RBF approximation by partition of unity for valuation of options under exponential Lévy processes

International audienceThe prices of some European and American-style contracts on assets driven by a class of Markov processes containing, in particular, L\'{e}vy processes of pure jump type with infinite jump activity, are obtained numerically, as solutions of the partial integro-differential equations (PIDEs) they satisfy. This paper overcomes the ill-conditioning inherent in global meshfree methods by using localized RBF approximations known as the RBF partition of unity (RBF-PU) method for (PIDEs) arising in option pricing problems in L\'{e}vy driven assets. Then, Crank-Nicolson, LeapFrog (CNLF) is applied for time discretization. We treat the local term using an implicit step, and the nonlocal term using an explicit step, to avoid the inversion of the nonsparse matrix. For dealing with early exercise feature of American option and solving free boundary problem we use the implicit-explicit method combined with a penalty method. Efficiency and practical performance are demonstrated by numerical experiments for pricing European and American contracts. Suggested reviewers Mohan Kadalbajoo, Mehdi Dehghan, simon hubbert Submission Files Included in this PDF File Name [File Type] cover letter.pdf [Cover Letter] Research Highlights.pdf [Highlights] Levy_Fereshtian.pdf [Manuscript File] Submission Files Not Included in this PDF File Name [File Type] Paper_Levy.zip [LaTeX Source File] To view all the submission files, including those not included in the PDF, click on the manuscript title on your EVISE Homepage, then click 'Download zip file'

Localized kernel-based approximation for pricing financial options under regime switching jump diffusion model

In this paper, we consider European and American option pricing problems under regime switching jump diffusion models which are formulated as a system of partial integro-differential equations (PIDEs) with fixed and free boundaries. For free boundary problem arising in pricing American option, we use operator splitting method to deal with early exercise feature of American option. For developing a numerical technique we employ localized radial basis function generated finite difference (RBF-FD) approximation to overcome the ill-conditioning and high density issues of discretized matrices. The proposed method leads to linear systems with tridiagonal and diagonal dominant matrices. Also, in this paper the convergence and consistency of the proposed method are discussed. Numerical examples presented in the last section illustrate the robustness and practical performance of the proposed algorithm for pricing European and American options. Published by Elsevier B.V. on behalf of IMACS