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

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  • Luca Bagnato
  • Valerio Potì
  • Maria Zoia

Abstract

In this paper, we will tackle the issue of accounting for skewness and potentially severe excess kurtosis of the empirical distribution of a random variable of interest by adjusting a parent leptokurtic distribution, using orthogonal polynomials. We will show that the polynomial shape adapter that allows the transformation from a given parent to a target distribution is a linear combination of the orthogonal polynomials associated to the former with coefficients depending on the difference between the moments of these two distributions. A recent work (Zoia, Commun Stat Theory Methods 39(1):52–64, 2010 ) has shown how to adjust the normal density by using Hermite polynomials but this application is suitable only for series with moderate kurtosis (lower than 5). This is why we provide two other parent distributions, the logistic and the hyperbolic secant which, once polynomially adjusted, can be used to reshape series with higher degrees of kurtosis. We will apply these results for modelling heavy-tailed and skewed distributions of financial asset returns by using both the conditional and unconditional approaches. We empirically demonstrate the advantages of using the polynomially adapted distributions in place of popular alternatives. Copyright Springer-Verlag Berlin Heidelberg 2015

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  • Luca Bagnato & Valerio Potì & Maria Zoia, 2015. "The role of orthogonal polynomials in adjusting hyperpolic secant and logistic distributions to analyse financial asset returns," Statistical Papers, Springer, vol. 56(4), pages 1205-1234, November.
    • Handle: RePEc:spr:stpapr:v:56:y:2015:i:4:p:1205-1234
    DOI: 10.1007/s00362-014-0633-3
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    Cited by:

    1. Brenda Castillo-Brais & Ángel León & Juan Mora, 2022. "Estimating Value-at-Risk and Expected Shortfall: Do Polynomial Expansions Outperform Parametric Densities?," Mathematics, MDPI, vol. 10(22), pages 1-17, November.
    2. Vacca, Gianmarco & Zoia, Maria Grazia & Bagnato, Luca, 2022. "Forecasting in GARCH models with polynomially modified innovations," International Journal of Forecasting, Elsevier, vol. 38(1), pages 117-141.
    3. Zoia, Maria Grazia & Biffi, Paola & Nicolussi, Federica, 2018. "Value at risk and expected shortfall based on Gram-Charlier-like expansions," Journal of Banking & Finance, Elsevier, vol. 93(C), pages 92-104.
    4. León, Ángel & Ñíguez, Trino-Manuel, 2021. "The transformed Gram Charlier distribution: Parametric properties and financial risk applications," Journal of Empirical Finance, Elsevier, vol. 63(C), pages 323-349.
    5. Vacca, Gianmarco & Zoia, Maria Grazia, 2019. "Kurtosis analysis in GARCH models with Gram–Charlier-like innovations," Economics Letters, Elsevier, vol. 183(C), pages 1-1.
    6. Piero Quatto & Gianmarco Vacca & Maria Grazia Zoia, 2021. "Modeling Portfolios with Leptokurtic and Dependent Risk Factors," Papers 2106.04218, arXiv.org.
    7. Kaczmarzyk Jan, 2018. "Forecasting Currency Risk in an Enterprise Using the Monte Carlo Simulation," Financial Sciences. Nauki o Finansach, Sciendo, vol. 23(4), pages 50-62, December.
    8. Quatto, Piero & Vacca, Gianmarco & Zoia, Maria Grazia, 2021. "A new copula for modeling portfolios with skewed, leptokurtic and high-order dependent risk factors," The North American Journal of Economics and Finance, Elsevier, vol. 58(C).
    9. Maria Grazia Zoia & Gianmarco Vacca & Laura Barbieri, 2020. "Modeling Multivariate Financial Series and Computing Risk Measures via Gram–Charlier-Like Expansions," Risks, MDPI, vol. 8(4), pages 1-21, November.

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