Short Notes on (Implied) Volatility

1. At the risk of stating the obvious, implied volatility is nothing but the quoting of price under B-S framework. The use of it for anything else (i.e. hedging) is a recipe for disaster.
2. In light of this, the purpose of having a stochastic volatility model is not primarily to have better fits to market smiles (though that is important too), but rather:
• To allow for consistent hedging (especially vega hedging);
• To prescribe some dynamics to the evolution of the smile^.
3. In the B-S framework, if one long an option and continuously delta-hedge, the payoff would be 0.5*S*S*Gamma*(realized var. - implied var.)*dt. What about under a stochastic volatility model? Suppose that the variance process is also stochastic and we only delta hedge w.r.t. the stock price. If the variance process is dv = a dt + b dX, then the delta-hedged P&L would have a residual of b*Vv*dX, thanks to the additional source of randomness.

^ Local volatility model is somewhat helpful for the first purpose (hedging). It however produces smile dynamics that are at odd with observation. See the discussion in the original SABR paper.

 Reference: Rebonato, "Volatility and Correlation: The Perfect Hedger and the Fox" Ch. 6
                  Wilmott, "Paul Wilmott on Quantitative Finance" Ch. 12