Product after giving half cooked story, says give me estimate.

Welcome to devrant! and that is the way it works. Estimate how long it will take the PM to finish writing the spec, and add that to your estimate+how long it will take you to implement the current spec+how long you think it will take to implement the missing spec+meetings times about the spec+1 week and multiply the result by the incompetent factor of you PM. the lowset I had was a 1.5 factor.