I've been under the impression that an option's delta represents, among other things, the likelihood that that option expires ITM. If that's the case, wouldn't it stand to reason that an option whose strike price is equal to the underlying SP would always have a delta of 50%? I usually see that such ATM options have deltas in the 42-58% range, but shouldn't it by definition be nearly exactly 50%? Another Q I suspect is related to my misunderstanding above: why isn't the delta curve symmetrical on both sides of the strike price? For example, with an underlying SP of $50, why aren't the deltas of, say, a $48 Call and a $52 call (or at least their absolute values) idential? IOW, with a $50 underlying, isn't it equally likely that at expiry the underlying is either below $48 or above $52? And if so, shouldn't the absolute values of their deltas be the same? (All of the above is predicated on my understanding that an option's delta does indeed represent the likelihood that it expires ITM...if that's not actually true, I'd like to understand what I'm getting wrong.)

Interest and Dividend have an effect in the model of moving the current stock price to account for those flows and changing the delta to reflect the implied stock price.

Think of it absent rates or include rates as the ATM forward. The ATM (forward-strike) option is a pick-em proposition; there is an equal chance of expiring ITM or OTM. Why does an ATM digital have a theo-val of 50/100?

As destriero pointed out, BS pricing is based on the forward price, not current spot. You are correct that delta represents the probability of expiring ITM. Also, should try to think of normal distribution in returns, so lognormal in price such that delta(102 call) > delta(98 put)

As a side point, ATM forward options will not have a perfectly 50-delta either, instead will have a slight bias toward the call.

I am just saying that even a perfectly ATMF option (i.e. exactly accounting for the rates and divs) would have a small call bias due to the lognormality and that bias would grow with higher vol.

Unlimited upside; bound to zero. No negative values therefore the call bias. Look at the deltas on low-priced tickers with high vola.

Delta is the rate of change of the option price relative to changes in the underlying. Always always always. Delta is a handy approximation for the other three roles in which it is commonly used. [Hedge ratios and P(ITM)]. The P(ITM) role is the weakest, as there is no direct link. The BSM model weakens as DTE tends to zero, and 'lumpy' outcomes are a result. Use of the BSM delta for P(ITM) demonstrates this weakness most prominently: compare ATM deltas for far-off expiries to those close to expiration. There is an MIT video on YouTube which derives equivalencies between delta and P(ITM) -- if you've got questions, that'd make a great place to start. ... HA! Found it: