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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:

91G20 Derivative securities (option pricing, hedging, etc.)
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35K65 Degenerate parabolic equations

Citations:

Zbl 0064.19802
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