Asymptotic approximations for the distribution of the product of correlated normal random variables

TL;DR

This paper derives asymptotic formulas for the distribution, tail probabilities, and risk measures of the product of two correlated normal variables with non-zero means, aiding in statistical and financial risk analysis.

Contribution

It introduces novel asymptotic approximations for the distribution and risk measures of the product of correlated normal variables with non-zero means.

Findings

Asymptotic formulas for the probability density function

Approximate tail probabilities and quantiles

Risk measures like value at risk and tail value at risk

Abstract

We obtain asymptotic approximations for the probability density function of the product of two correlated normal random variables with non-zero means and arbitrary variances. As a consequence, we deduce asymptotic approximations for the tail probabilities and quantile functions of this distribution, as well as an asymptotic approximation for the widely used risk measures value at risk and tail value at risk.