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Suppose I have 2 stocks $S_{1}$ and $S_{2}$: \begin{align} & dS_{1}=rS_{1}dt+\sigma_{1}S_{1}dB_{1}\\ & dS_{2}=rS_{2}dt+\sigma_{2}S_{2}dB_{2}\\ & dB_{1}dB_{2}=\rho dt \end{align} Then I have a option A with payoff $(S_{1}+S_{2}-K)^{+}$ and another option B with payoff $(S_{1}-S_{2}-K)^{+}$

Question: I want to know the rigorous proof of the relationship between option A/B and correlation $\rho$? Or you may tell me where I can find the proof?

Intuitively: when $\rho$ is increasing, will move aligned with each other, then we can think that $S_{1}+S_{2}$ will become larger, and $S_{1}+S_{2}$ will become smaller, the option A price will be bigger, Option B price will be smaller. So we think price of A is a increasing function of $\rho$ and price of $B$ is a decreasing function of $\rho$.

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why would you think S1+S2 will increase with larger p? In fact, empirically S1+S2 should be lower with higher correlations. Correlations generally increase when the market as a whole decreases in value dragging with it S1 and S2. But trying to find some more rigorous backup than just my claim. But keep in mind for a start that such basket options are highly sensitive to 2nd and higher order risks such as volatility skew, cross gamma risks and the like. This may help: opalconsulting.ch/dataa/248de.pdf –  Matt Wolf Mar 20 '13 at 7:59
    
Formally, I am think it is call option written on $S_{1}+S_{2}$ and $Vol(S_{1}+S_{2}) \approx \sqrt{\sigma_{1}^2+\sigma_{2}^2+2*\rho*\sigma_{1}*\sigma_{2}}$, so when $\rho$ is increasing, then $Vol(S_{1}+S_{2})$ will become bigger, the price will be bigger. –  nkhuyu Mar 20 '13 at 8:26
    
you only consider vol effects. You also need to consider the much more prevalent relationship between asset price correlations and the signage of such asset price returns. –  Matt Wolf Mar 20 '13 at 8:29
    
nkhuyu prices the option in the framework of correlated GBM and constant parameters. This is idealized and there such phenomenons don't exist. Reality is probably different but first we should understand the ideal world. In the idealized world only vol and correlations exist. Of course when trading one should not forget about reality. –  Richard Mar 20 '13 at 12:54
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