A well system includes a tubing string and a subsurface safety valve interconnected with the tubing string and including a housing that defines a central flow passageway and includes a flapper pivotable within the central flow passageway between closed and open positions. A flow tube is positioned within the central flow passageway and engageable with the flapper to move the flapper to the open position. An actuation piston is operatively coupled to the flow tube, a balance piston is operatively coupled to the flow tube, and a lock-open piston is engageable with an actuator sleeve operatively coupled to the flow tube. A lock-open tool is positionable within the central flow passageway to convey hydraulic pressure into a lock-open piston bore and thereby actuate the lock-open piston to an actuated position that moves the flow tube and permanently locks the flapper in the open position.