Explicit RKF-Compact Scheme for Pricing Regime Switching American Options with Varying Time Step
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In this research work, an explicit Runge-Kutta-Fehlberg time integration with a fourth-order compact finite difference scheme in space is employed for solving the regime-switching pricing model. First, we recast the free boundary problem into a system of nonlinear partial differential equations with a multi-fixed domain. We further introduce a transformation based on the square root function with a fixed free boundary from which a high order analytical approximation is obtained for computing the derivative of the optimal exercise boundary in each regime. The high order analytical approximation is achieved by the method of extrapolation. As such, it enables us to employ fourth-order spatial discretization and an adaptive time integration with Dirichlet boundary conditions for obtaining the numerical solution of the asset option, option Greeks, and the optimal exercise boundary for each regime. In the set of equations, Hermite interpolation with Newton basis is used to estimate the coupled assets options and option Greeks. A numerical experiment is carried out with two- and four-regimes examples and results are compared with the existing methods. The results obtained from the numerical experiment show that the present method provides better performance in terms of computational speed and more accurate solutions with a large step size.
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