Abstract
This paper provides a review of the Fractal Market Hypothesis (FMH) focusing on financial times series analysis. In order to put the FMH into a broader perspective, the Random Walk and Efficient Market Hypotheses are considered together with the basic principles of fractal geometry. After exploring the historical developments associated with different financial hypotheses, an overview of the basic mathematical modelling is provided. The principal goal of this paper is to consider the intrinsic scaling properties that are characteristic for each hypothesis. In regard to the FMH, it is explained why a financial time series can be taken to be characterised by a 1/t1−1/γ scaling law, where γ>0 is the Lévy index, which is able to quantify the likelihood of extreme changes in price differences occurring (or otherwise). In this context, the paper explores how the Lévy index, coupled with other metrics, such as the Lyapunov Exponent and the Volatility, can be combined to provide long-term forecasts. Using these forecasts as a quantification for risk assessment, short-term price predictions are considered using a machine learning approach to evolve a nonlinear formula that simulates price values. A short case study is presented which reports on the use of this approach to forecast Bitcoin exchange rate values.
Highlights
The principal purpose of this paper is to provide a review of the Fractal Market Hypothesis (FMH) which is a hypothesis for analysing financial time series based on the principles of fractal geometry, the self-affine properties of stochastic fields
In the 1960s and 1970s, these concepts were developed further into a broader framework through the work of Eugene Fama, for example, who studied market dynamics and developed Bachelier’s ideas beyond the model of independent increments. He subsequently formed the basis of a financial model that would later come to be known as the Efficient Market Hypothesis (EMH) [26]
For the applications of fractal geometry in financial analysis, we are typically interested in a time series, a digital signal of time, which has a topological dimension of n = 1
Summary
The principal purpose of this paper is to provide a review of the Fractal Market Hypothesis (FMH) which is a hypothesis for analysing financial time series based on the principles of fractal geometry, the self-affine properties of stochastic fields. This paper has been composed for readers who have no or little prior knowledge of fractal geometry or the principles of financial time series modelling, risk assessment analysis and future price prediction. For this reason, this paper provides a short introduction to fractal geometry and an overview on the mathematical modelling of financial signals. Following a brief introduction on how probability theory was first introduced to the study of market movements and the analysis that went before, we introduce some of the major theoretical developments in the modelling of financial times series. The foundations for the mathematical models associated with these concepts are considered later on in the paper
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