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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.
MSC:
91G60Numerical methods in mathematical finance
65M70Spectral, collocation and related methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
91G20Derivative securities