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Financial modeling with heavy‐tailed stable distributions

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The aim of this article was to give an accessible introduction to stable distributions for financial modeling. There is a real need to use better models for financial returns because the normal (or bell curve/Gaussian) model does not capture the large fluctuations seen in real assets. Stable laws are a class of heavy‐tailed probability distributions that can model large fluctuations and allow more general dependence structures. WIREs Comput Stat 2014, 6:45–55. doi: 10.1002/wics.1286 This article is categorized under: Applications of Computational Statistics > Computational Finance Statistical and Graphical Methods of Data Analysis > Multivariate Analysis Statistical and Graphical Methods of Data Analysis > Robust Methods
Graphs of standardized S(α, β = 0.3, 1, 0; 0) (0‐parameterization, top panel) and S(α, β = 0.3, 1, 0; 1) (1‐parameterization, bottom panel) densities for a range of α values. The distributions are skewed right because β > 0. Note that the shapes are the same in both plots for a given (α,β) pair, but the different parameterizations have different shifts. With the 1‐parameterization, arbitrarily small changes in α or β can have a large effect on the location of the mode.
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Contour lines of some bivariate stable distributions. (a) Independent components with α = 1.1, (b) elliptical contours with α = 1.7, (c) discrete spectral measure with 5‐point masses and α = 1.3, and (d) discrete spectral measure with 3‐point masses and α = 0.75.
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Statistical and Graphical Methods of Data Analysis > Robust Methods
Statistical and Graphical Methods of Data Analysis > Multivariate Analysis
Applications of Computational Statistics > Computational Finance

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