Option pricing for barrier options under a regime-switching mixed fractional Brownian motion with jumps model

This paper presents a novel approach for deriving closed-form solutions to evaluate the discrete barrier options under the mixed fractional Brownian motion (MFBM) with jumps. Due to the introduction of MFBM, our model exhibits long-range dependence compared to the classical regime-switching Lévy model, allowing for a better capture of the behavior of financial markets. For the feasibility of the analytical method, the Z-transform method is introduced to convert the recursive procedure produced by discrete monitoring patterns of barrier options into continuous integral equations. To approximate the option value, we employ the Fourier cosine series expansion (COS) method. Additionally, in the context of regime-switching environments, the Fourier transform technique is employed to the partial integro-differential equations (PIDEs) of option values to compute the conditional characteristic function of the underlying asset process. The numerical experiments validate the theoretical framework, demonstrating that the proposed methodology offers greater advantages compared to Monte Carlo simulations in terms of computational efficiency and numerical stability across diverse market scenarios.