Developing Fractal Ranges- Part 3

Up the learning curve

Now that I have some ideas around basic computing of the Hurst exposnent using varying parameters and inputs, I would lke to take a step back and think about the bigger picture of the models architecture. Here I would like to lay out some potential building blocks and how we could apply them to model asset prices.

Hurst Exponent

As discussed in more detail in part 2, the Hurst exponent can give an indication on the tendencies of price over various time scales. I can get an idication if an asset is more likely to trend or revert to the mean for example. However, one of the challenges is deciding what type of data to feed the calculation to get the most useful measure.

Second Derivatives

Another likely building block is rate of change data. However, not just determining the general slope or "momentum" but looking at the second derivative or "bending" of the momentum. What I am really measuring is the acceleration. For asset prices, this could indicate acceleration in the rate of change, which may be predictive of future movements.

These two equations are a good starting point; but it may be worth investigating more advanced methods like Savit-Gam....

First Derivative (Momentum)

ΔP(t) = P(t) - P(t-1)

Second Derivative (Acceleration)

Δ²P(t) = ΔP(t) - ΔP(t-1)

Combining the Hurst exponent along with the acceleration may provide a useful measure of when an asset is both in a trending state as well as accelerating in that trend-

Another approach would be to look for mean reverting assets that reflected a deceleration or stalling. This could indicate a likely reversion opportunity as a trader.

Generators for fractals

Fractals are patterns that look similar at any scale. Like a snowflafe, zooming in on any part reveals a picture similar to the whole snowflake. The video "Powers of Ten," which was released by Charles and Ray Eames in 1977 is known for its unique portrayal of relative scales. It starts with a view of a couple having a picnic and then zooms out and in by powers of ten to show the vastness of the cosmos and the microscopic world.

Powers of Ten

Fractals can be generated using Iterated Function Systems (IFS). An IFS can be thought of as a shrinking function. For example if you have a picture and you would like to create a smaller version of that picture. You could take the original picture and apply a "shrinking" rule to it.

In an IFS there is a set of shrinking rules and each rule makes the picture smaller, but in a specific way. In technical terms these are called contraction mappings. These rules are iterated over and over applying the rules each time. Since the rules are different, the picture gets smaller but in different ways based on the underlying rules.

Perhaps the hurst exponent could be used to help generate fractal patterns that are more similar to historical data. A lot of possibilities to explore around these ideas.

Possible generator structure

A good visualization is to think of a simple snowflake. This is the base shape or base fractal. I can then parameterize the Fractal. This makes the snowflake more flexible; instead of having one fixed shape, I can change it based on my needs. For example:

Changing the size of the snowflake
Changing its "sharpness"
Adding Glitter (Noise)

Incorporating the second order derivatives of price, volume and volatility and the hurst exponent could be structured in this way:

Applying ROCs

Changing Size (Scale of the Fractal):

Price: If the second derivative of price (acceleration/deceleration of momentum) is large in magnitude, it could mean a more significant market event, leading to a larger fractal. Conversely, a small magnitude could lead to a smaller fractal.
Volume: A large second derivative of volume indicates rapidly increasing or decreasing trading activity, which could similarly be translated to the scale of the fractal.
Volatility: A sudden spike in volatility's second derivative could result in larger fractal patterns, representing market uncertainty.

Changing Sharpness (Roughness of the Fractal):

Price: If the second derivative of price is frequently changing signs, it indicates a lot of fluctuations, leading to a rougher fractal. Stability in the sign (either continuously positive or negative) might lead to smoother patterns.
Volume: Rapid changes in the second derivative of volume could lead to a fractal with sharper and more jagged edges, representing inconsistent trading activity.
Volatility: Sharp changes in the second derivative of volatility will translate to a more chaotic and rough fractal.

Adding Glitter (Noise):

Price: If there's a lot of "noise" or random movements in price that don't necessarily follow the broader market trend, you can sprinkle these random changes into the fractal, like glitter on a snowflake.
Volume: Occasional, unpredictable spikes in volume could be translated as noise in our fractal.
Volatility: Random and short-lived spikes in volatility can also be represented as noise.

Applying Hurst

When applying the Hurst Exponent to fractal generators, I can adjust the complexity and depth of the fractal patterns. A higher Hurst value, suggesting persistence, might lead to more intricate and deep fractals. In contrast, a lower value can result in simpler and shallow patterns.

By comparing the information from the Hurst Exponent with the acceleration or deceleration metrics derived from second derivatives, I hope to discern when current market momentum is likely to continue or revert. For instance, a strong persistent trend (H > 0.5) combined with a positive acceleration in price could indicate a very strong upward trend.

Puting it all together

By considering the Hurst Exponent alongside the second derivatives of price, volume, and volatility, I hope to get a more nuanced picture:

The second derivatives tell me about the immediate acceleration or deceleration in price, volume, and volatility. The Hurst Exponent informs me about the long-term memory or trend of the time series.

In conclusion, applying the Hurst Exponent within this framework offers a more holistic view of market dynamics, bridging the gap between traditional time series analysis and modern techniques that leverage second derivatives and fractals.