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It's common to just set all zeroes equal to the minimum nonzero observation in your dataset.


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One interesting fact that explains a lot at least to me is that implied vol has to do with only one option price. (One data point) Whereas local volatility is not determined from one unique call price in particular, but rather by many data points. To make things simple consider 4 neighbor data points in the surface. From this four is derived one local vol. ...


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I'll answer both of your questions in one go: Your ideas are correct. If the Black-Scholes model was true, the implied volatility surface would be flat but it is not in real life. Thus, the geometric Brownian motion as stock price model is misspecified and we need more sophisticated models (sto vol, jumps etc), in particular if we want to price more ...


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The market will quote Call and Put options prices within a bid-ask spread. In order to imply the volatility, one may choose to use the bid, the ask, or the mid. Although the mid is a better idea in general, there is no right choice. The point is that there is always a spread in the implied volatility. Now, the Put-Call parity only holds within the a spread. ...


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