Journal of Optimization Theory and Applications

, Volume 179, Issue 1, pp 311–331 | Cite as

Power Penalty Approach to American Options Pricing Under Regime Switching

  • Kai ZhangEmail author
  • Xiaoqi Yang


This work aims at studying a power penalty approach to the coupled system of differential complementarity problems arising from the valuation of American options under regime switching. We introduce a power penalty method to approximate the differential complementarity problems, which results in a set of coupled nonlinear partial differential equations. By virtue of variational inequality theory, we establish the unique solvability of the system of differential complementarity problems. Moreover, the convergence property of this power penalty method in an appropriate infinite-dimensional space is explored, where an exponential convergence rate of the power penalty method is established and the monotonic convergence of the penalty method with respect to the penalty parameter is shown. Finally, some numerical experiments are presented to verify the convergence property of the power penalty method.


American option pricing Regime switching Differential complementarity problem Power penalty method Convergence analysis 

Mathematics Subject Classification

65N12 65K10 91B28 



The authors would like to thank the anonymous referees and the editor for their helpful comments and suggestions toward the improvement of this paper. Kai Zhang wishes to thank the support from the MOE Project of Key Research Institute of Humanities and Social Sciences at Universities (Grant No. 14JJD790041). Project 11001178 is partially supported by National Natural Science Foundation of China.


  1. 1.
    Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cont, R.: Volatility clustering in financial markets: empirical facts and agent-based models. Long Mem. Econ. 2, 289–309 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Andersen, T.G., Bondarenko, O., Todorov, V., Tauchen, G.: The fine structure of equity-index option dynamics. J. Econom. 187(2), 532–546 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Wilmott, P.: Paul Wilmott on Quantitative Finance. Wiley, New York (2000)zbMATHGoogle Scholar
  5. 5.
    Buffington, J., Elliott, R.J.: American options with regime switching. Int. J. Theor. Appl. Finance 5(05), 497–514 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dai, M., Zhang, Q., Zhu, Q.J.: Trend following trading under a regime switching model. SIAM J. Financ. Math. 1(1), 780–810 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Achdou, Y., Pironneau, O.: Computational Methods for Option Pricing. Frontiers in Applied Mathematics, SIAM, Philadelphia (2005)CrossRefzbMATHGoogle Scholar
  8. 8.
    Jaillet, P., Lamberton, D., Lapeyre, B.: Variational inequalities and the pricing of American options. Acta Applicandae Mathematicae 21(3), 263–289 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Yang, H.: A numerical analysis of American options with regime switching. J. Sci. Comput. 44(1), 69–91 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rubinov, A.M., Yang, X.: Lagrange-Type Functions in Constrained Non-convex Optimization, vol. 85. Kluwer Academic Publishers, Boston (2003)zbMATHGoogle Scholar
  11. 11.
    Wang, S., Yang, X.Q., Teo, K.L.: Power penalty method for a linear complementarity problem arising from American option valuation. J. Optim. Theory Appl. 129(2), 227–254 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Zhang, K., Yang, X., Teo, K.L.: Convergence analysis of a monotonic penalty method for American option pricing. J. Math. Anal. Appl. 348(2), 915–926 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kwok, Y.K.: Mathematical Models of Financial Derivatives. Springer, Berlin (1998)zbMATHGoogle Scholar
  14. 14.
    Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)CrossRefzbMATHGoogle Scholar
  15. 15.
    Wang, S.: A novel fitted finite volume method for the Black–Scholes equation governing option pricing. IMA J. Numer. Anal. 24(4), 699–720 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Zhang, K., Teo, K.L., Swartz, M.: A robust numerical scheme for pricing American options under regime switching based on penalty method. Comput. Econ. 3(4), 463–483 (2014)CrossRefGoogle Scholar
  17. 17.
    Sun, Z., Liu, Z., Yang, X.: On power penalty methods for linear complementarity problems arising from American option pricing. J. Glob. Optim. 63(1), 165–180 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Shenzhen Audencia Business SchoolShenzhen UniversityShenzhenChina
  2. 2.Department of Applied MathematicsHong Kong Polytechnic UniversityHung HomHong Kong

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