The role of orthogonal polynomials in adjusting hyperpolic secant and logistic distributions to analyse financial asset returns

Bagnato, Luca; Potì, Valerio; Zoia, Maria Grazia

doi:10.1007/s00362-014-0633-3

Cited by 19 publications

(18 citation statements)

References 16 publications

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“…This paper extends recent insights (see [3,5,6]) on the issue of tailoring distributions in order to account for over-kurtosis. Starting from a given spherical distribution, the orthogonal polynomials of its related modular variable are provided and used to design the intended distribution shape adapter.…”

Section: Discussionsupporting

confidence: 61%

“…So far, this method-which can be viewed as an inheritance of the Gram-Charlier expansion-has proved effective when applied to distributions to account for possibly severe kurtosis and skewness (e.g., [5,6]). In this paper, we gain further insight into the matter and work out a similar orthogonal-polynomial-based approach to increasing the values of the moments-in particular, the fourth-within a multivariate spherical framework.…”

Section: Introductionmentioning

confidence: 99%

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The Role of Orthogonal Polynomials in Tailoring Spherical Distributions to Kurtosis Requirements

Bagnato¹,

Faliva²,

Zoia³

2016

Symmetry

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This paper carries out an investigation of the orthogonal-polynomial approach to reshaping symmetric distributions to fit in with data requirements so as to cover the multivariate case. With this objective in mind, reference is made to the class of spherical distributions, given that they provide a natural multivariate generalization of univariate even densities. After showing how to tailor a spherical distribution via orthogonal polynomials to better comply with kurtosis requirements, we provide operational conditions for the positiveness of the resulting multivariate Gram-Charlier-like expansion, together with its kurtosis range. Finally, the approach proposed here is applied to some selected spherical distributions.

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Section: Discussionsupporting

confidence: 61%

Section: Introductionmentioning

confidence: 99%

The Role of Orthogonal Polynomials in Tailoring Spherical Distributions to Kurtosis Requirements

Bagnato¹,

Faliva²,

Zoia³

2016

Symmetry

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“…This latter approach, which has the advantage of allowing for greater flexibility in fitting empirical distributions, is the one we have followed in this paper. Recently, Zoia (2010); Bagnato et al (2015)) have proposed a method to account for excess kurtosis of a density based on its polynomial transformation through its associated orthogonal polynomials. In the Gaussian case, these polynomials are the Hermite ones and the polynomially modified density is known as Gram-Charlier expansion.…”

Section: Introductionmentioning

confidence: 99%

Value at risk and expected shortfall based on Gram-Charlier-like expansions

Zoia

Biffi

Nicolussi

2018

Journal of Banking & Finance

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The reliability of risk measures for financial portfolios crucially rests on the availability of sound representations of the involved random variables. The trade-off between adherence to reality and specification parsimony can find a fitting balance in a technique that "adjust" the moments of a density function by making use of its associated orthogonal polynomials. This approach rests on the Gram-Charlier expansion of a Gaussian law which, allowing for leptokurtosis to an appreciable extent, makes the resulting random variable a tail-sensitive density function. In this paper we determine the density of sums of leptokurtic normal variables duly adjusted for excess kurtosis by means of their Gram-Charlier expansions based on Hermite polynomials. The resultant density can be effectively used to represent a portfolio return and as such proves suitable for computing some risk measures such as Value at Risk and expected short fall. An application to a portfolio of financial returns is used to provide evidence of the effectiveness of the proposed approach.

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“…For a given set of basis functions, a different ordering will generate a different set of polynomials. We use the basis functions x n = x n 1 1 • • • x n q q . But others are sometimes appropriate: see Withers (2009) for an example.…”

Section: Introductionmentioning

confidence: 99%

Orthogonal polynomials generated by random vectors

Withers¹,

Nadarajah

2016

Communications in Statistics - Theory and Methods

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Every random q-vector with finite moments generates a set of orthonormal polynomials.These are generated from the basis functions x n = x n 1 1 • • • x n q q using Gram-Schmidt orthogonalization. One can cycle through these basis functions using any number of ways. Here, we give results using minimum cycling. The polynomials look simpler when centered about the mean of X, and still simpler form when X is symmetric about zero. This leads to an extension of the multivariate Hermite polynomial for a general random vector symmetric about zero. As an example, the results are applied to the multivariate normal distribution.

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The role of orthogonal polynomials in adjusting hyperpolic secant and logistic distributions to analyse financial asset returns

Cited by 19 publications

References 16 publications

The Role of Orthogonal Polynomials in Tailoring Spherical Distributions to Kurtosis Requirements

The Role of Orthogonal Polynomials in Tailoring Spherical Distributions to Kurtosis Requirements

Value at risk and expected shortfall based on Gram-Charlier-like expansions

Orthogonal polynomials generated by random vectors

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