That's pretty much the question. It is a must to say "Yes, I do" for such kind of question or you can answer just as "I do".
"I do" is definitely a grammatically correct answer. So are "Why do you ask?" and "None of your business" and "That answer is available (binary only) upon payment of the proper license fee" and "That's not important now" and "My hair is on fire!" All of these are grammatically correct. Whether they are appropriate answers is a different question, depending on the rest of the conversation.
The two-word answer "I do" is even the best answer when the question is more like "Who owns this tiger?" or "Who has the key to the tiger food locker?"