I went through your question. It would be better for me if you sent a photograph of the lesion. Whatever, are you sure that it is a haemangioma, not a vascular malformation? Have you done doppler ultrasonography or MRI or biopsy? In case of vascular malformation the lesion would be flat with normal temperature and grows rapidly. It needs pulsed dye laser as a treatment under the guidence of a dermatologist. But haemangioma is always raised, warm, grows slowly, you can feel with your finger easily. The common form is the capillary or strawberry haemangioma. It is not present at birth, usually begins to appear after 14 days and continues for about 1 year, then starts to involute. When you see its size is reducing, color becoming fade, becoming contracted(fibrosis) then you can be sure it is resolving. Most of the haemangioma requires no treatment; if there is any residual scar, simple pulsed dye laser is enough. There are some medical treatment like intralesional steroid / propranolol / vincristine to reduce the size . Surgical option like embolization, if it becomes life threatening or causing disfiguration. For pulsed dye laser (PDL) you need a dermatologist. For other treatment you should visit a dermatologist or a surgeon. Cost depends on the size, duration of treatment, it can only be told by respective physician. But as your child is only 4 months, first be sure about the diagnosis and if it is surely a haemangioma, wait and watch. Don’t interfere if it is reducing in size until it becomes a permanent scar, then visit a dermatologist for PDL. For PDL you need not stay in hospital, other forms of treatment are also provided as outdoor basis, need no admission unless there is serious bleeding or ulceration.