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One thing I can't understand clearly is why there is so much focus on the volatility smile. Given my knowledge of the Black and Scholes model, this is what I get:

People use the volatility smile as a forecast of future volatility. They assume that the market is 100% efficient and everyone would engage in dynamic hedging if they could or that everyone agree that the Black and Scholes model gives the 'fair' price for the option, whatever it means to them. Therefore any difference between the market price and the BS price shouldn't be interpreted as an opportunity for arbitrage or a presence of model risk, but what 'the market' is saying the BS parameters should be, in special the volatility the market is predicting for the future time until the maturity of the European option.

Are there many issues with this interpretation? If not, why not to estimate the volatility directly by using a stochastic model like Heston instead of working with so many layers of complexity and assumptions? Could you indicate good papers trying to assess the accuracy of the implied volatility as a predictor of future volatility?

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Look at the B-S parameters for the dynamics of the stock.

$\frac{dS}{S} = \mu dt + \sigma dt$

$\sigma$ is independent of strike in the B-S model, which means all derivatives priced assuming these dynamics should have the same volatility. This clearly is not the case given the existence of smile and skew. You can't assume the BS model produces the "fair" price when its assumptions are violated right at the start.

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    $\begingroup$ This is not what I assume. This is what I assume that people assume. You are right, the volatility is constant for all strike prices under the standard BS model. The question is why people analyse this volatility curve so much and how they use it. If they trust in the standard BS model, they should be making profit from arbitrage. If they want to incorporate a stochastic volatility, they should use an alternative model to account for it. So what is the point of obtaining those implied volatilities? $\endgroup$
    – John
    May 11, 2015 at 18:22
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    $\begingroup$ Quoting options in terms of B-S parameters is extremely convenient due to the model's explicit solution and ability to solve for parameters relatively easily. Since every firm most likely has their own option pricing models it is nice to have a common language when quoting volatility in the market. $\endgroup$
    – eltigrechino
    May 11, 2015 at 18:40
  • $\begingroup$ @John implied volatility is the price in the market. It tells us immediately what the market thinks about the volatility as of today. A stochastic volatility model models the volatility in the future. $\endgroup$
    – SmallChess
    May 11, 2015 at 23:53

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