Assessing the Physical Probability of an ATM Caplet Finishing In the Money

Summary

The document asks how likely an at-the-money caplet is to finish in the money under real-world conditions. The questioner compares a Black-formula estimate based on the forward rate, strike, volatility, and maturity with a qualitative view: when forward curves slope upward, a future spot rate might be closer to today’s spot rate than to the current forward, potentially making the caplet less likely to finish in the money under the physical measure. They ask whether a probability around their calculated estimate is plausible and note that a synthetic market-data simulation produces lower probabilities.

The response does not offer a practitioner benchmark. It frames the issue as whether the forward rate is an unbiased forecast of the eventual realized rate, and recommends examining historical time series for bias. Any detected bias would need to be assessed against the risk and return of attempting to trade it. The discussion distinguishes pricing probabilities from real-world probabilities but supplies no historical sample, empirical estimate, or evidence that a persistent bias exists.

Key ideas

• An at-the-money caplet’s in-the-money probability depends on whether the question concerns a pricing measure or real-world outcomes.
• A Black-formula probability is based on market inputs and does not by itself establish the physical probability of payoff.
• An upward-sloping forward curve motivates the question of whether future realized rates tend to fall below current forward rates.
• Historical rate data can be used to test whether forwards are biased estimates of subsequent realized rates.
• A possible forecast bias matters for trading only when its risk and return are considered; the discussion provides no empirical benchmark.

Tags

# Likelihood of a caplet ending in the money


# Likelihood of a caplet ending in the money












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.

## Answer by Mark Joshi (score 1)

https://quant.stackexchange.com/a/19145

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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