# A Review of the Fractal Market Hypothesis for Trading and Market Price Prediction

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## Abstract

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## 1. Introduction

#### 1.1. On the Study of Risk

#### 1.2. Basic Technical Issues in Financial Analysis

- The Price Filter Approach where the strategy is to buy or sell after some price increases or decreases by some pre-defined percentage.
- Moving Average Approach, where a buy or sell is generated after the price moves above or below a longer-term rolling average.
- Support and Resistance Approach, which is based upon the principle that most trends begin when a price of a commodity breaks out from a fixed price range. This type of strategy seeks to buy or sell when the price rises above or below a local maximum.
- Channel Breakout Approach, which is defined as a region within which a high price (taken over a number of periods) is within a pre-defined percentage of the corresponding low price (over the periods considered).

#### 1.3. Financial Time Series Analysis

#### 1.4. Structure of the Paper

## 2. The Random Walk Hypothesis

## 3. The Efficient Market Hypothesis

#### 3.1. The Modern Portfolio Theory

- financial returns do not follow Gaussian distributions;
- correlations between asset classes are not fixed, but instead, vary depending on external events, especially in times of crises.

#### 3.2. The Black–Scholes Model

#### 3.3. Value and Limitations of the Efficient Market Hypothesis

- (i)
- price increments exhibit statistical stationarity in which samples of data taken over equal time increments can be superimposed onto each other in a statistical sense;
- (ii)
- the scaling of prices can be suitably re-scaled such that they too, can be superimposed onto each other in a statistical sense;
- (iii)
- price changes are statistically time independent.

## 4. An Overview of Fractal Geometry

#### 4.1. Self-Similar Functions

#### 4.2. Self-Affine Structures

#### 4.3. The Mandelbrot Set

#### 4.4. Multi-Fractals

#### 4.5. Self-Affine Functions and Fractional Calculus

## 5. The Fractal Market Hypothesis

#### 5.1. Nonlinear Dynamics and Chaos Theory

#### 5.2. Fractals and Finance

#### 5.3. Black Swans

- (i)
- it is an outlier which lies outside the realm of regular expectations, where nothing in the past points to its possibility;
- (ii)
- it carries an extreme impact;
- (iii)
- in spite of its unlikelihood, human nature leads to the concoction of explanations after the fact, thereby attempting to make an event explainable and predictable [99].

#### 5.4. Non-Gaussian Distributional Analysis

- (i)
- peaked with long-tails—Leptokurtic;
- (ii)
- non-symmetric—skewed;
- (iii)
- statistically non-stationary in regard to both (i) and (ii).

## 6. Mathematical Models for Financial Hypotheses

#### 6.1. The Random Walk Hypothesis

#### 6.2. The Efficient Market Hypothesis

#### 6.3. The Fractal Market Hypothesis

#### 6.3.1. Memory Function for the FMH

#### 6.3.2. Asymptotic Rate Equation

## 7. Case Study: Trend Analysis and Price Forecasting of Cryptocurrencies

#### 7.1. Long-Term Prediction

#### 7.2. Short-Term Market Price Prediction

- the inclusion of a larger number of the functions available with the TuringBot, for example, including ‘Other functions, Logical functions and History functions’;
- the effect of varying the period (window size) used to compute the formulae, and the accuracy of future predictions for a varying number of future projections;
- the correlation between the predictive accuracy, and the memory function, based on an estimate of $\alpha $ for different sets of price data;
- the effect of smoothing the data (for a specific period) prior to formulae evolution on the predictive accuracy of the formulae.

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Example of a three-dimensional random walk simulated using the Matlab Econometrics Toolbox [21].

**Figure 2.**The Cantor Set for six iterations giving ‘fractal dust’ with a fractal dimension of $0.6309$.

**Figure 3.**The standard Mandelbrot set (left) for ${i}^{2}=-1$ and the nonstandard set (right) for ${i}^{2}=+1$. Both sets are computed for $x\in [-2,1]$ (horizontal axis) and $iy\in [-1.5i,1.5i]$ (vertical axis) for a ${10}^{3}\times {10}^{3}$ grid and 100 iterations [66].

**Figure 5.**Simulation of a financial time series ${u}_{n}$ based on the Random Walk Hypothesis (below) using Equation (12) for $N=10,000$ and a stationary zero-mean Gaussian-distributed stochastic field ${r}_{n}$ (above) for the price differences (where only the first 3000 samples are displayed).

**Figure 6.**Left: Comparison of a Lévy distribution (red) for $\gamma <2$ when $p\left(x\right)\sim 1/\mid x{\mid}^{1+\gamma}$ with a standard Gaussian distribution (blue) when $\gamma =2$. Right: Comparison of a Gaussian distribution with the distribution for price changes of a Cryptocurrency (Bitcoin–USD exchange rates). The Figure on the right hand side illustrates the incompatibility of the distribution for price changes (grey bars and blue line) with a normal distribution (red line) and its similarity with a Lévy distribution which is peaked at the centre and has significant bars in the tail [142].

**Figure 7.**Comparisons of a financial signal simulation based on the RWH when $\gamma =1$ (above), the FMH when $\gamma =1.5$ (centre) and the EMH when $\gamma =2$ (lower plot).

**Figure 8.**Trend analysis of daily opening BTS–USD from 17-12-2020 to 30-11-2021 after normalisation. The BVR (

**left**) and the LVR (

**right**) are computed using the function Backtester(40,35,350) given in [14]. In both cases, the normalised BTS–USD data are plotted after application of a moving average filter (red lines). The green lines shows the post-filtered BVR and the LVR signals. The zero-crossing positions (blue lines) indicate the points in time where a change in the trend of the filtered signals takes place.

**Figure 9.**The normalised 5-day rolling average of the opening BTS–USD prices from 17-12-2020 to 30-11-2021 (red line), the corresponding Volatility of the signal (green line), and the Lévy index (blue line) computed using a moving window with a period of 5.

**Figure 10.**Example of daily price predictions (over ten days) for the opening values of the daily BTS–USD using the TuringBot to evolve formulas that sequentially predict a future price, one-day at a time, using the previous 40-day prices. The actual price data are given by the blue line and the sequential single day predictions are given by the red line for BTC–USD exchange price values from 21 November 2021 to 30 November 2021.

**Figure 11.**Example screenshot of the TuringBot Symbolic Regression Software Graphical User Interface [151] showing the evolution of formulas to simulate the 40-day BTS–USD price values.

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**MDPI and ACS Style**

Blackledge, J.; Lamphiere, M.
A Review of the Fractal Market Hypothesis for Trading and Market Price Prediction. *Mathematics* **2022**, *10*, 117.
https://doi.org/10.3390/math10010117

**AMA Style**

Blackledge J, Lamphiere M.
A Review of the Fractal Market Hypothesis for Trading and Market Price Prediction. *Mathematics*. 2022; 10(1):117.
https://doi.org/10.3390/math10010117

**Chicago/Turabian Style**

Blackledge, Jonathan, and Marc Lamphiere.
2022. "A Review of the Fractal Market Hypothesis for Trading and Market Price Prediction" *Mathematics* 10, no. 1: 117.
https://doi.org/10.3390/math10010117