2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/216714We prove that the perpetual American put option price of level dependent volatility model with compound Poisson jumps is convex and is the classical solution of its associated quasi-variational inequality, that it is $C^2$ except at the stopping boundary and that it is $C^1$ everywhere (i.e. the smooth pasting condition always holds).Optimization and ControlPricing of Securities62L15; 60J75text