Quintic B-spline collocation approach for solving generalized Black–Scholes equation governing option pricing

Abstract

We develop a numerical algorithm for solving generalized Black–Scholes partial differential equation, which arises in European option pricing. The method comprises the horizontal method of lines for time integration and 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) to discretize in temporal direction and the quintic B-spline collocation method in uniform spatial direction. The convergence analysis and stability of proposed method are discussed in detail, it is justifying that the approximate solution converges to the exact solution of orders O(k+h3) for the back-ward Euler method and O(k2+h3) for the Crank–Nicolson method, where k and h are mesh sizes in the time and space directions, respectively. The proposed method is also shown to be unconditionally stable. This scheme applied on some test examples, the numerical results illustrate the efficiency of the method and confirm the theoretical behavior of the rates of convergence. Results shown by this method are found to be in good agreement with the known exact solutions. The produced results are also seen to be more accurate than some available results given in the literature.