Finance-Informed Operator Learning for Option Pricing with Quantum-Compatible Realizations

Pricing European options under local volatility requires solving a partial differential equation whose coefficients change with every recalibration, and practitioners repeatedly need not only prices but also their sensitivities to the underlying across spot–time surfaces for many strike and maturity configurations. Neural surrogates are an efficient alternative, approximating the solution operator directly so that model evaluation replaces repeated numerical solves. Near expiry, however, the solution loses regularity, and a network trained on prices alone can misstate the curvature needed for hedging or return prices outside the no-arbitrage bounds. We propose a finance-informed Deep Operator Network (FI-DeepONet) that decomposes the solution operator into a closed-form reference price and an additive learned correction. The framework is finance-informed in that financial structure enters the representation and the output map rather than the loss: the reference uses the strike-line integrated variance—the time integral of the squared local volatility evaluated at the strike, which captures the leading near-expiry curvature singularity, and a smooth monotone admissibility layer enforces pointwise no-arbitrage price bounds, at the cost of piecewise rather than global smoothness in the spot variable. A deterministic post-processing step then corrects the predicted grid after inference and is evaluated separately from the learned operator. We establish a short-maturity estimate and differentiated asymptotics for the exact correction, together with identities relating errors in the learned correction to pricing and sensitivity errors, as well as PDE-residual errors. On in-distribution (ID) tests, the model reduces the global relative price error by nearly an order of magnitude over vanilla and physics-informed DeepONet baselines across independent initializations. On out-of-distribution (OOD) tests, generalization is moderate when the input parameters lie outside the training range but within the same parametric class of local-volatility functions. We also test the trained model on real market data using index option quotes and find that it remains accurate without market-specific network retraining, though it does not improve on the analytic formula given the same volatility input. We further give a quantum-compatible realization of the trained operator, in which selected linear maps are either compiled exactly or approximated within a restricted diagonal–orthogonal family before supervised adaptation.