A parametrized barycentric approximation for inverse problems with application to the Black–Scholes formula

Abstract

We introduce a method to construct a bivariate rational approximation particularly suited to accurately and compactly represent the inverse of a bivariate function. At the core of the method is a parametrized barycentric form of which the involved unknowns are determined from a sequence of univariate approximations. Our exposition focuses on the inversion of the Black–Scholes formula, yielding an accurate expression for the implied volatility. We demonstrate that our result significantly improves the accuracy of existing bivariate approximations, and, based on S&P 500 Index option data, we show that the accuracy gain proves to be practically relevant. Using the obtained coefficients, included in this article, the approach can easily be implemented.