The Chebyshev method for the implied volatility

Kathrin Glau, Paul Herold, Dilip B. Madan and Christian Pötz

Abstract

The implied volatility is a crucial element in any financial toolbox, since it is used to both quote and hedge options as well as for model calibration. In contrast to the Black–Scholes formula, its inverse, the implied volatility is not explicitly avail- able, and numerical approximation is required. In this paper, we propose a bivariate interpolation of the implied volatility surface based on Chebyshev polynomials. This yields a closed-form approximation of the implied volatility, which is easy to implement and to maintain. We prove a subexponential error decay. This allows us to obtain an accuracy close to machine precision with polynomials of a low degree. We compare the performance of our chosen method in terms of runtime and accuracy with the most common reference methods. In contrast to existing interpolation methods, our method is able to compute the implied volatility for all relevant option data. We use numerical experiments to confirm this results in a considerable increase in efficiency, especially for large data sets.