Deep learning-enhanced calibration for nonlinear mixed stochastic models in European option pricing

This paper studies European option pricing in a regime-switching Heston-Hull-White framework. The model combines stochastic volatility, stochastic interest rates, and a Markov-chain-driven regime-switching mechanism, allowing part of the market dynamics to vary across different states. Under the risk-neutral measure, a semi-analytical pricing formula for European options is derived. To address the calibration difficulty caused by the nonlinear and high-dimensional pricing map, a deep-learning-aided residual-surrogate calibration procedure is introduced. The neural network is not used to replace the semi-analytical pricing formula; instead, it learns a local residual surrogate around representative parameter values and assists the subsequent parameter search. Final option prices are recomputed from the original RS-HHW pricing formula under the refined parameters. The proposed DL-RS-HHW model is examined using SSE 50 ETF option data and compared with the Heston, Bates, HHW, and RS-HHW benchmarks. The empirical results show that DL-RS-HHW reduces both in-sample and out-of-sample pricing errors under the RMSE and MAE criteria. Training and optimization diagnostics indicate stable calibration, while Diebold-Mariano tests support the statistical significance of the pricing-error reductions. Pesaran-Timmermann tests provide further evidence on directional accuracy. These results suggest that residual-surrogate-assisted calibration improves the empirical calibration of semi-analytical hybrid option pricing models.