of Leonardo Pisano’s Book of Calculation; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2003. [Google Scholar]
- Cooper, J.M.; Hutchinson, D.S. Plato: Completed Works; Hackett: Indianapolis, USA, 1997; Available online: https://hackettpublishing.com/complete-works (accessed on 9 August 2024).
- Anderson, B.D.O.; Ye, M. Recent Advances in the Modelling and Analysis of Opinion Dynamics on Influence Networks. Int. J. Autom. Comput. 2019, 16, 129–149. [Google Scholar] [CrossRef]
- Dixsaut, M. Platon Le Désir de Comprendre; Vrin-Bibliothèque des Philosophies: Paris, France, 2003. [Google Scholar]
- Goles, E.; Olivos, J. Periodic behaviour of generalized threshold functions. Discret. Math. 1980, 30, 187–189. [Google Scholar] [CrossRef]
- Song, C.; Havlin, S.; Makse, H.A. Self-similarity of complex networks. Nature 2005, 433, 392–395. [Google Scholar] [CrossRef] [PubMed]
- Polhill, J.G.; Hare, M.; Bauermann, T.; Anzola, D.; Palmer, E.; Salt, D.; Antosz, P. Using Agent-Based Models for Prediction in Complex and Wicked Systems. J. Artif. Soc. Soc. Simul. 2021, 24, 2. [Google Scholar] [CrossRef]
- Herzog, L. Are Financial Markets Epistemically Efficient? In The Philosophy of Money and Finance; Sandberg, J., Warenski, L., Eds.; Oxford University Press: Oxford, UK, 2024; pp. 91–110. [Google Scholar] [CrossRef]
- Boualem, A.; Fouchal, H.; Ayaida, M.; De Runz, C. Fibonacci tiles strategy for optimal coverage in IoT networks. Ann. Telecommun. 2022, 77, 331–344. [Google Scholar] [CrossRef]
- Grassi, L.B.; Marins, D.P.A.; Paim, J.F.; Palaoro, L.B.; Segatto, M.E.V.; Paiva, M.H.M. Fibonacci solar tree: Mathematical modeling of the solar incidence as a function of its constructive variables. Renew. Energy 2024, 229, 120646. [Google Scholar] [CrossRef]
- Tatabhatla, V.M.R.; Agarwal, V.; Agarwal, A.; Singh, A.K. Reduced partial shading effect and enhancement of performance metrics using a Fibonacci based algorithm. Int. J. Numer. Model. 2024, 37, e3242. [Google Scholar] [CrossRef]
n | Pascal’s Trade Chains | Fibonacci n-Letter Chains | Extension (Fibonacci n-Letter Chains in Unit Points) | Norm |
---|---|---|---|---|
1 | 1, 1 | 1 | 1 | 1 |
2 | 1, 2, 1 | 1, 1 | 2, 1 | 3 |
3 | 1, 3, 3, 1 | 1, 2 | 3, 4 | 7 |
4 | 1, 4, 6, 4, 1 | 1, 3, 1 | 4, 9, 2 | 15 |
5 | 1, 5, 10, 10, 5, 1 | 1, 4, 3 | 5, 16, 9 | 30 |
6 | 1, 6, 15, 20, 15, 6, 1 | 1, 5, 6, 1 | 6, 25, 24, 3 | 58 |
7 | 1, 7, 21, 35, 35, 21, 7, 1 | 1, 6, 10, 4 | 7, 36, 50, 16 | 109 |
8 | 1, 8, 28, 56, 70, 56, 28, 8, 1 | 1, 7, 15, 10, 1 | 8, 49, 90, 50, 4 | 201 |
9 | 1, 9, 36, 84, 126, 126, 84, 36, 9, 1 | 1, 8, 21, 20, 5 | 9, 64, 147, 120, 25 | 365 |
10 | 1, 10, 45, 120, 210, 252, 210, … | 1, 9, 28, 35, 15, 1 | 10, 81, 224, 245, 90, 5 | 655 |
11 | 1, 11, 55, 165, 330, 462, 462, … | 1, 10, 36, 56, 35, 6 | … | … |
12 | 1, 12, 66, 220, 495, 792, 924, … | 1, 11, 45, 84, 70, 21, 1 | … | … |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Maloumian, N. Wandering Drunkards Walk after Fibonacci Rabbits: How the Presence of Shared Market Opinions Modifies the Outcome of Uncertainty. Entropy 2024, 26, 686. https://doi.org/10.3390/e26080686
Maloumian N. Wandering Drunkards Walk after Fibonacci Rabbits: How the Presence of Shared Market Opinions Modifies the Outcome of Uncertainty. Entropy. 2024; 26(8):686. https://doi.org/10.3390/e26080686
Chicago/Turabian StyleMaloumian, Nicolas. 2024. "Wandering Drunkards Walk after Fibonacci Rabbits: How the Presence of Shared Market Opinions Modifies the Outcome of Uncertainty" Entropy 26, no. 8: 686. https://doi.org/10.3390/e26080686