One question: when asking for the correlation of a basket, a trader told me 50% whereas I expected him to give me asset pairwise correlations (i.e. the correlation matrix). What does this 50% mean please ?
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Let us consider a basket $B$ with components $S_1,\dots,S_n$ : $$B(t) = \sum_{i=1}^nw_iS_i(t)$$ At time $t$, each component has standard deviation $\sigma_i$, $i \in \{1,\dots,n\}$, and pairwise correlations are $\rho_{ij}$, $i \not= j$. Thus: $$\sigma_B^2=\sum_{i=1}^nw_i^2\sigma_i^2+2\sum_{i=1}^n\sum_{1=j}^iw_iw_j\sigma_i\sigma_j\rho_{ij}$$ The implied basket correlation $\rho_B$ is defined as: $$\rho_B=\frac{\sigma_B^2-\sum_{i=1}^nw_i^2\sigma_i^2}{2\sum_{i=1}^n\sum_{1=j}^iw_iw_j\sigma_i\sigma_j}$$ It can be interpreted as an "average" pairwise correlation between the components of the basket, and that's what the 50% stands for.
answered Jun 4, 2018 at 10:46
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$\begingroup$ No worries. Don't hesitate to accept it @Cedric_W. $\endgroup$ Jun 4, 2018 at 11:08
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2$\begingroup$ Another way to say it is: if all the correlations $\rho_{ij}$ were equal to a common value $\rho_B$ then this value could be found by the formula above. $\endgroup$– Alex CJun 5, 2018 at 23:32