European Option Pricing and Deep Learning-Aided Calibration Under Heston–Hull–White with a Stochastic Long-Run Variance Mean

This paper studies European option pricing in a Heston–Hull–White model with a stochastic long-run variance mean. Within this setting, a semi-analytical pricing formula is derived using a change of numeraire together with characteristic-function and Fourier-transform methods. The model is examined with SSE 50ETF option data from 1 January 2024 to 1 January 2025 through rolling-window calibration and a deep-learning-guided joint calibration scheme. The empirical results show that the proposed model reduces overall pricing errors relative to several standard benchmark models, while the deep-learning hybrid specification delivers the strongest out-of-sample pricing performance among the models considered.