Rannacher time-marching with orthogonal spline collocation method for retrieving the discontinuous behavior of hedging parameters

Abstract

In this paper, we extensively study the orthogonal spline collocation method known as the spline collocation at Gauss points with Rannacher’s time-marching scheme for free boundary value option pricing problems. Such financial problems commonly feature non-smooth payoff functions that cause inaccuracies in approximating the solution and its derivatives. As a result, unlike the problems with the smooth initial data, the quadratic convergence is not realized by the Crank-Nicolson time-stepping scheme for these problems. Furthermore, the non-smoothness in the initial condition leads to severe degradation in the convergence rates and spurious oscillations near the discontinuity. The rationale is that classical schemes strongly rely on the smoothness of the initial data. A rigorous time-marching scheme referred to as Rannacher time-stepping scheme is introduced for the American option’s price diagnosed by a linear complementarity problem to smoothen the data. Moreover, with careful analysis, second and fourth orders of convergence are established for the present scheme in temporal and spatial directions, respectively. The numerical results for three test problems are presented in tables and graphs to validate the theory. These results show that the present scheme achieves higher accuracy and sufficiently restores the expected behavior.