In the sentence
Both options compile and run without problems.
what is the correct version of verb: compile or compiles (run or runs) ?
On one hand we’re talking about two subjects, thus plural ending needs no -s. On other hand we’re talking about each option specifically, thus “option runS”.
The subject of the sentence (actually the head of the noun phrase that serves as the subject) is the plural options. In some circumstances, a nominally plural subject will take a singular verb, but that happens when the many of that subject is considered as a unit. So we can say
The group is meeting tomorrow.
Even though group means more than one person, the verb is singular because we think of the group acting as a single unit. But here, as you note, the the word both emphasizes that we’re considering two options that compile, each presumably on their own.
Two separate entities, so a plural verb. You use the singular verb in
Each option compiles.
because each means each one, or one at a time.