Delta is different from the probability of an option expiring In The Money (ITM); and here is the surprisingly simple mathematical proof.

According to Black Scholes,

where

Since σ and T are positive values, and that N(x) is a monotonically increasing function, it is logical that

which then means

and we can see that the probability of a call expiring in the money must be less than its delta. The same logic applies to put options as well.

*The above also explains why delta of ATM calls are usually a bit greater than 0.5.

From the math, it is apparent that this difference between delta and P(ITM) varies with respect to volatility and time to expiration, and becomes significant with far dated options in volatility contango.

According to Black Scholes,

where

σ = implied volatility

T = time to expiration

Since σ and T are positive values, and that N(x) is a monotonically increasing function, it is logical that

which then means

and we can see that the probability of a call expiring in the money must be less than its delta. The same logic applies to put options as well.

*The above also explains why delta of ATM calls are usually a bit greater than 0.5.

From the math, it is apparent that this difference between delta and P(ITM) varies with respect to volatility and time to expiration, and becomes significant with far dated options in volatility contango.

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