Bifurcation analysis of a single-group asset flow model
Authors:
H. Merdan, G. Caginalp and W. C. Troy
Journal:
Quart. Appl. Math. 74 (2016), 275-296
MSC (2010):
Primary 91B25, 91B26, 91B50, 34D20, 37G15; Secondary 91G80, 91G99, 34C60, 37N40
DOI:
https://doi.org/10.1090/qam/1418
Published electronically:
March 21, 2016
MathSciNet review:
3505604
Full-text PDF Free Access
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Abstract: We study the stability and Hopf bifurcation analysis of an asset pricing model that is based on the model introduced by Caginalp and Balenovich, under the assumption of a fixed amount of cash and stock in the system. First, we analyze stability of equilibrium points. Choosing the momentum coefficient as a bifurcation parameter, we also show that Hopf bifurcation occurs when the bifurcation parameter passes through a critical value. Analytical results are supported by numerical simulations. A key conclusion for economics and finance is the existence of periodic solutions in the absence of exogenous factors for an interval of the bifurcation parameter, which is the trend-based (or momentum) coefficient.
References
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- Drew Fudenberg and Jean Tirole, Game theory, MIT Press, Cambridge, MA, 1991. MR 1124618
- Jack K. Hale and Hüseyin Koçak, Dynamics and bifurcations, Texts in Applied Mathematics, vol. 3, Springer-Verlag, New York, 1991. MR 1138981, DOI 10.1007/978-1-4612-4426-4
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- D. Kahneman and A. Tversky, Prospect theory: an analysis of decision making under risk, Econometrica 47 (1979), 263–291.
- L. Lopes, Between hope and fear: the psychology of risk, Adv. Exp. Soc. Psychol. 20 (1987), 255–295.
- Xiangao Li, Shigui Ruan, and Junjie Wei, Stability and bifurcation in delay-differential equations with two delays, J. Math. Anal. Appl. 236 (1999), no. 2, 254–280. MR 1704582, DOI 10.1006/jmaa.1999.6418
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- R. May, Stability and Complexity in Model Ecosystem, Princeton University Press, Princeton, NJ, 1973.
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- H. Merdan and H. Cakmak, Liquidity effect on the asset price forecasting, Journal of Nonlinear Systems and Applications (2012), 82-87.
- Hande Akkocaoğlu, Hüseyin Merdan, and Canan Çelik, Hopf bifurcation analysis of a general non-linear differential equation with delay, J. Comput. Appl. Math. 237 (2013), no. 1, 565–575. MR 2966929, DOI 10.1016/j.cam.2012.06.029
- J. M. Poterba and L. H. Summers, Mean reversion in stock prices: Evidence and implications, Journal of Financial Economics 22 (1988), 27-59.
- H. Shefrin, A Behavioral Approach to Asset Pricing, Elsevier, London, 2005.
- H. Shefrin and M. Statman, The disposition to sell winners too early and ride losers too long: Theory and evidence, Journal of Finance 40 (1985), 777-790.
- V. L. Smith, G. L. Suchanek and A. W. Williams, Bubbles, crashes and endogenous expectations in experimental spot asset markets, Econometrica 56 (1988), 1119-1151.
- A. Tversky and D. Kahneman, Judgment under uncertainty: Heuristics and biases Science 185 (1974), 1109-1186.
- Radouane Yafia, Hopf bifurcation in differential equations with delay for tumor-immune system competition model, SIAM J. Appl. Math. 67 (2007), no. 6, 1693–1703. MR 2350003, DOI 10.1137/060657947
- D. S. Watson and M. Getz, Price Theory and Its Uses, 5th edition, University Press of America, Lanham, MD, 1993.
- P. Wilmott, Paul Wilmott Introduces Quantitative Finance, John Wiley & Sons, 2007.
References
- L. J. S. Allen, An Introduction to Mathematical Biology, Pearson/Prentice Hall, 2007.
- J. Bélair and S. A. Campbell, Stability and bifurcations of equilibria in a multiple-delayed differential equation, SIAM J. Appl. Math. 54 (1994), no. 5, 1402–1424. MR 1293106 (95j:34100), DOI 10.1137/S0036139993248853
- Z. Bodie, A. Kane and A. J. Marcus, Investments, 7th edition, McGraw-Hill Education, Boston, 2008.
- G. Caginalp and D. Balenovich, Market oscillations induced by the competition between value-based and trend-based investment strategies, Appl. Math. Finance 1 (1994), 29–164.
- G. Caginalp and D. Balenovich, Asset flow and momentum: deterministic and stochastic equations, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 357 (1999), no. 1758, 2119–2133. MR 1712407 (2000e:91067), DOI 10.1098/rsta.1999.0421
- G. Caginalp and G. B. Ermentrout, A kinetic thermodynamics approach to the psychology of fluctuations in financial markets, Appl. Math. Lett. 3 (1990), 17-19.
- G. Caginalp and G. B. Ermentrout, Numerical studies of differential equations related to theoretical financial markets, Appl. Math. Lett. 4 (1991), no. 1, 35–38. MR 1088032, DOI 10.1016/0893-9659(91)90118-F
- G. Caginalp and M. DeSantis, Nonlinearity in the dynamics of financial markets, Nonlinear Anal. Real World Appl. 12 (2011), no. 2, 1140–1151. MR 2736200 (2011i:91155), DOI 10.1016/j.nonrwa.2010.09.008
- G. Caginalp and M. DeSantis, Multi-group asset flow equations and stability, Discrete Contin. Dyn. Syst. Ser. B 16 (2011), no. 1, 109–150. MR 2799544 (2012c:91257), DOI 10.3934/dcdsb.2011.16.109
- G. Caginalp and M. Desantis, Stock price dynamics: nonlinear trend, volume, volatility, resistance and money supply, Quant. Finance 11 (2011), no. 6, 849–861. MR 2806968 (2012e:91315), DOI 10.1080/14697680903220356
- G. Caginalp and H. Merdan, Asset price dynamics with heterogeneous groups, Phys. D 225 (2007), no. 1, 43–54. MR 2304913 (2007k:91130), DOI 10.1016/j.physd.2006.09.036
- G. Caginalp, D. Porter, and V. Smith, Initial cash/asset ratio and asset prices: an experimental study, Proc. Natl. Acad. Sci. USA 95 (1998), no. 2, 756–761 (electronic). MR 1613147, DOI 10.1073/pnas.95.2.756
- K. D. Daniel, D. Hirshleifer, and A. Subrahmanyam, Investor psychology and security market under and overreaction, Journal of Finance 53 (1998), 1839-1885.
- M. DeSantis, D. Swigon, and G. Caginalp, Nonlinear dynamics and stability in a multigroup asset flow model, SIAM J. Appl. Dyn. Syst. 11 (2012), no. 3, 1114–1148. MR 3022062, DOI 10.1137/120862211
- R. C. Dorf and R. H. Bishop, Modern Control Systems, 11th edition, Pearson Prentice-Hall, Upper Saddle River, NJ, 2008.
- A. Duran, Stability analysis of asset flow differential equations, Appl. Math. Lett. 24 (2011), no. 4, 471–477. MR 2749729, DOI 10.1016/j.aml.2010.10.044
- R. Frisch and H. Holme, The characteristic solutions of a mixed difference and differential equation occurring in economic dynamics, Econometrica 3 (1935), 225–239.
- D. Fudenberg and J. Tirole, Game theory, MIT Press, Cambridge, MA, 1991. MR 1124618 (92m:90001)
- J. K. Hale and H. Koçak, Dynamics and bifurcations, Texts in Applied Mathematics, vol. 3, Springer-Verlag, New York, 1991. MR 1138981 (93e:58047), DOI 10.1007/978-1-4612-4426-4
- J. M. Henderson and R. E. Quant, Microeconomic Theory, A Mathematical Approach, 3rd edition, McGraw-Hill, 1980.
- N. Kaldor, A classificatory note on the determinateness of equilibrium, The Review of Economic Studies 1 (1934), 122-136.
- D. Kahneman and A. Tversky, Prospect theory: an analysis of decision making under risk, Econometrica 47 (1979), 263–291.
- L. Lopes, Between hope and fear: the psychology of risk, Adv. Exp. Soc. Psychol. 20 (1987), 255–295.
- X. Li, S. Ruan, and J. Wei, Stability and bifurcation in delay-differential equations with two delays, J. Math. Anal. Appl. 236 (1999), no. 2, 254–280. MR 1704582 (2001a:34131), DOI 10.1006/jmaa.1999.6418
- M. Mackey and L. Glass, Oscillations and chaos in physiological control systems, Science 197 (1977), 287–289.
- R. May, Stability and Complexity in Model Ecosystem, Princeton University Press, Princeton, NJ, 1973.
- H. Merdan and M. Alisen, A mathematical model for asset pricing, Appl. Math. Comput. 218 (2011), no. 4, 1449–1456. MR 2831653, DOI 10.1016/j.amc.2011.06.028
- H. Merdan and H. Cakmak, Liquidity effect on the asset price forecasting, Journal of Nonlinear Systems and Applications (2012), 82-87.
- H. Akkocaoğlu, H. Merdan, and C. Çelik, Hopf bifurcation analysis of a general non-linear differential equation with delay, J. Comput. Appl. Math. 237 (2013), no. 1, 565–575. MR 2966929, DOI 10.1016/j.cam.2012.06.029
- J. M. Poterba and L. H. Summers, Mean reversion in stock prices: Evidence and implications, Journal of Financial Economics 22 (1988), 27-59.
- H. Shefrin, A Behavioral Approach to Asset Pricing, Elsevier, London, 2005.
- H. Shefrin and M. Statman, The disposition to sell winners too early and ride losers too long: Theory and evidence, Journal of Finance 40 (1985), 777-790.
- V. L. Smith, G. L. Suchanek and A. W. Williams, Bubbles, crashes and endogenous expectations in experimental spot asset markets, Econometrica 56 (1988), 1119-1151.
- A. Tversky and D. Kahneman, Judgment under uncertainty: Heuristics and biases Science 185 (1974), 1109-1186.
- R. Yafia, Hopf bifurcation in differential equations with delay for tumor-immune system competition model, SIAM J. Appl. Math. 67 (2007), no. 6, 1693–1703 (electronic). MR 2350003, DOI 10.1137/060657947
- D. S. Watson and M. Getz, Price Theory and Its Uses, 5th edition, University Press of America, Lanham, MD, 1993.
- P. Wilmott, Paul Wilmott Introduces Quantitative Finance, John Wiley & Sons, 2007.
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Additional Information
H. Merdan
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 — and — Department of Mathematics, TOBB University of Economics and Technology, 06560-Ankara, Turkey
Address at time of publication:
Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
MR Author ID:
729206
Email:
merdan@pitt.edu, merdan@etu.edu.tr
G. Caginalp
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email:
caginalp@pitt.edu
W. C. Troy
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email:
troy@math.pitt.edu
Keywords:
Asset price dynamics,
stability of price dynamics,
Hopf bifurcation,
price trend,
momentum,
market dynamics,
liquidity,
periodic solutions
Received by editor(s):
August 15, 2014
Published electronically:
March 21, 2016
Additional Notes:
The first author was supported by TUBITAK (The Scientific and Technological Research Council of Turkey)
Article copyright:
© Copyright 2016
Brown University