Scaling, stability and distribution of the high-frequency returns of the Ibex35 index

Abstract

Abstract. In this paper we perform a statistical analysis of the high-frequency returns of the Ibex35 Madrid stock exchange index. We find that its probability distribution seems to be stable over different time scales, a stylized fact observed in many different financial time series. However, an in-depth analysis of the data using maximum likelihood estimation and different goodness-of-fit tests rejects the Lévy-stable law as a plausible underlying probabilistic model. The analysis shows that the Normal Inverse Gaussian distribution provides an overall fit for the data better than any of the other subclasses of the family of the Generalized Hyperbolic distributions and certainly much better than the Lévy-stable laws. Furthermore, the right (resp. left) tail of the distribution seems to follow a power-law with exponent α ≈ 4.60 (resp. α ≈ 4.28). Finally, we present evidence that the observed stability is due to temporal correlations or non-stationarities of the data.

Regime-Switching And Levy Jump Dynamics In Option-Adjusted Spreads

“…Other studies have also shown this distribution's superior empirical fit to other asset classes [54,39,26,27]. Using IBEX35 data, [60] find that the Normal Inverse Gaussian distribution provides an overall fit for the data better than any of the other subclasses of Generalized Hyperbolic distributions and much better than the Lévy-stable laws. More recently, Rachev et al [55] have deployed the NIG distribution, together with other statistical machinery, to study and (in their words) resolve such well-known 'puzzles' as (i) Predictability of asset returns (ii) The Equity Premium, and (iii) The Volatility Puzzle.…”

Highly flexible distributions to fit multiple frequency financial returns

“…It seems reasonable to set it as the benchmark. Many studies, however, highlight the limited power of the normal distribution to describe financial data dynamics [13,14,17], and propose many alternatives such as the NIG or the skewed t, which are also compared to the normal.…”

“…We estimate the models via maximum likelihood as suggested by Suárez-García and Gómez-Ullate [14] and Clauset et al [22], and compare them using the Kolmogorov-Smirnov (K-S) test, the Anderson-Darling (A-D) test, and the Bayesian information criterion per observation (BIC o ). 2 The lower the value of any criterion, the better the fit [23].…”

“…In recent studies [6][7][8], the SGT is simply compared to the normal or other low-flexible distribution; BenSaïda [9] has studied its efficiency into detecting the real distribution of a given data via extensive simulations, and developed a closed-form distribution and quantile functions essential to improve the implementation of the value-at-risk (VaR) in financial risk analysis [6,7]. Similarly, applications on the GH are carried essentially on the nested versions [10][11][12][13][14], and scarcely on the general form [15][16][17].…”

Hidden circulation fluctuations periodicity of index daily return

“…Analysis shows that the kinematics and dynamics of the fluctuation of return and price in stock markets cover physics, biology, and nonlinear science, vibration theory, sociology, economics and human psychology, etc. [1][2][3][4][5][6][7].…”