On the scaling of the distribution of daily price fluctuations in the Mexican financial market index

Alfonso, L.; Mansilla, Ricardo; Terrero-Escalante, César A.

doi:10.1016/j.physa.2012.01.023

Cited by 10 publications

(14 citation statements)

References 27 publications

(46 reference statements)

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“…Further, for this index the Lévy-stable law provides a much better fit than the NIG distribution [12]. Similar results have been observed for the IPC mexican index: the hypothesis of stability could not be rejected at 5% confidence level while the hypothesis of NIG distributed daily log-returns was clearly rejected [2]. However, according to our findings, the Lévy-stable model is not the best option for modeling the high-frequency returns of the Ibex35 since the NIG distribution is a much better candidate.…”

Section: Discussionsupporting

confidence: 79%

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

Suárez-García

Gómez‐Ullate

2013

Physica A: Statistical Mechanics and its Applications

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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.

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Section: Discussionsupporting

confidence: 79%

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

Suárez-García

Gómez‐Ullate

2013

Physica A: Statistical Mechanics and its Applications

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“…This began with the work of Mandelbrot in 1963 [16], who showed that returns of cotton prices are more accurately modelled by an α-stable density with α = 1.7 than with a normal density, followed by the work of Fama in 1965 [8] arriving at a similar conclusion for daily returns of the Dow Jones Industrial Average. More recent investigations include α-stable behavior in Mexican financial markets [1], as well as financial modeling by more general Lévy processes [17,9]. The implications of heavy tailed behavior in financial processes cannot be underestimated in practice; Wilmott [39] gives the following example based on daily data of the S&P 500 from 1980-2004.…”

Section: Fractional Diffusion For Relative Stock Performance Many Tymentioning

confidence: 99%

“…The same is true for the multidimensional kernels G + and M + ν by linearity of the Fourier transform, and for G × and M × ν by Fubini's theorem. 1 In the machine learning literature, authors such as [31] describe this Fourier transform as the spectral density in the context of Bochner's theorem on stationary kernels, and write it in the equivalent form, up to Fourier transform convention:…”

mentioning

confidence: 99%

Machine Learning of Space-Fractional Differential Equations

Gulian¹,

Raissi²,

Perdikaris³

et al. 2019

SIAM J. Sci. Comput.

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Data-driven discovery of "hidden physics" -i.e., machine learning of differential equation models underlying observed data -has recently been approached by embedding the discovery problem into a Gaussian Process regression of spatial data, treating and discovering unknown equation parameters as hyperparameters of a "physics informed" Gaussian Process kernel. This kernel includes the parametrized differential operators applied to a prior covariance kernel. We extend this framework to the data-driven discovery of linear space-fractional differential equations. The methodology is compatible with a wide variety of space-fractional operators in R d and stationary covariance kernels, including the Matérn class, and allows for optimizing the Matérn parameter during training. Since fractional derivatives are typically not given by closed-form analytic expressions, the main challenges to be addressed are a user-friendly, general way to set up fractional-order derivatives of covariance kernels, together with feasible and robust numerical methods for such implementations. Making use of the simple Fourier-space representation of space-fractional derivatives in R d , we provide a unified set of integral formulas for the resulting Gaussian Process kernels. The shift property of the Fourier transform results in formulas involving d-dimensional integrals that can be efficiently treated using generalized Gauss-Laguerre quadrature.The implementation of fractional derivatives has several benefits. First, the method allows for discovering models involving fractional-order PDEs for systems characterized by heavy tails or anomalous diffusion, while bypassing the analytical difficulty of fractional calculus. Data sets exhibiting such features are of increasing prevalence in physical and financial domains. Second, a single fractional-order archetype allows for a derivative term of arbitrary order to be learned, with the order itself being a parameter in the regression. As a result, even when used for discovering integer-order equations, the proposed method has several benefits compared to previous works on data-driven discovery of differential equations; the user is not required to assume a "dictionary" of derivatives of various orders, and directly controls the parsimony of the models being discovered. We illustrate our method on several examples, including fractional-order interpolation of advection-diffusion and modeling relative stock performance in the S&P 500 with α-stable motion via a fractional diffusion equation.

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“…. , 100 and (σ x , σ y ) = (1, 1), (1, 2), (1,5), (2, 2), (2, 5), (5,5). For the bootstrap confidence interval, we took B = 500, as in Section 4.…”

Section: Simulation Study For Interval Estimators Of Rmentioning

confidence: 99%

“…According to O'Reilly and Rueda [28], times in supermarket sales, edge turbulence of fusion devices, network traffic behavior in switched Ethernet systems, modeling individual behavior in a large marine predator, evolutionary programming using mutations, distribution of marks in high school, fractal structures, models for fish locomotion, distribution of economical indices, south Spain seismic series, geophysical data analysis, supermarket sales, velocity difference in systems of vortex elements, currency exchange market, random field models for geological heterogeneity, structural reorganization in rice piles, and wave scattering from self-affine surfaces. Three of the most recent applications relate to daily price fluctuations in the Mexican financial market index (Alfonso et al [1]), observations of anomalous diffusion (Sagi et al, 2012), and bistable systems (Srokowski [33]). …”

Section: Introductionmentioning

confidence: 99%