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Nonlocal problems have been used to model very different applied scientific phenomena which involve the fractional Laplacian when one looks at the Lévy processes and stochastic interfaces. This paper deals with the nonlocal problems on a bounded domain where the stiffness matrices of the resulting systems are Toeplitz plus tridiagonal or far from being diagonally dominant as occurs when dealing with linear finite element approximations. By exploiting a weakly diagonally dominant Toeplitz property of the stiffness matrices, the optimal convergence of the two-grid method is well established in [Fiorentino and Serra-Capizzano, SIAM J. Sei. Comput., 17 (1996), pp. 1068-1081; Chen and Deng, SIAM J. Matrix Ano,I. Appl., 38 (2017), pp. 869-890], and there are still questions about the best ways to define the coarsening and interpolation operators when the stiffness matrix is far from being weakly diagonally dominant [Stuben, J. Comput. Appl. Math., 128 (2001), pp. 281-309]. In this work, using spectral indications from our analysis of the involved matrices, the simple (traditional) restriction operator and prolongation operator are employed in order to handle general algebraic systems which are neither Toeplitz nor weo,kly d,io,gono,lly domino,nt corresponding to the fractional Laplacian kernel and the constant kernel, respectively. We focus our efforts on providing the detailed proof of the convergence of the two-grid method for such situations. Moreover, the convergence of the full multigrid is also discussed with the constant kernel. The numerical experiments are performed to verify the convergence with only O(NlogN) complexity by the fast Fourier transform, where N is the number of the grid points. [ABSTRACT FROM AUTHOR] more...

This paper applies an algorithm for the convolution of compactly supported Legendre series (the CONLeg method) (cf. Hale and Townsend, An algorithm for the convolution of Legendre series. SIAM J. Sci. Comput., 2014, 36, A1207–A1220), to pricing European-type, early-exercise and discrete-monitored barrier options under a Lévy process. The paper employs Chebfun (cf. Trefethen et al., Chebfun Guide, 2014 (Pafnuty Publications: Oxford), Available online at: ) in computational finance and provides a quadrature-free approach by applying the Chebyshev series in financial modelling. A significant advantage of using the CONLeg method is to formulate option pricing and option Greek curves rather than individual prices/values. Moreover, the CONLeg method can yield high accuracy in option pricing when the risk-free smooth probability density function (PDF) is smooth/non-smooth. Finally, we show that our method can accurately price options deep in/out of the money and with very long/short maturities. Compared with existing techniques, the CONLeg method performs either favourably or comparably in numerical experiments. [ABSTRACT FROM AUTHOR] more...