Article Contents
Article Contents

# Speculative behavior and chaotic asset price dynamics: On the emergence of a bandcount accretion bifurcation structure

• * Corresponding author: anastasiia.panchuk@gmail.com
• We study a simple financial market model with interacting chartists and fundamentalists that may give rise to multiband chaotic attractors. In particular, asset prices fluctuate erratically around their fundamental values, displaying a significant bull and bear market behavior. An in-depth analytical and numerical study of our model furthermore reveals the emergence of a new bifurcation structure, a phenomenon that we call a bandcount accretion bifurcation structure. The latter consists of regions associated with chaotic dynamics only, the boundaries of which are not defined by homoclinic bifurcations, but mainly by contact bifurcations of particular type where two distinct critical points of certain ranks coincide.

Mathematics Subject Classification: Primary: 37E05, 37G35; Secondary: 37N40, 39A33.

 Citation:

• Figure 1.  (a) Schematic representation of the $( \varepsilon, \mu)$ parameter plane; (b)–(g) Sample plots of function $f$ and its absorbing intervals

Figure 2.  (a) 2D bifurcation diagram in the $( \varepsilon, \mu)$ parameter plane of $f$, with $a = 1.25$. (b) Close up of the rectangular area marked in (a)

Figure 3.  1D bifurcation diagram corresponding to the blue arrow in Figure 2(a), with $\varepsilon = 0.72$ and $a = 1.25$. The critical points $\ell_{{i}}$, $\mathit{m}^{-}_{{i}}$, $\mathit{m}^{+}_{{i}}$ and $\mathit{r}_{{i}}$ of different ranks are shown by blue, green, orange and red lines, respectively

Figure 4.  Plot of function $f$ and formation of gaps in the upper part $B^R$ of the chaotic attractor $\mathcal{Q}$

Figure 5.  Plot of function $f$ and formation of gaps in the lower part $B^L$ of the chaotic attractor $\mathcal{Q}$

Figure 6.  (a) 2D bifurcation diagram in the $( \varepsilon, \mu)$ parameter plane of $f$, with $a = 1.25$, related to a triangular shaped "jut" of chaoticity region $\mathcal{C}_{2+1}$. (b) Close up of the rectangular area marked in (a)

Figure 7.  1D bifurcation diagram corresponding to the blue arrow marked "1" in Figure 6(b), with $\varepsilon = 1.37$ and $a = 1.25$. The critical points $\ell_{{i}}$, $\mathit{m}^{-}_{{i}}$, $\mathit{m}^{+}_{{i}}$ and $\mathit{r}_{{i}}$ of different ranks are shown by blue, green, orange and red lines, respectively

Figure 8.  Plot of function $f$, with $a = 1.25$, $\varepsilon = 1.37$ and $\mu = \tilde{\mu} = -3.44$. Pink and dark-red lines show the related sequences of preimages of zero

Figure 9.  Plot of function $f$, with $a = 1.25$, $\varepsilon = 1.37$ and $\mu = \tilde{\mu} = -3.454$. Pink and dark-red lines show the related sequences of preimages of zero

Figure 10.  1D bifurcation diagram corresponding to the blue arrow marked "2" in Figure 6(b), with $\varepsilon = 1.34$ and $a = 1.25$. The critical points $\ell_{{i}}$, $\mathit{m}^{-}_{{i}}$, $\mathit{m}^{+}_{{i}}$ and $\mathit{r}_{{i}}$ of different ranks are shown by blue, green, orange and red lines, respectively

Figure 11.  1D bifurcation diagram corresponding to the blue arrow marked "3" in Figure 6(b), with $\varepsilon = 1.32$ and $a = 1.25$. The critical points $\ell_{{i}}$, $\mathit{m}^{-}_{{i}}$, $\mathit{m}^{+}_{{i}}$ and $\mathit{r}_{{i}}$ of different ranks are shown by blue, green, orange and red lines, respectively

Figure 12.  The functioning of the financial market model. The panel depicts the dynamics of Figure 9 in the time domain. The parameters are $\varepsilon = 1.37$, $a = 1.25$, and $\mu = -3.454$

Figure 13.  (a) 2D bifurcation diagram in the $( \varepsilon, \mu)$ parameter plane of $f$, with $a = 1.1$. (b) Close up of the parallelogram area marked by the dark-red line in (a)

•  [1] M. Anufriev and J. Tuinstra, The impact of short-selling constraints on financial market stability in a heterogeneous agents model, J. Econ. Dyn. Control, 37 (2013), 1523-1543.  doi: 10.1016/j.jedc.2013.04.015. [2] V. Avrutin, L. Gardini, M. Schanz and I. Sushko, Bifurcations of chaotic atttractors in one-dimensional piecewise smooth maps, Int. J. Bif. Chaos, 24 (2014), 1440012, 10pp. doi: 10.1142/S0218127414400124. [3] V. Avrutin, L. Gardini, I. Sushko and F. Tramontana, Continuous and Discontinuous Piecewise-smooth One-Dimensional Maps: Invariant Sets and Bifurcation Structures, World Scientific, Singapore, 2019. doi: 10.1142/8285. [4] A. Beja and M. Goldman, On the dynamic behaviour of prices in disequilibrium, Journal of Finance, 34 (1980), 235-247. [5] W. Brock, C. Hommes and F. Wagener, More hedging instruments may destabilize markets, J. Econ. Dyn. Control, 33 (2009), 1912-1928.  doi: 10.1016/j.jedc.2009.05.004. [6] W. A. Brock and C. H. Hommes, Heterogeneous beliefs and routes to chaos in a simple asset pricing model, J. Econ. Dyn. Control, 22 (1998), 1235-1274.  doi: 10.1016/S0165-1889(98)00011-6. [7] C. Chiarella, The dynamics of speculative behaviour, Annals of Operations Research, 37 (1992), 101-123.  doi: 10.1007/BF02071051. [8] R. H. Day and W. Huang, Bulls, bears and market sheep, J. Econ. Behav. Organization, 14 (1990), 299-329.  doi: 10.1016/0167-2681(90)90061-H. [9] P. De Grauwe, H. Dewachter and M. Embrechts, Exchange Rate Theory: Chaotic Models of Foreign Exchange Markets, Blackwell, Oxford, 1993. [10] F. Dercole and D. Radi, Does the "uptick rule" stabilize the stock market? insights from adaptive rational equilibrium dynamics, Chaos Solitons Fract., 130 (2020), 109426, 19pp. doi: 10.1016/j. chaos. 2019.109426. [11] R. Dieci and X. -Z. He, Heterogeneous agent models in finance, in Handbook of Computational Economics: Heterogeneous Agent Modeling (eds. C. Hommes and B. LeBaron), North-Holland, Amsterdam, 2018, 257–328. [12] D. Farmer and S. Joshi, The price dynamics of common trading strategies, J. Econ. Behav. Organization, 49 (2002), 149-171.  doi: 10.1016/S0167-2681(02)00065-3. [13] J. K. Galbraith, A Short History of Financial Euphoria, Penguin Books, London, 1994. [14] L. Gardini, D. Radi, N. Schmitt, I. Sushko and F. Westerhoff, Currency Manipulation and Currency Wars: Analyzing the Dynamics of Competitive Central Bank Interventions, Mimeo, University of Urbino, 2020. [15] L. Gardini, D. Radi, N. Schmitt, I. Sushko and F. Westerhoff, Exchange Rate Dynamics and Central Bank Interventions: On the (de)Stabilizing Nature of Targeting Long-run Fundamentals Interventions, Mimeo, University of Urbino, 2020. [16] X.-Z. He and F. Westerhoff, Commodity markets, price limiters and speculative price dynamics, J. Econ. Dyn. Control, 29 (2005), 1577-1596.  doi: 10.1016/j.jedc.2004.09.003. [17] F. Hilker and F. Westerhoff, Preventing extinction and mass outbreaks in irregularly fluctuating populations, American Naturalist, 170 (2007), 232-241. [18] C. H. Hommes,  Behavioral Rationality and Heterogeneous Expectations in Complex Economic Systems, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9781139094276. [19] W. Huang and  R. H. Day,  Chaotically switching bear and bull markets: The derivation of stock price distributions from behavioral rules, in Nonlinear Dynamics and Evolutionary Economics (eds. R. H. Day and P. Chen), Oxford University Press, Oxford, 1993. [20] W. Huang and H. Zheng, Financial crisis and regime-dependent dynamics, J. Econ. Behav. Organization, 82 (2012), 445-461. [21] W. Huang, H. Zheng and W.-M. Chia, Financial crises and interacting heterogeneous agents, J. Econ. Dyn. Control, 34 (2010), 1105-1122.  doi: 10.1016/j.jedc.2010.01.013. [22] J. Jungeilges, E. Maklakova and T. Perevalova, Asset price dynamics in a "bull and bear market", Struct. Change Econ. Dynamics, 56 (2021), 117-128. [23] J. Jungeilges, E. Maklakova and T. Perevalova, Stochastic sensitivity of bull and bear states, J. Econ. Interact. Coord. (2021, forthcoming). [24] C. Kindleberger and R. Aliber, Manias, Panics, and Crashes: A History of Financial Crises, Wiley, New Jersey, 2011. doi: 10.1057/9780230536753. [25] T. Lux, Herd behaviour, bubbles and crashes, Economic Journal, 105 (1995), 881-896.  doi: 10.2307/2235156. [26] A. Panchuk, I. Sushko and F. Westerhoff, A financial market model with two discontinuities: Bifurcation structures in the chaotic domain, Chaos, 28 (2018), 055908, 21pp. doi: 10.1063/1.5024382. [27] N. Schmitt, J. Tuinstra and F. Westerhoff, Side effects of nonlinear profit taxes in a behavioral market entry model: abrupt changes, coexisting attractors and hysteresis problems, J. Econ. Behav. Organization, 135 (2017), 15-38. [28] N. Schmitt and F. Westerhoff, On the bimodality of the distribution of the S & P 500's distortion: Empirical evidence and theoretical explanations, J. Econ. Dyn. Control, 80 (2017), 34-53.  doi: 10.1016/j.jedc.2017.05.002. [29] J. Segura, F. Hilker and D. Franco, Degenerate period adding bifurcation structure of one-dimensional bimodal piecewise linear maps, SIAM Journal on Applied Mathematics, 80 (2020), 1356-1376.  doi: 10.1137/19M1251023. [30] R. Shiller,  Irrational Exuberance, Princeton University Press, Princeton, 2015. [31] I. Sushko and L. Gardini, Degenerate bifurcations and border collisions in piecewise smooth 1D and 2D maps, Int. J. Bif. Chaos, 20 (2010), 2045-2070.  doi: 10.1142/S0218127410026927. [32] I. Sushko, L. Gardini and V. Avrutin, Nonsmooth one-dimensional maps: Some basic concepts and definitions, J. Differ. Equ. Appl., 22 (2016), 1816-1870.  doi: 10.1080/10236198.2016.1248426. [33] I. Sushko, F. Tramontana, F. Westerhoff and V. Avrutin, Symmetry breaking in a bull and bear financial market model, Chaos Solitons Fract., 79 (2015), 57-72.  doi: 10.1016/j.chaos.2015.03.013. [34] F. Tramontana, F. Westerhoff and L. Gardini, The bull and bear market model of Huang and Day: Some extensions and new results, J. Econ. Dyn. Control, 37 (2013), 2351-2370.  doi: 10.1016/j.jedc.2013.06.005. [35] F. Westerhoff, Samuelson's multiplier-accelerator model revisited, Applied Economics Letters, 13 (2006), 89-92.  doi: 10.1080/13504850500390663. [36] F. Westerhoff and  R. Franke,  Agent-based models for economic policy design: Two illustrative examples, in The Oxford Handbook of Computational Economics and Finance (eds. S.-H. Chen, M. Kaboudan and Y.-R. Du), Oxford University Press, Oxford, 2018. [37] F. Westerhoff and R. Dieci, The effectiveness of Keynes-Tobin transaction taxes when heterogeneous agents can trade in different markets: A behavioral finance approach, J. Econ. Dyn. Control, 30 (2006), 293-322.  doi: 10.1016/j.jedc.2004.12.004. [38] E. Zeeman, On the unstable behaviour of stock exchanges, Journal of Mathematical Economics, 1 (1974), 39-49.  doi: 10.1016/0304-4068(74)90034-2.

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