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Re your first question: Use the implied volatility $\sigma_{imp}(X,\tau)$ for strike $X$ and expiry $\tau$. The option price, and hence the implied volatility, is driven by the options markets. Your option model should first and foremost be able to replicate observed option prices (hence, you plug in implied vols).


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In the following, I am assuming the BS73 model and I assume that "ATM" means $$ S = Xe^{-r\tau} $$ The pricing formula for a European call then becomes $$ \tag{1} O\propto N\left(+\frac{1}{2}\sigma\sqrt{\tau}\right)-N\left(-\frac{1}{2}\sigma\sqrt{\tau}\right) $$ times some scaling factor which is irrelevant for our purpose. Clearly, $$ Vega\equiv\frac{\...


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Based on your computation, you can observe that the $N’$ term is always positive, between 0 and 0.4. As $\sigma$ is always positive, you can focus on the $-d_2$ term. When $d_2 > 0$, i.e. call is ITM, delta has a negative sensitivity to volatility ; conversely for OTM call. That is in line with your remark.


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This is a good question. See my answer to a question here The point is that under Black-Scholes (and also many SV models) not only European prices but also American options prices are homogeneous of degree 1 in strike and spot as the optimal exercise time does not affect the homogeneity property in strike and spot price. Hence also for American options ...


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