Next quarter the stock goes up another $20K so you owe an additional 26%*20,000= $5,200. But you are stuck with an illiquid investment that have a very difficult time selling until their is an IPO or a take over.
Someone can correct me if I'm wrong - but I believe I only pay the AMT when I exercise. As the stock continues to move up each quarter, I'm not paying tax on it. I only pay tax when I sell again, and that AMT that I previously paid is applied as a tax credit when I liquidate.
I wouldn't be continually paying tax on the gains. I pay tax upfront on the AMT when I exercise, then I pay the difference on the gains from that moment when I sell.
Yes you have it exactly correct you pay AMT on the difference between the exercise size price and the fair market value. After you sell you get an AMT tax credit which can be applied to future AMT taxes, or portion (AFAIK something like $3k/year) of it can be applied ordinary income when your ordinary tax liability exceeds your AMT liability. From personal experience I can say that I ran up 100K+ in AMT tax credits during the late 90s from exercising the stock options. I retired in 2000 and it took until 2010 to exhaust the credit, and would have been longer except for Congress passed a temporary law that made it easier to get AMT credits back.
I can also say I had one friend (and I read/heard about scores of other horror stories) in 1999/2000 tech bubble did essentially what you are proposing. Exercised stock options in a start up prior to an IPO, paid a lot of money in AMT taxes in 1999 (which they often funded by borrowing), by the time 2000 came the IPO never happened, cause the bubble in burst. In my friends case the IPO happened before the bubble but as the Engineering VP he had to wait a year for the lock up period and the stock price went from $60 to $2. He then is left with a huge tax bill but the asset is basically worthless.
In the scenario I specified where the stock goes up another 20K in the few months, the good news about borrowing money to exercise now is you do temporarily save the 26%* 20K=$5,200. The bad news is you still have to come up with $26,000 to pay the AMT tax. Plus you don't save the full $5,200 because when you do sell the stock you have to pay either 15% of 19% tax on the 20K long term gain. So your true tax savings are only $5,200-($3,000-$3,800)= $1,400 to $2,200. So in effect you are risking $26,000 in AMT tax today to save a couple thousand dollars in a few years.
The point I was trying to make is you really want to sit down with tax planning software in hand, and see how much total taxes you save in 3-5 years, by exercising now as oppose to waiting until you have an actual liquidity event like an IPO. Run several scenarios of liquidity events and the subsequent stock value including the case where the start up goes broke.
I will say that interaction between how the alternative minimum tax system works and the regular tax system works is very complicated stuff. It is quite possible worth sitting done with a CPA if we are talking about 100K plus gains. It is also very possible the CPA has alternative suggestion.