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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 ...


Since the volatility is not changing, we can assume that the only change is the underlying asset price $S$. Then \begin{align*} C(S+\Delta) &\approx C(S) + Delta \times\Delta +\frac{1}{2} Gamma \times \Delta^2 \\ &=11.50 + 0.58 \times 0.5 + \frac{1}{2}\times 2 \times (0.5)^2\\ &=12.04. \end{align*}


On the topic of your second paragraph, the author below is the authority on precisely that topic. Start at page 19


If you want to know what Greeks the market assigns to an option, i.e. the market implied Greeks, then you would use the implied volatility. And that is what traders like to look at.


For an ease of argument assume there are no discrete dividends and interest rates are non-stochastic and identically zero. Consider a delta-hedged log-contract, which is essentially equivalent to a variance swap in this case. Since this contract can statically be replicated by traded options, you can think of variance itself as a traded asset. This means the ...


Consider the graph of price vs implied volatility of an at-the-money call option. At 0 volatility, the price is zero, as with zero vol the spot remains constant and finishes at the strike for zero payoff. For low volatilities, there is a famous approximation that call value is about $0.4 S \sigma \sqrt T.$ That gives the graph initially increasing ...

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