# Exponential family techniques in the lognormal left tail, with applications to portfolio VaR

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.