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

Abstract

The prices of European and American-style contracts on assets driven by Markov processes satisfy partial integro-differential equations (PIDEs). In particular, this holds true for assets driven by Lévy processes, which are very popular in mathematical finance. We focus below on Lévy processes whose jump part has infinite (small jumps) activity, in which case the resulting singular kernel is rather difficult to treat numerically [1,23]. We illustrate, using the CGMY example, that this challenge may be successfully met by using a combination of: a) the Asmussen-Rosinsky approximation of the small jumps by Brownian motion; b) localized RBF approximations known as the RBF partition of unity (RBF-PU), which overcome the ill-conditioning inherent in global mesh-free methods; c) Crank-Nicolson Leap-Frog (CNLF) for time discretization. d) treating the local term using an implicit step, and the nonlocal term using an explicit step (to avoid the inversion of the nonsparse matrix)Finally, to deal with free boundary problems associated to American options, we combine the implicit-explicit method with a penalty method.Efficiency and practical performance are demonstrated by several numerical experiments for pricing European and American contracts.