Fuzzy transform; fuzzy partition; implied volatitlity

Estimation of the implied volatility surface is a crucial step for pricing of illiquid or non-listed options. For its estimation we use market data of traded liquid options. In order to obtain smooth surface we mostly use some nonparametric regression approach, though some combination of market data can lead to the inconsistency of the estimated values (i.e. leading to arbitrage opportunities). In this contribution we focus on a novel nonparametric regres- sion approach based on fuzzy transform technique. The original definition is extended into multivariate higher degree F-transform so that we are able to smooth (2+1)D discrete data. Proposed procedure allows us to calculate also partial derivatives of the estimated function easily. Extending the optimization problem by suitable constraint we can prevent most of the sources of arbitrage related to option valuation with implied volatility smiles.