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

Paper: arXiv 2301.10044

Authors: Kenichiro Shiraya, Tomohisa Yamakami

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.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces a mathematically sound framework using corrected multivariate Hermite polynomial expansions to solve the positivity problem in copula construction. The application to FX cross-currency volatility smiles demonstrates clear real-world utility, supported by empirical comparisons on major currency pairs.

Research Flowchart

  flowchart TD
    A["Research Goal: Construct valid copula via Hermite expansion to estimate FX cross volatility smile"] --> B["Inputs: Market FX option volatility data (EUR-USD, USD-JPY, EUR-JPY)"]
    B --> C["Methodology: Expand joint density using multivariate Hermite polynomials"]
    C --> D["Correction Process: Apply density corrections to eliminate negative values & ensure valid copula"]
    D --> E["Model Evaluation: Calibrate parameters & fit volatility smile across currency pairs"]
    E --> F["Outcomes: Successfully reproduces EUR-JPY smile and handles complex correlations better than standard copulas"]