Exponential Family Techniques for the Lognormal Left Tail

Let $X$ be lognormal$(\mu,\sigma^2)$ with density $f(x)$, let $\theta>0$ and define ${L}(\theta)=E e^{-\theta X}$. We study properties of the exponentially tilted density (Esscher transform) $f_\theta(x) =e^{-\theta x}f(x)/{L}(\theta)$, in particular its moments, its asymptotic form as $\theta\to\infty$ and asymptotics for the Cram\'er function; the asymptotic formulas involve the Lambert W function. This is used to provide two different numerical methods for evaluating the left tail probability of lognormal sum $S_n=X_1+\cdots+X_n$: a saddlepoint approximation and an exponential twisting importance sampling estimator. For the latter we demonstrate the asymptotic consistency by proving logarithmic efficiency in terms of the mean square error. Numerical examples for the c.d.f.\ $F_n(x)$ and the p.d.f.\ $f_n(x)$ of $S_n$ are given in a range of values of $\sigma^2,n,x$ motivated from portfolio Value-at-Risk calculations.