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A little confused, I consulted the Wilmott forums for guidance on how I can interpret vega/vomma. Another user's post reminded me that the Black-Scholes model assumes that the underlying has constant volatility, so vega is an out-of-model concept. If Black-Scholes assumes that volatility does not change, then would this not render vega/vomma calculations useless since they are based from the Black-Scholes formula? The vega calculation seems odd because one gauges how the option value changes with volatility, but then Black-Scholes assumes volatility does not change. Going a step further, the idea of vomma in a Black-Scholes world seems odd because if volatility does not change, then surely vega would not change?

I understand that Black-Scholes makes some questionable assumptions, but if we price options in a Black-Scholes world, then it seems peculiar that these metrics have come about as a result of the Black-Scholes formula. I would really appreciate any insight.

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    $\begingroup$ They are still useful as a measure of model risk. All models are wrong, some are useful. $\endgroup$ – experquisite Feb 4 '16 at 18:04
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On the topic of your second paragraph, the author below is the authority on precisely that topic. Start at page 19

https://www8.gsb.columbia.edu/leadership/sites/leadership/files/Is%20economics.pdf

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You are onto something, it is inconsistent to be calculating vega with Black-Scholes considering it assumes that volatility is constant.

Black-Scholes is not a good for modeling option prices/implied volatility. It's a very good intuitive model (like the CAPM), and a good way of organizing thoughts, but it is not an accurate depiction of reality. If it were, you wouldn't be seeing volatility smiles (it would be a noisy horizontal line instead).

If you're looking into modeling implied volatility, you should look into model free implied volatility which is based on risk neutral densities.

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2220067

This is about implied volatility. Modeling realized volatility is another story.

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