Constructing Copulas Using Corrected Hermite Polynomial Expansion for Estimating Cross Foreign Exchange Volatility

TL;DR

This paper introduces a novel method for constructing copulas using corrected Hermite polynomial expansions to better model complex correlation structures, and applies it to estimate cross currency volatility smiles in the foreign exchange market.

Contribution

It proposes a new copula construction method based on corrected Hermite polynomial expansions, enabling better modeling of complex correlations in financial data.

Findings

The proposed copula accurately reproduces the volatility smile of cross currency pairs.

Numerical experiments validate the effectiveness of the proposed copula compared to existing methods.

The method captures daily volatility fluctuations with fixed higher-order parameters.

Abstract

Copulas are used to construct joint distributions in many areas. In some problems, it is necessary to deal with correlation structures that are more complicated than the commonly known copulas. A finite order multivariate Hermite polynomial expansion, as an approximation of a joint density function, can handle complex correlation structures. However, it does not construct copulas because the density function can take negative values. In this study, we propose a method to construct a copula based on the finite sum of multivariate Hermite polynomial expansions by applying corrections to the joint density function. Furthermore, we apply this copula to estimate the volatility smile of cross currency pairs in the foreign exchange option market. This method can easily reproduce the volatility smile of cross currency pairs by appropriately adjusting the parameters and following the daily volatility fluctuations even if the higher-order parameters are fixed. In the numerical experiments, we compare the estimation results of the volatility smile of EUR-JPY with those of USD-JPY and EUR-USD for the proposed and other copulas, and show the validity of the proposed copula.