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Do options on options (i.e. compound options) trade on derivatives exchanges? Or are they only traded over-the-counter? If they do trade on exchanges, which exchanges do they trade on?

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No.(*)

Some equity option exchanges trade complex options, which are securities consisting of multiple individual option 'legs' or stocks, but not options where the underlying security is also an option.

(*) Exchanges do from time to time try new types of instrument hoping they will get sufficient interest, so perhaps at one point on the floor of an exchange they gave it a try, but even if it was allowed, there may be insufficient liquidity to for people to want to trade it.

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