I was trying to price SPY options and found that the implied stock price from out of the money options differ from deep in the money options. I understand that this is because of early exercise option for the dividend for deep in the money options. My question is, what is the mathematically correct way to determine the stock price for cusp options which are in between exercise and not exercise? In other words, how do I determine if a certain option contains x% exercise-ness and 1-x% not?

are these american style options ? You could calculate the "fugit" which is the expected life and if it less than the option term then the probability of exercise is non-zero

I understand that, but I want to quantify that "non-zero." When I looked at SPY options last, the implied stock price is a gradual drop off, not a binary exercise or not exercise. That gradualness is what I want to quantify.

which options were you looking at in particular ? Puts or calls ?

Let's say SPY Dec 18 118 Strike. Right now, the C is 3.26/30 and the P is 1.29/32 vs. 120.25 which the means of the market implies a forward price of 119.97. Clearly this doesn't completely account for the non-exercise case, which would have a forward price of 120.25 - expected dividend = ~119.61 (what is shown in deep in the money options) or the exercise case which would have a forward price of just 120.25 (this is shown in far out of the money case). It implies it has a % of exercisability in it and a % of not.

why would the forward price be 120.25 for the "non-exercise" case ?do you know if there are any dividends between now and expiry ?

There is one. Bloomberg predicts it would be $0.64. If you are not allowed to exercise. The stock will drop by the div amount. In the exercise early case, the forward is the pre-div price which is 120.25.

hiI played around with my option pricer. It does look like the SPYs go ex on 17th of Dec which is the expiry of your option. The fugit is not going to be useful here as it will be the same as the option expiry.However, the probability of exercise seems to be roughly given by the call delta. for example, the 118C has a delta of 68. The European call is worth 2.95 and the American call is worth 3.30. the difference is 35c which is about the delta * the dividend (64c). If we look at a deeper ITM option say 110, then using a vol of 20 the European call is worth 9.77 and the American call 10.36 with almost 100 delta. the difference is almost the dividend.