Predicting stock exchange developments?

Since the mid-1990s a team of ULg researchers has applied a mathematical model to certain financial phenomena. This model has the special feature of integrating notions of chaos and unpredictability into a determinist system. Professors Jacques Bair and Gentiane Haesbroeck have jointly written a publication offering an overall synopsis of this research (1).

Mathematical modelling has earned its spurs over recent decades. More and more reliable, it has enabled the description, and at a lower cost, of varied and complex phenomena in a summarising, intelligible and interpretable way. Whether they have their origins in disciplines as diverse as climatology, oceanography, medicine, biology or astrophysics, to mention just these examples.

In the exact sciences, after numerous approximations, retouches, improvements in basic equations and tests, the models have offered descriptions which are sufficiently close to reality, both from a descriptive point of view, which has allowed a synopsis and interpretation of information which is otherwise too large and numerous, as well as from a predictive perspective. The value of their use no longer needs to be demonstrated, and they have been able to establish their authority and establish their importance within current research programmes.

Inevitably the capacity to describe past phenomenon which are otherwise unintelligible, and more widely still to predict more or less reliably are two attractive abilities. And certain models have thus rapidly been applied to the human sciences, and more particularly to the economy and the world of finance, as coveted as it is obscure.

How many works of fiction, films or novels, have imagined a miracle recipe for getting rich in Wall Street? How many economists have dreamed of being able to predict major stock market crashes, and being established as heroes in the pantheon of capitalism? And what if mathematical models could render these works of fiction and dreams obsolete?

Between the linearity of models and the chaos of life

In the middle of the 1990s scientists from different countries were looking to develop determinist models which could be applied to the chaotic world of finance. At the University of Liège one research group was built around Professor Jacques Bair: it took a very close interest in the subject and was one of the first to transmit this new mode of thinking into francophone Belgium. ‘With Professor Crama we created the GEMME (Groupe d’Etude des Mathématiques, du Management et de l’Economie). We were supported by many fledgling researchers, at the time, and were very active on the question of the integration of mathematics in the world of the economy, and we published a whole series of articles, notably with Professor Haesbroeck.’

The starting hypothesis postulated that it was possible to integrate random variations into a determinist model and bring to light certain regularities within it. The first equations used were relatively simple. They concerned linear functions, accepting the existence of a proportional link, a cause and effect relation between several points situated at equal distance over time. ‘These equations allow regular and exponential notions, such as the theory of compound interest, to be modelled,’ explains the researcher. ‘But they do not allow other much more unpredictable financial realties to be envisaged, such as the evolution of a market index or the formation of a product’s price.’

To take up a citation by Ian Stewart, quoted by Professor Bair, ‘the science of today shows us that nature is ruthlessly non linear.’ The idea which followed was to start from a non linear discrete dynamic model, in other words a model presented (in the most simple cases) in the form of Y_t+1 = f(Y_t). According to this equality one can calculate the value of Y_t+1 at time t+1 on the basis of this Y_t at a previous time t, thanks to the f function being considered, which is not linear but makes a parameter intervene. Such a model enables a notion of chaos to be embraced within a determinist system, and thus allows predictable and unpredictable magnitudes to be modelled.

Chaos can be characterised thanks to properties of which the main ones are the density, which defines the ensemble of the variations of the function used, the sensitiveness to the initial conditions, or the butterfly effect, which allows us to understand that a slight modification of the initial condition can generate considerable changes, and the order. This latter notion enables us to see that ‘ordered structures can appear within fluctuations which seem random.’

This equation has enabled several phenomena to be modelled, such as the evolution of a market index or the establishment of prices. In the latter case, a simply linear equation allowed a monopoly market to be brought to light, in which a business company can raise prices at its leisure. By integrating an extra parameter the reality becomes more unpredictable. This parameter, competition, given in the example offered by the article, has a weakening effect, a negative feedback, which thus introduces a non-linearity, and regulates the evolution of a product’s price. ‘We noticed that, depending on the variation of this new parameter, the predictions were either purely and simply chaotic, or that a regularity was established within the chaos.’

Applied to finance, this type of model, called a logistic model, very obviously has many limitations. It does not take into account all the factors which govern the reality of the evolution of an economic magnitude, whether it is a market index or the establishment of the price of a product. It does however allow a certain regularity to become apparent within the magnitudes studied, and to predict that certain results will occur. But the model does not show when they will take place. ‘Certain colleagues have all the same succeeded in predicting a stock market crash. But then again, there have been so many crashes,’ cautions the mathematician. ‘How do we know that it wasn’t purely a matter of luck? In any case today we have put that to one side. We have done the rounds with it, mathematically speaking. From a general point of view, economic and human phenomena are so complex that I don’t know if we will one day really be able to model them reliably. Nevertheless, from a philosophical perspective, it was valuable to show that it was possible to model, to bring to the forefront certain regularities which seemed a priori completely random, that we could consider integrating the notion of chaos into a purely determinist mathematical model, and to do so in integrating and varying an extra parameter. It is for that reason that we wrote the article ‘Chaos Models in Economics’.’

Full steam ahead towards statistics

If it seems difficult to reliably model economic phenomena subject to numerous unpredictable factors, such as human actions (speculation, fears in times of crisis, demographic changes, political influences, etc.) or natural disasters (which can have a brutal influence on exchange rates or raw materials, for example), numerous mathematicians are still working to approach reality ever closer, but are leaving behind determinist models and more and more taking the direction of statistical or probability models, which take more into account notions of chaos and randomness. In this area, even today and after forty years of existence the Black and Scholes model seems to be the authority. It is used in all the financial markets. This model enables a modelling of stock exchanges according to several conditions and parameters, determining the theoretical value of an option whilst taking into account that this value at time t is the product of a purely stochastic process.

Models and future perspectives for mathematicians

In times of crisis even the boldest are twice as prudent. And the shrewd advice of seasoned mathematicians can prove useful. If there is indeed one point which can be predicted without a model, it is that mathematicians can more and more picture prolific careers outside of teaching. The models applied to the economy are not always reliable, but they allow shadowy areas to be looked into with some seriousness and allow them to be seen more clearly. Insurance companies which work in actuarial research groups, the banks and the financial markets are making greater and greater use of the expertise of mathematicians. More generally the discipline can open outwards to a variety of areas, such as sport, music, agronomy, etc. sparking great interest and revealing to students who have little interest in mathematics vocations which have been well hidden up until now. Jacques Blair has understood this, and is today devoting his time at the end of his career to didactics and rendering his passion accessible to a wider public, notably in working with the French periodical, Tangente.

(1) Bair J, Haesbroeck G., Modèles chaotiques en économie, Tangent sup., Prévoir pour décider, POLE, 2012.