# American Institute of Mathematical Sciences

March  2013, 18(2): 403-415. doi: 10.3934/dcdsb.2013.18.403

## A mesoscopic stock market model with hysteretic agents

 1 Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, United Kingdom 2 Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA 22030, United States 3 Department of Economics, University of Strathclyde, Sir William Duncan Building, 130 Rottenrow, Glasgow G4 0GE

Received  October 2011 Revised  June 2012 Published  November 2012

Following the approach of [22], we derive a system of Fokker-Planck equations to model a stock-market in which hysteretic agents can take long and short positions. We show numerically that the resulting mesoscopic model has rich behaviour, being hysteretic at the mesoscale and displaying bubbles and volatility clustering in particular.
Citation: Michael Grinfeld, Harbir Lamba, Rod Cross. A mesoscopic stock market model with hysteretic agents. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 403-415. doi: 10.3934/dcdsb.2013.18.403
##### References:
 [1] Y. Amihud and H. Mendelson, Trading mechanism and stock returns: An empirical investigation,, J. Finance, 42 (1987), 533. doi: 10.1111/j.1540-6261.1987.tb04567.x. Google Scholar [2] J. Bect, H. Baili and G. Fleury, Generalized Fokker-Planck equation for piecewise-diffusion processes with boundary hitting resets,, in, (2006), 1360. Google Scholar [3] B. Biais, "The Organization of Financial Markets,", Rotman School's Distinguished Lecture Series, (2008). Google Scholar [4] B. Biais, L. Glosten and C. Spratt, Market microstructure: A survey of microfoundations, empirical results and policy implications,, J. Financial Markets, 8 (2005), 217. doi: 10.1016/j.finmar.2004.11.001. Google Scholar [5] I. Brocas and J. D. Carrillo, From perception to action: An economic model of brain processes,, mimeo., (2010). Google Scholar [6] J. S. Chang and G. Cooper, A practical difference scheme for Fokker-Planck equations,, J. Comp. Phys., 6 (1970), 1. Google Scholar [7] S. Cordier, L. Pareschi and C. Piatecki, Mesoscopic modelling of financial markets,, J. Stat. Phys., 134 (2009), 161. Google Scholar [8] R. Cross, V. Kozyakin, B. O'Callaghan, A. Pokrovskii, and A. Pokrovskiy, Periodic sequences of arbitrage: A tale of four currencies,, Metroeconomica, 62 (2011), 1. Google Scholar [9] R. Cross, M. Grinfeld and H. Lamba, A mean-field model of investor behaviour,, J. Phys. Conf. Ser., 55 (2006), 55. Google Scholar [10] R. Cross, M. Grinfeld and H. Lamba, Hysteresis and economics,, IEEE Control Systems Magazine, 29 (2009), 30. Google Scholar [11] R. Cross, M. Grinfeld, H. Lamba and T. Seaman, A threshold model of investor psychology,, Physica A, 354 (2005), 463. Google Scholar [12] R. Cross, M. Grinfeld, H. Lamba and T. Seaman, Stylized facts from a threshold-based heterogeneous agent model,, Eur. J. Phys. B, 57 (2007), 213. doi: 10.1140/epjb/e2007-00108-5. Google Scholar [13] B. Düring and G. Toscani, Hydrodynamics from kinetic models of conservative economies,, Physica A, 384 (2007), 493. Google Scholar [14] M. D. Evans and R. K. Lyons, How is the macro news transmitted to exchange rates?,, J. Financial Econ., 88 (2008), 26. Google Scholar [15] J. Gyntelberg, M. Loretan, T. Subhanij and E. Chan, Private information, stock markets and exchange rates,, BIS Discussion paper, (2009). Google Scholar [16] L. Harris, "Trading and Exchange,", Oxford University Press, (2003). Google Scholar [17] F. de Jong and B. Rindi, "The Microstructure of Financial Markets,", Cambridge University Press, (2009). Google Scholar [18] D. B. Keim and A. Madhavan, The costs of institutional equity trades: an overview,, Financial Analysis J., 54 (1998), 50. Google Scholar [19] H. Lamba, A queueing theory description of cascades in financial markets and fat-tailed price returns,, Euro. Physics J. B, 77 (2010), 297. doi: 10.1140/epjb/e2010-00248-5. Google Scholar [20] H. Lamba and T. Seaman, Rational expectations, psychology and inductive learning via moving thresholds,, Physica A, 387 (2008), 3904. doi: 10.1016/j.physa.2008.01.061. Google Scholar [21] R. Naes and S. Skjeltorp, Is the market microstructure of stock markets important?,, Norges Bank Econ. Bull., 77 (2006), 123. Google Scholar [22] A. Omurtag and L. Sirovich, Modeling a large population of traders: Mimesis and stability,, J. Econ. Behav. Organiz., 61 (2006), 562. Google Scholar [23] B. Park and V. Petrosian, Fokker-Planck equations of stochastic acceleration: A study of numerical methods,, Astrophys. J. Supp. Ser., 103 (1996), 255. doi: 10.1086/192278. Google Scholar [24] , "Triennial Central Bank Survey: Report on Global Foreign Exchange Market in 2010,'', BIS, (2010). Google Scholar

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##### References:
 [1] Y. Amihud and H. Mendelson, Trading mechanism and stock returns: An empirical investigation,, J. Finance, 42 (1987), 533. doi: 10.1111/j.1540-6261.1987.tb04567.x. Google Scholar [2] J. Bect, H. Baili and G. Fleury, Generalized Fokker-Planck equation for piecewise-diffusion processes with boundary hitting resets,, in, (2006), 1360. Google Scholar [3] B. Biais, "The Organization of Financial Markets,", Rotman School's Distinguished Lecture Series, (2008). Google Scholar [4] B. Biais, L. Glosten and C. Spratt, Market microstructure: A survey of microfoundations, empirical results and policy implications,, J. Financial Markets, 8 (2005), 217. doi: 10.1016/j.finmar.2004.11.001. Google Scholar [5] I. Brocas and J. D. Carrillo, From perception to action: An economic model of brain processes,, mimeo., (2010). Google Scholar [6] J. S. Chang and G. Cooper, A practical difference scheme for Fokker-Planck equations,, J. Comp. Phys., 6 (1970), 1. Google Scholar [7] S. Cordier, L. Pareschi and C. Piatecki, Mesoscopic modelling of financial markets,, J. Stat. Phys., 134 (2009), 161. Google Scholar [8] R. Cross, V. Kozyakin, B. O'Callaghan, A. Pokrovskii, and A. Pokrovskiy, Periodic sequences of arbitrage: A tale of four currencies,, Metroeconomica, 62 (2011), 1. Google Scholar [9] R. Cross, M. Grinfeld and H. Lamba, A mean-field model of investor behaviour,, J. Phys. Conf. Ser., 55 (2006), 55. Google Scholar [10] R. Cross, M. Grinfeld and H. Lamba, Hysteresis and economics,, IEEE Control Systems Magazine, 29 (2009), 30. Google Scholar [11] R. Cross, M. Grinfeld, H. Lamba and T. Seaman, A threshold model of investor psychology,, Physica A, 354 (2005), 463. Google Scholar [12] R. Cross, M. Grinfeld, H. Lamba and T. Seaman, Stylized facts from a threshold-based heterogeneous agent model,, Eur. J. Phys. B, 57 (2007), 213. doi: 10.1140/epjb/e2007-00108-5. Google Scholar [13] B. Düring and G. Toscani, Hydrodynamics from kinetic models of conservative economies,, Physica A, 384 (2007), 493. Google Scholar [14] M. D. Evans and R. K. Lyons, How is the macro news transmitted to exchange rates?,, J. Financial Econ., 88 (2008), 26. Google Scholar [15] J. Gyntelberg, M. Loretan, T. Subhanij and E. Chan, Private information, stock markets and exchange rates,, BIS Discussion paper, (2009). Google Scholar [16] L. Harris, "Trading and Exchange,", Oxford University Press, (2003). Google Scholar [17] F. de Jong and B. Rindi, "The Microstructure of Financial Markets,", Cambridge University Press, (2009). Google Scholar [18] D. B. Keim and A. Madhavan, The costs of institutional equity trades: an overview,, Financial Analysis J., 54 (1998), 50. Google Scholar [19] H. Lamba, A queueing theory description of cascades in financial markets and fat-tailed price returns,, Euro. Physics J. B, 77 (2010), 297. doi: 10.1140/epjb/e2010-00248-5. Google Scholar [20] H. Lamba and T. Seaman, Rational expectations, psychology and inductive learning via moving thresholds,, Physica A, 387 (2008), 3904. doi: 10.1016/j.physa.2008.01.061. Google Scholar [21] R. Naes and S. Skjeltorp, Is the market microstructure of stock markets important?,, Norges Bank Econ. Bull., 77 (2006), 123. Google Scholar [22] A. Omurtag and L. Sirovich, Modeling a large population of traders: Mimesis and stability,, J. Econ. Behav. Organiz., 61 (2006), 562. Google Scholar [23] B. Park and V. Petrosian, Fokker-Planck equations of stochastic acceleration: A study of numerical methods,, Astrophys. J. Supp. Ser., 103 (1996), 255. doi: 10.1086/192278. Google Scholar [24] , "Triennial Central Bank Survey: Report on Global Foreign Exchange Market in 2010,'', BIS, (2010). Google Scholar
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