This is correct. To put some concrete numbers on it, I'm in my 6th year at a startup. In 2016, I spent $6,400 to exercise ~50,000 options that had a total FMV of ~$60,000. This is going to add over $7,000 to my 2016 taxes due to AMT.
If you're single and your income is over $115K, or if you're married and your total income is over $150K, there will be tax implications for exercising your options, unless you exercise them when the spread is $0.
That's not quite how AMT works. AMT is not added to your taxes, it's an entirely alternative tax system (hence the A in AMT). You'll pay the larger of the two values (AMT vs ordinary tax), and given the taxation rates, you may not have made enough to actually trigger AMT. The rate for AMT is lower than ordinary tax, and there's an exemption for the first $N (where N varies based on several factors).
I understand how AMT works; these numbers are from actually calculating my 2016 taxes. With the exercise, I'm paying over $7,000 more than I otherwise would have.
Well, you MAY owe AMT; it's not certain, and depends on your specific financial situation. Also if you pay AMT then you are eligible for an AMT tax credit which can offset the taxes you have to pay later on when you sell the shares.
I think the bottom line is that ISOs are far better than NQSOs in terms of your own tax liability.
This is also what I found when I researched this a couple years ago. I'm a little concerned that TFA and most of the comments here seem to have it the other way around ("do I need to go file an amended tax return?"). But indeed, according to [0], "The tax benefit is that on exercise the individual does not have to pay ordinary income tax (nor employment taxes) on the difference between the exercise price and the fair market value of the shares issued (however, the holder may have to pay U.S. alternative minimum tax instead)."
For NQSO this is true, for ISO this is false. The exercise of an ISO grant is not treated as ordinary income.