Sums $S_n=X_1+...+X_n$ of lognormals arises in a wide variety of disciplines such as engineering, economics, insurance or finance, and are often employed in modeling across the sciences. The right lognormal tail $P(S_n>y)$ is heavy-tailed and typically analyzed by subexponential techniques.
The left tail $P(S_n<z)$ is of interest for example in portfolio VaR valculations .
The typical tool would be applying saddlepoint or large deviations techniques. This faces, however, the problem that the Laplace transform $L(\theta)=Ee^{-\theta X}$ is not explicit.
We present an approximation for $L(\theta)$ in terms of the Lambert $W$ function.
This is used to describe the shape of the exponentially tilted distribution $\tilde P(X\in dx)=e^{-\theta x}P(X\in dx)/L(\theta)$ and to derive a saddlepoint type approximation for $P(S_n<z)$.
Also related importance sampling algorithms are presented.
Numerical examples are presented in a range of parameters that we consider realistic for portfolio VaR calculations.