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

摘要

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.