Financial Noise

The conventional view of stock markets and their behavior is that the price of a stock reflects all the information available on that stock. In this model, any variations in price would correspond to the arrival of new information.

Some of that information is predictable, and some isn't. If the market works effectively and efficiently, predictable information is already accounted for in the price. In other words, a rise or fall has already taken place on the basis of predictions. When the news finally arrives, it can no longer affect the price. Hence, price fluctuations occur only in response to truly new pieces of information--the parts that aren't predictable based on available data.

It makes sense, then, to model fluctuations in stock prices as a random (or Markov) process, which proceeds in independent increments. Human ingenuity, fear, intuition, fallibility, and incompetence, however, conspire to complicate the situation.

Stock-market price fluctuations pictured as a one-dimensional random walk.

Economist Thomas Lux of the University of Bonn in Germany and electrical engineer Michele Marchesi of the University of Cagliari in Italy unveil some of those complexities in the Feb. 11 Nature. Their mathematical analyses demonstrate that, in the realm of the stock market, information in does not necessarily lead to correspondingly rational behavior out. Far from being a fundamentally rational enterprise, such a market is inevitably influenced by the herd behavior of their participants.

The researchers based their financial-market computer simulations on models of physical systems made up of a large number of interacting units, or agents. The virtual traders were divided into two groups. One group followed the conventional premise that the participating traders expect the price to follow the so-called fundamental value of the asset, buying (or selling) when they believe the actual market price is below (or above the fundamental value). Instead of making decisions based on real information on the predicted future value of a stock, the second group--called noise traders--attempted to identify trends and patterns and considered the behavior of other traders as key sources of information.

In an additional refinement, Lux and Marchesi also distinguished between optimistic and pessimistic noise traders. "It is important for the resulting market operations whether a noise trader believes in a rising or declining market," the researchers note. "Optimists will buy additional units of the asset, whereas pessimists will sell part of their actual holdings of the asset."

The model allowed individuals to move from one group to another in response to changes in the fundamental value of an asset and to price fluctuations resulting from the agents' market operations. "Individuals react to certain economic forces by changing their behavior with a certain probability," Lux and Marchesi remark.

In an efficient, rational market, the magnitude of stock price changes would reflect the scale of the inputs that influence the price. In other words, the movements of prices are an immediate and unbiased reflection of incoming news about future earnings prospects. This is often not what actually happens, however.

Lux and Marchesi show that factoring in the behavior of the various types of noise traders destabilizes the market. Indeed, such activity typically leads to periods of extreme price volatility. Those fluctuations have little or nothing to do with changes in the inputs that normally influence price. They arise from the interaction of traders with different beliefs and market strategies.

"The alternation between tranquil and turbulent periods comes about through the changes of agents between groups," Lux and Marchesi say. "In particular, in periods of high volatility we also find a large fraction of agents in the noise trader group. Theoretical analysis shows that a critical value for the number of noise traders exists where the system loses its stability."

References:

Bouchard, J.-P., P. Cizeau, L. Lalloux, and M. Potters. 1999. Mutual attractions: Physics and finance. Physics World (January):25.

Case, J. 1998. The modeling and analysis of financial time series. American Mathematical Monthly 105(May):412.

Lux, T., and M. Marchesi. 1999. Scaling and criticality in a stochastic multi-agent model of a financial market. Nature 397(Feb. 11):493.

Mandelbrot, B.B. 1999. A multifractal walk down Wall Street. Scientific American (February):70.

Mantegna, R.N., and Stanley, E. 1995. Scaling behaviour in the dynamics of an economic index. Nature 376(July 6):46.

Peterson, I. 1998. The Mathematical Tourist: New and Updated Snapshots of Modern Mathematics. New York: W.H. Freeman.

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Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.