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[[I work in a startup and naturally have been granted a bunch of Nonstatutory
Stock Options. Since one must pay tax at exercise for NSOs, I am considering
exercising my vested options ASAP in order to minimize my tax burden.]]

Ok...most likely not a bad plan.

[[This company is not (yet?) public, so how do I determine the FMV of these
shares? I've been reading IRS Pub 560 (pp. 8-9) but am confused about how to
determine FMV in this case.]]

It CAN get confusing...But forget Pub 560, and instead read IRS Pub 525. It'll give you a better understanding of the issus.

I'll help you is a part of Pub 560 that is of interest to you...but make sure to go to the IRS web site and download/read the entire publication (since you'll have to refer back and forth to the various sections). After your reading, if you have any other specific questions, I'll be happy to try and answer 'em here.
Nonstatutory Stock Options

If you are granted a nonstatutory stock option, the amount of income to include and the time to include it depend on whether the fair market value of the option can be readily determined. The fair market value of an option can be readily determined if it is actively traded on an established market.

The fair market value of an option that is not traded on an established market can be readily determined only if all of the following conditions exist:

1) You can transfer the option,
2) You can exercise the option immediately in full,
3) The option or the property subject to the option is not subject to any condition or restriction (other than a condition to secure payment of the purchase price), that has a significant effect on the fair market value of the option, and
4) The fair market value of the option privilege can be readily determined.

The option privilege for an option to buy is the opportunity to benefit during the option's exercise period from any increase in the value of property subject to the option without risking any capital. For example, if during the exercise period, the fair market value of stock subject to an option is greater than the option's exercise price, a profit may be realized by exercising the option and immediately selling the stock at its higher value. The option privilege for an option to sell is the opportunity to benefit during the exercise period from a decrease in the value of the property subject to the option.

If you receive a nonstatutory stock option that has a readily determined fair market value at the time it is granted to you, the option is treated like other property received as compensation. See Restricted Property Received for Services, earlier, for rules on how much income to include and when to include it. However, you cannot choose to include any part of the value of a nonstatutory stock option in your income in the year of the transfer.

If the fair market value of the option is not readily determined at the time it is granted to you (even if it is determined later), you do not have income until you transfer or exercise the option. When you exercise this kind of option, the amount to include in your income is the difference between the amount you pay for the property and its fair market value when it becomes substantially vested.

Your basis in the property you acquire under the option is the amount you pay for it plus any amount you must include in your gross income under this rule. For more information on restricted property, see Restricted Property Received for Services, earlier.

If you transfer this kind of option in an arm's-length transaction, you must include in your income the money or other property you received for the transfer, as if you had exercised the option.

Hope this helps...
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