Risk Management of Variable Annuity Guarantees via Sobolev-Trained Neural Networks

Hedging a book of variable annuity (VA) guarantees requires both fair values and risk sensitivities (Greeks), recomputed across the whole portfolio as markets move. Nested stochastic Monte Carlo, the industry standard, is accurate but far too slow for real-time use; fast neural-network surrogates trained only on prices solve the speed problem but reintroduce the accuracy one — their derivatives are frequently wrong in sign or an order of magnitude off, making them unusable for hedging. This paper closes that gap by pairing Adjoint Algorithmic Differentiation (AAD), which yields exact pathwise Greeks at negligible extra simulation cost, with Sobolev training, which folds those Greeks directly into the network's loss alongside price labels. Across all nine standard VA guarantee types (GMAB, GMDB, GMMB, GMIB, GMWB, and their combinations), price-only training is shown to make Greek accuracy worse than a naive constant predictor for most riders (negative R²), while Sobolev training recovers dollar-delta R² > 0.974 and interest-rate Greek R² > 0.956 across the board, at only a small cost to price fit — while pricing a contract and computing its full risk profile in under a millisecond, five to six orders of magnitude faster than nested Monte Carlo or bump-and-revalue. The simulation engine (ActuaLib, C++17 with AAD), PyTorch training code, and full training/test datasets are released openly to make real-time, portfolio-level hedging analytics for variable annuities reproducible and practically achievable.