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with what likelihood would one expect an ATM caplet to end up in the money? Just as a very rough guess, from real world experience.

When I consider N(d2) from the Black formula, for spot = strike = 4%, vola = 20%, T = 1, tenor = 12m, I get something around 46.*%.

On the other hand, when I think about it qualitatively: Under most market conditions, the forward rate curve is increasing. As a very, very rough argument, we would expect the spot rate 1 year into the future to be similar to today's spot rate rather than today's 1 year forward rate, i.e. to be lower than today's ATM strike. Thus, under the physical measure, the caplet should be less likely to end up in the money than end up out of the money. Does this roughly conform with a number around 46%?

For my thesis, I am using a simulation model that tries to create synthetic realizations of market data (in a complicated procedure that would go too far to explain here). In this synthetic world, I get ITM-likelihoods much lower than the above and I am wondering about real world estimates from a practitioner's point of view.

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I would say that the real issue here is do you think the forward rate an is unbiased estimate of the actual rate? and if not why not? The statistical test would then be to take historical time series and see where the bias lies. If there truly is a non-trivial bias, what would be the risk-return trade-off of trying to exploit it?

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