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Here are two questions related to implied volatilities.

a) The set up here is for an European option. We can get its implied volatility smile from calibration, the question is why could we also observe 'volatility smile' in the market?

b) The second question is for the Heston stochastic volatility model. We use the Heston model to calibrate the implied volatilities of an European option for short term and long term maturities. The results shows clearly it is better fit with long maturity. Why? (Here the Heston model implied volatilities are plotted against the market implied volatilities)

The full answer is not necessarily needed here, some indications or references can also help.

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  • $\begingroup$ options are quoted in implied vol $\endgroup$ – MJ73550 Jun 3 '16 at 15:32
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  1. The option price and the implied volatility is a one-to-one relationship. At one time, there are many options trading in the market, with varying strike $K$ and time to maturity. If you can try to find the relationship between volatility smiled and $K$ or $t_{maturity}$, you will find the smile.

  2. There is no correct answer to this, and there is one possible explanation. Stocks volatility can be decomposed into small daily fluctuations and big sudden movements. The big sudden movements usually comes from fresh news recently, which are very unpredictable. The small daily fluctuations are relatively predictable, especially accumulated in the long run. That said, it is generally much harder to predict near term volatility than long term volatility. On the other hand, heston model assumes a mean-reverting process for the volatility. In the long run, this could be good model because we do find the volatility to be mean-reverting, but in the short run, the noises are just too big.

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