Numerical pricing of American options under two stochastic factor models with jumps using a meshless local Petrov-Galerkin method

摘要

The most recent update of financial option models is American options understochastic volatility models with jumps in returns (SVJ) and stochasticvolatility models with jumps in returns and volatility (SVCJ). To evaluatethese options, mesh-based methods are applied in a number of papers but it iswell-known that these methods depend strongly on the mesh properties which isthe major disadvantage of them. Therefore, we propose the use of the meshlessmethods to solve the aforementioned options models, especially in this work weselect and analyze one scheme of them, named local radial point interpolation(LRPI) based on Wendland's compactly supported radial basis functions(WCS-RBFs) with C6, C4 and C2 smoothness degrees. The LRPI method which is aspecial type of meshless local Petrov-Galerkin method (MLPG), offers severaladvantages over the mesh-based methods, nevertheless it has never been appliedto option pricing, at least to the very best of our knowledge. These schemesare the truly meshless methods, because, a traditional non-overlappingcontinuous mesh is not required, neither for the construction of the shapefunctions, nor for the integration of the local sub-domains. In this work, theAmerican option which is a free boundary problem, is reduced to a problem withfixed boundary using a Richardson extrapolation technique. Then theimplicit-explicit (IMEX) time stepping scheme is employed for the timederivative which allows us to smooth the discontinuities of the options'payoffs. Stability analysis of the method is analyzed and performed. In fact,according to an analysis carried out in the present paper, the proposed methodis unconditionally stable. Numerical experiments are presented showing that theproposed approaches are extremely accurate and fast.