Multiple Shooting Method for Solving Black-Scholes Equation

In this paper, the Black-Scholes (B-S) model for the pricing of the European and the barrier call options are considered, which yields a partial differential problem. First, A numerical technique based on Crank-Nicolson (C-N) method is used to discretisize the time domain. Consequently, the partial differential equation will be converted to a system of an ordinary differential equation (ODE). Then, the multiple shooting method combined with Lagrange polynomials is utilized to solve the ODEs. Regarding the convergence order of the approximate solution which normally decreases due to the non-smooth properties of the option's payoff (at the strike price), in this study, the equipped C-N scheme with variable step size strategy is applied for the time discretization. As a result, the variable step size strategy prevents the error propagation by controlling the error at each time step and increases the computational speed by raising the step size in the smooth points of the domain. In order to implement the variable step size, an algorithm is presented. In addition, the stability of the presented method is analyzed. The extracted numerical results represent the accuracy and efficiency of the proposed method.