We present examples of agent-based and stochastic models of competition and business processes in economics and finance. We start from as simple as possible models, which have microscopic, agent-based, versions and macroscopic treatment in behavior. Microscopic and macroscopic versions of herding model proposed by Kirman and Bass diffusion of new products are considered in this contribution as two basic ideas. Further we demonstrate that general herding behavior can be considered as a background of nonlinear stochastic model of financial fluctuations.
In the recent decades there were many attempts to create an agent-based model for the financial markets, yet no model so far is realistic enough and tractable to be considered as an ideal model @cite_29 . One of the best examples of realistic models is so-called Lux and Marchesi model @cite_20 , which is heavily based on the behavioral economics ideas mathematically put down as utility functions for the agents in the market, thus it is considered to be very reasonable and realistic @cite_29 . Yet this model is too complex, namely it has many parameters and complex agent interaction mechanics, to be analytically tractable. Another example of a very complex agent-based model would be Bornholdt's spin model @cite_17 @cite_7 , which is based on a certain interpretation of the well-known Ising model (for the details on the original model see any handbook on statistical physics (ex., @cite_1 )).
@cite_7 @cite_29 @cite_1 @cite_20 @cite_17
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We present an overview of some representative Agent-Based Models in Economics. We discuss why and how agent-based models represent an important step in order to explain the dynamics and the statistical properties of financial markets beyond the Classical Theory of Economics. We perform a schematic analysis of several models with respect to some specific key categories such as agents' strategies, price evolution, number of agents, etc. In the conclusive part of this review we address some open questions and future perspectives and highlight the conceptual importance of some usually neglected topics, such as non-stationarity and the self-organization of financial markets. 1. What is Statistical Mechanics? 2. Random walks and emergent properties 3. Temperature and equilibrium 4. Phase-space dynamics and ergodicity 5. Entropy 6. Free Energies 7. Quantum statistical mechanics 8. Calculation and computation 9. Order parameters, broken symmetry, and topology 10. Correlations, response, and dissipation 11. Abrupt phase transitions 12. Continuous phase transitions APPENDIX: FOURIER METHODS This paper reports statistical analyses performed on simulated data from a stochastic multi-agent model of speculative behaviour in a financial market. The price dynamics resulting from this artificial market process exhibits the same type of scaling laws as do empirical data from stock markets and foreign exchange markets: (i) one observes scaling in the probability distribution of relative price changes with a Pareto exponent around 2.6, (ii) volatility shows significant long-range correlation with a self-similarity parameter H around 0.85. This happens although we assume that news about the intrinsic or fundamental value of the asset follows a white noise process and, hence, incorporation of news about fundamental factors is insufficient to explain either of the characteristics (i) or (ii). As a consequence, in our model, the main stylised facts of financial data originate from the working of the market itself and one need not resort to scaling in unobservable extraneous signals as an explanation of the source of scaling in financial prices. The emergence of power laws can be explained by the existence of a critical state which is repeatedly approached in the course of the system's development. A simple spin model is studied, motivated by the dynamics of traders in a market, where expectation bubbles and crashes occur. The dynamics is governed by interactions, which are frustrated across different scales: while ferromagnetic couplings connect each spin to its local neighborhood, an additional coupling relates each spin to the global magnetization. This new coupling is allowed to be anti-ferromagnetic. The resulting frustration causes a metastable dynamics with intermittency and phases of chaotic dynamics. The model reproduces main observations of real economic markets as power-law distributed returns and clustered volatility.