A Non-parametric Method for Approximating Joint Densities and Copula Functions for Financial Markets

When we measure the market risk of a portfolio with multiple of risk factors, we, sometimes implicitly, deal with the risk factors\u27 joint distribution. However, only a few methods are available to render tractable forms of multivariate distributions for risk aggregation.This paper discusses approximation techniques using the Hermite expansion for marginal and joint density functions. These techniques (or expansion methods) approximate probability density functions by a sum of Hermite polynomials multiplied by the associated weight function. The advantage of the use of expansion methods is that they only require the moments of the target distributions up to some nite degree, assuming they exist.The biggest shortcoming of the expansion methods is their poor approximation quality. This paper introduces techniques to redeem this problem, and considers application to risk aggregation. We also approximate joint density functions and show that expansion methods are applicable to approximating conditional expectations and copula density functions. Numerical examples for bivariate cases show that our approximations can capture characteristics of original observations. Such techniques may facilitate the further investigation of non-linear dependence structures among risk factors in the nancial markets