To sigmoid-based functional description of the volatility smile

TL;DR

This paper introduces a sigmoid-based static parameterization of the implied volatility surface that fits market data well, guarantees arbitrage-free calibration, and enables stable interpolation and extrapolation of volatility and probability density surfaces.

Contribution

A novel sigmoid-based parameterization method for implied volatility surfaces that ensures arbitrage-free calibration and robust surface construction.

Findings

The proposed model fits market implied volatilities effectively.

It guarantees arbitrage-free conditions at all calibration nodes.

Numerical tests show improved fit quality over existing models.

Abstract

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.