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relationship between euro or dollar futures and eurodollar pair

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The daily volume in 6E (futures) is easy to follow because of central exchanges, the daily volume is roughly 250B$ and the open interest is 525B$ (directional bets)

For the FX market this is much more difficult to say, due to the lack of a central exchange. Fx is also largely handled bilateral between banks, but the volume in EUR/USD FX is much higher than the futures.

There is not a single answer to your question, in some cases a party could decide to build up a position (hedge or speculative) with futures, and will push the future in a leading role, in other cases, it could be the opposite, so for me the one is not leading to the other, it depends on the moment in time.

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