Good day all!

We know that the delta of a long ATM call option must be around +0.5. 

The at-the-moneyness means that the current stock price S equals the strike price K, which means the ln(S/K) in the Black-Scholes formula for d1 becomes ZERO. All well there. 

In order for N(d1) to be 0.5, d1 should be ZERO (from the standard normal distribution). We already know that ln(S/K) is ZERO. Therefore, the numerator and the denominator in d1 become (r + 0.5 * sigma^2) * T and sigma * sqrt(T) respectively.

I used risk free rate r = 5% (annualized), sigma = 0.8% (annualized), and T = 0.25 years, but really my question is for any values (except T = ZERO) that can be substituted in the formula for d1. In this specific case, d1 computes to ~3.16 and N(d1) therefore is close to 1 (0.99). 

I am lost as to why N(d1) doesn't compute to anywhere close to 0.5, given this is an ATM call option. Thanks for educating.

This actually comes back to your previous option pricing question. The important ATM concept in this context is wrt the forward price. In this case, you need more than a three standard deviation negative price move to overcome the drift (which may look extreme, but your vol is tiny). The N(d1) from the ATM forward call is something like 0.5008, probably more in line with intuition.

That was amazing! I re-computed the N(d1) using ln((S*exp(-r * T)) / K) instead of ln(S/K) and the delta is now 0.492 =~ 0.5!

Thanks Bearish -- that really helped!

So is it the case that when anyone mentions money-ness in the options space, it's ALWAYS with respect to the forward and not the spot price? Or are there exceptions?

No, sadly, people’s language is pretty sloppy on this topic. It is generally helpful to specify outright whether you are talking about ATM spot or forward. Of course, if the volatility is 20% rather than 0.8%, the difference will be a lot less significant

Thanks much! As a general rule when computing N(d1), should I use the spot price or the forward price i.e. ln(S/K) or ln((S*exp(-rT))/K)?

P.S. That volatility of 0.8% is real! That's the S&P500 volatility of daily returns over the past 30 days, where I computed returns as computed as ln(P_t+1 / P_t). And I just realized I made the dumb mistake of not annualizing it by multiplying it with sqrt(252)! Volatility is annualized in the Black-Scholes formula!