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You can think of delta for calls (-delta for puts) as the first order approximation to probability of expiring in the money. If you subtract this probability from 100%, you'll have the probability of expiring worthless. If you want more exact probability, there are algorithms to construct a pdf from IV skew, and calculate probability from there.
Typically one only thinks about inflation delta in the context of an inflation derivatives portfolio. Then it is the sensitivity to a 1bp change in the zero coupon inflation rate for each maturity. As others have mentioned, regular bonds are sensitive to inflation. However we typically describe that risk as a risk to nominal interest rates, rather than ...
Inflation delta is typically used in the field of fixed income and is defined as sensitivity of the the present value of cash flows to the changes in the inflation curve. In business it is used to measure the inflation risk in the portfolio relative to the liabilities.
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