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I think I have figured this out. The key to the understanding is to think of the options' vegas as "key-strike vegas" compared to the var swap/replication portfolio's vega, which is analogous to "key rate durations of a bond portfolio" to the total effective duration of the portfolio.

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The variance swap's Vega that is equal to the variance notional refers to the realized variance. The Black-Scholes vega refers to the market implied volatility. Now if you want, you can estimate the realized variance at expiry from the volatility of the options (for instance taking the atm variance arbitrarily), and that's often what people do. But that's ...

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You could use beta ($\beta$) to get a very crude approximation of the standard deviation ($\sigma$) of the stock. $$\beta_s = \rho_{s,m} \frac{\sigma_s}{\sigma_m}$$ Where the subscript $s$ stands for a stock and the subscript $m$ for the market. $\sigma_s$ is known. You could assume an industry wide correlation ($\rho$) value for stocks in a specific ...

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For starters, one can argue they provide a better fit to the distribution of asset returns than a Normal distribution simply because stable distributions allow for more degrees of freedom. I had a discussion with a very well-known financial mathematician on the subject of using stable distributions as the return process for derivatives pricing, and his ...

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