orig. post on Quora Dan Walter's Answer
This is actually possible under two scenarios. One of these is very common.
The first scenarios happens all the time. If the company does a reverse split (the number of outstanding shares is reduced and the stock price is increased) then the strike price of options will usually go up accordingly. This is very common when a company does an IPO. Often it is required to get the publicly tradeable stock to a price range that is consistent with a typical offering. For example imagine you have 50,000,000 outstanding shares and your value is $150M. Your stock price would be $3. But, you would need that price to be $15-30 for a typical IPO. With a value of $150M you would need to have only 10,000,000 shares outstanding to have a "per share" price of $15. So you would do a reverse stock split and everything associated with the price, including historical grant prices, would also change.
The second scenario is far less common. Most companies determine the strike price on grant date for both tax and accounting reasons. The company can, however, choose to grant equity while setting that date to price the options to a date (or event) into the future. This may be done for performance-based stock options, there are a few types that work like this. That being said, in the US, this is very uncommon.
for more on Performance-base Equity, check out this matrix. 2012 Performance Equity Compensation Design and Use Matrix