This paper is devoted to the price–storage dynamics in natural gas markets. A novel stochastic path-dependent volatility model is introduced with path dependence in both price volatility and storage increments. Model calibrations are conducted for both the price and storage dynamics. Further, we discuss the pricing problem of discrete-time swing options using the dynamic programming principle, and propose a deep-learning-based method for this non-Markovian setting. A numerical algorithm is provided, and convergence analysis results are given for the deep-learning approach. The methodologies developed for calibration, numerical pricing and deeplearning approximation are broadly applicable, and we expect them to support further advances in the modeling and risk management of energy markets.