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What are the advantages of buying basket equity swaps derivative compared to single equity swap? Will correlation play a role in basket equity swap?

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Correlation will play an indirect role. Begin by noting that, if correlation is 100%, then your basket swap will behave the same as a swap on a single equity (from a modeling point of view). Thus your risk-management systems and derivatives will care about the correlation. For pure valuation of the swap, there's no role for correlation. Valuation is basically just mark-to-market.

Generally, equity swaps are used to trade in long-short positions that are unavailable or impractical from a regulatory point of view. For example, you might be avoiding dividend taxes or trading foreign securities. If you have a position that you intend to treat as a unit, comprised of a basket of securities, it makes sense to combine the whole thing into one swap to keep transaction costs low.

Note that basket options are considered correlation plays.

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