Advertisement

Computational and Applied Mathematics

, Volume 37, Issue 3, pp 2381–2398 | Cite as

A two-grid penalty method for American options

  • Tatiana P. Chernogorova
  • Miglena N. Koleva
  • Radoslav L. Valkov
Article
  • 82 Downloads

Abstract

In this paper we consider the pricing of American options, governed by a partial differential complementarity problem. The differential problem is first approximated by a semi-linear PDE using two distinct penalty approaches which are well known in computational finance. We then initiate the two-grid algorithm by solving the nonlinear problem on a coarse grid and further the linearized in the interpolated coarse-grid solution problem on a fine grid. By means of the maximum principle the algorithm is shown to be of fourth order convergence rate in space. Numerical experiments verify the presented two-grid approach where we draw some interesting conclusions.

Keywords

Penalty method Finite difference Two-grid Newton method Maximum principle 

Mathematics Subject Classification

35R35 65M06 

Notes

Acknowledgements

Authors would like to thank the anonymous reviewers for their valuable and constructive comments that greatly contributed to improve the quality of the paper.

References

  1. Axelsson O (1993) On mesh independence and Newton-type methods. Appl Math 38(4):249–265MathSciNetzbMATHGoogle Scholar
  2. Deng J, Tao Z, Zhang T (2016) Iterative penalty finite element methods for the steady incompressible magnetohydrodynamic problem. Comput Appl Math 1–21. doi: 10.1007/s40314-016-0323-y
  3. Duffy DJ (2013) Finite difference methods in financial engineering: a partial differential equation approach. Wiley, ChichesterGoogle Scholar
  4. Forsyth PA, Vetzal KR (2002) Quadratic convergence for valuing American options using a penalty method. SIAM J Sci Comput 23(6):2095–2122MathSciNetCrossRefzbMATHGoogle Scholar
  5. Grossmann C, Roos H-G (2007) Numerical treatment of partial differential equations. Springer, BerlinCrossRefzbMATHGoogle Scholar
  6. Gyulov TB, Valkov RL (2016) American option pricing problem transformed on finite interval. Int J Comput Math 93(5):821–836MathSciNetCrossRefzbMATHGoogle Scholar
  7. Haentjens T, in ’t Hout KJ (2012) Alternating direction implicit finite difference schemes for the Heston–Hull–White partial differential equation. J Comp Finance 16(1):83–110CrossRefGoogle Scholar
  8. Howison S, Reisinger C, Witte JH (2013) The effect of nonsmooth payoffs on the penalty approximation of American options. SIAM J Financ Math 4(1):539–574MathSciNetCrossRefzbMATHGoogle Scholar
  9. Jaillet P, Lamberton D, Lapeyre B (1990) Variational inequalities and the pricing of American options. Acta Appl Math 21(3):263–289MathSciNetCrossRefzbMATHGoogle Scholar
  10. Jiang L (2005) Mathematical modeling and methods of option pricing. World Scientific, Singapore, p 329CrossRefzbMATHGoogle Scholar
  11. Khaliq AQM, Voss DA, Kazmi SHK (2006) A linearly implicit predictor-corrector scheme for pricing American options using a penalty method approach. J Bank Finance 30(2):489–502CrossRefGoogle Scholar
  12. Koleva M, Valkov R (2016) Two-grid algorithms for pricing American options by a penalty method. Proc Algoritmy 2016:275–284Google Scholar
  13. Kovalov P, Linetsky V, Marcozzi M (2007) Pricing multi-asset American options: a finite element method-of-lines with smooth penalty. J Sci Comp 33(3):209–237MathSciNetCrossRefzbMATHGoogle Scholar
  14. Lauko M, Ševčovič D (2010) Comparison of the numerical and analytical approximations of the early exercise boundary of American options. ANZIAM J 51:430–448MathSciNetCrossRefzbMATHGoogle Scholar
  15. Martín-Vaquero J, Khaliq AQM, Kleefeld B (2014) Stabilized explicit Runge–Kutta methods for multi-asset American options. Comput Math Appl 67(6):1293–1308MathSciNetCrossRefzbMATHGoogle Scholar
  16. Moradipour M, Yousefi SA (2016) Using a meshless kernel-based method to solve the Black–Scholes variational inequality of American options. Comput Appl Math 1–13. doi: 10.1007/s40314-016-0351-7
  17. Nielsen BFr, Skavhaug O, Tveito A (2002) Penalty and front-fixing methods for the numerical solution of American option problems. J Comput Finance 5(4):69–98CrossRefGoogle Scholar
  18. Nielsen BFr, Skavhaug O, Tveito A (2008) Penalty methods for the numerical solution of American multi-asset option problems. J Comput Appl Math 222(1):3–16MathSciNetCrossRefzbMATHGoogle Scholar
  19. Samarskii AA (2001) The theory of difference schemes. Marcel Dekker, BaselCrossRefzbMATHGoogle Scholar
  20. Ševčovič D, Stehlıková B, Mikula K (2011) Analytical and numerical methods for pricing financial derivatives. Nova Science Publishers, New YorkGoogle Scholar
  21. Vulkov LG, Zadorin AI (2010) Two-grid algorithms for an ordinary second order equation with exponential boundary layer in the solution. Int J Numer Anal Model 7(3):580–592MathSciNetzbMATHGoogle Scholar
  22. Wang J, Forsyth P (2008) Maximal use of central differencing for Hamilton–Jacobi–Bellman PDEs in finance. SIAM J Numer Anal 46(3):1580–1601MathSciNetCrossRefzbMATHGoogle Scholar
  23. Xu J (1996) Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J Numer Anal 33(5):1759–1777MathSciNetCrossRefzbMATHGoogle Scholar
  24. Zhang K, Wang S (2011) Convergence property of an interior penalty approach to pricing American option. J Ind Manag Optim 7(2):435MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017

Authors and Affiliations

  • Tatiana P. Chernogorova
    • 1
  • Miglena N. Koleva
    • 2
  • Radoslav L. Valkov
    • 3
  1. 1.Department of Numerical Methods and AlgorithmsSofia UniversitySofiaBulgaria
  2. 2.Department of MathematicsRuse UniversityRuseBulgaria
  3. 3.Department of Mathematics and Computer ScienceUniversity of AntwerpAntwerpBelgium

Personalised recommendations