Local weak form meshless techniques based on the radial point interpolation (RPI) method and local boundary integral equation (LBIE) method to evaluate European and American options

摘要

For the first time in mathematical finance field, we propose the local weakform meshless methods for option pricing; especially in this paper we selectand analysis two schemes of them named local boundary integral equation method(LBIE) based on moving least squares approximation (MLS) and local radial pointinterpolation (LRPI) based on Wu's compactly supported radial basis functions(WCS-RBFs). LBIE and LRPI schemes are the truly meshless methods, because, atraditional non-overlapping, continuous mesh is not required, either for theconstruction of the shape functions, or for the integration of the localsub-domains. In this work, the American option which is a free boundaryproblem, is reduced to a problem with fixed boundary using a Richardsonextrapolation technique. Then the $\theta$-weighted scheme is employed for thetime derivative. Stability analysis of the methods is analyzed and performed bythe matrix method. In fact, based on an analysis carried out in the presentpaper, the methods are unconditionally stable for implicit Euler (\theta = 0)and Crank-Nicolson (\theta = 0.5) schemes. It should be noted that LBIE andLRPI schemes lead to banded and sparse system matrices. Therefore, we use apowerful iterative algorithm named the Bi-conjugate gradient stabilized method(BCGSTAB) to get rid of this system. Numerical experiments are presentedshowing that the LBIE and LRPI approaches are extremely accurate and fast.