I am 19 and just started taking finasteride and I was wondering if the lack of dht will affect my bone development?

Finasteride at 1mg daily shouldn't affect your bone density, but thorough evaluation and tracking by your physician is important.

Finasteride use by men is not associated with the actual maturation of bones during development and maturation. It is however, associated with an increased risk of osteoporosis later in life. These studies have mostly been done with men using the 5 mg and the data on the 1 mg dose (commonly used for hair loss) is not entirely known. You need not be concerned about bone maturation. However, if you have concerns about your bones or you have a history of osteoporosis or at at increased risk of bone mineral loss, please speak to your physician.

REFERENCELin WL, et al. A population-based nested case-control study: the use of 5-alpha-reductase inhibitors and the increased risk of osteoporosis diagnosis in patients with benign prostate hyperplasia. Clin Endocrinol (Oxf). 2015.

Finasteride reduces DHT but the amount of the recommended does does not impact bone growth. At 19, your bones should be reasonably developed