How do we estimate the pre-market opening price?

As others have said, what you're asking would be the million-dollar question so to speak!

There are so many variables that go into such a calculation, and MANY of them are ones to which you don't have access, or even if you did, would necessarily know how to apply to the stock. Some of them are contextual, meaning they may apply more in one set of circumstances than in another, so it's difficult at best and impossible at worst, almost like trying to predict where a particular raindrop is going to land.

All you can do is watch the buy/sell ratio and volume trends to get a general idea of which direction a stock is likely to go, but even that can be just guesswork. What can ruin all of your hard work? A well-timed press release from the target company or an analyst just before trading begins, or some sort of external news story that gets a lot of investor attention. All the math and models in the world won't save you when that happens - man times news affects stock more on emotion than facts. Ever heard the saying, "Buy on rumor and sell on fact"?

Not to sound like a smart aleck, but if you figure out this particular riddle, don't tell ANYONE unless you're charging mightily for the answer, because it'll make you incredibly rich!

I interpret the question as seeking an informational prediction, not a profit opportunity. That is, you want to "predict" the premarket open of an individual stock in the same sense that the overnight S&P 500 futures "predict" the premarket open of the SPY ETF. This will account for information known to the market at the time of the prediction, but will not attempt to predict any subsequent moves (which in an efficient market will have zero mean).

So I think others' mentions of "everyone would be doing it" and "million-dollar question" are off-point. Everyone could do it and many already are. There's no paradox because it's not a way to beat the market.

So here's what I think can be done:

1. Some US stocks like Microsoft trade on the Frankfurt exchange, which for most of the year opens at 3:00a New York time. Those prices can be converted from euros to dollars to get an indicative New York premarket open.

2. If a Frankfurt price is not available, an estimate can be made from correlated stocks and indexes. You could do a historical regression of the premarket open of the stock against, say, the available price moves as of 3:30a for similar stocks (US or foreign) that do trade in Europe, and futures on indexes that the stock belongs to. An efficient market does not prevent this from working, since it is not a prediction for making money. You'd be predicting the change from the previous close at a time when you can no longer trade at the previous close.