Exponential Family Techniques for the Lognormal Left Tail

TL;DR

This paper explores exponential family techniques for analyzing the left tail of the lognormal distribution, providing new asymptotic formulas, numerical methods for tail probability estimation, and demonstrating their efficiency in financial risk applications.

Contribution

It introduces novel asymptotic formulas involving Lambert W, and develops saddlepoint and importance sampling methods for lognormal tail estimation, with proven efficiency.

Findings

Asymptotic formulas involve Lambert W function.

Two numerical methods for tail probability estimation are proposed.

Importance sampling estimator is logarithmically efficient.

Abstract

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.