Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-25T01:56:50.068Z Has data issue: false hasContentIssue false
  • English
  • Français

A Discontinuous Galerkin Method for Pricing American Options Under the Constant Elasticity of Variance Model

Published online by Cambridge University Press:  24 March 2015

David P. Nicholls  and
Andrew Sward
David P. Nicholls*
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, USA
Andrew Sward
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, USA
*
*Corresponding author. Email addresses: davidn@uic.edu (D. P. Nicholls), swardandphi@gmail.com (A. Sward)
Get access

Abstract

The pricing of option contracts is one of the classical problems in Mathematical Finance. While useful exact solution formulas exist for simple contracts, typically numerical simulations are mandated due to the fact that standard features, such as early-exercise, preclude the existence of such solutions. In this paper we consider derivatives which generalize the classical Black-Scholes setting by not only admitting the early-exercise feature, but also considering assets which evolve by the Constant Elasticity of Variance (CEV) process (which includes the Geometric Brownian Motion of Black-Scholes as a special case). In this paper we investigate a Discontinuous Galerkin method for valuing European and American options on assets evolving under the CEV process which has a number of advantages over existing approaches including adaptability, accuracy, and ease of parallelization.

Keywords

91G2091G6065M70Option pricingPDE methodsCEV processDiscontinuous Galerkin method
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 3 , March 2015 , pp. 761 - 778
Copyright
Copyright © Global-Science Press 2015 

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]Achdou, Y. and Pironneau, O.. Computational methods for option pricing, volume 30 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005.Google Scholar
[2]Becker, S.. The constant elasticity of variance model and its implications for option pricing. J. Finance, 35:661–673, 1980.Google Scholar
[3]Burden, R. and Faires, J. D.. Numerical analysis. Brooks/Cole Publishing Co., Pacific Grove, CA, sixth edition, 1997.Google Scholar
[4]Brennan, M.J. and Schwartz, E. S.. The valuation of American put options. Journal of Finance, 32:449–462, 1977.Google Scholar
[5]Cont, R., Lantos, N., and Pironneau, O.. A reduced basis for option pricing. SIAM J. Financial Math., 2:287–316, 2011.Google Scholar
[6]Cox, J.. The constant elasticity of variance option pricing model. J. Portfolio Management, 22:15–17, 1996.Google Scholar
[7]Deville, O.M., Fischer, P. F., and Mund, E. H.. High-order methods for incompressible fluid flow, volume 9 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, 2002.Google Scholar
[8]Emanuel, D. and MacBeth, J.. Further results on the constant elasticity of variance call option pricing model. J. Financial and Quantitative Anal., 17:533–554, 1982.Google Scholar
[9]Foufas, G. and Larson, M.. Valuing Asian options using the finite element method and duality techniques. Journal of Computational and Applied Mathematics, 222(1):377–427, 2008.Google Scholar
[10]Glasserman, P.. Monte Carlo methods in financial engineering, volume 53 of Applications of Mathematics (New York). Springer-Verlag, New York, 2004. Stochastic Modelling and Applied Probability.Google Scholar
[11]Higham, J.D.. An introduction to financial option valuation. Cambridge University Press, Cambridge, 2004.Google Scholar
[12]Hsu, L.Y., Lin, T. I., and Lee, C. F.. Constant elasticity of variance (CEV) option pricing model: integration and detailed derivation. Math. Comput. Simulation, 79(1):60–71, 2008.CrossRefGoogle Scholar
[13]Hozman, J.. Discontinous Galerkin method for the numerical solution of option pricing. In APLIMAT 2012 (11th International Conference), Slovak University of Technology in Bratslava, 2012.Google Scholar
[14]Hull, J.. Options, Futures, and Other Derivatives. Prentice Hall, 2012.Google Scholar
[15]Hesthaven, J.S. and Warburton, T.. Nodal discontinuous Galerkin methods, volume 54 of Texts in Applied Mathematics. Springer, New York, 2008. Algorithms, analysis, and applications.Google Scholar
[16]Knessl, C. and Xu, M.. On a free boundary problem for an American put option under the CEV process. Appl. Math. Lett., 24(7):1191–1198, 2011.Google Scholar
[17]Lu, R. and Hsu, Y.-H.. Valuation of standard options under the constant elasticity of variance model. International Journal of Business and Economics, 4(2):157–165, 2005.Google Scholar
[18]Lo, F.C., Tang, H. M., Ku, K. C., and Hui, C. H.. Valuing time-dependent CEV barrier options. J. Appl. Math. Decis. Sci., pages Art. ID 359623, 17, 2009.Google Scholar
[19]Vos, E.P.J., Sherwin, S. J., and Kirby, R. M.. From h to p efficiently: implementing finite and spectral/hp element methods to achieve optimal performance for low- and high-order discretisations. J. Comput. Phys., 229(13):5161–5181, 2010.Google Scholar
[20]Wilmott, P., Howison, S., and Dewynne, J.. The mathematics of financial derivatives. Cambridge University Press, Cambridge, 1995.Google Scholar
[21]Willyard, M.. Adaptive Spectral Element Methods To Price American Options. PhD thesis, Florida State University, 2011.Google Scholar
[22]Wong, H.Y. and Zhao, J.. An artificial boundary method for American option pricing under the CEV model. SIAM J. Numer. Anal., 46(4):2183–2209, 2008.Google Scholar
[23]Zhu, P.S.. An exact and explicit solution for the valuation of American put options. Quantitative Finance, 6(3):229–242, 2006.Google Scholar