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The Gram–Charlier expansion of a target probability density, (Formula presented.), is an (Formula presented.) -convergent series (Formula presented.) in terms of a reference density (Formula presented.) and its orthonormal polynomials (Formula presented.). We implement this for the density of a regime-switching Lévy process at a given time horizon T. The main step is the evaluation of moments of all orders of (Formula presented.) in terms of model primitives, for which we give a matrix-exponential representation. A number of numerical examples, in part involving pricing of European options, are presented. The traditional choice of (Formula presented.) as normal with the same mean and variance as (Formula presented.) only works for the regime-switching Black–Scholes model. Outside the scope of Black–Scholes, (Formula presented.) is typically taken as a normal inverse Gaussian. A similar analysis is given for time-changed Lévy processes modelling stochastic volatility.

One of the main topics in the theory of orthogonal polynomials is the study of their asymptotics. Various types of asymptotics corresponding to orthogonal polynomials can be studied, which provide valuable information on polynomials with sufficiently large degrees. In particular, the Mehler-Heine formulas provide the local asymptotics of the polynomials that are properly scaled, and establish a limit relationship between the polynomials and other types of functions. In this article, it was possible to write the modified Charlier polynomials as a linear combination of two consecutive Charlier polynomials. Combining the results obtained with the Mehler-Heine formula of the Charlier polynomials, the one corresponding to the modified ones was obtained., Uno de los temas principales en la teoría de los polinomios ortogonales es el estudio de sus asintóticas. Se pueden estudiar varios tipos de asintóticas correspondientes a los polinomios ortogonales, las cuales brindan información valiosa sobre los polinomios con grados suficientemente grandes. En particular, las fórmulas de Mehler-Heine, proporcionan las asintóticas locales de los polinomios que se escalan adecuadamente, y establecen una relación límite entre los polinomios y otros tipos de funciones. En el presente artículo se logró escribir los polinomios modificados de Charlier como combinación lineal de dos polinomios consecutivos de Charlier. Combinando los resultados obtenidos con la fórmula de Mehler-Heine de los polinomios de Charlier, se obtuvo la correspondiente a los modificados.

One of the main topics in the theory of orthogonal polynomials is the study of their asymptotics. Various types of asymptotics corresponding to orthogonal polynomials can be studied, which provide valuable information on polynomials with sufficiently large degrees. In particular, the Mehler-Heine formulas provide the local asymptotics of the polynomials that are properly scaled, and establish a limit relationship between the polynomials and other types of functions. In this article, it was possible to write the modified Charlier polynomials as a linear combination of two consecutive Charlier polynomials. Combining the results obtained with the Mehler-Heine formula of the Charlier polynomials, the one corresponding to the modified ones was obtained., Uno de los temas principales en la teoría de los polinomios ortogonales es el estudio de sus asintóticas. Se pueden estudiar varios tipos de asintóticas correspondientes a los polinomios ortogonales, las cuales brindan información valiosa sobre los polinomios con grados suficientemente grandes. En particular, las fórmulas de Mehler-Heine, proporcionan las asintóticas locales de los polinomios que se escalan adecuadamente, y establecen una relación límite entre los polinomios y otros tipos de funciones. En el presente artículo se logró escribir los polinomios modificados de Charlier como combinación lineal de dos polinomios consecutivos de Charlier. Combinando los resultados obtenidos con la fórmula de Mehler-Heine de los polinomios de Charlier, se obtuvo la correspondiente a los modificados.

This paper develops an approach based on Gram–Charlier-like expansions for modeling financial series to take in due account features such as leptokurtosis. A Gram–Charlier-like expansion adjusts the moments of interest of a given distribution via its own orthogonal polynomials. This approach, formerly adopted for univariate series, is here extended to a multivariate context by means of spherical densities. Previous works proposed the Gram–Charlier of the multivariate Gaussian, obtained by using Hermite polynomials. This work shows how polynomial expansions of an entire class of spherical laws can be worked out with the aim of obtaining a wide set of leptokurtic multivariate distributions. A Gram–Charlier-like expansion is a distribution characterized by an additional parameter with respect to the parent spherical law. This parameter, which measures the increase in kurtosis due to the polynomial expansion, can be estimated so as to make the resulting distribution capable of describing the empirical kurtosis found in the data. An application of the Gram–Charlier-like expansions to a set of financial assets proves their effectiveness in modeling multivariate financial series and assessing risk measures, such as the value at risk and the expected shortfall

In this paper we study an extension of the Gram–Charlier (GC) density in Jondeau and Rockinger (2001) which consists of a Gallant and Nychka (1987) transformation to ensure positivity without parameter restrictions. We derive its parametric properties such as unimodality, cumulative distribution, higher-order moments, truncated moments, and the closed-form expressions for the expected shortfall (ES) and lower partial moments. We obtain the analytic th order stationarity conditions for the unconditional moments of the TGARCH model under the transformed GC (TGC) density. In an empirical application to asset return series, we estimate the tail index; backtest the density, VaR and ES; and implement a comparative analysis based on Hansen’s skewed-t distribution. Finally, we present extensions to time-varying conditional skewness and kurtosis, and a new class of mixture densities based on this TGC distribution.

In this paper we study an extension of the Gram–Charlier (GC) density in Jondeau and Rockinger (2001) which consists of a Gallant and Nychka (1987) transformation to ensure positivity without parameter restrictions. We derive its parametric properties such as unimodality, cumulative distribution, higher-order moments, truncated moments, and the closed-form expressions for the expected shortfall (ES) and lower partial moments. We obtain the analytic th order stationarity conditions for the unconditional moments of the TGARCH model under the transformed GC (TGC) density. In an empirical application to asset return series, we estimate the tail index; backtest the density, VaR and ES; and implement a comparative analysis based on Hansen’s skewed-t distribution. Finally, we present extensions to time-varying conditional skewness and kurtosis, and a new class of mixture densities based on this TGC distribution.

This paper proposes risk measures for bank solvency by accurately measuring the solvency risk components. These measures consider the minimum regulatory solvency levels and banks’ risk appetite level and risk profile. For this purpose, we used semi-nonparametric statistics to model stylized facts of the risk distribution, particularly the high-order moments of the Solvency Decline Rate, the Tier Decline Rate, and the Portfolio Growth Rate variables. Additionally, these risk measures can be used to measure the risk of regulatory intervention and to define policies that establish the minimum solvency levels required by banking regulators by estimating the Quantile Risk Metrics. As a case study, we collected data on the solvency indicators of the Colombian banking system, which adapts to the standards established by the Basel Committee. According to the results, the liquidity injection measures implemented in response to the needs generated by the COVID-19 pandemic led to an increase in the levels of the risk portfolio in the Colombian banking system, which exceeded the 99th percentile of the probability distribution of monthly portfolio value changes.

The approach based on polynomially-modified distributions, known as Gram–Charlier-like (GCl) expansions, has been proven effective to account for both excess kurtosis and skewness of financial data. In this paper, we examine GARCH models with innovations distributed as GCl expansions (GC-GARCH). The kurtosis gluts ascribable to both time-varying volatility and GCl distributed GARCH innovationsis evaluated. Furthermore, a ‘‘kurtosis targeting’’ approach is devised to estimate the kurtosis of GCl innovations. This leads to GC-GARCH models tailored to fit the kurtosis requirements of financial data

The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson−Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution.

The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson−Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution.

Since financial series are usually heavy-tailed and skewed, research has formerly considered well-known leptokurtic distributions to model these series and, recently, has focused on the technique of adjusting the moments of a probability law by using its orthogonal polynomials. This paper combines these approaches by modifying the moments of the convoluted hyperbolic-secant (CHS). The resulting density is a Gram-Charlier-like (GC-like) expansion capable to account for skewness and excess kurtosis. Multivariate extensions of these expansions are obtained on an argument using spherical distributions. Both the univariate and multivariate (GC-like) expansions prove to be effective in modelling heavy-tailed series and computing risk measures.

Spherical distributions arise quite naturally as multivariate versions of univariate (even) densities and prove useful in several applications. Likewise their univariate counterparts, they may not always meet the kurtosis requirements of empirical evidence. This paper devises a methodological approach which duly reshapes spherical distributions to match kurtosis requirements to due extent. This approach is tailored to the family of power-raised hyperbolic secant laws and hinges on Gram–Charlier-like expansions via second-degree orthogonal polynomials

Strong stability preserving (SSP) coefficients govern the maximally allowable step-size at which positivity or contractivity preservation of integration methods for initial value problems is guaranteed. In this paper, we show that the task of computing linear SSP coefficients of explicit one-step methods is, to a certain extent, equivalent to the problem of characterizing positive quadratures with integer nodes with respect to Poisson-Charlier measures. Using this equivalence, we provide sharp upper and lower bounds for the optimal linear SSP coefficients in terms of the zeros of generalized Laguerre orthogonal polynomials. This in particular provides us with a sharp upper bound for the optimal SSP coefficients of explicit Runge-Kutta methods. Also based on this equivalence, we propose a highly efficient and stable algorithm for computing these coefficients, and their associated optimal linear SSP methods, based on adaptive spectral transformations of Poisson-Charlier measures. The algorithm possesses the remarkable property that its complexity depends only on the order of the method and thus is independent of the number of stages. Our results are achieved by adapting and extending an ingenious technique by Bernstein in his seminal work on absolutely monotonic functions. Moreover, the techniques introduced in this work can be adapted to solve the integer quadrature problem for any positive discrete multi-parametric measure supported on $\mathbb{N}$ under some mild conditions on the zeros of the associated orthogonal polynomials., Comment: 50 pages, 5 figures

Spherical distributions arise quite naturally as multivariate versions of univariate (even) densities and prove useful in several applications. Likewise their univariate counterparts, they may not always meet the kurtosis requirements of empirical evidence. This paper devises a methodological approach which duly reshapes spherical distributions to match kurtosis requirements to due extent. This approach is tailored to the family of power-raised hyperbolic secant laws and hinges on Gram–Charlier-like expansions via second-degree orthogonal polynomials.

This paper offers a new approach to modeling the distribution of a portfolio composed of either asset returns or insurance losses. To capture the leptokurtosis, which is inherent in most financial series, data are modeled by using Gram-Charlier (GC) expansions. Since we are interested in operating with several series simultaneously, the distribution of the sum of GC random variables is derived. This latter turns out to be a tail-sensitive density, suitable for modeling the distribution of a portfolio return-losses and, accordingly, can be conveniently adopted for computing risk measures such as the value at risk and the expected shortfall as well as some performance measures based on its partial moments. The closed form expressions of these risk measures are derived for cases when the density of a portfolio is the sum of GC expansions, either with the same or different kurtosis. An empirical application of this approach to a portfolio of financial asset indexes provides evidence of the comparative effectiveness of this technique in computing risk measures, both in and out of the sample period.

The paper devises a family of leptokurtic bell-shaped distributions which is based on the hyperbolic secant raised to a positive power, and bridges the Laplace and Gaussian laws on asymptotic arguments. Moment and cumulant generating functions are then derived and represented in terms of polygamma functions. The behaviour of shape parameters, namely kurtosis and entropy, is investigated. In addition, Gram–Charlier-type (GCT) expansions, based on the aforementioned distributions and their orthogonal polynomials, are specified, and an operational criterion is provided to meet modelling requirements in a possibly severe kurtosis and skewness environment. The role played by entropy within the kurtosis ranges of GCT expansions is also examined.

The reliability of risk measures of 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 essentially 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 via their Gram-Charlier expansions based on Hermite polynomials. The aforesaid density can be properly used to compute some risk measures such as the Value at Risk and the expected short fall. An application to a portfolio of financial returns provides evidence of the effectiveness of the proposed approach.

We derive closed-form expressions for risk measures based on partial moments by assuming the Gram-Charlier (GC) density for stock returns. As a result, the lower partial moment (LPM) measures can be expressed as linear functions on both skewness and excess kurtosis. Under this framework, we study the behavior of portfolio rankings with performance measures based on partial moments, that is, both Farinelli-Tibiletti (FT) and Kappa ratios. Contrary to previous results, significant differences are found in ranking portfolios between the Sharpe ratio and the FT family. We also obtain closed-form expressions for LPMs under the semi non-parametric (SNP) distribution which allows higher flexibility than the GC distribution.

We derive closed-form expressions for risk measures based on partial moments by assuming the Gram-Charlier (GC) density for stock returns. As a result, the lower partial moment (LPM) measures can be expressed as linear functions on both skewness and excess kurtosis. Under this framework, we study the behavior of portfolio rankings with performance measures based on partial moments, that is, both Farinelli-Tibiletti (FT) and Kappa ratios. Contrary to previous results, significant differences are found in ranking portfolios between the Sharpe ratio and the FT family. We also obtain closed-form expressions for LPMs under the semi non-parametric (SNP) distribution which allows higher flexibility than the GC distribution.

Compte rendu de l'ouvrage cité.

The article derives a novel Gram-Charlier A (GCA) Series based Extended Rule-of-Thumb (ExROT) for bandwidth selection in Kernel Density Estimation (KDE). There are existing various bandwidth selection rules achieving minimization of the Asymptotic Mean Integrated Square Error (AMISE) between the estimated probability density function (PDF) and the actual PDF. The rules differ in a way to estimate the integration of the squared second order derivative of an unknown PDF $(f(\cdot))$, identified as the roughness $R(f''(\cdot))$. The simplest Rule-of-Thumb (ROT) estimates $R(f''(\cdot))$ with an assumption that the density being estimated is Gaussian. Intuitively, better estimation of $R(f''(\cdot))$ and consequently better bandwidth selection rules can be derived, if the unknown PDF is approximated through an infinite series expansion based on a more generalized density assumption. As a demonstration and verification to this concept, the ExROT derived in the article uses an extended assumption that the density being estimated is near Gaussian. This helps use of the GCA expansion as an approximation to the unknown near Gaussian PDF. The ExROT for univariate KDE is extended to that for multivariate KDE. The required multivariate AMISE criteria is re-derived using elementary calculus of several variables, instead of Tensor calculus. The derivation uses the Kronecker product and the vector differential operator to achieve the AMISE expression in vector notations. There is also derived ExROT for kernel based density derivative estimator., Comment: 30 pages

info:eu-repo/semantics/published

If a probability distribution is sufficiently close to a normal distribution, its density can be approximated by a Gram/Charlier Series A expansion. In option pricing, this has been used to fit risk-neutral asset price distributions to the implied volatility smile, ensuring an arbitrage-free interpolation of implied volatilities across exercise prices. However, the existing literature is restricted to truncating the series expansion after the fourth moment. This paper presents an option pricing formula in terms of the full (untruncated) series and discusses a fitting algorithm, which ensures that a series truncated at a moment of arbitrary order represents a valid probability density. While it is well known that valid densities resulting from truncated Gram/Charlier Series A expansions do not always have sufficient flexibility to fit all market-observed option prices perfectly, this paper demonstrates that option pricing in a model based on these densities is as tractable as the (far less flexible) original model of Black and Scholes (1973), allowing non-trivial higher moments such as skewness, excess kurtosis and so on to be incorporated into the pricing of exotic options: Generalising the Gram/Charlier Series A approach to the multiperiod, multivariate case, a model calibrated to standard option prices is developed, in which a large class of exotic payoffs can be priced in closed form. Furthermore, this approach, when applied to a foreign exchange option market involving several currencies, can be used to ensure that the volatility smiles for options on the cross exchange rate are constructed in a consistent, arbitrage-free manner. © 2012 Elsevier B.V.

If a probability distribution is sufficiently close to a normal distribution, its density can be approximated by a Gram/Charlier Series A expansion. In option pricing, this has been used to fit risk-neutral asset price distributions to the implied volatility smile, ensuring an arbitrage-free interpolation of implied volatilities across exercise prices. However, the existing literature is restricted to truncating the series expansion after the fourth moment. This paper presents an option pricing formula in terms of the full (untruncated) series and discusses a fitting algorithm, which ensures that a series truncated at a moment of arbitrary order represents a valid probability density. While it is well known that valid densities resulting from truncated Gram/Charlier Series A expansions do not always have sufficient flexibility to fit all market-observed option prices perfectly, this paper demonstrates that option pricing in a model based on these densities is as tractable as the (far less flexible) original model of Black and Scholes (1973), allowing non-trivial higher moments such as skewness, excess kurtosis and so on to be incorporated into the pricing of exotic options: Generalising the Gram/Charlier Series A approach to the multiperiod, multivariate case, a model calibrated to standard option prices is developed, in which a large class of exotic payoffs can be priced in closed form. Furthermore, this approach, when applied to a foreign exchange option market involving several currencies, can be used to ensure that the volatility smiles for options on the cross exchange rate are constructed in a consistent, arbitrage-free manner. © 2012 Elsevier B.V.

This article considers the content, reception, and relevance of Louis Charlier's Essai sur le probleme theologique (1938). The core theme of this essay is both the nature and methodology of theology. In this respect, Charlier is participating in the theological selfreflection of the 1930's and 1940's. He opposed the strict Roman Neoscholasticism and pleaded for the full integration of history within theology. Together with M.-D. Chenu he is to be considered as the instigator of a thomistic "ressourcement", characterized by a plea for an open thomism. This first phase of the nouvelle theologie (new theology) was followed by a second phase, a theological "ressourcement", which pleads for a full return of theology to its sources (Bible, Fathers, liturgy) on the foundations of the first phase. During this second phase too (ca. 1942-1950), there were some positive but, more often, negative critiques on Charlier's Essai. Both the Essai of Charlier and Une ecole de theologie: le Saulchoir of Chenu (1937) aroused Roman irritation and in 1942 were placed on the Index.

This paper introduces a new family of multivariate distributions based on Gram-Charlier and Edgeworth expansions. This family encompasses many of the univariate semi-nonparametric densities proposed in financial econometrics as marginal of its different formulations. Within this family, we focus on the analysis of the specifications that guarantee positivity to obtain well-defined multivariate semi-nonparametric densities. We compare two different multivariate distributions of the family with the multivariate Edgeworth-Sargan, Normal, Student's t and skewed Student's t in an in- and out-sample framework for financial returns data. Our results show that the proposed specifications provide a quite reasonably good performance being so of interest for applications involving the modelling and forecasting of heavy-tailed distributions.

A study has been made of a Gram-Charlier matched filter for Shack-Hartmann sensing of wave-front slopes. The method is based on modeling the point-spread function by an expansion in terms of Gauss-Hermite polynomials. We present results for several subaperture/coherence area sizes both with and without CCD read noise. A more accurate estimation of the local slopes can be. achieved at low light levels in this way than with the standard first-moment estimator.

Convergent expansions are derived for three types of orthogonal polynomials: Charlier, Laguerre and Jacobi. The expansions have asymptotic properties for large values of the degree. The expansions are given in terms of functions that are special cases of the given polynomials. The method is based on expanding integrals in one or two points of the complex plane, these points being saddle points of the phase functions of the integrands.

Convergent expansions are derived for three types of orthogonal polynomials: Charlier, Laguerre and Jacobi. The expansions have asymptotic properties for large values of the degree. The expansions are given in terms of functions that are special cases of the given polynomials. The method is based on expanding integrals in one or two points of the complex plane, these points being saddle points of the phase functions of the integrands.

This report comprises two parts. The first part focuses on extending the well known Gram Charlier series expansion technique for univariate probability density functions to multivariate functions. The approach employed here is particularly attractive because it avoids explicit use of tensor analysis and multivariate Hermite polynomials. The desired expansion is accomplished via straightforward application of Kronecker products and matrix calculus formulae. The second part of this report addresses the problem of signal detection in the presence of background noise. A canonical representation of the likelihood ratio is derived that applies to a broad class of multivariate random signals embedded in Gaussian noise. The representation takes the form of an infinite series whose terms depend on the received measurement vector and the signal and noise statistics. It prescribes a solution to the binary detection problem in a variety of sonar applications.

Bibliogr., En appendice, entretien avec Jean Giraud