Volume 47, Issue 1
Spectral Method for the Black-Scholes Model of American Options Valuation

Haiming Song, Ran Zhang & Wen Yi Tian

J. Math. Study, 47 (2014), pp. 47-64.

Published online: 2014-03

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  • Abstract

In this paper, we devote ourselves to the research of numerical methods for American option pricing problems under the Black-Scholes model. The optimal exercise boundary which satisfies a nonlinear Volterra integral equation is resolved by a high-order collocation method based on graded meshes. For the other spatial domain boundary, an artificial boundary condition is applied to the pricing problem for the effective truncation of the semi-infinite domain. Then, the front-fixing and stretching transformations are employed to change the truncated problem in an irregular domain into a one-dimensional parabolic problem in [−1,1]. The Chebyshev spectral method coupled with fourth-order Runge-Kutta method is proposed for the resulting parabolic problem related to the options. The stability of the semi-discrete numerical method is established for the parabolic problem transformed from the original model. Numerical experiments are conducted to verify the performance of the proposed methods and compare them with some existing methods.

  • Keywords

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

  • AMS Subject Headings

35A35 90A09 65K10 65M12 65M60

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

songhm11@mails.jlu.edu.cn (Haiming Song)

zhangran@jlu.edu.cn (Ran Zhang)

twymath@gmail.com (Wen Yi Tian)

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@Article{JMS-47-47, author = {Song , Haiming and Zhang , Ran and Tian , Wen Yi }, title = {Spectral Method for the Black-Scholes Model of American Options Valuation}, journal = {Journal of Mathematical Study}, year = {2014}, volume = {47}, number = {1}, pages = {47--64}, abstract = {

In this paper, we devote ourselves to the research of numerical methods for American option pricing problems under the Black-Scholes model. The optimal exercise boundary which satisfies a nonlinear Volterra integral equation is resolved by a high-order collocation method based on graded meshes. For the other spatial domain boundary, an artificial boundary condition is applied to the pricing problem for the effective truncation of the semi-infinite domain. Then, the front-fixing and stretching transformations are employed to change the truncated problem in an irregular domain into a one-dimensional parabolic problem in [−1,1]. The Chebyshev spectral method coupled with fourth-order Runge-Kutta method is proposed for the resulting parabolic problem related to the options. The stability of the semi-discrete numerical method is established for the parabolic problem transformed from the original model. Numerical experiments are conducted to verify the performance of the proposed methods and compare them with some existing methods.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v47n1.14.03}, url = {http://global-sci.org/intro/article_detail/jms/9949.html} }
TY - JOUR T1 - Spectral Method for the Black-Scholes Model of American Options Valuation AU - Song , Haiming AU - Zhang , Ran AU - Tian , Wen Yi JO - Journal of Mathematical Study VL - 1 SP - 47 EP - 64 PY - 2014 DA - 2014/03 SN - 47 DO - http://dor.org/10.4208/jms.v47n1.14.03 UR - https://global-sci.org/intro/article_detail/jms/9949.html KW - American option pricing KW - Black-Scholes model KW - optimal exercise boundary KW - front-fixing KW - Chebyshev spectral method KW - Runge-Kutta method AB -

In this paper, we devote ourselves to the research of numerical methods for American option pricing problems under the Black-Scholes model. The optimal exercise boundary which satisfies a nonlinear Volterra integral equation is resolved by a high-order collocation method based on graded meshes. For the other spatial domain boundary, an artificial boundary condition is applied to the pricing problem for the effective truncation of the semi-infinite domain. Then, the front-fixing and stretching transformations are employed to change the truncated problem in an irregular domain into a one-dimensional parabolic problem in [−1,1]. The Chebyshev spectral method coupled with fourth-order Runge-Kutta method is proposed for the resulting parabolic problem related to the options. The stability of the semi-discrete numerical method is established for the parabolic problem transformed from the original model. Numerical experiments are conducted to verify the performance of the proposed methods and compare them with some existing methods.

Haiming Song, Ran Zhang & Wen Yi Tian. (2019). Spectral Method for the Black-Scholes Model of American Options Valuation. Journal of Mathematical Study. 47 (1). 47-64. doi:10.4208/jms.v47n1.14.03
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