GA flight instructor reported misunderstanding of ATC clearance resulted in a NMAC on base leg.

Narrative

I was in the traffic pattern with my student at ZZZ; practicing landings and takeoffs. The airport was quite busy; and the tower was constantly organizing the traffic flow in the pattern. My student and I were in the right downwind for Runway XX when we mistakenly thought we heard ATC say; 'cleared for the option Runway XX #2.' This was incorrect; we had not been cleared for the option but had misinterpreted this call; which had been for the airplane in the left downwind.We scanned for traffic (not thoroughly enough; however); and when we turned for the bar leg; we saw traffic coming in our direction and approximately 200 feet below us. We took evasive action to create more distance between us and the other airplane. They saw us as well and continued their descent to create more room between us. ATC gave us a heading to fly; and we were able to rejoin the flow of traffic without incident and land.A gentleman from ATC came down from the tower after the flight; and we had a conversation in which I admitted to being in the wrong. He stated that he deemed it not necessary to write a report and advised that I be more careful in the future. Even though ATC didn't write a report; I feel that it's my duty to document and fully report what happened; taking responsibility for the actions of my student and myself.