Do fractals represent a revolution in price forecasting, particularly in relation to “modern financial theory”?

Fractal geometry is an observable element in many elements of our environment, whether it is nature or a simple cauliflower. It can also be used to describe and understand financial markets. Benoît Mandelbrot was the first to talk about fractal geometry (fractals) in finance.

Indeed, since the early 1960s, he has been using fractals to analyze stock markets. Therefore, technical analysis, which is “the study of the evolution of the supply and demand of a financial asset based on its graphical representation in order to predict its future evolution”, sees the fractal approach gaining importance.


The technical analysis is based on probabilistic models while the fractal geometry approach is mainly empirical. Indeed, until now, market analysis has been based on a Brownian motion model and Gaussian problems.

There were, in particular, first of all, academic works developed from the 1900s onwards by Bachelier: he proposed to model the stock market prices by a Brownian movement. It is on the basis of his work that the model of modern market finance theory was built.

In this model “the price of a share is a continuous linear function per piece, each daily increase being a random variable according to a normal Law”, a model that underestimates the strong falls in prices observed at the thick tails of the distribution.

Hence his questioning by Benoît Mandelbrot. The latter, in the early 1960s, discovered that the price of cotton, like other elements, followed an irregular pattern and that it was therefore difficult to predict price changes. His theory therefore challenges the understanding of the “modern financial theory” that currently governs stock markets.

He then speaks of fractals, i. e. geometric figures formed by price variation. We can then ask ourselves if fractals constitute a revolution in price forecasting, particularly in relation to “modern financial theory”?

Therefore, after presenting the fractal model and describing the phenomenon, we will analyze its contribution compared to the current model used for market analysis and will discuss in a conclusion, the limits and future of fractal geometry analysis.

The fractal model in finance: What is a fractal?

A fractal is a figure formed by price variation: “it is a geometric object that can be cut into small pieces and each end of which has the same structure as the whole”. On the stock market, the geometric object is a very irregular curve that gives the price of a share as a function of time.

The curves are defined according to a “landscape” format of given width and height. Mandelbrot rejects the general theory that a stock price itself contains “all the information of the past and that therefore any variation in the price is a function of new information. As this information is unknown, the variations should be random and therefore follow a normal distribution. However, this is not the case and sudden price changes are more frequent than expected.

Fractal characterization can be established by the Elliott Wave approach, the fractal indicator and structural theory. First, we will talk about Elliott’s wave theory.

“The starting point of this theory is that markets evolve through a series of successive waves, regardless of the scale of observation of this market (from the minute to the very long term). This is called fractal process. The complete movement is composed of 5 waves.

3 of these waves are in the direction of movement, and two waves are in the opposite direction. The first, third, and fifth waves represent the impulsive form, the second and fourth waves the corrective form. Elliott waves are based on the fractal nature of stock market charts and break them down into successive waves.

Next comes the Fractal Indicator. “The Fractal indicator is a technical indicator that helps to detect peaks and valleys. This indicator is composed of 5 successive candles, with the highest price in the middle and left and right of this candle 2 highest lowest prices.

This indicates a peak in the market. The trend here is rather downward. And conversely the indicator also gives a signal when a trough is formed: a series of 5 consecutive candles, with the lowest price in the middle and left and right of this candle 2 lowest highest prices. This may indicate an upward trend.

Finally, the structural theory. This is the most recent theory developed for forecasting stock prices. Structural theory is a development of Elliott waves and fractal geometry (use of fractal analysis in stock price formation).

It is about decoding the fractal by the appearance of a structure in the graph. The theoretical model is based on the variation of the prices of a set of rules resulting from observation (e. g. Elliott Waves) and allows anticipation because it implies a certain determinism in the evolution of prices.

What has structural theory contributed to the traditional stock price forecasting model? First of all, there is the opposition to the classical model. Mandelbrot denounces by his contribution of fractals the mathematical tools that he considers unsuitable. Classical market theory says that stock prices follow a Gaussian representation. It considers that upward or downward variations are very rare and that strong variations are also rare.

However, Mandelbrot considers that the Krachs are proof that the courts do not follow a normal law. Indeed, for example, a variation of more than 10% of the Dow Jones is unlikely according to classical theory, yet in October 1987, the Dow Jones experienced a violent variation. Thus, for Mandelbrot, the classic model underestimates the risks of financial investments and leads to financial bubbles followed by crashes.

Mandelbrot then proposes the fractal model, a more realistic alternative that describes stock prices better than standard models. It provides a better understanding of markets and thus helps to avoid crises. “Fractals make it possible to simply study the prices and behaviour of a stock exchange, indicating its degree of variability by a single number, the fractal dimension of stock price curves. A number between 1 (the straight line) and 2 (the surface), which can help to better assess the risk of a sudden fall.

Then, on the other hand, the revolutionary contribution of fractals. Markets have proven not always to be efficient. Fractals confirm this theory, because fractals can only be used in an inefficient market. Indeed, the self-similarity that characterizes them, i.e. the “invariance” of the form whatever the scale, is impossible in a Brownian movement. In addition, they are not subject to random movement (which means that each point does not depend on the preceding point).

In addition, fractals are used to understand the complexity of markets, explain complex movements and regulate financial bubbles, and to detect financial bubbles. A bubble is characterized by the fact that prices deviate from the usual economic valuation due to the belief and speculation of buyers. Prices that move too consistently upwards or downwards are the characteristic of an anomaly reflecting behaviour that is too sheepish and therefore unstable.

Fractals are used to define and affirm whether we are witnessing a bubble. Associated with the Hurst coefficient, a fractal dimension of 1 corresponds to a Hurst coefficient greater than 0.5, i.e. we are witnessing an amplification of events, whether positive or negative, the probability of observing the same event later will be higher.

Indeed, “fractals make it possible to study the prices or behaviour of a stock exchange by indicating its degree of variability by a single number, the fractal dimension of stock price curves. A number between 1 (the straight line) and 2 (the surface), which helps to better assess the risk of a sudden fall. In this respect, fractals represent a real revolution!

In conclusion: what are the limits and advances? What future for fractals?

To say that markets are efficient, does not allow the use of fractals. But many people still think that markets are efficient, while the various crises call this efficiency into question. Fractals are not yet used and understood by most people.

Knowledge about them remains limited, despite the various works since 1962, the date of Mandelbrot’s first studies. These advocates will say that to get there, we have to start recognizing that markets are inefficient. However, few financiers question the criticisms and reservations about the standard financial theory known as “orthodox”. Moreover, many tools have begun to be developed to fill in the gaps and get closer to reality.

However, what Mandelbrot can be criticized for is that he only described the observable phenomena in the markets explaining how the markets are and not how they should be. Moreover, Mandelbrot says in his book “that much work remains to be done, and that other concepts must be explored as soon as possible”. It thus opens the way to another trend, the “multi-fractal” analysis.

This fractal model, unlike other financial models, requires very little data to provide much information: “it starts from the fundamental and sustainable facts of market functioning. It is economical, flexible, and imitates the market. Some believe that they may one day form the basis for much-needed financial market regulation. Nevertheless, Mandelbrot opened the debate on how to model financial markets while taking into account reality.



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