“My life seemed to be a series of events and accidents. Yet, when I look back, I see a pattern.”

So said Benoit Mandelbrot, mathematics legend and the father of fractal geometry. A self-described “wandering scientist,” pursuing what he called “unpredictable interests,” he moved across many disciplines at once to find new insights.  

The things he wanted to investigate were not of interest to anyone, so he spent much of his life as an outsider, seeking to extract an element of order from physical, mathematical, or social phenomena that were otherwise characterized by wild variability. The chaos and irregularity of the world, Mandelbrot believed, is something to be celebrated. 

Behind the veil of apparent complexity, Mandelbrot recognized that many of the seemingly disordered patterns of nature are highly ordered, following simple rules. He coined the term “fractals” in 1975 to describe a geometric shape that can be separated into parts, each of which is a reduced-scale version of the whole. 

For instance, the veins in leaves look like branches; branches look like tiny trees; and rocks look like miniature mountains. The same repeating patterns can be seen in snowflakes, rivers, broccoli flowers, and blood vessels. 

According to Mandelbrot, the geometry that describes the shape of coastlines and the distribution of galaxies elucidates the variability of markets. “The very heart of finance is fractal,” he said. The movements of a stock all look alike when the chart is enlarged or reduced so that it fits the same time and price scale.  

The patterns seen on the intra-day charts of minutes or hourly intervals are very similar to the patterns seen on the daily, weekly, monthly or even quarterly charts. This qualifies the charts as fractal curves, which opens the door to powerful mathematical and computer analysis.

Mandelbrot saw financial markets as an example of “wild randomness.” “Markets, like oceans, have turbulence,” he said. “Some days the change in markets is very small, and some days it moves in a huge leap. Only fractals can explain this kind of random change.” 

It is true that standard models deny the existence of dramatic market shifts; psychological complexity is overlooked even as traders act on fleeting expectations of what may or may not happen. “Sheer phantasms” as Mandelbrot put it.

Here are Mandelbrot’s five rules of market behavior. 

(1) Markets are risky: Extreme prices swings are the norm in financial markets, not aberrations that can be ignored. Price movements do not follow the well-mannered bell curve assumed by modern finance, they follow a more violent curve that makes an investors’ ride much bumpier. 

(2) Trouble runs in streaks: Big or small price changes are followed by more of the same kind. Market turbulence tends to cluster. Mayhem in one market spreads instantaneously to all others—and we only have vague notions of how this happens, or how to regulate it.

(3) Markets have a personality: Prices are not solely driven by real-world events, news, and people. When bankers, investors, speculators, and industrialists come together in a real marketplace, a special, new kind of dynamic emerges—greater than, and different from, the sum of the parts. 

(4) Markets mislead: The power of chance suffices to create spurious patterns and pseudo-cycles that appear predictable and bankable. But a financial market is particularly prone to such statistical mirages. Bubbles and crashes are inherent to markets. They are the inevitable consequence of the human need to find patterns in the patternless. 

(5) Market time is relative: There is what one may call a relativity of time in financial markets. Trading time speeds up the clock in periods of high volatility and slows it down in periods of stability.

These cold, hard facts will be built into sound trading strategies and sensible portfolios.  

If we think we can identify every catalyst and control or predict outcomes, we are setting ourselves up for a fall. Mandelbrot’s fundamental concept is that prices are not predictable, but their fluctuations can be described by the mathematical laws of chance. Therefore, their risk is measurable, and manageable. 

In his 2004 book, The (Mis)Behaviour of Markets, Mandelbrot wrote:

The precise market mechanism that links news to price, cause to effect, is mysterious and seems inconsistent. Threat of war: Dollar falls. Threat of war: Dollar rises. Which of the two will actually happen? After the fact, it seems obvious; in hindsight, fundamental analysis can be reconstituted and is always brilliant. But before the fact, both outcomes may seem equally likely. So how can one base an investment strategy and a risk profile entirely on this one dubious principle: I can know more than anybody else?

Fractals follow the mathematical principle of self-similarity, which means they are replicated at different scales and innumerable times, allowing for predictions about how the structure will behave in the future. 

Consider this chart: Glencore is at the top, while Coinbase is at the bottom. We spot an opportunity. 

Source: Koyfin

Similar to how Glencore’s IPO in May 2011 signaled the end of the commodities super cycle, Coinbase’s public listing in April 2021, when Bitcoin was worth more than $63,000, heralded the top in digital assets.

Glencore’s shares peaked at 559 pence on its market debut before dropping by 88 percent to as low as 66 pence during the 2015 commodity crash. 

“If enthusiasm surrounding its public offering coincided with the top in commodities,” we wrote at the time, “shouldn’t comparisons with Lehman and revulsion from a credit downgrade suggest that we are close to a bottom.”

In six months, Glencore’s stock price doubled. It was trading at 378 pence two years later. Glencore has finally regained its IPO price this year following another intervening decline of 70 percent (from 2018 to 2020).

Coinbase’s stock opened at $381, reached a high of $429 on the first trading day, and then dropped by 90 percent to just above $40 on May 8. What do you suppose will happen next after looking at the chart?

As we try to unscramble the complexity of markets, we are always seeking a measure of order from the apparent randomness. It is interesting to us how markets (mis)behave. As Mandelbrot said himself, “Beautiful, damn hard, increasingly useful. That’s fractals.”