Is there any evidence that an option delta approximates ITM expiry probability?

Question

Several sources (online and offline) that discuss the delta of a listed vanilla option, state that its delta is a (guesstimate?) of the probability of said option expiring ITM (in the BSM framework).

However, looking at the derivation of delta from the BS model (and its variants), it is not obvious (atleast to me), that the delta can be used as a proxy for the probability of ITM expiry. I want to know if there is any supporting evidence (theoretical or otherwise), that lends at least some credence to this assertion - or is it just an "old wives tale" ?

Answer 1

Actually the delta corresponds to risk-neutral probability of expiring in-the-money (up to a factor of carry cost). This is very different from the real-world probability of expiring in-the-money.

As to the derivation, if you write the risk-neutral expectation equation for in-the-money expiration, it comes to

$$ \int_{K}^{\infty} 1 \cdot p(S_\tau) dS_\tau $$ where $p(S_\tau)$ is the risk-neutral Black-Scholes probability density $$ \frac{n( d_2(S_0,S_\tau) )} {S_\tau \sigma \sqrt{\tau} }. $$ and the answer works out to the delta.

Note that this can be viewed as taking the derivative of the option pricing expectation equation $$ C=\int_{K}^{\infty} (S-K) \cdot p(S_\tau) dS_\tau $$ through the integral sign.