Sentence Correction

The Senate headed toward another critical hurdle in President Obama's bid to secure a sweeping Pacific-rim trade deal on which he has staked much of the prestige of the presidency.

the pronoun "he" has no antecedent noun. President Obama's is a possessive form.
Please correct me .

This is from an article posted in Washington post today.

Indeed, I also feel that "he" cannot refer to the possessive. Let us see what others say.

Technically, a subject pronoun such as he should not serve to stand in for a possessive such as Obama's.
The reason:
A possessive serves not as a noun but as a MODIFIER, and the referent for a subject pronoun must be a NOUN.
That said, virtually every reader of the sentence above would understand that the intended referent for he is Obama.
Further, pronoun ambiguity is a low-priority error on the GMAT.
A good rule of thumb:
If an answer choice includes a potentially ambiguous pronoun, do not immediately eliminate the answer choice.
Instead, look for an answer choice that avoids the potential ambiguity and is free of errors.