Given historical implied volatility and all other know variables (stock price, option strike price, option expiration date, dividend rate, interest rate) what is the best way to calculate the probability of an option being in the money at expiration?
N(d2) is near to the probability the option will expire in the money; I have a video showing how d2 is similar to distance to default in the Merton here on youtube.
N(d1) is the delta.
The technical issue is that N(d2) is a risk-neutral probability; the input in d2 is the riskfree rate, although the theory is more involved.
But, if you replace the riskfree rate with a realistic drift (mu) you have a reasonable estimate, however N(d2) of course assumes normally distributed log returns. So, as with BSM, your answer here still makes the limiting assumptions, namely normal log returns and constant volatility. (I don't know what "historical implied volaility" is: the input is a current, instantaneous volatility estimate, it can be historical or implied)