  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.
