# Forward Volatility vs Spot Volatility in Option Skew Models

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!

• the answer to the first question comes from Breeden Litzenberger. For the second, you want to hedge, i.e. you should cover expected cashflows. Thus, you should use the same quantity to which the pricing is exposed – Vitomir Jun 27 at 12:38

The use of forwards is just another method to look at the underlying. The Black-Scholes options model utilizes Spot and handles the carry as an interest rate in the model. On the other hand the Black Model uses forwards instead. Since the forward price would take into account the carry, both models should yield the same result if one is accounting for all the valuation parameters correctly. Being consistent in applying the same implied vol of the underlying that is used in the model is the key. If you are using the Black Scholes model, you would use the volatility of the spot underlying; if you are using Black, you would use the volatility of the forwards.

Your question is using some terminology incorrectly. Forward volatility refers to the volatility realized from t1 to t2 given that it's currently t0 and t0 < t1 < t2. What you are talking about is whether the moneyness of an option is expressed in relative to the spot or relative to the forward.

Which parametrization to pick is a choice; as long as you do all your calculations consistently, it will not matter. That being said, by working with the moneyness relative to the forward, it's directly clear what the vol is for an at-the-money (forward) option is (which might be useful in terms of using that as a backbone for marking your vols), while the moneyness of an at-the-mone spot option doesn't really tell you much if you don't know the values of other parameters that define the forward rate.