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

Multi-group asset flow equations and stability

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

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.
Citation: Gunduz Caginalp, Mark DeSantis. Multi-group asset flow equations and stability. Discrete & 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.  doi: 10.1016/S0304-405X(98)00027-0.  Google Scholar

[2]

Z. Bodie, A. Kane and A. J. Marcus, "Investments,", 7th edition, (2008).   Google Scholar

[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.  doi: 10.1016/0893-9659(90)90038-D.  Google Scholar

[4]

G. Caginalp and D. Balenovich, Asset flow and momentum: Deterministic and stochastic equations,, Philosophical Transactions of the Royal Society, 357 (1999), 2119.  doi: 10.1098/rsta.1999.0421.  Google Scholar

[5]

G. Caginalp and M. DeSantis, Stock price dynamics: Nonlinear trend, volume, volatility, resistance and money supply,, Quantitative Finance, ().   Google Scholar

[6]

G. Caginalp and V. Ilieva, The dynamics of trader motivations in asset bubbles,, Journal of Economic Behavior & Organization, 66 (2008), 641.  doi: 10.1016/j.jebo.2006.01.011.  Google Scholar

[7]

G. Caginalp and H. Merdan, Asset price dynamics with heterogeneous groups,, Physica D: Nonlinear Phenomena, 225 (2007), 43.  doi: 10.1016/j.physd.2006.09.036.  Google Scholar

[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.  doi: 10.1073/pnas.95.2.756.  Google Scholar

[9]

K. D. Daniel, D. Hirshleifer and A. Subrahmanyam, Investor psychology and security market under- and overreaction,, Journal of Finance, 53 (1998), 1839.  doi: 10.1111/0022-1082.00077.  Google Scholar

[10]

R. C. Dorf and R. H. Bishop, "Modern Control Systems,", 11th edition, (2008).   Google Scholar

[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.  doi: 10.1080/01630560802678598.  Google Scholar

[12]

A. Duran and G. Caginalp, Parameter optimization for differential equations in asset price forecasting,, Optimization Methods & Software, 23 (2008), 551.  doi: 10.1080/10556780801996178.  Google Scholar

[13]

W. Edwards, Conservatism in human information processing,, in, (1968), 17.   Google Scholar

[14]

D. Fudenberg and J. Tirole, "Game Theory,", Massachusetts Institute of Technology, (1991).   Google Scholar

[15]

M. Grinblatt and B. Han, Prospect theory, mental accounting, and momentum,, Journal of Financial Economics, 78 (2005), 311.  doi: 10.1016/j.jfineco.2004.10.006.  Google Scholar

[16]

J. M. Henderson and R. E. Quandt, "Microeconomic Theory, A Mathematical Approach,", 3rd edition, (1980).   Google Scholar

[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.  doi: 10.1111/0022-1082.00184.  Google Scholar

[18]

N. Jegadeesh and S. Titman, Returns to buying winners and selling losers: Implications for stock market efficiency,, Journal of Finance, 48 (1993), 65.  doi: 10.2307/2328882.  Google Scholar

[19]

N. Jegadeesh and S. Titman, Profitability of momentum strategies: An evaluation of alternative explanations,, Journal of Finance, 56 (2001), 699.  doi: 10.1111/0022-1082.00342.  Google Scholar

[20]

H. Merdan and M. Alisen, Asset price dynamics for a market involving more information about demand and supply,, preprint, ().   Google Scholar

[21]

H. Merdan and H. Cakmak, Liquidity effect on the asset price forecasting,, preprint, ().   Google Scholar

[22]

M. J. Osborne and A. Rubinstein, "A course in Game Theory,", Massachusetts Institute of Technology, (1994).   Google Scholar

[23]

J. M. Poterba and L. H. Summers, Mean reversion in stock prices: Evidence and implications,, Journal of Financial Economics, 22 (1988), 27.  doi: 10.1016/0304-405X(88)90021-9.  Google Scholar

[24]

H. Shefrin, "A Behavioral Approach to Asset Pricing,", Elsevier, (2005).   Google Scholar

[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.  doi: 10.2307/2327802.  Google Scholar

[26]

A. Shleifer and R. W. Vishny, The limits of arbitrage,, Journal of Finance, 52 (1997), 35.  doi: 10.2307/2329555.  Google Scholar

[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.  doi: 10.2307/1911361.  Google Scholar

[28]

A. Tversky and D. Kahneman, Judgment under uncertainty: Heuristics and biases,, Science, 185 (1974), 1109.  doi: 10.1126/science.185.4157.1124.  Google Scholar

[29]

D. S. Watson and M. Getz, "Price Theory and Its Uses,", 5th edition, (1993).   Google Scholar

[30]

P. Wilmott, "Paul Wilmott Introduces Quantitative Finance,", John Wiley & Sons, (2007).   Google Scholar

show all references

References:
[1]

N. Barberis, A. Shleifer and R. Vishny, A model of investor sentiment,, Journal of Financial Economics, 49 (1998), 307.  doi: 10.1016/S0304-405X(98)00027-0.  Google Scholar

[2]

Z. Bodie, A. Kane and A. J. Marcus, "Investments,", 7th edition, (2008).   Google Scholar

[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.  doi: 10.1016/0893-9659(90)90038-D.  Google Scholar

[4]

G. Caginalp and D. Balenovich, Asset flow and momentum: Deterministic and stochastic equations,, Philosophical Transactions of the Royal Society, 357 (1999), 2119.  doi: 10.1098/rsta.1999.0421.  Google Scholar

[5]

G. Caginalp and M. DeSantis, Stock price dynamics: Nonlinear trend, volume, volatility, resistance and money supply,, Quantitative Finance, ().   Google Scholar

[6]

G. Caginalp and V. Ilieva, The dynamics of trader motivations in asset bubbles,, Journal of Economic Behavior & Organization, 66 (2008), 641.  doi: 10.1016/j.jebo.2006.01.011.  Google Scholar

[7]

G. Caginalp and H. Merdan, Asset price dynamics with heterogeneous groups,, Physica D: Nonlinear Phenomena, 225 (2007), 43.  doi: 10.1016/j.physd.2006.09.036.  Google Scholar

[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.  doi: 10.1073/pnas.95.2.756.  Google Scholar

[9]

K. D. Daniel, D. Hirshleifer and A. Subrahmanyam, Investor psychology and security market under- and overreaction,, Journal of Finance, 53 (1998), 1839.  doi: 10.1111/0022-1082.00077.  Google Scholar

[10]

R. C. Dorf and R. H. Bishop, "Modern Control Systems,", 11th edition, (2008).   Google Scholar

[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.  doi: 10.1080/01630560802678598.  Google Scholar

[12]

A. Duran and G. Caginalp, Parameter optimization for differential equations in asset price forecasting,, Optimization Methods & Software, 23 (2008), 551.  doi: 10.1080/10556780801996178.  Google Scholar

[13]

W. Edwards, Conservatism in human information processing,, in, (1968), 17.   Google Scholar

[14]

D. Fudenberg and J. Tirole, "Game Theory,", Massachusetts Institute of Technology, (1991).   Google Scholar

[15]

M. Grinblatt and B. Han, Prospect theory, mental accounting, and momentum,, Journal of Financial Economics, 78 (2005), 311.  doi: 10.1016/j.jfineco.2004.10.006.  Google Scholar

[16]

J. M. Henderson and R. E. Quandt, "Microeconomic Theory, A Mathematical Approach,", 3rd edition, (1980).   Google Scholar

[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.  doi: 10.1111/0022-1082.00184.  Google Scholar

[18]

N. Jegadeesh and S. Titman, Returns to buying winners and selling losers: Implications for stock market efficiency,, Journal of Finance, 48 (1993), 65.  doi: 10.2307/2328882.  Google Scholar

[19]

N. Jegadeesh and S. Titman, Profitability of momentum strategies: An evaluation of alternative explanations,, Journal of Finance, 56 (2001), 699.  doi: 10.1111/0022-1082.00342.  Google Scholar

[20]

H. Merdan and M. Alisen, Asset price dynamics for a market involving more information about demand and supply,, preprint, ().   Google Scholar

[21]

H. Merdan and H. Cakmak, Liquidity effect on the asset price forecasting,, preprint, ().   Google Scholar

[22]

M. J. Osborne and A. Rubinstein, "A course in Game Theory,", Massachusetts Institute of Technology, (1994).   Google Scholar

[23]

J. M. Poterba and L. H. Summers, Mean reversion in stock prices: Evidence and implications,, Journal of Financial Economics, 22 (1988), 27.  doi: 10.1016/0304-405X(88)90021-9.  Google Scholar

[24]

H. Shefrin, "A Behavioral Approach to Asset Pricing,", Elsevier, (2005).   Google Scholar

[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.  doi: 10.2307/2327802.  Google Scholar

[26]

A. Shleifer and R. W. Vishny, The limits of arbitrage,, Journal of Finance, 52 (1997), 35.  doi: 10.2307/2329555.  Google Scholar

[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.  doi: 10.2307/1911361.  Google Scholar

[28]

A. Tversky and D. Kahneman, Judgment under uncertainty: Heuristics and biases,, Science, 185 (1974), 1109.  doi: 10.1126/science.185.4157.1124.  Google Scholar

[29]

D. S. Watson and M. Getz, "Price Theory and Its Uses,", 5th edition, (1993).   Google Scholar

[30]

P. Wilmott, "Paul Wilmott Introduces Quantitative Finance,", John Wiley & Sons, (2007).   Google Scholar

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