Continuous-time arbitrage-free pricing framework: assume GBM dS = μ S,dt + σ S,dW, form a delta-hedged portfolio Pi = V - Δ,S, apply Itô, and require the hedged portfolio earn the risk-free rate. The result is the BSM PDE dfracpartial Vpartial t + tfrac12σ^2 S^2 dfracpartial^2 Vpartial S^2 + r S dfracpartial Vpartial S - r V = 0, which by Feynman-Kac is equivalent to the risk-neutral expectation V = e^-rTmathbbE^mathbbQ[payoff]. The framework's assumptions (constant vol, no jumps, frictionless trading) are all violated in practice — its true contribution is the replication/hedging logic, not the price.