Bifurcation analysis of a single-group asset flow model

Merdan, H.; Caginalp, G.; Troy, W.

doi:10.1090/qam/1418

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

Abstract | References | Similar Articles | Additional Information

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

L. J. S. Allen, An Introduction to Mathematical Biology, Pearson/Prentice Hall, 2007.
Jacques Bélair and Sue Ann Campbell, Stability and bifurcations of equilibria in a multiple-delayed differential equation, SIAM J. Appl. Math. 54 (1994), no. 5, 1402–1424. MR 1293106, 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, 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
Gunduz Caginalp and Mark DeSantis, Nonlinearity in the dynamics of financial markets, Nonlinear Anal. Real World Appl. 12 (2011), no. 2, 1140–1151. MR 2736200, DOI 10.1016/j.nonrwa.2010.09.008
Gunduz Caginalp and Mark DeSantis, Multi-group asset flow equations and stability, Discrete Contin. Dyn. Syst. Ser. B 16 (2011), no. 1, 109–150. MR 2799544, 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, 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, DOI 10.1016/j.physd.2006.09.036
Gunduz Caginalp, David Porter, and Vernon Smith, Initial cash/asset ratio and asset prices: an experimental study, Proc. Natl. Acad. Sci. USA 95 (1998), no. 2, 756–761. 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.
Ahmet 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.
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
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.
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
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.
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.