Research on American Option Pricing Under the Heston Jump Diffusion Model—Based on Fourier Space Time-Stepping Method

American options are more complex to price than European options because they grant holders the right to exercise at any time before expiration, especially in realistic market environments that consider both stochastic volatility and asset price jumps. Therefore, this paper studies the pricing of American options under the Heston stochastic volatility model, incorporating the Merton jump-diffusion process. For this high-dimensional, nonlinear free boundary problem, this paper adopts the Fourier space time-stepping method for numerical solution. This method utilizes the characteristic function in Fourier space to implement time-stepping, effectively addressing computational difficulties caused by stochastic volatility and jump processes, and it determines the optimal exercise boundary by comparing the holding value with the immediate exercise value at each step. Numerical experiments show that the method is computationally stable and accurate, clearly capturing the early exercise premium and dynamic changes in the exercise boundary. Additionally, parameter sensitivity analysis reveals that the jump component significantly affects option value (with a premium of approximately 6.74%), highlighting the necessity of incorporating jump risk into pricing models. This work provides an effective numerical framework for American option pricing under stochastic volatility and jump environments, possessing both theoretical significance and practical application value.