Fractal analysis of Dow Jones industrial index returns (original) (raw)
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Mathematics
In this article, we propose a test of the dynamics of stock market indexes typical of the US and EU capital markets in order to determine which of the two fundamental hypotheses, efficient market hypothesis (EMH) or fractal market hypothesis (FMH), best describes market behavior. The article’s major goal is to show how to appropriately model return distributions for financial market indexes, specifically which geometric Brownian motion (GBM) and geometric fractional Brownian motion (GFBM) dynamic equations best define the evolution of the S&P 500 and Stoxx Europe 600 stock indexes. Daily stock index data were acquired from the Thomson Reuters Eikon database during a ten-year period, from January 2011 to December 2020. The main contribution of this work is determining whether these markets are efficient (as defined by the EMH), in which case the appropriate stock indexes dynamic equation is the GBM, or fractal (as described by the FMH), in which case the appropriate stock indexes dyn...
Application of the Fractal Market Hypothesis for Macroeconomics Time Series Analysis
ISAST Transactions on Electronic and Signal …, 2008
This paper explores the conceptual background to financial time series analysis and financial signal processing in terms of the Efficient Market Hypothesis. By revisiting the principal conventional approaches to market analysis and the reasoning associated with them, we develop a Fractal Market Hypothesis that is based on the application of non-stationary fractional dynamics using an operator of the type ∂ 2 ∂x 2 − σ q(t) ∂ q(t) ∂t q(t) where σ −1 is the fractional diffusivity and q is the Fourier dimension which, for the topology considered, (i.e. the onedimensional case) is related to the Fractal Dimension 1 < DF < 2 by q = 1 − DF + 3/2. We consider an approach that is based on the signal q(t) and its interpretation, including its use as a macroeconomic volatility index. In practice, this is based on the application of a moving window data processor that utilises Orthogonal Linear Regression to compute q from the power spectrum of the windowed data. This is applied to FTSE close-of-day data between 1980 and 2007 which reveals plausible correlations between the behaviour of this market over the period considered and the amplitude fluctuations of q(t) in terms of a macroeconomic model that is compounded in the operator above.
Fractal Dimensional Analysis in Financial Time Series
A predictability index for time series of a financial market vector consisting of chosen market parameters is suggested providing a measure of long range predictability of the market. It is based on fractional Brownian motion that includes Brownian motion as a particular case followed by the time series of financial market parameters. By analysing respective time series, these indices are computed for parameters like volatility, FII investments in the local market, IIP numbers, CPI numbers, Dow Jones Index, different stock market indices, currency rates, and gold prices.
Sttock Market Behavoir: A Fractal Analysis of Saudi Stock Exchange
International journal of business, 2014
The Saudi stock market is analyzed, using rescaled range analysis to estimate the fractal dimension of price returns and to test the Efficient Market Hypothesis. In order to determine the predictability of a time series, Hurst Exponent for each time series is measured and we find that Saudi market is not totally random during the time period under study. There exists long range dependene in Saudi stock market returns. For most instances, it is determined that the Saudi stock market returns comply with neither the weak form of the efficient market hypothesis nor the random walk assumption. Additionally, for completeness and as part of literature review we bring out Bachelier-Einstein?s absolute Brownian dynamics, and Samuelson-Merton models of Martingale with geometric Brownian dynamic structure of equations.
2003
Multi-fractal processes have been proposed as a new formalism for modeling the time series of returns in finance. The major attraction of these processes is their ability to generate various degrees of long memory in different powers of returns -a feature that has been found to characterize virtually all financial prices. Furthermore, elementary variants of multi-fractal models are very parsimonious formalizations as they are essentially one-parameter families of stochastic processes. The aim of this paper is to provide the characteristics of a causal multi-fractal model (replacing the earlier combinatorial approaches discussed in the literature), to estimate the parameters of this model and to use these estimates in forecasting financial volatility. We use the auto-covariances of log increments of the multi-fractal process in order to estimate its parameters consistently via GMM (Generalized Method of Moment). Simulations show that this approach leads to essentially unbiased estimates, which also have much smaller root mean squared errors than those obtained from the traditional 'scaling' approach. Our empirical estimates are used in out-of-sample forecasting of volatility for a number of important financial assets. Comparing the multi-fractal forecasts with those derived from GARCH and FIGARCH models yields results in favor of the new model: multi-fractal forecasts dominate all other forecasts in one out of four cases considered, while in the remaining cases they are head to head with one or more of their competitors.
Fractal Analysis of US Financial Markets The Hurst Exponent as an indicator of regime shift
Our research goal is to apply fractal analysis to financial time series to gain insights on the type of market, the persistence of the market type, i.e. the presence of long term memory, and to identify if a possible regime change is in the making (by using the Hurst exponent as a proxy for volatility) in order to have an edge for the application of 2 basic trading strategies: momentum trading and mean reversion trading. In addition, we want to verify the applicability of the Fractal Market Hypothesis (FMH) vs the Efficient Market Hypothesis (EMH). The principal method of fractal analysis we will employ is the R/S statistic, introduced by Hurst in 1951, to estimate the fractal dimension (and the Hurst exponent, H), in order to classify time series in 3 main buckets: mean reverting (low volatility, anti-persistent) for H < 0.5, trending (higher volatility, series has long term memory, or persistence) for H > 0.5, and for H = 0.5 (or close to it), the EMH regime.
On The Behavior of Malaysian Equities: Fractal Analysis Approach
2015
Fractal analyzing of continuous processes have recently emerged in literatures in various domains. Existence of long memory in many processes including financial time series have been evidenced via different methodologies in many literatures in past decade, which has inspired many recent literatures on quantifying the fractional Brownian motion (fBm) characteristics of financial time series. This paper questions the accuracy of commonly applied fractal analyzing methods on explaining persistent or antipersistent behavior of time series understudy. Rescaled range (R/S) and power spectrum techniques produce fractal dimensions for daily returns of twelve Malaysian stocks from the most well performed firms in Kuala Lumpur stock exchange. Zipf’s law generates linear and logarithmic power-law distribution plots to evaluate the validity of estimated fractal dimensions on prescribing persistent and antipersistent characteristics with less ambiguity. Findings of this study recommend a more t...
Fractal analysis of highly volatile markets: an application to technology equities
This paper examines technology equity price series using five self-affine fractal analysis techniques for estimating the Hurst exponent, Mandelbrot–Lévy characteristic exponent, and fractal dimension. Techniques employed are rescaled-range analysis, power-spectral density analysis, roughness-length analysis, the variogram or structure function method, and wavelet analysis. Evidence against efficient valuation supports the multifractal model of asset returns (MMAR) and disconfirms the weak form of the efficient market hypothesis (EMH). Strong evidence is presented for antipersistence of many technology equities, suggesting markets do not price all technology securities efficiently, or equally efficiently.
Portfolio selection and fractal market hypothesis: Evidence from the London stock exchange
Pamukkale University Journal of Engineering Sciences, 2023
It is well known that the models supporting the Modern Portfolio Theory (MPT) and the Efficient Market Hypothesis (EMH) are constructed in the framework of random walk theory. However, a large and growing literature criticizes those models. The Fractal Market Hypothesis (FMH) was proposed as an alternative hypothesis to EMH. The motivation of this study is Peters' [45,46] works that examine the portfolio selection case based on the non-normality framework. The aim of the study is to propose a new approach to theoretical framework of portfolio selection in terms of FMH. Daily observations of 92 stocks traded in London Stock Exchange are used to investigate the fractal behavior. Thus, the Hurst exponents as a means of indicator of a fractal structure are calculated for simulated portfolios. Results of the analysis show that the validity of MPT and EMH is questionable in London Stock Exchange. To examine the relationship between Hurst exponents (as a measure of risk) and returns, scattered diagrams are constructed for 5000 simulated portfolios. Existence of a pattern with a frontier is detected that may enable investors to optimize their portfolios. Further, The Hurst exponents of efficient frontier portfolios of Markowitz are calculated in order to investigate whether there is any linkage with the frontier of simulated portfolios. The results show that major deviations occur between these two frontiers. To understand these deviations, the Lyapunov exponents are suggested for detailed information. As a conclusion, it is recommended that investors should calculate an optimal solution with regards to the Hurst and Lyapunov exponents to maximize their returns.