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Given an underlying (e.g. future or an index), would options on that underlying trade in the same currency and on the same exchange?

For example, options on FTSE100 index itself would trade in GBP on a particular exchange. Options on SPX500 futures will trade on CME in USD?

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A quanto option is an option where the payoff will be denominated (and usually paid) in a different currency.

For emerging markets, options are often denominated in a non-convertible currency (KRW, INR, TWD). For the Kospi denominated in KRW, you have futures on KRW denominated options in Eurex (cash settled in EUR), this is called a Compo option (you have a EUR/KRW fx exposure on the premium).

For INR and TWD, you have futures on INR/TWD denominated indexes (nifty/msci tw) traded, in that case 1 point of the index has a US dollar value, this is a Quanto future/option (you don't have any USD/TWD or USD/INR fx exposure).

An other example for futures would be NIKKEI, futures are available in JPY on SGX/JPX/CME. They are also available in USD on CME (no USD options however).

Over the counter you could also create an EUR option on a EUR Nikkei index (with a strike in Euros), I can't see a real life listed example however.

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