To sigmoid-based functional description of the volatility smile

• New static parametric fit of the implied volatility is proposed.
• The asymptotic behavior at wings follows theoretical results.
• Calibration procedure and interpolation/extrapolation algorithms are provided.
• Arbitrage free local volatility and implied density surfaces are constructed.
• Numerical results demonstrate that this model is more flexible than, e.g., SVI.

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.