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Well, if you assume $X$ has volatility $\sigma_X$ and $Y$ has volatility $\sigma_Y$, then $$\sigma_{X+Y} = \sqrt{ Var( X + Y) } = \sqrt{ \sigma_X^2+\sigma_Y^2 + 2 \sigma_X \sigma_Y \rho }$$ Then, you want to show $$\sigma_{X+Y} = \sqrt{ \sigma_X^2+\sigma_Y^2 + 2 \sigma_X \sigma_Y \rho } \leq \sigma_X + \sigma_Y$$ Squaring both sides: ...

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As always I recommend reading Rennie and Baxter for an introduction to option pricing that's not too technical and gives intuition about how it all works.

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From Ziegel (2013) : The risk of a financial position is usually summarized by a risk measure. As this risk measure has to be estimated from historical data, it is important to be able to verify and compare competing estimation procedures. In statistical decision theory, risk measures for which such verification and comparison is possible, are called ...

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I have never seen such an adjustment. While monthly data are irregularly sampled in time (in every way...calendar days, trading days, seconds, etc), that irregularity is likely to be a smaller effect than your choice of data frequency (monthly, weekly, daily data). That said, your question is intriguing because in other fields they do have to deal with ...

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