2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/208912We prove a sharp version of the Hopf boundary point lemma for Black-Scholes type equations. We also investigate the existence and the regularity of the spatial derivative of the solutions at the spatial boundary.10 pagesAnalysis of PDEsPricing of Securities35K65; 35B50text