Skip to main content
Erschienen in: Mathematics and Financial Economics 3/2019

29.01.2019

A macroscopic portfolio model: from rational agents to bounded rationality

verfasst von: Torsten Trimborn

Erschienen in: Mathematics and Financial Economics | Ausgabe 3/2019

Einloggen

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

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).

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Anhänge
Nur mit Berechtigung zugänglich
Literatur
1.
Zurück zum Zitat 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)MathSciNetMATHCrossRef 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)MathSciNetMATHCrossRef
3.
Zurück zum Zitat Beja, A., Goldman, M.B.: On the dynamic behavior of prices in disequilibrium. J. Financ. 35(2), 235–248 (1980)CrossRef Beja, A., Goldman, M.B.: On the dynamic behavior of prices in disequilibrium. J. Financ. 35(2), 235–248 (1980)CrossRef
4.
Zurück zum Zitat Bertsimas, D., Pachamanova, D.: Robust multiperiod portfolio management in the presence of transaction costs. Comput. Oper. Res. 35(1), 3–17 (2008)MathSciNetMATHCrossRef Bertsimas, D., Pachamanova, D.: Robust multiperiod portfolio management in the presence of transaction costs. Comput. Oper. Res. 35(1), 3–17 (2008)MathSciNetMATHCrossRef
5.
Zurück zum Zitat Breeden, D.T., Litzenberger, R.H.: Prices of state-contingent claims implicit in option prices. J. Bus. 51, 621–651 (1978)CrossRef Breeden, D.T., Litzenberger, R.H.: Prices of state-contingent claims implicit in option prices. J. Bus. 51, 621–651 (1978)CrossRef
7.
Zurück zum Zitat Brock, W.A., Hommes, C.H.: A rational route to randomness. Econom. J. Econom. Soc. 65, 1059–1095 (1997)MathSciNetMATH Brock, W.A., Hommes, C.H.: A rational route to randomness. Econom. J. Econom. Soc. 65, 1059–1095 (1997)MathSciNetMATH
8.
Zurück zum Zitat 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)MathSciNetMATHCrossRef 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)MathSciNetMATHCrossRef
9.
Zurück zum Zitat Brown, D.J., Lewis, L.M.: Myopic economic agents. Econom. J. Econom. Soc. 49, 359–368 (1981)MathSciNetMATH Brown, D.J., Lewis, L.M.: Myopic economic agents. Econom. J. Econom. Soc. 49, 359–368 (1981)MathSciNetMATH
10.
Zurück zum Zitat Camacho, E.F., Alba, C.B.: Model Predictive Control. Springer, Berlin (2013) Camacho, E.F., Alba, C.B.: Model Predictive Control. Springer, Berlin (2013)
12.
Zurück zum Zitat Chiarella, C., He, X.-Z.: Heterogeneous beliefs, risk and learning in a simple asset pricing model. Comput. Econ. 19(1), 95–132 (2002)MATHCrossRef Chiarella, C., He, X.-Z.: Heterogeneous beliefs, risk and learning in a simple asset pricing model. Comput. Econ. 19(1), 95–132 (2002)MATHCrossRef
13.
Zurück zum Zitat Conlisk, J.: Why bounded rationality? J. Econ. Lit. 34(2), 669–700 (1996) Conlisk, J.: Why bounded rationality? J. Econ. Lit. 34(2), 669–700 (1996)
14.
Zurück zum Zitat Cont, R., Bouchaud, J.-P.: Herd behavior and aggregate fluctuations in financial markets. Macroecon. Dyn. 4(02), 170–196 (2000)MATHCrossRef Cont, R., Bouchaud, J.-P.: Herd behavior and aggregate fluctuations in financial markets. Macroecon. Dyn. 4(02), 170–196 (2000)MATHCrossRef
15.
Zurück zum Zitat Day, R.H., Huang, W.: Bulls, bears and market sheep. J. Econ. Behav. Organ. 14(3), 299–329 (1990)CrossRef Day, R.H., Huang, W.: Bulls, bears and market sheep. J. Econ. Behav. Organ. 14(3), 299–329 (1990)CrossRef
16.
Zurück zum Zitat Duffie, D., Zame, W.: The consumption-based capital asset pricing model. Econom. J. Econom. Soc. 57, 1279–1297 (1989)MathSciNetMATH Duffie, D., Zame, W.: The consumption-based capital asset pricing model. Econom. J. Econom. Soc. 57, 1279–1297 (1989)MathSciNetMATH
17.
Zurück zum Zitat 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)CrossRef 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)CrossRef
18.
Zurück zum Zitat Fama, E.F.: The behavior of stock-market prices. J. Bus. 38(1), 34–105 (1965)CrossRef Fama, E.F.: The behavior of stock-market prices. J. Bus. 38(1), 34–105 (1965)CrossRef
19.
Zurück zum Zitat Franke, R., Westerhoff, F.: Structural stochastic volatility in asset pricing dynamics: estimation and model contest. J. Econ. Dyn. Control 36(8), 1193–1211 (2012)MathSciNetMATHCrossRef Franke, R., Westerhoff, F.: Structural stochastic volatility in asset pricing dynamics: estimation and model contest. J. Econ. Dyn. Control 36(8), 1193–1211 (2012)MathSciNetMATHCrossRef
20.
Zurück zum Zitat 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) 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)
21.
Zurück zum Zitat 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)CrossRef 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)CrossRef
22.
Zurück zum Zitat 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)MathSciNetMATHCrossRef 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)MathSciNetMATHCrossRef
23.
Zurück zum Zitat Grune, L., Rantzer, A.: On the infinite horizon performance of receding horizon controllers. IEEE Trans. Autom. Control 53(9), 2100–2111 (2008)MathSciNetMATHCrossRef Grune, L., Rantzer, A.: On the infinite horizon performance of receding horizon controllers. IEEE Trans. Autom. Control 53(9), 2100–2111 (2008)MathSciNetMATHCrossRef
24.
Zurück zum Zitat Hommes, C.H.: Modeling the stylized facts in finance through simple nonlinear adaptive systems. Proc. Natl. Acad. Sci. 99(suppl 3), 7221–7228 (2002)CrossRef Hommes, C.H.: Modeling the stylized facts in finance through simple nonlinear adaptive systems. Proc. Natl. Acad. Sci. 99(suppl 3), 7221–7228 (2002)CrossRef
25.
Zurück zum Zitat Hommes, C.H.: Heterogeneous agent models in economics and finance. Handb. Comput. Econ. 2, 1109–1186 (2006)CrossRef Hommes, C.H.: Heterogeneous agent models in economics and finance. Handb. Comput. Econ. 2, 1109–1186 (2006)CrossRef
26.
Zurück zum Zitat 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) 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)
27.
Zurück zum Zitat Kahneman, D.: Maps of bounded rationality: psychology for behavioral economics. Am. Econ. Rev. 93(5), 1449–1475 (2003)CrossRef Kahneman, D.: Maps of bounded rationality: psychology for behavioral economics. Am. Econ. Rev. 93(5), 1449–1475 (2003)CrossRef
28.
Zurück zum Zitat Kahneman, D., Tversky, A.: Prospect theory: an analysis of decision under risk. Econom. J. Econom. Soc. 47, 263–291 (1979)MathSciNetMATH Kahneman, D., Tversky, A.: Prospect theory: an analysis of decision under risk. Econom. J. Econom. Soc. 47, 263–291 (1979)MathSciNetMATH
29.
Zurück zum Zitat Kirk, D.E.: Optimal Control Theory: An Introduction. Springer, Berlin (1970) Kirk, D.E.: Optimal Control Theory: An Introduction. Springer, Berlin (1970)
30.
Zurück zum Zitat Kirman, A.: The crisis in economic theory. Rivista Italiana Degli Economisti 16(1), 9–36 (2011) Kirman, A.: The crisis in economic theory. Rivista Italiana Degli Economisti 16(1), 9–36 (2011)
32.
Zurück zum Zitat Levy, M., Levy, H., Solomon, S.: A microscopic model of the stock market: cycles, booms, and crashes. Econ. Lett. 45(1), 103–111 (1994)MATHCrossRef Levy, M., Levy, H., Solomon, S.: A microscopic model of the stock market: cycles, booms, and crashes. Econ. Lett. 45(1), 103–111 (1994)MATHCrossRef
33.
Zurück zum Zitat Lintner, J.: Security prices, risk, and maximal gains from diversification. J. Financ. 20(4), 587–615 (1965) Lintner, J.: Security prices, risk, and maximal gains from diversification. J. Financ. 20(4), 587–615 (1965)
34.
Zurück zum Zitat Lo, A.W.: The adaptive markets hypothesis: market efficiency from an evolutionary perspective. J Portf Manag 30th Anniv. 30(5), 15–29 (2004) Lo, A.W.: The adaptive markets hypothesis: market efficiency from an evolutionary perspective. J Portf Manag 30th Anniv. 30(5), 15–29 (2004)
35.
Zurück zum Zitat Lux, T.: Herd behaviour, bubbles and crashes. Econ. J. 105, 881–896 (1995)CrossRef Lux, T.: Herd behaviour, bubbles and crashes. Econ. J. 105, 881–896 (1995)CrossRef
36.
Zurück zum Zitat 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) 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)
37.
Zurück zum Zitat Lux, T., Marchesi, M.: Scaling and criticality in a stochastic multi-agent model of a financial market. Nature 397(6719), 498–500 (1999)CrossRef Lux, T., Marchesi, M.: Scaling and criticality in a stochastic multi-agent model of a financial market. Nature 397(6719), 498–500 (1999)CrossRef
38.
Zurück zum Zitat Malkiel, B.G.: The efficient market hypothesis and its critics. J. Econ. Perspect. 17(1), 59–82 (2003)CrossRef Malkiel, B.G.: The efficient market hypothesis and its critics. J. Econ. Perspect. 17(1), 59–82 (2003)CrossRef
39.
Zurück zum Zitat Mantel, R.R., et al.: On the characterization of aggregate excess demand. J. Econ. Theory 7(3), 348–353 (1974)CrossRef Mantel, R.R., et al.: On the characterization of aggregate excess demand. J. Econ. Theory 7(3), 348–353 (1974)CrossRef
40.
Zurück zum Zitat Markowitz, H.: Portfolio selection. J. Financ. 7(1), 77–91 (1952) Markowitz, H.: Portfolio selection. J. Financ. 7(1), 77–91 (1952)
41.
Zurück zum Zitat Merton, R.C.: Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Econ. Stat. 51, 247–257 (1969)CrossRef Merton, R.C.: Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Econ. Stat. 51, 247–257 (1969)CrossRef
42.
Zurück zum Zitat 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) 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)
43.
Zurück zum Zitat 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) 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)
44.
Zurück zum Zitat 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)MathSciNetMATHCrossRef 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)MathSciNetMATHCrossRef
45.
Zurück zum Zitat Mossin, J.: Equilibrium in a capital asset market. Econom.: J. Econom. Soc. 34, 768–783 (1966)CrossRef Mossin, J.: Equilibrium in a capital asset market. Econom.: J. Econom. Soc. 34, 768–783 (1966)CrossRef
46.
Zurück zum Zitat Niehans, J.: The international allocation of savings with quadratic transaction (or risk) costs. J. Int. Money Financ. 11(3), 222–234 (1992)CrossRef Niehans, J.: The international allocation of savings with quadratic transaction (or risk) costs. J. Int. Money Financ. 11(3), 222–234 (1992)CrossRef
47.
Zurück zum Zitat Odean, T.: Volume, volatility, price, and profit when all traders are above average. J. Financ. 53(6), 1887–1934 (1998)CrossRef Odean, T.: Volume, volatility, price, and profit when all traders are above average. J. Financ. 53(6), 1887–1934 (1998)CrossRef
48.
Zurück zum Zitat 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) 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)
49.
Zurück zum Zitat Rubinstein, M.: The valuation of uncertain income streams and the pricing of options. Bell J. Econ. 7, 407–425 (1976)MathSciNetCrossRef Rubinstein, M.: The valuation of uncertain income streams and the pricing of options. Bell J. Econ. 7, 407–425 (1976)MathSciNetCrossRef
50.
Zurück zum Zitat Sharpe, W.F.: Capital asset prices: a theory of market equilibrium under conditions of risk. J. Financ. 19(3), 425–442 (1964) Sharpe, W.F.: Capital asset prices: a theory of market equilibrium under conditions of risk. J. Financ. 19(3), 425–442 (1964)
51.
Zurück zum Zitat Shiller, R.J.: From efficient markets theory to behavioral finance. J. Econ. Perspect. 17(1), 83–104 (2003)CrossRef Shiller, R.J.: From efficient markets theory to behavioral finance. J. Econ. Perspect. 17(1), 83–104 (2003)CrossRef
52.
Zurück zum Zitat Simon, H.A.: A behavioral model of rational choice. Q. J. Econ. 69, 99–118 (1955)CrossRef Simon, H.A.: A behavioral model of rational choice. Q. J. Econ. 69, 99–118 (1955)CrossRef
54.
Zurück zum Zitat Sontag, E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems, vol. 6. Springer, Berlin (2013) Sontag, E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems, vol. 6. Springer, Berlin (2013)
55.
Zurück zum Zitat Stanley, H.E.: Phase Transitions and Critical Phenomena. Clarendon Press, Oxford (1971) Stanley, H.E.: Phase Transitions and Critical Phenomena. Clarendon Press, Oxford (1971)
56.
Zurück zum Zitat Treynor, J.L.: Market value, time, and risk. (1961) Treynor, J.L.: Market value, time, and risk. (1961)
57.
Zurück zum Zitat 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) 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.
Zurück zum Zitat Trimborn, T., Pareschi, L., Frank, M.: Portfolio optimization and model predictive control: a kinetic approach. arXiv preprint arXiv:1711.03291, (2017) Trimborn, T., Pareschi, L., Frank, M.: Portfolio optimization and model predictive control: a kinetic approach. arXiv preprint arXiv:​1711.​03291, (2017)
59.
Zurück zum Zitat Walras, L.: Études d’économie politique appliquée:(Théorie de la production de la richesse sociale). In: Rouge, F. (ed.), Paris (1898) Walras, L.: Études d’économie politique appliquée:(Théorie de la production de la richesse sociale). In: Rouge, F. (ed.), Paris (1898)
60.
Metadaten
Titel
A macroscopic portfolio model: from rational agents to bounded rationality
verfasst von
Torsten Trimborn
Publikationsdatum
29.01.2019
Verlag
Springer Berlin Heidelberg
Erschienen in
Mathematics and Financial Economics / Ausgabe 3/2019
Print ISSN: 1862-9679
Elektronische ISSN: 1862-9660
DOI
https://doi.org/10.1007/s11579-019-00235-z

Weitere Artikel der Ausgabe 3/2019

Mathematics and Financial Economics 3/2019 Zur Ausgabe