# It is All About Fractals – Tech Stocks and The LPPLS Model (Remastered)

Romain/ octobre 16, 2020/ Finance, Science/ 3 comments

The concept of fractal is fascinating. But it is also pretty abstract.

If you read papers or books written by French mathematician Benoit Mandelbrot (aka the father of fractals), you may find meaningful things about nature, geometrical properties and even market prices dynamics. Indeed, Mandelbrot showed a long time ago that the behavior of asset returns is not distributed following a normal law, even if in 2020, many finance professionals still use risk models assuming Gaussian distributions, such as the Black-Scholes option pricing model or VaR calculation.

Nevertheless, Mandelbrot’s works are not necessarily easy to assimilate, and you may realize that it is complicated to formulate a rigorous definition of the fractal concept. In fact, many definitions can be found online, but Mandlebrot disagreed with many of them. He summarized the concept as “beautiful, damn hard, and increasingly useful”, but one could argue that it is not very helpful.

An interesting illustration of the fractal dimension concept was more recently put forward by French physicist Didier Sornette in his worth-reading book *Why Stock Markets Crash: Critical Events in Complex Financial Systems*. According to Sornette, “the fractal dimension *d* quantifies precisely how the relative length *L(ε**)* changes with the ruler length *ε *(which we also call ‘resolution’).”

In other words, fractal objects can be regarded as objects that display similar properties whatever the “resolution”. Such a concept is also known as *scale-invariance*.

## “Priming the Pump”

Capital markets perfectly fit with that definition, as the price of a stock or a bond exhibits similar patterns, whether you consider an intra-day time horizon, a week, several months, or many years. Of course, experts in technical analysis have known that for decades, but the interesting part is that more complexed dynamics seem to replicate themselves on very different time horizons.

For instance, the action of Western authorities to support asset prices since the Lehman Brothers crisis can be described as follows:

- Provide liquidity in order to enable sellers to reduce their risk in good conditions.
- Ease monetary conditions so as to incite more participants to cease investment opportunities and force a short squeeze.
- Keep an optimistic tone, whatever the situation, from an economic or political perspective.

The fourth step, which is more or less independent from the government or central bank action, is the fact that it tends to lead to overconfidence in the financial system, i.e. the formation of a speculative bubble.

What is the relation with fractals?

On a long-term horizon, this is what we have seen since the bottom of 2009. This is also what happened on a smaller period, from December 2018 to February 2020, and finally from last March to August. But more interestingly, it also happened recently on a very short-term horizon (i.e. from mid-September to last Monday). Exactly the same process, from an unofficial securities protection strategy to market euphoria.

## Stocks Mania and Preliminary Signs of Avalanches

More people have become familiar with the log-periodicity power law singularity (LPPLS) model, as I presented a simulation in July suggesting that a rupture would happen on the Nasdaq at the beginning of September (see It is All About Waves – Tech Stocks and The Log-Periodicity Power Law Singularity Model). Note that the Python package used for that simulation was developed by Joshua Nielsen and can be easily retrieved on GitHub.

Since then, I have received many emails of people wondering if the model could help us to understand what will come next. As I explained in a previous post, it is unclear whether the model would help us to understand if the correction of September is a critical rupture with the respect to the 2009-2020 bull run, or just a pullback with more extreme levels coming after that (see Attack of the Clones – A Tech Bubble Story).

However, I realized a few days ago that their questions could be examined through a new angle. What if, instead of running the model on the same multi-year period again, we chose to focus on a smaller time horizon (i.e. a few of days)?

As already explained before, if stock prices are fractal objects, then the same dynamics could exist on different horizons. Said differently, the LPPLS model should work independently of the length of the period.

That intuition lead to a new simulation on Nasdaq 100 future contract as markets became out of control last Monday:

The conclusion of the LPPLS model was that a correction might happen before the open of Tuesday. And everyone can see that it was once again pretty accurate (see chart below).

Note that the model does not tell us what comes next. Authorities have proved that they are experts in markets pumping, so everything is possible. But an endogenous rupture imply that the dynamics has been broken and that markets have become weaker. In other words, the cost of the protection strategy is increasing at an exponential rate.

## Markets, Narratives, and Sentiment Propagation

What is exactly that critical time detected by the model? Why are markets so vulnerable at that particular moment?

Remember that it is all about fractals. Capital markets can be modeled as complex networks connecting investors to one another. Participants tend to have different opinion, and different narratives compete with each other.

Like any other network, markets are vulnerable to bad news, which could be regarded as virus that propagate through the system. The more investors are connected to the same hub, meaning the more they share the same opinion on prices, the more vulnerable the system is.

Of course, bad news could be external events, like a real virus for instance, a natural disaster, or whatever. But they could also be endogenous events. This is more complicated but really interesting.

When the market reaches such a euphoric moment, everyone is highly interconnected, and almost all investors obey to the same dominant narrative (i.e. “stocks always go up”). Sornette and other econophysics researchers have shown that the market becomes a fractal object (or a scale-free network), with the same structure and the same opinion on stocks whether you consider it as a whole or if you focus only on a smaller part.

At that moment, prices become their own fundamentals, which could explain why so many people become passionate about technical analysis. But this is a very dangerous situation. Indeed, volumes tend to be low, and as soon as a large seller come to the market (i.e. a significant fund willing to take some profit), then fundamentals become suddenly negative, resulting in a brutal negative feedback loop that propagates very quickly through the entire system.

The outcome for the market is a brutal sell-off and a rupture of the bullish dynamic.

## It is All About Cats?

This is what we saw on September 3^{rd}, and more recently on October 13^{th}.

Once again, it is very complicated to anticipate what will happen next. But at least, those two theoretical predictions of the LPPLS model led to nice cat ears (please note that I am not saying that there is any market signal here):

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