Complex Systems Approach To Economics

M. Bartolozzi, D. B. Leinweber and A.W. Thomas

Econophysics

Recently, the complex dynamics of the stock market has captured the interest of many physicists. This new branch of research, known by the name of ECONOPHYSICS, studies the stock market as a complex self-interacting system. The building blocks of the system are not particles or atoms but human agents, whose decisions are taken according to external factors and personal intuition.

Stock Market and Physical Systems

A physical approach to the study of stock market dynamics is justified by a great number of analogies with physical systems.

Price Fluctuations and Turbulence

The fluctuations of the longitudinal velocity (below left) in a turbulent flow (left) show the same intermittent behaviour of the price fluctuations in the stock market (below right).

The theoretical distribution of the price returns, r(t)=ln P(t) – ln P(t-1), is of interest. Broad tails (d) in the probability distribution function (pdf), related to extreme events, emphasize the difference from the classical Gaussian distribution of the efficient market. The new field of SUPERSTATISTICS, using a dynamical-system approach, appears to be able to explain this peculiar aspect of price fluctuations.

In this approach, the dynamics of the fluctuations are modelled as a stochastic Langevin process as follows

where G(t) is a Gaussian noise. The broad wings in the distribution are given by the fluctuations of the strength coefficients ãt and ót, that become stochastic variables with a distribution of their own.

Crashes and Earthquakes

A power law increase of the price, P(t)~(t-tc)-a, with superimposed accelereting oscillations has been detected before the biggest market crashes, such as the one in 1987 (right). This phenomenon is similar to the power law increase of the seismic activity before a major earthquake (below right). The result of a devastating earthquake in Guatemala. The market crash is a financial “equivalent.”

The market crashes are investigated using the concept of DISCRETE SCALE INVARIANCE (DSI). Within this approach the market is considered to be close to a phase transition just before the crash and the exponent of the power law related to the crash is not a real number but rather a complex number, giving rise to log-periodic oscillations. In this framework the price, P(t), before the crash can be approximated as

Critical Self-Organisation In The Market

Power laws are a common feature, not only of the stock market but of many complex systems: solar flares, mass extinctions, earthquakes and, in general, systems that undergo a phase transition at a critical point. In complex systems, where the dynamics are determined by the interaction of many interacting elements, the critical point is reached without the fine tuning of an external parameter. For this reason, these systems are believed to be in a SELF-ORGANIZED CRITICAL (SOC) state. The characteristic feature of systems exhibithing SOC is the avalanche dynamics needed to preserve the critical state. This is similar to what happens in an hourglass while trying to keep the slope of the sand constant.

The challenging problem of identifying avalanches in the stock market is resolved using a new tool from turbulence theory: the wavelet transform. Similar to a Fourier transform with a variable time horizon, this technique enables one to distinguish between high activity and quiet periods based on the wavelet-power coefficients (g).

Stock-Market Simulation

Another important branch of research is related to the understanding the microscopic dynamics of the stock market using numerical simulations. As an example we show a cellular automata model in which traders, represented by spins, are distributed on clusters over a two dimensional grid.

The trading dynamics is determined by a stochastic heat-bath dynamics. This simple model is able to reproduce most of the characteristic features of a real stock market, as shown for the time series (below left) and for the distribution of returns (below right).

Prospectives

The physics methods proposed to study the dynamics of a complex system such as the stock market can bring a new light on this fascinating area and have a real-world impact. In particular