Although Black \& Scholes is no longer considered as a valuable financial model, the implied volatility function is considered an object of primary interest, particular in empirical finance. The present paper proposes a methodology to generate volatility surfaces -- i.e. volatility as a function of the strike price and time to maturity -- while giving precise error estimates. This is done with no reference to model parameters but rather the absolute log of the option price and to its strike. Further to the main results of Section 5 the paper proposes applications to specific examples which include Heston and Lévy models.