Introducing fractal dimension algorithms to calculate the Hurst exponent of financial time series

Abstract

Abstract. In this paper, three new algorithms are introduced in order to explore long memory in financial time series. They are based on a new concept of fractal dimension of a curve. A mathematical support is provided for each algorithm and its accuracy is tested for different length time series by Monte Carlo simulations. In particular, in the case of short length series, the introduced algorithms perform much better than the classical methods. Finally, an empirical application for some stock market indexes as well as some individual stocks is presented.

After notes on self-similarity exponent for fractal structures

“…It is worth pointing out that such a dimension leads to fruitful applications and results for those interested in the calculation the self-similarity exponent. The main references this work is based on are [1][2][3][4][5]. Next, we brie y describe how the present article is organized.…”

A new topological indicator for chaos in mechanical systems

“…On the other hand, it has also been applied to study nonlinear and chaotic systems. Self-similar processes often arise in this kind of systems, where the (box) dimension is linked with the self-similarity exponent (also called Hurst exponent) through the formula H = 2 − d. In this paper, we will study a new fractal dimension for a curve or a random process that was previously introduced in [23], where it was proved a theoretical connection between this novel dimension and the Hurst exponent for self-similar processes. In addition to that, we would like also to point out that this fractal dimension allowed there to contribute the so-called FD algorithms, which were proved empirically to be valid for Hurst exponent calculation purposes.…”

“…In the present work, we apply the fractal dimension definition for a curve introduced in [23] (based on a fractal dimension model contributed previously in [7]) to provide an algorithm which allows its estimation.…”

“…In this way, some definitions of fractal dimension for a fractal structure have been already studied in [7][8][9]11] and applied in [10,23], to quote some of them.…”

“…Moreover, in [23], fractal dimension III was used to define the fractal dimension for a curve with respect to a fractal structure induced on its image set. In that paper, some theoretical results were contributed in order to estimate the fractal dimension of a curve.…”