Advanced Mathematical Models & Applications

In this paper, we propose a novel fractional-order (FO) framework for bond pricing that extends classical stochastic interest-rate models by replacing the standard first-order temporal derivative with the Caputo operator. By integrating Lie- symmetry methods, we derive explicit invariant solutions for the resulting nonlocal pricing partial differential equation (PDE), capturing long-range memory effects and anomalous diffusion behaviors observed in real financial markets. We establish well-posedness of the fractional model by reformulating it as an equivalent integral equation and applying the Banach fixed-point theorem to prove existence and uniqueness of solutions in an appropriate Banach space. Employing advanced Lyapunov techniques, we further derive novel Mittag-Leffler stability (MLS) criteria that guarantee both local and global stability of the equilibrium solution. To assess practical implications, we implement an efficient finite-difference scheme combining the L1 approximation for the Caputo derivative with central spatial discretization and the Thomas algorithm and present two representative numerical case studies. Our results demonstrate that the fractional-order model not only preserves positivity and mean reversion but also yields higher late-time bond prices compared to its integer-order counterpart, offering a unified, analytically tractable, and empirically faithful approach to interest-rate derivative valuation.