European option pricing under the approximate fractional Heston jump–diffusion model with a stochastic long-term mean

To address the limitations of the traditional Heston model, which fails to capture asset price jumps, long-range dependence, and the term structure of implied volatility and variance swap curves, this study proposes an enhanced approximate fractional jump-diffusion Heston model with a stochastic long-term mean. The model incorporates an approximate fractional Brownian motion, which not only captures long-range dependence but also addresses the interpretability issues commonly associated with fractional Brownian motion in financial applications. Additionally, a stochastic long-term mean is introduced to model the evolving volatility term structure, and an asymmetric double-exponential jump process is used to characterize asset price jumps. This triple enhancement provides a more realistic representation of market dynamics. We derive an approximate analytical solution for European call options, offering significant computational efficiency and practical utility. Compared to Monte Carlo simulations, the analytical solution reduces computational costs and enables rapid parameter calibration, making it highly suitable for real-world applications. Numerical analyses confirm the model’s convergence and stability, while empirical results demonstrate its superior pricing accuracy over both He’s model and the traditional Heston model. • Proposed approximate fractional Heston jump-diffusion model with stochastic long-term mean. • AFBM captures long memory; stochastic long-term mean describes volatility term structure. • Derived closed-form pricing formula for European options, validated via Monte Carlo simulation. • Numerical analysis and empirical studies confirmed model validity and superiority.