Multifractional Properties of Stock Indices Decomposed by Filtering Their Pointwise Hölder Regularity

Abstract

We propose a decomposition of financial time series into Gaussian subsequences characterized by a constant Hölder exponent. In (multi)fractal models this condition is equivalent to the subsequences themselves being stationarity. For the different subsequences, we study the scaling of the variance and the bias that is generated when the Hölder exponent is re-estimated using traditional estimators. The results achieved by both analyses are shown to be strongly consistent with the assumption that the price process can be modeled by the multifractional Brownian motion, a nonstationary process whose Hölder regularity changes from point to point.

Stochastic 2-microlocal analysis

“…This process and its generalizations have been the subject of various studies in recent years [2][3][4][5][6]. It is also currently used as a model in applications such as traffic engineering [7] or financial analysis [8].…”

Pointwise Regularity Exponents and Well-Behaved Residuals in Stock Markets

“…In particular, taking into account twenty-three works, [5] conclude that the estimates are considerably different and range in the interval [0.41, 0.70] (remind that H belongs to the interval [0,1]). Other analyses ( [5,6,15,16]) show that the functional parameter fluctuates around ½, rarely overstays for long periods above this threshold and displays on the contrary large downward movements when market crashes occur. This behavior seems to be strongly consistent with how actual financial markets do operate.…”

“…By construction: (a) the process is able to capture at the same time a very irregular local behavior and long range dependence; (b) relation (6) indicates that the MPRE behaves locally as an fBm. This is quite evident from Figure 3, which displays a sample path of MPRE: the random functional parameter H(t) is shown in panel (a), panel (b) displays the trajectory of the MPRE, and panel (c) reproduces the process increments.…”

“…From a practitioner viewpoint, many reasons justify the use of the MPRE as a model of the financial dynamics; detailed discussions can be found in [5], [6], [7], [8] and [9]. Here we restrict ourselves to mention the capability the process has to (a) provide a rationale for the trading Pointwise Regularity Exponents and Well-Behaved Residuals in Stock Markets Sergio Bianchi and Alexandre Pantanella mechanism, and (b) replicate the stylized facts observed in finance.…”

Estimating serial correlation and self-similarity in financial time series—A diversification approach with applications to high frequency data

“…Other approaches (e.g. Bianchi and Pianese [35] and Bianchi et al [36]) work appropriately with less data points, however, the methodology of these authors is parametric and imposes ex-ante limits on the shape of the parameter to be estimated.…”