Exploring Gaussian radial basis function integrals for weight generation with application in financial option pricing

Abstract

We introduce a novel numerical method via a class of radial basis function-produced finite difference solvers, applicable to both interpolation and partial differential equation (PDE) problems. The method leverages integrals of the Gaussian kernel, introducing new weights for problem-solving. Analytical solutions to approximate the derivatives of a function are derived and computed on a stencil with both non-uniform and uniform distances. Our observations indicate that the analytical weights exhibit greater stability compared to the numerical weights when addressing problems. In the final step, we use the derived formulations to solve a multi-dimensional option pricing problem in finance. The results demonstrate that our proposed numerical method outperforms in terms of numerical accuracy across grids of different sizes. Given the multi-dimensional nature of the dealing model, which involves handling a basket of assets, our approach becomes particularly relevant for assessing and managing financial risks.