Identification of the local speed function in a Lévy model for option pricing

Abstract

We propose a non-parametric stable calibration method based on Tikhonov regularization for the local speed function in a local Lévy model. The jump term in this model introduces an integral operator into the classic Black-Scholes partial differential equation such that the associated model calibration to observed option prices can be treated as a parameter identification problem for a partial integrodifferential equation. This problem is shown to be ill-posed and thus requires regularization. It is proven that nonlinear Tikhonov regularization is a stable and convergent method for this problem. Furthermore, convergence rate results are established under an abstract source condition. Finally the theoretical results are underpinned by numerical illustrations including a real-world example. © 2008 Rocky Mountain Mathematics Consortium.