An integration preconditioning method for solving option pricing problems

In this paper, we present an integration preconditioning method to solve multi-asset option pricing problems modelled by the well-known Black-Scholes equation. This integration preconditioning technique helps transform the partial differential equations into integral equations and contribute to a well-conditioned system. It benefits the calculation from avoiding the ill-posedness of numerical derivatives approximation in solving problems modelled by partial differential equations. Two kinds of interpolation approximations: quadrature formulas and radial basis functions (RBFs) are adopted. The integration preconditioning method improves both the accuracy and stability when compared with the traditional direct differential methods. Besides, while combining with the integral operator, the RBFs are more free to select the value of shape parameters. All the introduced benefits are investigated and verified by numerical results.