My question is regarding Volatility Skew Models and their inputs. I have noticed that a vast majority of models take as an input the forward of the underlying (even in the case of stocks - where the eventual exercising of the option leads to delivery of stocks, not forwards on stocks). For example, some local volatility and Parametrization models seem to care about the moneyness defined by the forward ln(F/k) rather than the spot ln(S/k). That is, forward volatilities instead of spot volatilities - naturally they produce different implied vols for each strike. My questions then become the following:
1) Why are forwards used? Arent spot volatilities essentially what are important for volatility trading in the sense of implied vs realized?
2) Which volatility is the correct one to use for hedging purposes? For example: if the implied volatility suggested by the forward volatility is 10%, and the spot volatility model suggests 12% , and the underlying realized 11% - will i make a profit on this? (Lets assume a Black Scholes Framework of hedging for this example).
Could it also be possible that my understanding of Forward and Spot volatility is wrong and i have misunderstood the jargon? If so please let me know!
Thank you for your help in advance.