American Institute of Mathematical Sciences

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doi: 10.3934/dcdsb.2021117

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

Anastasiia Panchuk 1,, and  Frank Westerhoff 2,

1. 

Institute of Mathematics NASU, 3 Tereshchenkivska Str., 01601 Kyiv, Ukraine
2. 

Department of Economics, University of Bamberg, Feldkirchenstrasse 21, 96045 Bamberg, Germany

* Corresponding author: anastasiia.panchuk@gmail.com

Received  October 2020 Revised  February 2021 Published  April 2021

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.

Keywords: Financial markets, asset price dynamics, chartists and fundamentalists, multiband chaotic attractors, contact bifurcation for critical points, bandcount accretion bifurcation structure.
Mathematics Subject Classification: Primary: 37E05, 37G35; Secondary: 37N40, 39A33.
Citation: Anastasiia Panchuk, Frank Westerhoff. Speculative behavior and chaotic asset price dynamics: On the emergence of a bandcount accretion bifurcation structure. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021117
References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

A. Beja and M. Goldman, On the dynamic behaviour of prices in disequilibrium, Journal of Finance, 34 (1980), 235-247.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

C. Chiarella, The dynamics of speculative behaviour, Annals of Operations Research, 37 (1992), 101-123.  doi: 10.1007/BF02071051.  Google Scholar

[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.  Google Scholar

[9]

P. De Grauwe, H. Dewachter and M. Embrechts, Exchange Rate Theory: Chaotic Models of Foreign Exchange Markets, Blackwell, Oxford, 1993. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[13]

J. K. Galbraith, A Short History of Financial Euphoria, Penguin Books, London, 1994. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[17]

F. Hilker and F. Westerhoff, Preventing extinction and mass outbreaks in irregularly fluctuating populations, American Naturalist, 170 (2007), 232-241.   Google Scholar

[18] C. H. Hommes, Behavioral Rationality and Heterogeneous Expectations in Complex Economic Systems, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9781139094276.  Google Scholar
[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.   Google Scholar
[20]

W. Huang and H. Zheng, Financial crisis and regime-dependent dynamics, J. Econ. Behav. Organization, 82 (2012), 445-461.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[23]

J. Jungeilges, E. Maklakova and T. Perevalova, Stochastic sensitivity of bull and bear states, J. Econ. Interact. Coord. (2021, forthcoming). Google Scholar

[24]

C. Kindleberger and R. Aliber, Manias, Panics, and Crashes: A History of Financial Crises, Wiley, New Jersey, 2011. doi: 10.1057/9780230536753.  Google Scholar

[25]

T. Lux, Herd behaviour, bubbles and crashes, Economic Journal, 105 (1995), 881-896.  doi: 10.2307/2235156.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30] R. Shiller, Irrational Exuberance, Princeton University Press, Princeton, 2015.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

F. Westerhoff, Samuelson's multiplier-accelerator model revisited, Applied Economics Letters, 13 (2006), 89-92.  doi: 10.1080/13504850500390663.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

A. Beja and M. Goldman, On the dynamic behaviour of prices in disequilibrium, Journal of Finance, 34 (1980), 235-247.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

C. Chiarella, The dynamics of speculative behaviour, Annals of Operations Research, 37 (1992), 101-123.  doi: 10.1007/BF02071051.  Google Scholar

[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.  Google Scholar

[9]

P. De Grauwe, H. Dewachter and M. Embrechts, Exchange Rate Theory: Chaotic Models of Foreign Exchange Markets, Blackwell, Oxford, 1993. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[13]

J. K. Galbraith, A Short History of Financial Euphoria, Penguin Books, London, 1994. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[17]

F. Hilker and F. Westerhoff, Preventing extinction and mass outbreaks in irregularly fluctuating populations, American Naturalist, 170 (2007), 232-241.   Google Scholar

[18] C. H. Hommes, Behavioral Rationality and Heterogeneous Expectations in Complex Economic Systems, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9781139094276.  Google Scholar
[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.   Google Scholar
[20]

W. Huang and H. Zheng, Financial crisis and regime-dependent dynamics, J. Econ. Behav. Organization, 82 (2012), 445-461.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[23]

J. Jungeilges, E. Maklakova and T. Perevalova, Stochastic sensitivity of bull and bear states, J. Econ. Interact. Coord. (2021, forthcoming). Google Scholar

[24]

C. Kindleberger and R. Aliber, Manias, Panics, and Crashes: A History of Financial Crises, Wiley, New Jersey, 2011. doi: 10.1057/9780230536753.  Google Scholar

[25]

T. Lux, Herd behaviour, bubbles and crashes, Economic Journal, 105 (1995), 881-896.  doi: 10.2307/2235156.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30] R. Shiller, Irrational Exuberance, Princeton University Press, Princeton, 2015.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

F. Westerhoff, Samuelson's multiplier-accelerator model revisited, Applied Economics Letters, 13 (2006), 89-92.  doi: 10.1080/13504850500390663.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

Figure 1.  (a) Schematic representation of the $ ( \varepsilon, \mu) $ parameter plane; (b)–(g) Sample plots of function $ f $ and its absorbing intervals
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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)
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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
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Figure 4.  Plot of function $ f $ and formation of gaps in the upper part $ B^R $ of the chaotic attractor $ \mathcal{Q} $
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Figure 5.  Plot of function $ f $ and formation of gaps in the lower part $ B^L $ of the chaotic attractor $ \mathcal{Q} $
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$ \mathcal{C}_{2+1} $. (b) Close up of the rectangular area marked in (a)">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} $. (b) Close up of the rectangular area marked in (a)
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$ \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 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
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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
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Figure 9.  Plot of function $ f $, with $ a = 1.25 $, $ \varepsilon = 1.37 $ and $ \mu = \tilde{\mu} = -3.454 $. Pink and dark-red lines show the related sequences of preimages of zero
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$ \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 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
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$ \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 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
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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 $
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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)
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