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A novel fitted finite volume method for the Black-Scholes equation governing option pricing. (English) Zbl 1147.91332
The author proposes and analyses a novel fitted volume numerical method for a degenerate partial differential equation, Black-Scholes-type equation, governing option pricing. The fitting technique is based on the idea proposed by D. N. de G. Allen and R. V. Southwell [Q. J. Mech. Appl. Math. 8, 129–145 (1955; Zbl 0064.19802)]. The author shows that the system matrix of the discretization scheme is an M-matrix, so that the discretization is monotonic. Then it is formulated as a Petrov-Galerkin finite element method to establish the stability of the method with respect to a discrete energy norm. Author shows that the error of the numerical solution in the energy norm is bounded by O(h), where h denotes the mesh parameter. Numerical experiments are performed to demonstrate the effectiveness of the method.

MSC:
91B28Finance etc. (MSC2000)
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
65M15Error bounds (IVP of PDE)
35K65Parabolic equations of degenerate type