Concerning economic indicators, is there a built-in function enabling to automatically get the released "actual" value on python at the exact moment when it is made available on Eikon ?
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Yes, streaming market data is available through Eikon Data APIs library v1.1.2 or higher. See the tutorial titled "Streaming API - How to retrieve real-time streaming prices". Alternatively use RDP Library (which includes Eikon Data APIs library as a module). See the article titled "Consume Realtime data with Refinitiv Data Platform".