No you should not assume he knows the language if you do not get it from your race or background. If there is no rule giving you the language you do not have it.
Keep in mind that your character need not come from the Underdark so he might in fact not know the language. (Surface Drow are a thing in some settings.) He might also come from the Underdark and never bothered to learn the language of the "lower races" living there. (In some settings Drow are quite racist.)
If you have a valid reason to know the language you might bargain with your DM to get it. You might give up one of your languages but this would be hard as you only have Common and Elvish. Giving up Common would be bad as you would probably need it to talk to the other group member and giving up Elvish would also be strange. This would basically mean creating your own custom background. This is legit but needs the OK of the DM and should not result in advantages for you.
So you should take a look what you could deal in to get Undercommon if you really want to have it.
If your DM has a nice day and you have a valid reason to know Undercommon he might give it to you for free of course. But having something of to bargain would help.
I would only hand out the language for free if I do not plan to go to the Underdark any time soon with my campaign. That would reduce the language to a style thing witch would be for free.