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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 in mathematical finance 65M70 Spectral, collocation and related methods (IVP of PDE) 65M12 Stability and convergence of numerical methods (IVP of PDE) 91G20 Derivative securities
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