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Exponential Family Techniques for the Lognormal Left Tail

By Søren Asmussen, Jens Ledet Jensen and Leonardo Rojas-Nandayapa
Thiele Research Reports
No. 01, June 2014
Abstract:

Let $X$ be lognormal$(\mu,\sigma^2)$ with density $f(x)$, let $\theta > 0$ and define $L(\theta)=\mathbb{E}\mathrm{e}^{-\theta X}$. We study properties of the exponentially tilted density (Esscher transform) $f_\theta(x) =\mathrm{e}^{-\theta x}f(x)/L(\theta)$, in particular its moments, its asymptotic form as $\theta\to\infty$ and asymptotics for the saddlepoint $\theta(x)$ determined by $\mathbb{E}[X\mathrm{e}^{-\theta X}]/L(\theta)=x$. 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 lognormal sum $S_n=X_1+\cdots+X_n$: a saddlepoint approximation and an exponential twisting 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 $\sigma^2,n,x$ motivated from portfolio Value-at-Risk calculations.

Keywords: Lognormal distribution, Cramér function, Esscher transform, exponential change of measure, Laplace transform, Laplace method, saddlepoint approximation, Lambert W function, rare event simulation, importance sampling, VaR.

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