You will owe New York state income tax on your stock options but not New York City tax.
New York says:
When an individual has a change of resident status for New York City during the taxable year, the amount of New York City compensation received from stock options, restricted stock, or stock appreciation rights in the taxable year that is included in the individual's federal gross income (as either ordinary income or capital gain income) is also dependent on the individual's resident status at the time that the compensation is recognized for federal income tax purposes.
If the compensation is recognized during the New York City resident period, the entire amount of the compensation recognized for federal income tax purposes (both the compensation element and any appreciation in the value of the stock after the exercise date) is includable in New York City source income.
However, if the compensation is recognized during the nonresident period, the compensation is not subject to New York City tax.
TSB-M-07(7)I Income Tax
