Developing Fractal Ranges- Part 1

Why the Fractal Range?

Fractals are omnipresent, from the branching patterns of broccoli to the way light interacts within a prism. The world is constructed on a framework that is simultaneously beautiful and ineffable. If you haven't explored the work of Robert Edward Grant, it's worth a look.

It's logical, then, to assume that we might find similar natural patterns in markets. The Misbehavior of Markets by Benoit Mandelbrot and Richard L. Hudson echoes this sentiment and is frequently cited by Keith. I highly recommend this book; it has enriched my perspective on market modeling and will be a reference for my future endeavors.

Fractals, Mandelbrot, and Rate of Change

Mandelbrot observed that financial market data exhibited fractal characteristics. That is, data patterns recurred at different scales, akin to a fractal.

Fractal Hunting with Hurst Exponents

Building on the foundational knowledge of fractals, the Hurst exponent, often denoted by the letter H, emerges as a pivotal tool to measure and detect fractals. Here's a simple analogy:

Imagine observing a drunken person's walk; there are three possible scenarios:

Random Walk (H=0.5): The individual stumbles aimlessly, with no discernible pattern to their steps. In financial terms, this mirrors purely random price movements.
Trending (H>0.5): The person is inclined to move in a particular direction, say, towards a taco truck. Their path, though not straight, generally leads them to the truck. In trading, this suggests that an upward-trending asset might continue its ascent.
Mean Reverting (H<0.5): Here, the person tends to gravitate back to a starting point, like a lamppost, after wandering away. In the context of stocks, this resembles prices that revert to a "normal" level after reaching extremes.

The Hurst exponent helps categorize a series (like stock prices) based on these behaviors:

If H is close to 0.5, the series resembles random noise.
If H is significantly above 0.5, the series is trending.
If H is significantly below 0.5, the series is mean-reverting.

Applications to Financial Time Series Data

Fractals represent patterns that recur at varying scales. Envision zooming into a tree's image; its branches mirror the tree's overall structure. This essence captures fractals.

Given that fractals are repetitive patterns across scales, the initial step involves analyzing data over different time windows or "lags". I'll begin with the trade, trend, and tail durations used in Hedgeye, aiming to gauge the series' behavior over each duration. Employing the Hurst exponent, I anticipate quantifying the series' tendencies across time frames.

If the time series exhibits fractal characteristics, the Hurst exponent's value should remain consistent across scales. This consistency would suggest the series possesses fractal attributes.

Final Thoughts

The Hurst exponent might prove instrumental in identifying fractals and their behaviors. When such behavior remains consistent across time scales, it underscores the series' fractal nature.

One avenue worth exploring is Fourier analysis, which could decompose the time series into its constituent cycles, aiding Hurst analysis. However, this might be a diversion, and revisiting Mandelbrot's book could offer clarity. Another strategy might involve using data as inputs for generator objects to produce a combined signal.

Ideally, I'd like to combine both methods with a Markov Chain Monte Carlo (MCMC) simulation to predict probable price ranges. I'll delve deeper into these concepts in my ongoing series on crafting fractal ranges for trading in financial markets.

Warm regards,