## 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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### References:

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