The selling price, after the 10% discount, is  120 - 0.1*120 = 108 dollars.


It still provides the profit of 20%.


Hence, for the buying price P we have this equation

    1.2P = 108,


which gives

    P = 108/1.2 = 90 dollars.


ANSWER.  The price which the merchant pays for each watch is $90.