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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: ...


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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