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.
Highlights
Nowadays, various models and methodologies are available to calculate theoretical values of options, including the famous Black–Scholes model as well as stochastic volatility models such as the Heston or the Stein and Stein models
WG denotes the wavelet-Galerkin method combined with the Crank–Nicolson scheme with Richardson extrapolation as discussed in this paper, FD denotes the finite difference method from [21], FPI denotes the method from [19], which is based on fixed point iterations and the Crank–Nicolson scheme, and IMEX denotes the explicit–implicit method from [20], which is a combination of the cubic spline collocation method and the backward difference method
The basis is adapted to homogeneous Dirichlet boundary conditions, wavelets have three vanishing moments, and the wavelets on the levels i and j are orthogonal both with respect to the
Summary
Various models and methodologies are available to calculate theoretical values of options, including the famous Black–Scholes model as well as stochastic volatility models such as the Heston or the Stein and Stein models. It is important that the basis is well-conditioned because the condition numbers of discretization matrices depend on condition numbers of the basis and small condition numbers of system matrices guarantee the stability of computation and influences the number of iterations of iterative methods used for the numerical solution of the resulting system Due to these interesting properties, the wavelet basis proposed in this paper can be used in many applications such as the numerical solution of various types of operator equations using the wavelet-Galerkin method, an adaptive wavelet method or a collocation method. In comparison with methods from [19,20,21], the proposed method requires a smaller number of degrees of freedom to obtain a sufficiently accurate solution
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