Rational Spectral Collocation Method for Solving Black-Scholes and Heston Equations

In this paper, we raise a new method for numerically solving the partial differential equations (PDEs) of the Black-Scholes and Heston models, which play an important role in financial option pricing theory. Our proposed method is based on the rational spectral collocation method and the contour integral method. The presence of discontinuities in the first-order derivative of the initial condition of the PDEs prevents the spectral method from achieving high accuracy. However, the rational spectral method excels in overcoming this drawback. So we discretize the spatial variables of PDEs by rational spectral method, which yields a system of ordinary differential equations. Then we solve it by the numerical inverse Laplace transform using contour integral method. It is very important to select an appropriate parameters in the contour integral method, we revise the optimal parameters proposed by Trefethen and Weideman (Math Comput 76(259):1341-1356, 2007) in hyperbolic contour to control the effect of roundoff error. During solving the independent shifted linear systems, preconditioned Krylov subspace iteration is used to improve computational efficiency. We also compare the numerical results obtained from our proposed method with those obtained from the finite difference and spectral methods, showing its high accuracy and efficiency in pricing various financial options, including those mentioned above.