Multi-group asset flow equations and stability

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July 2011, 16(1): 109-150. doi: 10.3934/dcdsb.2011.16.109

Multi-group asset flow equations and stability

Gunduz Caginalp ^1, and Mark DeSantis ^2,

1.	Mathematics Department, University of Pittsburgh, Pittsburgh, PA 15260, United States
2.	University of Pittsburgh, Department of Mathematics 301 Thackeray Hall, Pittsburgh, PA 15260, United States

Received April 2010 Revised June 2010 Published April 2011

Abstract
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We consider a two-group asset flow model of a financial instrument with one group focused on price trend, the other on value. We prove the existence of both stable and unstable regions for the system of differential equations and show that a strong motivation based on (particularly recent) price trend is associated with instability. Numerical computations using a set of typical parameters describe precise regions of stability and instability. A precise limiting connection between the discrete and differential equations is also established.

Keywords: Underreaction, overreaction, efficient market, market dynamics., finite assets.

Mathematics Subject Classification: Primary: 91G99, 34D20, 91B25, 91B26, 91B5.

Citation: Gunduz Caginalp, Mark DeSantis. Multi-group asset flow equations and stability. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 109-150. doi: 10.3934/dcdsb.2011.16.109

References:

[1]	N. Barberis, A. Shleifer and R. Vishny, A model of investor sentiment, Journal of Financial Economics, 49 (1998), 307-343. doi: 10.1016/S0304-405X(98)00027-0.
[2]	Z. Bodie, A. Kane and A. J. Marcus, "Investments," 7^th edition, McGraw-Hill Education, Boston, 2008.
[3]	G. Caginalp and B. Ermentrout, A kinetic thermodynamics approach to the psychology of fluctuations in financial markets, Applied Mathematics Letters (4), 3 (1990), 17-19. doi: 10.1016/0893-9659(90)90038-D.
[4]	G. Caginalp and D. Balenovich, Asset flow and momentum: Deterministic and stochastic equations, Philosophical Transactions of the Royal Society, 357 (1999), 2119-2133. doi: 10.1098/rsta.1999.0421.
[5]	G. Caginalp and M. DeSantis, Stock price dynamics: Nonlinear trend, volume, volatility, resistance and money supply, Quantitative Finance, to appear.
[6]	G. Caginalp and V. Ilieva, The dynamics of trader motivations in asset bubbles, Journal of Economic Behavior & Organization, 66 (2008), 641-656. doi: 10.1016/j.jebo.2006.01.011.
[7]	G. Caginalp and H. Merdan, Asset price dynamics with heterogeneous groups, Physica D: Nonlinear Phenomena, 225 (2007), 43-54. doi: 10.1016/j.physd.2006.09.036.
[8]	G. Caginalp, D. Porter and V. L. Smith, Initial cash/asset ratio and asset prices: An experimental study, Proceedings of the National Academy of Sciences, 95 (1998), 756-761. doi: 10.1073/pnas.95.2.756.
[9]	K. D. Daniel, D. Hirshleifer and A. Subrahmanyam, Investor psychology and security market under- and overreaction, Journal of Finance, 53 (1998), 1839-1885. doi: 10.1111/0022-1082.00077.
[10]	R. C. Dorf and R. H. Bishop, "Modern Control Systems," 11^th edition, Pearson Prentice-Hall, Upper Saddle River, NJ, 2008.
[11]	A. Duran, Sensitivity analysis of asset flow differential equations and volatility comparison of two related variables, Numerical Functional Analysis and Optimization, 30 (2009), 82-97. doi: 10.1080/01630560802678598.
[12]	A. Duran and G. Caginalp, Parameter optimization for differential equations in asset price forecasting, Optimization Methods & Software, 23 (2008), 551-574. doi: 10.1080/10556780801996178.
[13]	W. Edwards, Conservatism in human information processing, in "Formal Representation of Human Judgment" (eds. B. Kleinmutz), John Wiley and Sons, New York (1968), 17-52.
[14]	D. Fudenberg and J. Tirole, "Game Theory," Massachusetts Institute of Technology, Cambridge, 1991.
[15]	M. Grinblatt and B. Han, Prospect theory, mental accounting, and momentum, Journal of Financial Economics, 78 (2005), 311-339. doi: 10.1016/j.jfineco.2004.10.006.
[16]	J. M. Henderson and R. E. Quandt, "Microeconomic Theory, A Mathematical Approach," 3^rd edition, McGraw-Hill, 1980.
[17]	H. Hong and J. C. Stein, A unified theory of underreaction, momentum trading, and overreaction in asset markets, Journal of Finance, 54 (1999), 2143-2184. doi: 10.1111/0022-1082.00184.
[18]	N. Jegadeesh and S. Titman, Returns to buying winners and selling losers: Implications for stock market efficiency, Journal of Finance, 48 (1993), 65-91. doi: 10.2307/2328882.
[19]	N. Jegadeesh and S. Titman, Profitability of momentum strategies: An evaluation of alternative explanations, Journal of Finance, 56 (2001), 699-720. doi: 10.1111/0022-1082.00342.
[20]	H. Merdan and M. Alisen, Asset price dynamics for a market involving more information about demand and supply, preprint, TOBB University of Economics and Technology.
[21]	H. Merdan and H. Cakmak, Liquidity effect on the asset price forecasting, preprint, TOBB University of Economics and Technology.
[22]	M. J. Osborne and A. Rubinstein, "A course in Game Theory," Massachusetts Institute of Technology, Cambridge, 1994.
[23]	J. M. Poterba and L. H. Summers, Mean reversion in stock prices: Evidence and implications, Journal of Financial Economics, 22 (1988), 27-59. doi: 10.1016/0304-405X(88)90021-9.
[24]	H. Shefrin, "A Behavioral Approach to Asset Pricing," Elsevier, London, 2005.
[25]	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. doi: 10.2307/2327802.
[26]	A. Shleifer and R. W. Vishny, The limits of arbitrage, Journal of Finance, 52 (1997), 35-55. doi: 10.2307/2329555.
[27]	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. doi: 10.2307/1911361.
[28]	A. Tversky and D. Kahneman, Judgment under uncertainty: Heuristics and biases, Science, 185 (1974), 1109-1186. doi: 10.1126/science.185.4157.1124.
[29]	D. S. Watson and M. Getz, "Price Theory and Its Uses," 5^th edition, University Press of America, Lanham, MD, 1993.
[30]	P. Wilmott, "Paul Wilmott Introduces Quantitative Finance," John Wiley & Sons, 2007.

show all references

References:

[1]	N. Barberis, A. Shleifer and R. Vishny, A model of investor sentiment, Journal of Financial Economics, 49 (1998), 307-343. doi: 10.1016/S0304-405X(98)00027-0.
[2]	Z. Bodie, A. Kane and A. J. Marcus, "Investments," 7^th edition, McGraw-Hill Education, Boston, 2008.
[3]	G. Caginalp and B. Ermentrout, A kinetic thermodynamics approach to the psychology of fluctuations in financial markets, Applied Mathematics Letters (4), 3 (1990), 17-19. doi: 10.1016/0893-9659(90)90038-D.
[4]	G. Caginalp and D. Balenovich, Asset flow and momentum: Deterministic and stochastic equations, Philosophical Transactions of the Royal Society, 357 (1999), 2119-2133. doi: 10.1098/rsta.1999.0421.
[5]	G. Caginalp and M. DeSantis, Stock price dynamics: Nonlinear trend, volume, volatility, resistance and money supply, Quantitative Finance, to appear.
[6]	G. Caginalp and V. Ilieva, The dynamics of trader motivations in asset bubbles, Journal of Economic Behavior & Organization, 66 (2008), 641-656. doi: 10.1016/j.jebo.2006.01.011.
[7]	G. Caginalp and H. Merdan, Asset price dynamics with heterogeneous groups, Physica D: Nonlinear Phenomena, 225 (2007), 43-54. doi: 10.1016/j.physd.2006.09.036.
[8]	G. Caginalp, D. Porter and V. L. Smith, Initial cash/asset ratio and asset prices: An experimental study, Proceedings of the National Academy of Sciences, 95 (1998), 756-761. doi: 10.1073/pnas.95.2.756.
[9]	K. D. Daniel, D. Hirshleifer and A. Subrahmanyam, Investor psychology and security market under- and overreaction, Journal of Finance, 53 (1998), 1839-1885. doi: 10.1111/0022-1082.00077.
[10]	R. C. Dorf and R. H. Bishop, "Modern Control Systems," 11^th edition, Pearson Prentice-Hall, Upper Saddle River, NJ, 2008.
[11]	A. Duran, Sensitivity analysis of asset flow differential equations and volatility comparison of two related variables, Numerical Functional Analysis and Optimization, 30 (2009), 82-97. doi: 10.1080/01630560802678598.
[12]	A. Duran and G. Caginalp, Parameter optimization for differential equations in asset price forecasting, Optimization Methods & Software, 23 (2008), 551-574. doi: 10.1080/10556780801996178.
[13]	W. Edwards, Conservatism in human information processing, in "Formal Representation of Human Judgment" (eds. B. Kleinmutz), John Wiley and Sons, New York (1968), 17-52.
[14]	D. Fudenberg and J. Tirole, "Game Theory," Massachusetts Institute of Technology, Cambridge, 1991.
[15]	M. Grinblatt and B. Han, Prospect theory, mental accounting, and momentum, Journal of Financial Economics, 78 (2005), 311-339. doi: 10.1016/j.jfineco.2004.10.006.
[16]	J. M. Henderson and R. E. Quandt, "Microeconomic Theory, A Mathematical Approach," 3^rd edition, McGraw-Hill, 1980.
[17]	H. Hong and J. C. Stein, A unified theory of underreaction, momentum trading, and overreaction in asset markets, Journal of Finance, 54 (1999), 2143-2184. doi: 10.1111/0022-1082.00184.
[18]	N. Jegadeesh and S. Titman, Returns to buying winners and selling losers: Implications for stock market efficiency, Journal of Finance, 48 (1993), 65-91. doi: 10.2307/2328882.
[19]	N. Jegadeesh and S. Titman, Profitability of momentum strategies: An evaluation of alternative explanations, Journal of Finance, 56 (2001), 699-720. doi: 10.1111/0022-1082.00342.
[20]	H. Merdan and M. Alisen, Asset price dynamics for a market involving more information about demand and supply, preprint, TOBB University of Economics and Technology.
[21]	H. Merdan and H. Cakmak, Liquidity effect on the asset price forecasting, preprint, TOBB University of Economics and Technology.
[22]	M. J. Osborne and A. Rubinstein, "A course in Game Theory," Massachusetts Institute of Technology, Cambridge, 1994.
[23]	J. M. Poterba and L. H. Summers, Mean reversion in stock prices: Evidence and implications, Journal of Financial Economics, 22 (1988), 27-59. doi: 10.1016/0304-405X(88)90021-9.
[24]	H. Shefrin, "A Behavioral Approach to Asset Pricing," Elsevier, London, 2005.
[25]	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. doi: 10.2307/2327802.
[26]	A. Shleifer and R. W. Vishny, The limits of arbitrage, Journal of Finance, 52 (1997), 35-55. doi: 10.2307/2329555.
[27]	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. doi: 10.2307/1911361.
[28]	A. Tversky and D. Kahneman, Judgment under uncertainty: Heuristics and biases, Science, 185 (1974), 1109-1186. doi: 10.1126/science.185.4157.1124.
[29]	D. S. Watson and M. Getz, "Price Theory and Its Uses," 5^th edition, University Press of America, Lanham, MD, 1993.
[30]	P. Wilmott, "Paul Wilmott Introduces Quantitative Finance," John Wiley & Sons, 2007.

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