We (I) do not know exactly the type(s) of options that have been granted to Musk. In general, stock options have an exercise price based on a number of supposedly independent factors.
You are correct that as long as he does not
exercise any of the options granted to him, there is no tax due. But any taxes that will be due upon exercising them will depend upon whether he ponies up the cash personally to purchase the stock at the exercise price or whether he leverages the transactions by immediately selling enough shares to get the cash to buy the shares he wants. Even if he ponies up the $$$, there still might be some tax due if there is a difference in the current market versus the market value when the options were granted.
So, if the exercise price is $150/share and he wants one million shares, then he either pays into Tesla $150 million (no taxes due possibly) or he has a simultaneous purchase and sale of an amount in excess of the $150 million so that he winds up with the stock.
Then, of course it depends whether these are qualified or non-qualified options. Most are reported on a W-2 with mandatory withholding (Social Security/Medicare) and voluntary withholding (income taxes.)
If he does exercise some of his options and retains the stock, he will have some income taxes due anyway. It remains to be seen if California will have its grubby hands out to get its share of any taxes as I believe these options were granted when he was still a California resident. We have some pretty stringent rules about the timing of reporting gains when a taxpayer moves out of state.
****Caveat: this area is something I dealt with a long time ago. Perhaps my memory is blurred or confused
. S**t happens after four decades slogging along in this arena. And if he has personal losses on his tax returns, he might have made a §83 election to be taxed currently on the unexercised options as he bears a "substantial risk of forfeiture." This election establishes basis for gain or loss purposes if he does exercise the options.