Exponential Family Techniques for the Lognormal Left Tail

Let X be lognormal( μ , σ 2 ) with density f ( x ); let θ > 0 and define L ( θ ) = E e − θX . We study properties of the exponentially tilted density (Esscher transform) f θ ( x ) = e − θ x f ( x )/ L ( θ ), in particular its moments, its asymptotic form as θ → ∞ and asymptotics for the saddlepoint θ ( x ) determined by E [ X e − θX ] / L ( θ ) = 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 σ 2 , n and x motivated by portfolio value‐at‐risk calculations.