Why Volatility Smiles Challenge Black–Scholes Assumptions
Summary
The document questions whether an option’s implied volatility smile should be read as a forecast of future volatility. Its reply points to a basic limitation of Black–Scholes: under its assumptions, volatility is independent of strike, so options on the same underlying and maturity should not require different volatilities. A smile or skew therefore signals that the model’s assumptions do not describe observed option prices consistently across strikes.
This challenges the idea that a market price differing from a Black–Scholes price is simply the market’s prediction of future volatility, or that Black–Scholes necessarily gives a fair price. The exchange does not compare implied volatility with realized volatility, assess forecasting accuracy, or explain how to price options under alternative models such as Heston. It offers a concise conceptual correction rather than empirical evidence or a complete treatment of volatility forecasting.
Key ideas
- In Black–Scholes, volatility does not vary by strike.
- A volatility smile or skew shows that Black–Scholes assumptions do not fit option prices across strikes.
- An implied volatility should not automatically be treated as a forecast of future realized volatility.
- The exchange gives no empirical test of implied volatility’s forecasting accuracy.
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Full text
# Why is the volatility smile important
# Why is the volatility smile important
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?
## Answer by eltigrechino (score 1)
https://quant.stackexchange.com/a/17751
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.