Derivative-Informed Operator Learning for Finance: On-the-Fly Greeks, Surfaces, Hedging, and Control

Financial decision systems require fast surrogate models for pricing, calibration, hedging, XVA, stress testing, and portfolio optimization. Standard neural surrogates reproduce prices or risk quantities, but downstream tasks depend as much on derivatives: deltas, vegas, curve and credit-spread sensitivities, exposure and objective gradients. We formulate a derivative-informed operator-learning framework in which the learned map -- a neural operator, random-feature operator, or finite-dimensional surrogate -- is trained both to match a high-fidelity pricing or risk operator and to match directional Fr\'echet derivatives generated on the fly. The framework combines operator learning, adjoint algorithmic differentiation, tangent sensitivity equations, random sketching of Jacobian actions, and no-arbitrage constraints. We derive error bounds showing derivative accuracy controls local stress errors, hedging error, and optimizer instability, and that discrete-time hedging error is also governed by second-order (gamma) accuracy. A Black--Scholes network over eight seeds shows a tuned derivative weight cuts vega error by 40\% and delta error by 15\% while modestly improving prices, but not an unsupervised second-order Greek. Heston and Bates random-feature experiments reduce stochastic-volatility and jump-parameter sensitivity errors by 60--76\%. A random-feature DeepONet/Galerkin operator mapping instantaneous-volatility curves to dense price surfaces reduces out-of-sample JVP error by 44\% and price RMSE by 23\% over eight seeds; it also shows derivative consistency alone does not remove no-arbitrage violations, so economic constraints must be imposed explicitly. The framework gives a disciplined route from value-only surrogates to derivative-aware engines that output differentiable instruments for hedging, risk, and control.