The paper proposes Finance-Informed Neural Network (FINN), a novel framework for option pricing and hedging that integrates financial theory directly into the learning objective. Instead of training on observed option prices, FINN learns through a self-supervised replication objective based on dynamic hedging, ensuring economic consistency by construction. The authors prove theoretically that minimizing replication error recovers the arbitrage-free pricing operator and yields economically meaningful sensitivities. Empirically, FINN accurately recovers classical Black-Scholes prices and performs robustly in stochastic volatility environments, including the Heston model, while remaining stable in settings where analytical solutions are unavailable. Fundamental pricing relationships such as put-call parity emerge endogenously. When applied to implied-volatility surface reconstruction, FINN produces surfaces that are consistently closer to observed market-implied volatilities than those obtained from Heston calibrations, indicating superior out-of-sample adaptability and reduced structural bias. Importantly, FINN extends beyond liquid option markets: it can be trained directly on historical spot prices to construct coherent option prices and Greeks for assets with no listed options. More broadly, FINN defines a new paradigm for financial pricing, in which prices are learned from replication and risk-control principles rather than inferred from parametric assumptions or direct supervision on option prices. By reframing option pricing as the learning of a pricing operator rather than the fitting of prices, FINN offers practitioners a practical and scalable tool for pricing, hedging, and risk management across both established and emerging financial markets.