Fractals in AI Based Trading

Benoit Mandelbrot, the French Mathematician, determined that only fractals are capable of capturing the reality of markets.

This is an abstract object used to describe and simulate natural occurring systems. Mandelbrot explains it as, “A fractal is a shape made of parts similar to the whole in some way.” This means as one zooms into the object, the smaller parts resemble the larger ones.

Fractals are similar to many naturally occurring forms such as trees, DNA, lighting bolts, ocean waves, among others. For markets, we need to focus on ocean waves.

As wind blows across a fluid surface, it causes the fluid to form waves. In the case of ocean waves there are aspects of randomness due to the fact that waves different in height, shape, and duration. With no real way to predict any of these factors.

Waves are classified as a stochastic process. Meaning that they are objects made up of random variables. Stochastic processes are used as mathematical models for systems that appear to be very random in nature.

Markets, like ocean waves, have turbulence that can be modeled with fractals. One day a person can visit a beach and see calm smooth water with little change. A few days later the same water will be violent with fast moving waves crashing onto the shore. Markets do the same thing.

Fractals are formed by recursion. Here you take a simple equation, calculate it, utilize the answer to re-calculate it and repeat in a never ending loop.

Before we get back to markets there must be a discussion on the Coastline Paradox. This is the concept that scale matters. When measuring a coastline with a ruler (12 inches long) one would get a much larger total than if they used a rod one meter long and smaller still if they used an even longer pole.

The reason is the amount of irregularity lost at the larger scale. Consider a stock chart zoomed out to one year. It appears as a smooth wave like shape. Zoom into that chart at one month and the smoothness disappears. It is now a series of tiny up and down bumps instead of a smooth line. Change the scale again to one day and the bumps become more visible and much more dramatic.

Even thought the chart becomes more detailed the more it is zoomed, the shape is similar over the course of a year, month, or even day. Fractal Geometry provides the math to study such patterns that remain the same through scales of time change. A phenomenon known as invariances in mathematics.

Fractals do not predict market prices. They do provide estimates of the probability of what the market might do in the future. Giving us a prediction of volatility instead of price. This is due to the fact that volatility clusters. Big jumps come just before other big jumps. Small change comes just before small change.

Predicting volatility is similar to predicting the weather. Both are largely correlated with recent events. However, sometimes it rains when it was suppose to be sunny.

Price movements are probably unpredictable and do not conform to a bell shaped distribution. That is we cannot say something like, “this stock will be up $2 by Monday.” Yet prices are not mutually independent or randomly distributed. Meaning we can predict volatility and AI should be able to do this even better.

The most exciting use of AI in trading right now is discovering fractal type patterns in the prices of exchange traded assets. These may be as distinct as fingerprints.

Funds are already attempting this with Deep Learning and other types of AI systems. Fractal based patterning can be used for stock trading but could be much better suited to options trading where volatility prediction is the main objection. Maybe even replacing the current pricing model of Black-Scholes.

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