Efficient and accurate algorithms for the computation and inversion of the incomplete gamma function ratios

Algorithms for the numerical evaluation of the incomplete gamma function ratios $P(a,x)=\gamma(a,x)/\Gamma(a)$ and $Q(a,x)=\Gamma(a,x)/\Gamma(a)$ are described for positive values of $a$ and $x$. Also, inversion methods are given for solving the equations $P(a,x)=p$, $Q(a,x)=q$, with $0<p,q<1$. Both the direct computation and the inversion of the incomplete gamma function ratios are used in many problems in statistics and applied probability. The analytical approach from earlier literature is summarized, and new initial estimates are derived for starting the inversion algorithms. The performance of the associated software to our algorithms (the Fortran 90 module IncgamFI) is analyzed and compared with earlier published algorithms

On the computation of moments of the partial non-central chi-squared distribution function

Properties satisfied by the moments of the partial non-central chi-square distribution function, also known as Nuttall Q-functions, and methods for computing these moments are discussed in this paper. The Nuttall Q-function is involved in the study of a variety of problems in different fields, as for example digital communications.Comment: 6 page

Efficient algorithms for the inversion of the cumulative central beta distribution

Accurate and efficient algorithms for the inversion of the cumulative central beta distribution are described. The algorithms are based on the combination of a fourth-order fixed point method with good non-local convergence properties (the Schwarzian-Newton method), asymptotic inversion methods and sharp bounds in the tails of the distribution function.The authors acknowledge financial support from Ministerio de Economía y Competitividad, project MTM2012-34787. NMT thanks CWI, Amsterdam, for scientific support

The Asymptotic and Numerical Inversion of the Marcum Q-Function

The generalized Marcum functions appear in problems of technical and scientific areas such as, for example, radar detection and communications. In mathematical statistics and probability theory these functions are called the noncentral gamma or the noncentral chi-squared cumulative distribution functions. In this paper, we describe a new asymptotic method for inverting the generalized Marcum Q-function and for the complementary Marcum P-function. Also, we show how monotonicity and convexity properties of these functions can be used to find initial values for reliable Newton or secant methods to invert the function. We present details of numerical computations that show the reliability of the asymptotic approximations

Computation of the Marcum Q-function

Methods and an algorithm for computing the generalized Marcum $Q-$function ($Q_{\mu}(x,y)$) and the complementary function ($P_{\mu}(x,y)$) are described. These functions appear in problems of different technical and scientific areas such as, for example, radar detection and communications, statistics and probability theory, where they are called the non-central chi-square or the non central gamma cumulative distribution functions. The algorithm for computing the Marcum functions combines different methods of evaluation in different regions: series expansions, integral representations, asymptotic expansions, and use of three-term homogeneous recurrence relations. A relative accuracy close to $10^{-12}$ can be obtained in the parameter region $(x,y,\mu) \in [0,\,A]\times [0,\,A]\times [1,\,A]$, $A=200$, while for larger parameters the accuracy decreases (close to $10^{-11}$ for $A=1000$ and close to $5\times 10^{-11}$ for $A=10000$).Comment: Accepted for publication in ACM Trans. Math. Soft

Algorithms for improving the efficiency of CEV, CIR and JDCEV option pricing models

The non-central chi-square distribution function has extensive use in the field of Mathematical Finance. To a great extent, this is due to its involvement in the constant elasticity of variance (hereafter, CEV) option pricing model of Cox (1975), in the term structure of interest rates model of Cox et al. (1985a) (hereafter, CIR), and the jump to default extended CEV (hereafter, JDCEV) framework of Carr and Linetsky (2006). Efficient computation methods are required to rapidly price complex contracts and calibrate financial models. The processes with several parameters, like the CEV or JDCEV models that we will address are examples of where this is important, since in this case the pricing problem (for many strikes) is used inside an optimization method. With this work we intend to test recent developments concerning the efficient computation of the non-central chi-square distribution function in the context of these option pricing models. We will give particular emphasis to the recent developments presented in the work of Gil et al. (2012), Gil et al. (2013), Dias and Nunes (2014), and Gil et al. (2015). For each option pricing model, we will define reference data-sets compatible with the most common combination of values used in pricing practice, following a framework that is similar to the one presented in Larguinho et al. (2013). We will conclude by offering novel analytical solutions for the JDCEV delta hedge ratios for the recovery parts of the put.A distribuição de probabilidade chi-quadrado não-central tem sido alvo de vasta utilização no domínio da Matematica Financeira, em grande parte devido à sua utilização no modelo constant elasticity of variance (doravante, CEV) de Cox (1975), no term structure of interest rates model de Cox et al. (1985a) e no modelo jump to default extended CEV (doravante, JDCEV) de Carr and Linetsky (2006). Metodos de cálculo eficientes e rápidos são de especial relevancia na calibração de modelos para a determinação do preço de contratos financeiros complexos. Os modelos CEV, CIR e JDCEV sao exemplos de modelos com diversos parametros que, quando usados em contexto de determinação do preço de opções com vários preçoss de exercício, mostram como esta optimização e fundamental. Com este trabalho pretendemos testar os mais recentes desenvolvimentos no calculo eficiente da distribuição de probabilidade nao-central chi-quadrado, no contexto dos modelos de cálculo de preço de opções mencionados anteriormente. Daremos enfase aos recentes desenvolvimentos apresentados nos trabalhos de Gil et al. (2012), Gil et al. (2013), Dias and Nunes (2014) e de Gil et al. (2015). Para cada um dos modelos, definiremos um conjunto de parametros de referencia compativel com as combinações mais usadas na prática, seguindo uma metodologia similiar a usada em Larguinho et al. (2013). Concluímos com a derivação de novas soluções analíticas para os racios de delta hedging no modelo JDCEV

Numerical aspects of special functions

This paper describes methods that are important for the numerical evaluation of certain functions that frequently occur in applied mathematics, physics and mathematical statistics. This includes what we consider to be the basic methods, such as recurrence relations, series expansions (both convergent and asymptotic), and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. Examples will be given on the use of special functions in certain problems from mathematical physics and mathematical statistics (integrals and series with special functions)

Numerical aspects of special functions

This paper describes methods that are important for the numerical evaluation of certain functions that frequently occur in applied mathematics, physics and mathematical statistics. This includes what we consider to be the basic methods, such as recurrence relations, series expansions (both convergent and asymptotic), and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. Examples will be given on the use of special functions in certain problems from mathematical physics and mathematical statistics (integrals and series with special functions)