Exponential Family Techniques for the Lognormal Left Tail

Let X be lognormal(,sigma(2)) with density f(x); let theta > 0 and define L(theta )=Ee(-theta X). We study properties of the exponentially tilted density (Esscher transform) f(x) = e(-x theta)f(x)/L(theta ), in particular its moments, its asymptotic form as and asymptotics for the saddlepoint theta (x) determined by E[Xe-theta X]/L(theta )=x. The asymptotic formulas involve the Lambert W function. The established relations are used to provide two different numerical methods for evaluating the left tail probability of the sum of lognormals S-n=X-1++X-n: a saddlepoint approximation and an exponential tilting importance sampling estimator. For the latter, we demonstrate logarithmic efficiency. Numerical examples for the cdf F-n(x) and the pdf f(n)(x) of S-n are given in a range of values of sigma(2),n and x motivated by portfolio value-at-risk calculations.