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.
Keywords: American option pricing; Black-Scholes model; optimal exercise boundary; front-fixing; Chebyshev spectral method; Runge-Kutta method.