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Does realized inflation or expected future inflation determine how much money needs to be placed into the escrow account each day?

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I think this is not an inflation specific question, isn't it. The same question can apply to interest swap. And I assume by inflation swap, you mean zero coupon inflation swap(ZCIS).

The margin is determined by the mark to market (MTM) of the contact. In the case of ZCIS, it will need a model, i.e., inflation curve model, for valuating the final fixing and thus the MTM. The good news is that the models across the street are pretty much similar. The difference would probably be smaller than the bid/ask spread.

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