Broken Blood Vessel

My daughter has had a broken blood vessel under her eye for the last 8 months and it does not seem to be going away as the doctor told us it would. In fact it seems redder lately. Is there anything we can do besides expense laser surgery? There was never a trauma or any visual problems. It is just a red spot under her eye, kind of on the lower lid almost on her upper cheek. It spiders out, but it is very small. We never called an ophthalmologist; but my mom told me it was a broken blood vessel because she has had them before. And my daughter’s pediatrician said it should go away in a few months but it hasn't.