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

Søren Asmussen & Jens Ledet Jensen
(Department of Mathematics, Aarhus University)
ASE-event
Torsdag, 9 oktober, 2014, at 14:15-15:00, in Koll. G3 (1532-218)
Abstrakt:

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.

Kontaktperson: Søren Asmussen