The Chebyshev method for the implied volatility

The implied volatility is a crucial element of any financial toolbox, since it is used for quoting and the hedging of options as well as for model calibration. We propose a bivariate interpolation of the implied volatility... [ view full abstract ]

The implied volatility is a crucial element of any financial toolbox, since it is used for quoting and the hedging of options as well as for model calibration. 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. Numerical experiments confirm a considerable increase in efficiency, especially for large data sets.