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You often hear about the volatility smile. Is that something that occurs within a standard Black Scholes model, with the usual formula for the price of a call?

If we were to observe some call price (e.g., 15), and then derived implied volatilities from the BS formula depending on different strike prices but fixed maturity (i.e, maturity = 1 and strike goes from 80 to 140?), would we then see a smile?

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If we were to observe some call price (e.g., 15), and then derived implied volatilities from the BS formula depending on different strike prices but fixed maturity (i.e, maturity = 1, and strike goes from 80 to 140??), would we then see a smile?

Yes. Market prices for various strikes and a given maturity often have higher implied volatilities from the Black-Scholes model away from at-the-money.

It is not accounted for in the Black-Scholes model in the fact that volatility is not a function of strike, so volatility is assumed to be constant across strikes, but the market does not price options that way.

I don't know that a quantitative theory has ever been proven; I've always just assumed that people are willing to pay slightly more for options deep in or out of the money based on their strategy, but I have no evidence to base that theory on.

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The Black-Scholes model was based on assuming lognormal stock price fluctuations with a constant volatility. However, the modern practice is to use the Black-Scholes formula not as a prediction but merely as a parametrization of option prices, where the observed price of a given option at a given time translates to a "local" implied volatility (IV). Thus, when the resulting IV varies over strikes and over time, it is parametrizing a breakdown of the original Black-Scholes assumptions. Predicting an option price is tautologically reframed as predicting the corresponding IV. At most we can say that if the original Black-Scholes assumptions are roughly right, the IV is roughly constant.

The volatility smile is an expected result of stock price fluctuations with heavier tails than lognormal. That is, sudden very large moves up or down due to news or sentiment shifts, though rare, are less rare than the lognormal distribution indicates. (See the work of Nassim Taleb.) In the absence of a replacement for Black-Scholes that actually models the more accurate distribution, we patch it up by noting that the further out-of-the-money options are priced as if the underlying lognormal volatility were higher (a proxy for the heavier tails).

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