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A cubic B-spline collocation method for a numerical solution of the generalized Black-Scholes equation. (English) Zbl 1255.91431
Summary: The uniform cubic B-spline collocation method is implemented to find the numerical solution of the generalized Black-Scholes partial differential equation. We use the horizontal method of lines to discretize the temporal variable and the spatial variable by means of a θ-method, θ[1/2,1] (θ=1 corresponds to the back-ward Euler method and θ=1/2 corresponds to the Crank-Nicolson method), and a cubic B-spline collocation method on uniform meshes, respectively. The method corresponding to θ=1 is shown to be unconditionally stable and first order accurate with respect to the time variable and second order accurate with respect to the space variable while the method corresponding to θ=1/2 is shown to be unconditionally stable and second order accurate with respect to both the variables. Finally, the numerical examples demonstrate the stability and accuracy of the method.
91G60Numerical methods in mathematical finance
65M70Spectral, collocation and related methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
91G20Derivative securities