Tail behavior of sums and differences of log-normal random variables

TL;DR

This paper derives precise tail asymptotics for sums and differences of correlated log-normal variables, providing tools for risk assessment, Monte Carlo variance reduction, and stress testing in finance.

Contribution

It offers explicit tail asymptotics based on a quadratic optimization problem, enabling improved probability estimates and risk management techniques for log-normal portfolios.

Findings

Derived sharp tail asymptotics for sums and differences of log-normal variables.

Proposed an efficient importance sampling estimator for the left tail.

Characterized the asymptotic behavior of Value at Risk as confidence approaches one.

Abstract

We present sharp tail asymptotics for the density and the distribution function of linear combinations of correlated log-normal random variables, that is, exponentials of components of a correlated Gaussian vector. The asymptotic behavior turns out to depend on the correlation between the components, and the explicit solution is found by solving a tractable quadratic optimization problem. These results can be used either to approximate the probability of tail events directly, or to construct variance reduction procedures to estimate these probabilities by Monte Carlo methods. In particular, we propose an efficient importance sampling estimator for the left tail of the distribution function of the sum of log-normal variables. As a corollary of the tail asymptotics, we compute the asymptotics of the conditional law of a Gaussian random vector given a linear combination of exponentials of its components. In risk management applications, this finding can be used for the systematic construction of stress tests, which the financial institutions are required to conduct by the regulators. We also characterize the asymptotic behavior of the Value at Risk for log-normal portfolios in the case where the confidence level tends to one.