The Complex Dynamics of a Simple Stock Market Model

Levy, Moshe; Persky, Nathan; Solomon, Sorin

doi:10.1142/s0129053396000082

Cited by 29 publications

(27 citation statements)

References 15 publications

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“…However, the mechanism for the emergence of a Gaussian shape is still different from its ori- . This is an example with a stock price development described as "chaotic" in [121]. However, it seems that the result is rather similar to pure randomness.…”

Section: Previous Resultsmentioning

confidence: 82%

Agent-based models of financial markets

et al. 2007

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This review deals with several microscopic ("agent-based") models of financial markets which have been studied by economists and physicists over the last decade: Kim-Markowitz, Levy-Levy-Solomon, ContBouchaud, Solomon-Weisbuch, Lux-Marchesi, Donangelo-Sneppen and Solomon-Levy-Huang. After an overview of simulation approaches in financial economics, we first give a summary of the DonangeloSneppen model of monetary exchange and compare it with related models in economics literature. Our selective review then outlines the main ingredients of some influential early models of multi-agent dynamics in financial markets (Kim-Markowitz, Levy-Levy-Solomon). As will be seen, these contributions draw their inspiration from the complex appearance of investors' interactions in real-life markets. Their main aim is to reproduce (and, thereby, provide possible explanations) for the spectacular bubbles and crashes seen in certain historical episodes, but they lack (like almost all the work before 1998 or so) a perspective in terms of the universal statistical features of financial time series. In fact, awareness of a set of such regularities (power-law tails of the distribution of returns, temporal scaling of volatility) only gradually appeared over the nineties. With the more precise description of the formerly relatively vague characteristics ( e.g. moving from the notion of fat tails to the more concrete one of a power-law with index around three), it became clear that financial markets dynamics give rise to some kind of universal scaling laws. Showing similarities with scaling laws for other systems with many interacting sub-units, an exploration of financial markets as multi-agent systems appeared to be a natural consequence. This topic was pursued by quite a number of contributions appearing in both the physics and economics literature since the late nineties. From the wealth of different flavors of multi-agent models that have appeared by now, we discuss the ContBouchaud, Solomon-Levy-Huang and Lux-Marchesi models. Open research questions are discussed in our concluding section.

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Section: Previous Resultsmentioning

confidence: 82%

Agent-based models of financial markets

et al. 2007

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“…One can think of the variable u as a weighted average wealth. Obviously for arbitrary functions c(w 1 , w 2 , ..., w N , t), the time evolution of u(t) and the functions w i (t) can be very eventfull [Levy, Persky and Solomon, 1996, Farmer 1999]. However we show below that the probability distribution of the relative wealth…”

Section: The Generalized Lotka -Volterra Modelmentioning

confidence: 80%

Power laws of wealth, market order volumes and market returns

Solomon

Richmond

2001

Physica A: Statistical Mechanics and its Applications

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Using the Generalised Lotka Volterra (GLV) model adapted to deal with muti agent systems we can investigate economic systems from a general viewpoint and obtain generic features common to most economies. Assuming only weak generic assumptions on capital dynamics, we are able to obtain very specific predictions for the distribution of social wealth. First, we show that in a 'fair' market, the wealth distribution among individual investors fulfills a power law. We then argue that 'fair play' for capital and minimal socio-biological needs of the humans traps the economy within a power law wealth distribution with a particular Pareto exponent α ∼ 3/2. In particular we relate it to the average number of individuals L depending on the average wealth: α ∼ L/(L − 1). Then we connect it to certain power exponents characterising the stock markets. We obtain that the distribution of volumes of the individual (buy and sell) orders follows a power law with similar exponent β ∼ α ∼ 3/2. Consequently, in a market where trades take place by matching pairs of such sell and buy orders, the corresponding exponent for the market returns is expected to be of order γ ∼ 2α ∼ 3. These results are consistent with recent experimental measurements of these power law exponents ([Maslov 2001] for β and[Gopikrishnan et al. 1999] for γ).

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“…By contrast, scaling behavior, power law distributions as well as spatial and temporal power law correlations in generic natural systems is still the subject of intense study [18][19][20][21][22][23][24][25][26][27][28][29][30][31].An approach that proved to be useful in the study of complex systems is to identify for each system the relevant elementary degrees of freedom and their interactions and to follow-up (by monitoring their computer simulation) the emergence in the system of the macroscopic collective phenomena [32]. This approach was applied to the study of multiscale dynamics in spin glasses [33] and stock market dynamics [34]. Using a generic class of models with a large number of interacting degrees of freedom, it was shown that macroscopic dynamics emerges under rather general conditions.…”

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confidence: 99%

“…Using a generic class of models with a large number of interacting degrees of freedom, it was shown that macroscopic dynamics emerges under rather general conditions. This dynamics exhibits power law scaling as well as intermittency [34][35][36]. These models turn out to be particularly suitable to describe systems such as stock market dynamics with many individual investors [37][38][39]41,44,42] where each system component describes a single investor (or stock [40]).…”

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confidence: 99%

Generic emergence of power law distributions and Lévy-Stable intermittent fluctuations in discrete logistic systems

et al. 1998

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The dynamics of generic stochastic Lotka-Volterra (discrete logistic) systems of the form [36] w i (t + 1) = λ(t)w i (t) + aw(t) − bw i (t)w(t) is studied by computer simulations. The variables w i , i = 1, ...N , are the individual system components andw(t) = 1 N i w i (t) is their average. The parameters a and b are constants, while λ(t) is randomly chosen at each time step from a given distribution. Models of this type describe the temporal evolution of a large variety of systems such as stock markets and city populations. These systems are characterized by a large number of interacting objects and the dynamics is dominated by multiplicative processes. The instantaneous probability distribution P (w, t) of the system components w i , turns out to fulfill a (truncated)The time evolution ofw(t) presents intermittent fluctuations parametrized by a truncated Lévy distribution of index α, showing a connection between the distribution of the w i 's at a given time and the temporal fluctuations of their average. Typeset using REVT E X 1 I. INTRODUCTION Power-law distributions have been observed in all domains of the natural sciences as well as in economics, linguistics and many other fields. Widely studied examples of power law distributions include the energy distribution between scales in turbulence [1], distribution of earthquake magnitudes [2], diameter distribution of craters and asteroids [3], the distribution of city populations [4,5], the distributions of income and of wealth [6-10], the size-distribution of business firms [11,12] and the distribution of the frequency of appearance of words in texts [4]. A related phenomenon is the fact that in a variety of systems the temporal fluctuations exhibit a scale invariant behavior in the form of (truncated) Lévy-stable distributions. Well known examples are the fluctuations in stock markets [7,13].Although systems which exhibit power-law distributions have been studied extensively in recent years there is no universally accepted framework which can explain the origin of the abundance and diversity of power-law distributions. One context in which the emergence of scaling laws and long range correlations in space and time is well understood is equilibrium statistical physics at the critical point [14][15][16][17]. By contrast, scaling behavior, power law distributions as well as spatial and temporal power law correlations in generic natural systems is still the subject of intense study [18][19][20][21][22][23][24][25][26][27][28][29][30][31].An approach that proved to be useful in the study of complex systems is to identify for each system the relevant elementary degrees of freedom and their interactions and to follow-up (by monitoring their computer simulation) the emergence in the system of the macroscopic collective phenomena [32]. This approach was applied to the study of multiscale dynamics in spin glasses [33] and stock market dynamics [34]. Using a generic class of models with a large number of interacting degrees of freedom, it was shown that macroscopic dyna...

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The Complex Dynamics of a Simple Stock Market Model

Cited by 29 publications

References 15 publications

Agent-based models of financial markets

Agent-based models of financial markets

Power laws of wealth, market order volumes and market returns

Generic emergence of power law distributions and Lévy-Stable intermittent fluctuations in discrete logistic systems

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