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How is the first price of u.s treasury futures decided? Does it depend on the last price of the previous future contract ?

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More generally, it sounds like you want to know how the initial settlement price of a futures contract is calculated. The exchange provides rules for each contract.

Sometimes it's by referencing an external value (eg, equity index futures, interest rate futures etc), or some average of trades done (eg, some commodity futures). This can mean in practice that there is no settlement price unless trades are done, and this is ok if there is no open interest at the end of the day, because there are no positions to value.

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