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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 \(\theta \)-method, \(\theta \in [1/2,1]\) (\(\theta =1\) corresponds to the back-ward Euler method and \(\theta =1/2\) corresponds to the Crank-Nicolson method), and a cubic B-spline collocation method on uniform meshes, respectively. The method corresponding to \(\theta =1\) 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 \(\theta =1/2\) 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:

91G60 Numerical methods (including Monte Carlo methods)
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
91G20 Derivative securities (option pricing, hedging, etc.)
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