Multinode Shepard Collocation Method for Pricing of Financial Derivatives

This paper explores the use of the multinode Shepard method for the numerical solution of the two-dimensional Black–Scholes equation. The proposed approach combines a spatial approximation based on the multinode Shepard operator with a temporal discretization based on the Backward Differentiation Formula. Numerical experiments assess the accuracy, robustness, and computational properties of the method. Particular attention is devoted to the behavior of the MS spatial discretization in the presence of the original non-smooth basket payoff and to its sensitivity to the geometry of the node set. The method is compared with both a global RBF collocation approach and a local sparse RBF-FD benchmark. Tests on uniform, Halton, and locally refined node sets show a clear separation between kink-dominated global errors and very high accuracy in smooth regions, together with a marked robustness with respect to changes in node geometry.