A macroscopic portfolio model: from rational agents to bounded rationality

  • Torsten TrimbornEmail author


We introduce a microscopic model of interacting financial agents, where each agent is characterized by two portfolios; money invested in bonds and money invested in stocks. Furthermore, each agent is faced with an optimization problem in order to determine the optimal asset allocation. Thus, we consider a differential game since all agents aim to invest optimal and we introduce the concept of Nash equilibrium solutions to ensure the existence of a solution. Especially, we denote an agent who solves this Nash equilibrium exactly a rational agent. As next step we use model predictive control to approximate the control problem. This enables us to derive a precise mathematical characterization of the degree of rationality of a financial agent. This is a novel concept in portfolio optimization and can be regarded as a general approach. In a second step we consider the case of a fully myopic agent, where we can solve the optimal investment decision of investors explicitly. We select the running cost to be the expected missed revenue of an agent which are determined by a combination of a fundamentalist and chartist strategy. Then we derive the mean field limit of the microscopic model in order to obtain a macroscopic portfolio model. The novelty in comparison to existent macroeconomic models in literature is that our model is derived from microeconomic dynamics. The resulting portfolio model is a three dimensional ODE system which enables us to derive analytical results. The conducted simulations reveal that the model shares many dynamical properties with existing models in literature. Thus, our model is able to replicate the most prominent features of financial markets, namely booms and crashes. In the case of random fundamental prices the model is even able to reproduce fat tails in logarithmic stock price return data. Mathematically, the model can be regarded as the moment model of the recently introduced mesoscopic kinetic portfolio model (Trimborn et al. in Portfolio optimization and model predictive con trol: a kinetic approach, arXiv:1711.03291, 2017).


Portfolio optimization Model predictive control Stock market Bounded rationality Crashes Booms 

JEL Classification

G11 G12 



Torsten Trimborn gratefully acknowledges funding by the Hans-Böckler-Stiftung.


  1. 1.
    Albi, G., Pareschi, L., Zanella, M.: Boltzmann-type control of opinion consensus through leaders. Philos. Trans. R. Soc. Lond. A: Math., Phys. Eng. Sci. 372(2028), 20140138 (2014)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Anand, K., Kirman, A., Marsili, M.: Epidemics of rules, information aggregation failure and market crashes. (2010)
  3. 3.
    Beja, A., Goldman, M.B.: On the dynamic behavior of prices in disequilibrium. J. Financ. 35(2), 235–248 (1980)Google Scholar
  4. 4.
    Bertsimas, D., Pachamanova, D.: Robust multiperiod portfolio management in the presence of transaction costs. Comput. Oper. Res. 35(1), 3–17 (2008)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Breeden, D.T., Litzenberger, R.H.: Prices of state-contingent claims implicit in option prices. J. Bus. 51, 621–651 (1978)Google Scholar
  6. 6.
    Bressan, A.: Noncooperative differential games. Milan J. Math. 79(2), 357–427 (2011)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Brock, W.A., Hommes, C.H.: A rational route to randomness. Econom. J. Econom. Soc. 65, 1059–1095 (1997)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Brock, W.A., Hommes, C.H.: Heterogeneous beliefs and routes to chaos in a simple asset pricing model. J. Econ. Dyn. Control 22(8), 1235–1274 (1998)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Brown, D.J., Lewis, L.M.: Myopic economic agents. Econom. J. Econom. Soc. 49, 359–368 (1981)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Camacho, E.F., Alba, C.B.: Model Predictive Control. Springer, Berlin (2013)Google Scholar
  11. 11.
    Chiarella, C.: The dynamics of speculative behaviour. Ann. Oper. Res. 37(1), 101–123 (1992)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Chiarella, C., He, X.-Z.: Heterogeneous beliefs, risk and learning in a simple asset pricing model. Comput. Econ. 19(1), 95–132 (2002)zbMATHGoogle Scholar
  13. 13.
    Conlisk, J.: Why bounded rationality? J. Econ. Lit. 34(2), 669–700 (1996)Google Scholar
  14. 14.
    Cont, R., Bouchaud, J.-P.: Herd behavior and aggregate fluctuations in financial markets. Macroecon. Dyn. 4(02), 170–196 (2000)zbMATHGoogle Scholar
  15. 15.
    Day, R.H., Huang, W.: Bulls, bears and market sheep. J. Econ. Behav. Organ. 14(3), 299–329 (1990)Google Scholar
  16. 16.
    Duffie, D., Zame, W.: The consumption-based capital asset pricing model. Econom. J. Econom. Soc. 57, 1279–1297 (1989)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Egenter, E., Lux, T., Stauffer, D.: Finite-size effects in monte carlo simulations of two stock market models. Phys. A: Stat. Mech. Appl. 268(1), 250–256 (1999)Google Scholar
  18. 18.
    Fama, E.F.: The behavior of stock-market prices. J. Bus. 38(1), 34–105 (1965)Google Scholar
  19. 19.
    Franke, R., Westerhoff, F.: Structural stochastic volatility in asset pricing dynamics: estimation and model contest. J. Econ. Dyn. Control 36(8), 1193–1211 (2012)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Golse, F.: On the dynamics of large particle systems in the mean field limit. Macroscopic and Large Scale Phenomena: Coarse Graining. Mean Field Limits and Ergodicity, pp. 1–144. Springer, Cham (2016)Google Scholar
  21. 21.
    Gros, D.: The effectiveness of capital controls: implications for monetary autonomy in the presence of incomplete market separation. Staff Pap. 34(4), 621–642 (1987)Google Scholar
  22. 22.
    Grüne, L., Pannek, J., Seehafer, M., Worthmann, K.: Analysis of unconstrained nonlinear mpc schemes with time varying control horizon. SIAM J. Control Optim. 48(8), 4938–4962 (2010)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Grune, L., Rantzer, A.: On the infinite horizon performance of receding horizon controllers. IEEE Trans. Autom. Control 53(9), 2100–2111 (2008)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Hommes, C.H.: Modeling the stylized facts in finance through simple nonlinear adaptive systems. Proc. Natl. Acad. Sci. 99(suppl 3), 7221–7228 (2002)Google Scholar
  25. 25.
    Hommes, C.H.: Heterogeneous agent models in economics and finance. Handb. Comput. Econ. 2, 1109–1186 (2006)Google Scholar
  26. 26.
    Jensen, M.C., Black, F., Scholes, M.S.: The capital asset pricing model: some empirical tests. In: M. C. Jensen (ed.) Studies in the Theory of Capital Markets, Praeger Publishers inc. (1972)Google Scholar
  27. 27.
    Kahneman, D.: Maps of bounded rationality: psychology for behavioral economics. Am. Econ. Rev. 93(5), 1449–1475 (2003)Google Scholar
  28. 28.
    Kahneman, D., Tversky, A.: Prospect theory: an analysis of decision under risk. Econom. J. Econom. Soc. 47, 263–291 (1979)zbMATHGoogle Scholar
  29. 29.
    Kirk, D.E.: Optimal Control Theory: An Introduction. Springer, Berlin (1970)Google Scholar
  30. 30.
    Kirman, A.: The crisis in economic theory. Rivista Italiana Degli Economisti 16(1), 9–36 (2011)Google Scholar
  31. 31.
    Lehmann, B.N.: Fads, martingales, and market efficiency. Q. J. Econ. 105(1), 1–28 (1990)Google Scholar
  32. 32.
    Levy, M., Levy, H., Solomon, S.: A microscopic model of the stock market: cycles, booms, and crashes. Econ. Lett. 45(1), 103–111 (1994)zbMATHGoogle Scholar
  33. 33.
    Lintner, J.: Security prices, risk, and maximal gains from diversification. J. Financ. 20(4), 587–615 (1965)Google Scholar
  34. 34.
    Lo, A.W.: The adaptive markets hypothesis: market efficiency from an evolutionary perspective. J Portf Manag 30th Anniv. 30(5), 15–29 (2004)Google Scholar
  35. 35.
    Lux, T.: Herd behaviour, bubbles and crashes. Econ. J. 105, 881–896 (1995)Google Scholar
  36. 36.
    Lux, T. et al: Stochastic behavioral asset pricing models and the stylized facts. Technical report, Economics working paper/Christian-Albrechts-Universität Kiel, Department of Economics, (2008)Google Scholar
  37. 37.
    Lux, T., Marchesi, M.: Scaling and criticality in a stochastic multi-agent model of a financial market. Nature 397(6719), 498–500 (1999)Google Scholar
  38. 38.
    Malkiel, B.G.: The efficient market hypothesis and its critics. J. Econ. Perspect. 17(1), 59–82 (2003)Google Scholar
  39. 39.
    Mantel, R.R., et al.: On the characterization of aggregate excess demand. J. Econ. Theory 7(3), 348–353 (1974)Google Scholar
  40. 40.
    Markowitz, H.: Portfolio selection. J. Financ. 7(1), 77–91 (1952)Google Scholar
  41. 41.
    Merton, R.C.: Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Econ. Stat. 51, 247–257 (1969)Google Scholar
  42. 42.
    Merton, R.C.: Optimum consumption and portfolio rules in a continuous-time model. In: Stochastic Optimization Models in Finance, pp. 621–661. Elsevier, Amsterdam (1975)Google Scholar
  43. 43.
    Michalska, H., Mayne, D.Q.: Receding horizon control of nonlinear systems. In: Proceedings of the 28th IEEE Conference on Decision and Control, 1989, pp. 107–108. IEEE, (1989)Google Scholar
  44. 44.
    Mitchell, J.E., Braun, S.: Rebalancing an investment portfolio in the presence of convex transaction costs, including market impact costs. Optim. Methods Softw. 28(3), 523–542 (2013)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Mossin, J.: Equilibrium in a capital asset market. Econom.: J. Econom. Soc. 34, 768–783 (1966)Google Scholar
  46. 46.
    Niehans, J.: The international allocation of savings with quadratic transaction (or risk) costs. J. Int. Money Financ. 11(3), 222–234 (1992)Google Scholar
  47. 47.
    Odean, T.: Volume, volatility, price, and profit when all traders are above average. J. Financ. 53(6), 1887–1934 (1998)Google Scholar
  48. 48.
    Ross, S.A.: The arbitrage theory of capital asset pricing. In: Handbook of the Fundamentals of Financial Decision Making: Part I, pp. 11–30. World Scientific, Singapore (2013)Google Scholar
  49. 49.
    Rubinstein, M.: The valuation of uncertain income streams and the pricing of options. Bell J. Econ. 7, 407–425 (1976)MathSciNetGoogle Scholar
  50. 50.
    Sharpe, W.F.: Capital asset prices: a theory of market equilibrium under conditions of risk. J. Financ. 19(3), 425–442 (1964)Google Scholar
  51. 51.
    Shiller, R.J.: From efficient markets theory to behavioral finance. J. Econ. Perspect. 17(1), 83–104 (2003)Google Scholar
  52. 52.
    Simon, H.A.: A behavioral model of rational choice. Q. J. Econ. 69, 99–118 (1955)Google Scholar
  53. 53.
    Sonnenschein, H.: Market excess demand functions. Econom.: J. Econom. Soc. 40, 549–563 (1972)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Sontag, E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems, vol. 6. Springer, Berlin (2013)Google Scholar
  55. 55.
    Stanley, H.E.: Phase Transitions and Critical Phenomena. Clarendon Press, Oxford (1971)Google Scholar
  56. 56.
    Treynor, J.L.: Market value, time, and risk. (1961)Google Scholar
  57. 57.
    Trimborn, T., Otte, P., Cramer, S., Beikirch, M., Pabich, E., Frank, M.: Sabcemm- a simulator for agent-based computational economic market models. arXiv preprint arXiv:1801.01811, (2018)
  58. 58.
    Trimborn, T., Pareschi, L., Frank, M.: Portfolio optimization and model predictive control: a kinetic approach. arXiv preprint arXiv:1711.03291, (2017)
  59. 59.
    Walras, L.: Études d’économie politique appliquée:(Théorie de la production de la richesse sociale). In: Rouge, F. (ed.), Paris (1898)Google Scholar
  60. 60.
    Zhou, W.-X., Sornette, D.: Self-organizing ising model of financial markets. Eur. Phys. J. B 55(2), 175–181 (2007)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.IGPM, RWTH AachenAachenGermany

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