New Quadrature-Based Approximations for the Characteristic Function and the Distribution Function of Sums of Lognormal Random Variables

Abstract

Characterizing the distribution of the sum of lognormal random variables (RVs) is still an open issue. This paper proposes simple and new quadrature-based approximations of the characteristic function (CF) and the cumulative distribution function (cdf) of the sum of independent and correlated lognormal RVs. Exploiting the recent Hermite-Gauss quadrature-based approximation, which is provided for the CF of a single lognormal RV, this paper proposes expressions that are given in terms of quadrature nodes and weights, as well as in terms of the parameters of individual lognormal RVs and the covariance matrix for the correlated RV case. More importantly, the developed expression for the cdf does not require prior knowledge of the actual cdf or the employment of specialized numerical integration methods. Numerical examples and comparisons with approximation techniques for the cdf of the sum found in the literature are provided. The examples show that while most of the known approximation techniques are valid for either small or large values of the abscissa, but not both, the proposed cdf formula provides reasonable approximation over a wide range of the abscissa. Furthermore, the study shows that as the actual cdf of the sum departs from the straight-line shape and its concavity increases, when plotted on a normal probability scale, even techniques that specialize in approximating the low-end tail of the cdf start to produce higher errors. The proposed formula continues to accurately approximate the actual cdf.