Which kind of persuasive speech is a response to argument made by another person and an attempt to defend your own position?

Refutation.

Explanation:

A refutation is a speech - or even a part of an essay - which addresses and disproves opposing arguments. That is, in this type of persuasive speech, you are replying to arguments made against your position in the matter. When responding, you try not only to discredit what the opposition says, but also to prove your opinion is the right one. Persuasive essays, in their turn, usually have a refutation paragraph or section in which the authors anticipate what others may say, and respond to those possible arguments so as to prove their opinions right.