Analytical computation of conditional moments in the extended Cox–Ingersoll–Ross process with regime switching: Hybrid PDE system solutions with financial applications

In this paper, we introduce a novel analytical approach for the computation of the $n$th conditional moments of an $m$-state regime-switching extended Cox–Ingersoll–Ross process driven by a continuous-time finite-state irreducible Markov chain. This approach is applicable for all integers $n\ge 1$ and $m\ge 1$, thereby ensuring wide-ranging utility. The key of our investigation is a complex hybrid system of inter-connected PDEs, derived through a utilization of the Feynman–Kac formula for regime-switching diffusion processes. Our exploration into the solutions of this hybrid PDE system culminates in the derivation of exact closed-form formulas for the conditional moments for diverse values of $n$ and $m$. Additionally, we study the asymptotic characteristics of the first conditional moments for the 2-state regime-switching Cox–Ingersoll–Ross process, particularly focusing on the effects of the symmetry inherent in the Markov chain’s intensity matrix and the implications of various parameter configurations. Highlighting the practicality of our methodology, we conduct Monte Carlo simulations to not only corroborate the accuracy and computational efficacy of our proposed approach but also to demonstrate its applicability to real-world applications in financial markets. A principal application highlighted in our study is the valuation of VIX futures and VIX options within a dynamic, mean-reverting, hybrid regime-switching framework. This exemplifies the potential of our analytical method to significantly impact contemporary financial modeling and derivative pricing.