Numerical Computation of the Lognormal Sum Distribution

This paper develops highly accurate numerical techniques for evaluating the mgf/ chf of a single lognormal variable and for computing the lognormal sum cdf. Complex integration techniques based on the steepest-descent integration are thus developed for evaluating the lognormal mgf/ chf. The saddlepoint of the integrand is explicitly expressed using the Lambert-W function. The optimal steepest-descent contour passing through the saddlepoint is then derived. Even a simple mid-point-rule-based integration technique can be used along this contour to evaluate the mgf/ chf at extremely high precision. A highly efficient, extremely accurate numerical method is then developed for evaluating the cdf of sum of independent lognormal variables. The cdf is expanded as an alternating series, on which the Epsilon algorithm for convergence acceleration is applied. This reduces the computational load significantly.