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 and 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), 533-555. doi: 10.1111/j.1540-6261.1987.tb04567.x.

[2]

J. Bect, H. Baili and G. Fleury, Generalized Fokker-Planck equation for piecewise-diffusion processes with boundary hitting resets, in "Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems," Kyoto, (2006), 1360-1367.

[3]

B. Biais, "The Organization of Financial Markets," Rotman School's Distinguished Lecture Series, University of Toronto, Toronto, 2008.

[4]

B. Biais, L. Glosten and C. Spratt, Market microstructure: A survey of microfoundations, empirical results and policy implications, J. Financial Markets, 8 (2005), 217-264. doi: 10.1016/j.finmar.2004.11.001.

[5]

I. Brocas and J. D. Carrillo, From perception to action: An economic model of brain processes, mimeo., Univ. of South Carolina, 2010.

[6]

J. S. Chang and G. Cooper, A practical difference scheme for Fokker-Planck equations, J. Comp. Phys., 6 (1970), 1-16.

[7]

S. Cordier, L. Pareschi and C. Piatecki, Mesoscopic modelling of financial markets, J. Stat. Phys., 134 (2009), 161-184.

[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), 1-45.

[9]

R. Cross, M. Grinfeld and H. Lamba, A mean-field model of investor behaviour, J. Phys. Conf. Ser., 55 (2006), 55-62.

[10]

R. Cross, M. Grinfeld and H. Lamba, Hysteresis and economics, IEEE Control Systems Magazine, 29 (2009), 30-43.

[11]

R. Cross, M. Grinfeld, H. Lamba and T. Seaman, A threshold model of investor psychology, Physica A, 354 (2005), 463-478.

[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), 213-218. doi: 10.1140/epjb/e2007-00108-5.

[13]

B. Düring and G. Toscani, Hydrodynamics from kinetic models of conservative economies, Physica A, 384 (2007), 493-506.

[14]

M. D. Evans and R. K. Lyons, How is the macro news transmitted to exchange rates?, J. Financial Econ., 88 (2008), 26-50.

[15]

J. Gyntelberg, M. Loretan, T. Subhanij and E. Chan, Private information, stock markets and exchange rates, BIS Discussion paper, Basel, Switzerland, August 2009.

[16]

L. Harris, "Trading and Exchange," Oxford University Press, Oxford, 2003.

[17]

F. de Jong and B. Rindi, "The Microstructure of Financial Markets," Cambridge University Press, New York, 2009.

[18]

D. B. Keim and A. Madhavan, The costs of institutional equity trades: an overview, Financial Analysis J., 54 (1998), 50-69.

[19]

H. Lamba, A queueing theory description of cascades in financial markets and fat-tailed price returns, Euro. Physics J. B, 77 (2010), 297-304. doi: 10.1140/epjb/e2010-00248-5.

[20]

H. Lamba and T. Seaman, Rational expectations, psychology and inductive learning via moving thresholds, Physica A, 387 (2008), 3904-3909. doi: 10.1016/j.physa.2008.01.061.

[21]

R. Naes and S. Skjeltorp, Is the market microstructure of stock markets important?, Norges Bank Econ. Bull., 77 (2006), 123-132.

[22]

A. Omurtag and L. Sirovich, Modeling a large population of traders: Mimesis and stability, J. Econ. Behav. Organiz., 61 (2006), 562-576.

[23]

B. Park and V. Petrosian, Fokker-Planck equations of stochastic acceleration: A study of numerical methods, Astrophys. J. Supp. Ser., 103 (1996), 255-267. doi: 10.1086/192278.

[24]

, "Triennial Central Bank Survey: Report on Global Foreign Exchange Market in 2010,'', BIS, (2010). 

show all references

References:
[1]

Y. Amihud and H. Mendelson, Trading mechanism and stock returns: An empirical investigation, J. Finance, 42 (1987), 533-555. doi: 10.1111/j.1540-6261.1987.tb04567.x.

[2]

J. Bect, H. Baili and G. Fleury, Generalized Fokker-Planck equation for piecewise-diffusion processes with boundary hitting resets, in "Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems," Kyoto, (2006), 1360-1367.

[3]

B. Biais, "The Organization of Financial Markets," Rotman School's Distinguished Lecture Series, University of Toronto, Toronto, 2008.

[4]

B. Biais, L. Glosten and C. Spratt, Market microstructure: A survey of microfoundations, empirical results and policy implications, J. Financial Markets, 8 (2005), 217-264. doi: 10.1016/j.finmar.2004.11.001.

[5]

I. Brocas and J. D. Carrillo, From perception to action: An economic model of brain processes, mimeo., Univ. of South Carolina, 2010.

[6]

J. S. Chang and G. Cooper, A practical difference scheme for Fokker-Planck equations, J. Comp. Phys., 6 (1970), 1-16.

[7]

S. Cordier, L. Pareschi and C. Piatecki, Mesoscopic modelling of financial markets, J. Stat. Phys., 134 (2009), 161-184.

[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), 1-45.

[9]

R. Cross, M. Grinfeld and H. Lamba, A mean-field model of investor behaviour, J. Phys. Conf. Ser., 55 (2006), 55-62.

[10]

R. Cross, M. Grinfeld and H. Lamba, Hysteresis and economics, IEEE Control Systems Magazine, 29 (2009), 30-43.

[11]

R. Cross, M. Grinfeld, H. Lamba and T. Seaman, A threshold model of investor psychology, Physica A, 354 (2005), 463-478.

[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), 213-218. doi: 10.1140/epjb/e2007-00108-5.

[13]

B. Düring and G. Toscani, Hydrodynamics from kinetic models of conservative economies, Physica A, 384 (2007), 493-506.

[14]

M. D. Evans and R. K. Lyons, How is the macro news transmitted to exchange rates?, J. Financial Econ., 88 (2008), 26-50.

[15]

J. Gyntelberg, M. Loretan, T. Subhanij and E. Chan, Private information, stock markets and exchange rates, BIS Discussion paper, Basel, Switzerland, August 2009.

[16]

L. Harris, "Trading and Exchange," Oxford University Press, Oxford, 2003.

[17]

F. de Jong and B. Rindi, "The Microstructure of Financial Markets," Cambridge University Press, New York, 2009.

[18]

D. B. Keim and A. Madhavan, The costs of institutional equity trades: an overview, Financial Analysis J., 54 (1998), 50-69.

[19]

H. Lamba, A queueing theory description of cascades in financial markets and fat-tailed price returns, Euro. Physics J. B, 77 (2010), 297-304. doi: 10.1140/epjb/e2010-00248-5.

[20]

H. Lamba and T. Seaman, Rational expectations, psychology and inductive learning via moving thresholds, Physica A, 387 (2008), 3904-3909. doi: 10.1016/j.physa.2008.01.061.

[21]

R. Naes and S. Skjeltorp, Is the market microstructure of stock markets important?, Norges Bank Econ. Bull., 77 (2006), 123-132.

[22]

A. Omurtag and L. Sirovich, Modeling a large population of traders: Mimesis and stability, J. Econ. Behav. Organiz., 61 (2006), 562-576.

[23]

B. Park and V. Petrosian, Fokker-Planck equations of stochastic acceleration: A study of numerical methods, Astrophys. J. Supp. Ser., 103 (1996), 255-267. doi: 10.1086/192278.

[24]

, "Triennial Central Bank Survey: Report on Global Foreign Exchange Market in 2010,'', BIS, (2010). 

[1]

Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215

[2]

Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic and Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009

[3]

Hongjie Dong, Yan Guo, Timur Yastrzhembskiy. Kinetic Fokker-Planck and Landau equations with specular reflection boundary condition. Kinetic and Related Models, 2022, 15 (3) : 467-516. doi: 10.3934/krm.2022003

[4]

John W. Barrett, Endre Süli. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers. Discrete and Continuous Dynamical Systems - S, 2010, 3 (3) : 371-408. doi: 10.3934/dcdss.2010.3.371

[5]

Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic and Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017

[6]

Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic and Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016

[7]

Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic and Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485

[8]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic and Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401

[9]

Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic and Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165

[10]

Krunal B. Kachhia. Comparative study of fractional Fokker-Planck equations with various fractional derivative operators. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 741-754. doi: 10.3934/dcdss.2020041

[11]

Luis Almeida, Federica Bubba, Benoît Perthame, Camille Pouchol. Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations. Networks and Heterogeneous Media, 2019, 14 (1) : 23-41. doi: 10.3934/nhm.2019002

[12]

Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic and Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056

[13]

Zeinab Karaki. Trend to the equilibrium for the Fokker-Planck system with an external magnetic field. Kinetic and Related Models, 2020, 13 (2) : 309-344. doi: 10.3934/krm.2020011

[14]

Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008

[15]

Roberta Bosi. Classical limit for linear and nonlinear quantum Fokker-Planck systems. Communications on Pure and Applied Analysis, 2009, 8 (3) : 845-870. doi: 10.3934/cpaa.2009.8.845

[16]

Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks and Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028

[17]

Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250

[18]

Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic and Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028

[19]

Yuan Gao, Guangzhen Jin, Jian-Guo Liu. Inbetweening auto-animation via Fokker-Planck dynamics and thresholding. Inverse Problems and Imaging, 2021, 15 (5) : 843-864. doi: 10.3934/ipi.2021016

[20]

Guillaume Bal, Benjamin Palacios. Pencil-beam approximation of fractional Fokker-Planck. Kinetic and Related Models, 2021, 14 (5) : 767-817. doi: 10.3934/krm.2021024

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (79)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]