Spectral Method for the Black-Scholes Model of American Options Valuation (Journal of Mathematical Study, 2014)
In this paper, we devote ourselves to the research of numerical methodsfor American option pricing problems under the Black-Scholes model. The optimalexercise boundary which satisfies a nonlinear Volterra integral equation is resolved bya high-order collocation method based on graded meshes. For the other spatial domainboundary, an artificial boundary condition is applied to the pricing problem for theeffective truncation of the semi-infinite domain. Then, the front-fixing and stretchingtransformations are employed to change the truncated problem in an irregular domaininto a one-dimensional parabolic problem in [−1,1]. The Chebyshev spectral methodcoupled with fourth-order Runge-Kutta method is proposed for the resulting parabolicproblem related to the options. The stability of the semi-discrete numerical method isestablished for the parabolic problem transformed from the original model. Numericalexperiments are conducted to verify the performance of the proposed methods andcompare them with some existing methods.