Pricing European and American options by radial basis point interpolation Jamal Amani Rad a , Kourosh Parand a , Luca Vincenzo Ballestra b,⇑ a Department of Computer Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Evin, P.O. Box 198396-3113, Tehran, Iran b Dipartimento di Economia, Seconda Università di Napoli, Corso Gran Priorato di Malta, 81043 Capua, Italy article info Keywords: Radial basis point interpolation Meshfree method Option pricing Black–Scholes Projected successive overrelaxation Penalty method abstract We propose the use of the meshfree radial basis point interpolation (RBPI) to solve the Black–Scholes model for European and American options. The RBPI meshfree method offers several advantages over the more conventional radial basis function approximation, nevertheless it has never been applied to option pricing, at least to the very best of our knowledge. In this paper the RBPI is combined with several numerical techniques, namely: an exponential change of variables, which allows us to approximate the option prices on their whole spatial domain, a mesh refinement algorithm, which turns out to be very suitable for dealing with the non-smooth options’ payoff, and an implicit Euler Richardson extrapolated scheme, which provides a satisfactory level of time accuracy. Finally, in order to solve the free boundary problem that arises in the case of American options three different approaches are used and compared: the projected successive overrelaxation method (PSOR), the Bermudan approximation, and the penalty approach. Numerical experiments are presented which demonstrate the computational efficiency of the RBPI and the effectiveness of the various techniques employed. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction Over the last thirty years, financial derivatives have raised increasing popularity in the markets. In particular, large vol- umes of options are traded everyday all over the world and it is therefore of great importance to give a correct valuation of these instruments. Options are contracts that give to the holder the right to buy (call) or to sell (put) an asset (underlying) at a previously agreed price (strike price) on or before a given expiration date (maturity). The majority of options can be grouped in two categories: European options, which can be exercised only at maturity, and American options, which can be exercised not only at maturity but also at any time prior to maturity. Options are priced using mathematical models that are often challenging to solve. In particular, the famous Black–Scholes model [1] yields explicit pricing formulae for some kinds of European options, including vanilla call and put, but for American options closed-form solutions are not available, and numerical approximations are needed. To this aim, the most common approaches are the finite difference/finite element/finite volume methods (see, e.g., [2–18]) and the binomial/ http://dx.doi.org/10.1016/j.amc.2014.11.016 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author. E-mail addresses: j.amanirad@gmail.com, j_amanirad@sbu.ac.ir (J.A. Rad), k_parand@sbu.ac.ir (K. Parand), lucavincenzo.ballestra@unina2.it (L.V. Ballestra). Applied Mathematics and Computation 251 (2015) 363–377 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc