stock market forecasting

chaos theory in the social sciences.

How do we time the market?

• Forecasting asset prices is a problem that has fascinated investors since the very advent of financial markets. Accurate predictions of the market movements imply fast and substantial capital gains. Attempts to forecast stock prices are numerous. The prediction methods have undergone a polarization in two main lines:

• The fundamental analysis is based on the study of balance sheets of companies and the analysis of evolution of companies' wealth emanating from events related to their development as new contracts, overseas market expansion, debt levels, monetary, and financial policies. This is the more formal theory, embraced by the academia.

• This line of thinking has persisted mainly because of the success of the portfolio theory by Markowitz and the CAPM theory, with their beta coefficient development . Both theories have enjoyed a great deal of popularity and are based on probability, statistics, and mathematical tools dealing with problem of uncertainty.

• These theories assume that the market movements correspond with the news arrival which are unforeseeable and make the price changes to follow a random walk model. A random movement implies that stock prices are equally likely to rise and fall.
  The main criticism of this approach relies on the observation of the existence of rallies and cracks of prices with do not follow a normal distribution as to be expected under a random walk scenario. These trends occur more often than the randomness would predict. Finally a large number of practitioners of this approach do not consider it prudent to bet about the future and limit their work to compare balance sheets and to minimize risks by creation of diversified portfolios.

• Of course, there are some who try to predict stock markets anyway. And accurate predictions are monopolized by well-trained individuals knowledgeable about the basic mechanisms of economy who have information that could influence the markets and the economy. The technique used by these people is basically estimation of the reaction of investors to the news. This prediction technique is encouraged by the success of post-diction , that is, the explanation of price oscillations a posteriori as a consequence of known facts. These kind of explanation constitute an ill posed theory in the Popper's sense. One cannot state that these explanations are true nor false: at most one can say if it sounds reasonable.

• This analysis denies the idea that the market owns a dynamic in itself, their followers are permanently looking for explanations for each movement. Sometimes, not finding any news that can account for a market fall they ascribe the move to other markets, which is a circular argument.

• There are two difficulties associated with using this technique:
  1) the news arrival is random, and 2) the quantification of information in terms of price is extremely complex. Because of these difficulties the technique has not been too successful.

• On the other side we have the technical analysis. This approach to study of market price variations has several branches, but is well defined by the chart analysis. It is more an art than a science. The main contribution of this approach is its recognition that today and future prices are in some sense linked to the past prices. The technical analysts use to identify the local maxima and minima of price evolution as values in which the investors begin to consider a stock as expensive or cheap. This way they define resistance to the prices where the market shows a change in the upward trend and supports where the change is on the downward trend. Then they draw lines linking the maxima and minima turning points separately and extrapolate them linearly. The typical technical prediction is to determine prices up to which a stock can rise in case of an up trend and the price up to which can fall otherwise. The trials to determine the direction of the movement has not been very successful.

• With the arrival of a greater computer power new methods are available in understanding of the market dynamics. The findings in chaotic systems, the studies of complex systems, and the dimensional shrinkage are new approaches to study of the problem.

• There are great numbers of diverse systems in nature that exhibit very complex, apparently random behaviors that can an appropriately described by simple equations.

• As an example of those systems consider the human heart. It's a cellular swarm being individuals very similar to each other although not identical, interacting altogether all the time.
  A first approach to a description of the system would be to build up a model of myocardial cellules and study their interaction. This approach would lead us to a set of thousand of coupled differential equations certainly difficult, if not impossible, to solve. However if one observes the heart as a whole, it possess an harmonic behavior, that can be described by few equations. This drastic reduction of the number of variables needed to describe a phenomenon is the ´dimensionality reduction'. This kinds of emergent cooperative behaviors are typical in systems driven by the aggregate of a lot of interacting individuals.

• Another example can be found in the clouds. The interaction of simple water molecules floating in the atmosphere is capable of expressing as macroscopic emergent shapes that we call clouds, and among which we can recognize a great diversity of structures that worth having different names and qualities. Again dimensionality reduction. There are a lot of examples of complex auto organized systems that allow analysis and certain predictive capacity on a reduced set of dimensions. The developed tools to attack this kind of problems can be easily adapted to study market dynamics. Making the assumption that the financial markets are complex auto organized systems composed by similar individuals each trying to maximize their income, we can hope that a description of low dimensionality may be suitable and that certain forecasting capacity is possible. Taking into account that the price reflects all the available information, price series should be enough to study the market. If we accept that the news arrive in a random fashion we cannot do any effort trying to predict them, and we must restrict our system, and consider that it is randomly shocked externally. That is our model of the system under study will try to catch the behavior of a partial version of the real market. As the system we are trying to describe is a complex one we must begin our model with a nonlinear system of differential equations of dimension greater than two and externally randomly driven. These systems usually exhibit chaos (hypersensitivity to initial conditions) as a generic behavior, which limits the prediction horizon to a short term even in the case that no news were known. On the other hand the same hypersensitivity of the chaotic systems might cause a drastic change in behavior in case of arrival of news driven our system to run by trajectories completely different to those to which it would follow if the perturbation would not be present.

• We assume that prices do drive prices themselves. If prices are high, we only can find sellers, if prices are low, only buyers. The problem is to know what is considered to be a high or a low price, and if this concepts evolve in time, which is its evolution. Market oscillations are the result of the detailed balance of the expectancies of each individual that makes it up. Those who buy, they do it thinking that prices will raise, and those who sell, that prices will fall. When the equilibrium between those forces is broken a trend shows up. And more investors, knowing that persistent movements do exist, act in consequence causing this movements to exaggerate. This mechanism comprise a series of interaction rules between agents that make up the basic unit to create a model of stock market dynamics, which should a posteriori be finely tuned to reality. We suppose that a great number of such agents conforms our system. We now hope that from this system to emergent global behavior that can be described by a nonlinear differential equations system of a low dimensionality. This process can be performed assuming a dimensionality and then let it free to vary while trying to minimize a certain error function. Beginning with the time series of prices, following a method proposed by F. Takens a multidimensional trajectory can be build up. Then we suppose that the observed evolution in this space is determined by a system driven by perturbations produced by the new arrival, so the complete evolution is:
  
  News-> Free evolution-> News-> Free evolution-> News-> Free evolution-> ...
  
  
  
• Our task is try to get the free evolution rules. Unhappily we cannot separate the data corresponding to free evolution from the driven one, we will consider that the time series is very polluted with external noise that we will try to clean. One possible way to achieve this task is to forget the details of the movement and keep only a coarse grained form of evolution which do not lose the main features of the series we are interested in. This could be done by keeping just the major turning points of the price evolution. Then we numerically look for a set of equations that can account for this behavior and we extrapolate this behavior to the near future.
  
  
• In so doing we can mention the main results obtained for the DJIA, the S&P500 and the Merval.
  
  
• A low dimensional description of the system is adequate.
  
• The system during the free evolution is a chaotic one and due to the
  driven forces the trajectories are always far from the attractor of the dynamics.
• The proportion of goals in the determination of market direction in five days into the future is about 60%, in an study of ten years of daily data. This result is very useful on the speculative grounds, as a tool to determine a better moment to get into a position or to get out of it.
• Finally we must remark the fully coincidence of the predictions over all the three mentioned indices marking a downturn on Oct'97 .