  We propose a new static parameterization of the implied volatility surface
which is constructed by using polynomials of sigmoid functions combined with
some other terms. This parameterization is flexible enough to fit market
implied volatilities which demonstrate smile or skew. An arbitrage-free
calibration algorithm is considered that constructs the implied volatility
surface as a grid in the strike-expiration space and guarantees a lack of
arbitrage at every node of this grid. We also demonstrate how to construct an
arbitrage-free interpolation and extrapolation in time, as well as build a
local volatility and implied pdf surfaces. Asymptotic behavior of this
parameterization is discussed, as well as results on stability of the
calibrated parameters are presented. Numerical examples show robustness of the
proposed approach in building all these surfaces as well as demonstrate a
better quality of the fit as compared with some known models.
