Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-08T10:33:59.182Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite Element and Discontinuous Galerkin Methods with Perfect Matched Layers for American Options

Part of: Mathematical finance

Published online by Cambridge University Press:  12 September 2017

Haiming Song ,
Kai Zhang  and
Yutian Li
Haiming Song
Affiliation:
Department of Mathematics, Jilin University, Changchun 130012, China
Kai Zhang*
Affiliation:
Department of Mathematics, Jilin University, Changchun 130012, China
Yutian Li*
Affiliation:
Department of Mathematics, and Institute of Computational and Theoretical Studies, Hong Kong Baptist University, Kowloon, Hong Kong
*
*Corresponding author. Email addresses:[email protected] (K. Zhang), [email protected] (Y. T. Li)
*Corresponding author. Email addresses:[email protected] (K. Zhang), [email protected] (Y. T. Li)
Get access

Abstract

This paper is devoted to the American option pricing problem governed by the Black-Scholes equation. The existence of an optimal exercise policy makes the problem a free boundary value problem of a parabolic equation on an unbounded domain. The optimal exercise boundary satisfies a nonlinear Volterra integral equation and is solved by a high-order collocation method based on graded meshes. This free boundary is then deformed to a fixed boundary by the front-fixing transformation. The boundary condition at infinity (due to the fact that the underlying asset's price could be arbitrarily large in theory), is treated by the perfectly matched layer technique. Finally, the resulting initial-boundary value problems for the option price and some of the Greeks on a bounded rectangular space-time domain are solved by a finite element method. In particular, for Delta, one of the Greeks, we propose a discontinuous Galerkin method to treat the discontinuity in its initial condition. Convergence results for these two methods are analyzed and several numerical simulations are provided to verify these theoretical results.

Keywords

American optionGreeksoptimal exercise boundaryperfectly matched layerfinite element methoddiscontinuous Galerkin method

MSC classification

Secondary: 91G60: Numerical methods (including Monte Carlo methods) 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 4 , November 2017 , pp. 829 - 851
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Adams, R. A., Sobolev Spaces, Academic Press, New York, 1975.Google Scholar
[2] Allegretto, W., Lin, Y. and Yang, H., Finite element error estimates for a nonlocal problem in American option valuation, SIAM J. Numer. Anal., 39 (2001), pp. 834–857.CrossRefGoogle Scholar
[3] Amin, K. and Khanna, A., Convergence of American option values from discrete-to continuous-time financial models, Math. Finance, 4 (1994), pp. 289–304.CrossRefGoogle Scholar
[4] Barone-Adesi, G. and Whaley, R., Efficient Analytical Approximation of American Option Values, J. Fin., 42 (1987), pp. 301–320.CrossRefGoogle Scholar
[5] Badea, L. and Wang, J., A new formulation for the valuation of American options, I: Solution uniqueness, in Proceedings of the 19th Daewoo Workshop in Analysis and Scientific Computing, Park, Eun-Jae and Lee, Jongwoo, eds., 2000, pp. 3–16.Google Scholar
[6] Badea, L. and Wang, J., A new formulation for the valuation of American options, II: Solution existence, in Proceedings of the 19th Daewoo Workshop in Analysis and Scientific Computing, Park, Eun-Jae and Lee, Jongwoo, eds., 2000, pp. 17–33.Google Scholar
[7] Berenger, J. P., A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), pp. 185–200.CrossRefGoogle Scholar
[8] Berenger, J. P., Three-dimensional perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 127 (1996), pp. 363–379.CrossRefGoogle Scholar
[9] Black, F. and Scholes, M., The pricing of options and corporate liabilities, J. Pol. Econ., 81 (1973), pp. 637–659.CrossRefGoogle Scholar
[10] Brennan, M. and Schwartz, E., The valuation of American put options, J. Fin., 32 (1977), pp. 449–462.CrossRefGoogle Scholar
[11] Brunner, H., Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, Cambridge, 2004.CrossRefGoogle Scholar
[12] Carr, P., Jarrow, R. and Myneni, R., Alternative characterizations of American put options, Math. Finance, 2 (1992), pp. 87–106.CrossRefGoogle Scholar
[13] Chen, J., Wang, D. S. and Wu, H. J., An adaptive finite element method with a modified perfectly matched layer formulation for diffraction gratings, Commun. Comput. Phys., 6 (2009), pp. 290–318.CrossRefGoogle Scholar
[14] Chen, Z. M. and Wu, H. J., An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures, SIAM. J. Numer. Anal., 41 (2003), pp. 799–826.CrossRefGoogle Scholar
[15] Chen, Z. M., Guo, B. Q. and Xiao, Y. M., An hp adaptive uniaxial perfectly matched layer method for Helmholtz scattering problems, Commun. Comput. Phys., 5 (2009), pp. 546–564.Google Scholar
[16] Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.Google Scholar
[17] Cox, J. C., Ross, S. A. and Rubinstein, M., Option pricing: A simplified approach, J. Fin. Econ., 7 (1979), pp. 229–263.CrossRefGoogle Scholar
[18] Evans, J. D., Kuske, R. and Keller, J. B., American options on assets with dividends near expiry, Math. Finance, 12 (2002), pp. 219–237.CrossRefGoogle Scholar
[19] Han, H. and Wu, X., A fast numerical method for the Black-Scholes equation of American options, SIAM J. Numer. Anal., 41 (2003), pp. 2081–2095.CrossRefGoogle Scholar
[20] Holmes, A. D. and Yang, H., A front-fixing finite element method for the valuation of American options, SIAM J. Sci. Comput., 30 (2008), pp. 2158–2180.CrossRefGoogle Scholar
[21] Hull, J., Fundamentals of Futures and Options Markets, 6th Revised ed, Prentice Hall, Upper Saddle River, 2007.Google Scholar
[22] Jaillet, P., Lamberton, D. and Lapeyre, B., Variational inequalities and the pricing of American options, Acta Appl. Math., 21 (1990), pp. 263–289.CrossRefGoogle Scholar
[23] Ju, N. and Zhong, R., An Approximate Formula for Pricing American Options, The Journal of Derivatives, 7 (1999), pp. 31–40.CrossRefGoogle Scholar
[24] Jiang, L., Mathematical Modeling and Methods of Option Pricing, World Scientific Press, Singapore, 2005.CrossRefGoogle Scholar
[25] Kim, I. J., The analytic valuation of American puts, Rev. Fin. Stud., 3 (1990), pp. 547–572.CrossRefGoogle Scholar
[26] Kwok, Y. K., Mathematical Models of Financial Derivatives, 2nd ed, Springer Finance, Berlin Heidelberg, 2008.Google Scholar
[27] Lantos, N. and Nataf, F., Perfectly matched layers for the heat and advection-diffusion equations, J. Comput. Phys., 229 (2010), pp. 9042–9052 CrossRefGoogle Scholar
[28] Lin, Y. P., Zhang, K. and Zou, J., Studies on some perfectly matched layers for one-dimensional time-dependent systems, Adv. Comput. Math., 30 (2009), pp. 1–35.CrossRefGoogle Scholar
[29] Larsson, S. and Thomee, V., Partial Differential Equations with Numerical Methods, Springer-Verlag Press, Berlin Heidelberg, 2003.Google Scholar
[30] Liang, C. and Xiang, X., Convergence of an anisotropic perfectly matched layer method for Helmholtz scattering problems, Numer. Math. Theory Methods Appl., 9 (2016), pp. 358–382.CrossRefGoogle Scholar
[31] Ma, J., Xiang, K. and Jiang, Y., An integral equation method with high-order collocation implementations for pricing American put options, Int. J. Econ. Finance, 2 (2010), pp. 102–112.CrossRefGoogle Scholar
[32] Nicholls, D. P. and Sward, A., A discontinuous Galerkin method for pricing American options under the constant elasticity of variance model, Commun. Comput. Phys., 17 (2015), pp. 761–778.CrossRefGoogle Scholar
[33] Riviere, B., Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, SIAM. Frontiers in Applied Mathematics, 2008.CrossRefGoogle Scholar
[34] Schwartz, E. S., The valuation of warrants: Implementing a new approach, J. Fin. Econ., 4 (1977), pp. 79–93.CrossRefGoogle Scholar
[35] Wu, X. and Zheng, W., An adaptive perfectly matched layer method for multiple cavity scattering problems, Commun. Comput. Phys., 19 (2016), pp. 534–558.CrossRefGoogle Scholar
[36] Zhang, R., Song, H. and Luan, N., A weak Galerkin finite element method for the valuation of American options, Front. Math. China, 9 (2014), pp. 455–476.CrossRefGoogle Scholar
[37] Zhang, K., Song, H. and Li, J., Front-fixing FEMs for the pricing of American options based on a PML technique, Appl. Anal., 94 (2015), pp. 903–931.CrossRefGoogle Scholar