The 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'.