Quadratic Spline Wavelets for Sparse Discretization of Jump–Diffusion Models

Abstract

This paper is concerned with a construction of new quadratic spline wavelets on a bounded interval satisfying homogeneous Dirichlet boundary conditions. The inner wavelets are translations and dilations of four generators. Two of them are symmetrical and two anti-symmetrical. The wavelets have three vanishing moments and the basis is well-conditioned. Furthermore, wavelets at levels i and j where i - j > 2 are orthogonal. Thus, matrices arising from discretization by the Galerkin method with this basis have O 1 nonzero entries in each column for various types of differential equations, which is not the case for most other wavelet bases. To illustrate applicability, the constructed bases are used for option pricing under jump–diffusion models, which are represented by partial integro-differential equations. Due to the orthogonality property and decay of entries of matrices corresponding to the integral term, the Crank–Nicolson method with Richardson extrapolation combined with the wavelet–Galerkin method also leads to matrices that can be approximated by matrices with O 1 nonzero entries in each column. Numerical experiments are provided for European options under the Merton model.