Journal of Scientific Computing

, Volume 79, Issue 1, pp 517–541 | Cite as

A Local Radial Basis Function Method for Pricing Options Under the Regime Switching Model

  • Hengguang Li
  • Reza MollapouraslEmail author
  • Majid Haghi


This paper is devoted to develop an efficient meshfree method based on radial basis functions (RBFs) to solve a system of partial differential equations arising from pricing options under the regime switching model. For global RBF methods, one of the major disadvantages is the computational cost and ill-conditioning associated with the dense linear systems that arise. So, we employ one of the local meshfree methods known as radial basis function based finite difference method. Then with an operator splitting method, sparse and well-conditioned system of complementarity problems are solved very fast for the American option. Also, the uniqueness of solution is proved for the discretized system of equations. Numerical examples presented in the last section illustrate the robustness and practical performance of the proposed algorithm for pricing European and American options.


Radial basis functions Finite difference Option pricing Regime switching model 



H. Li was supported in part by the NSF Grant DMS-1819041, and by the Wayne State University Career Development Chair Award.


  1. 1.
    Andersen, L.: Markov models for commodity futures: theory and practice. Quant. Finance 10(8), 831–854 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Babbin, J., Forsyth, P.A., Labahn, G.: A comparison of iterated optimal stopping and local policy iteration for American options under regime switching. J. Sci. Comput. 58(2), 409–430 (2014)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Ballestra, L.V., Pacelli, G.: Pricing European and American options with two stochastic factors: a highly efficient radial basis function approach. J. Econ. Dyn. Control 37(6), 1142–1167 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bansal, R., Zhou, H.: Term structure of interest rates with regime shifts. J. Finance 57(5), 1997–2043 (2002)Google Scholar
  5. 5.
    Bastani, A.F., Ahmadi, Z., Damircheli, D.: A radial basis collocation method for pricing American options under regime-switching jump-diffusion models. Appl. Numer. Math. 65, 79–90 (2013)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bayona, V., Moscoso, M., Carretero, M., Kindelan, M.: RBF-FD formulas and convergence properties. J. Comput. Phys. 229(22), 8281–8295 (2010)zbMATHGoogle Scholar
  7. 7.
    Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York (1994)zbMATHGoogle Scholar
  8. 8.
    Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Borici, A., Lüthi, H.-J.: Pricing American put options by fast solutions of the linear complementarity problem. In: Kontoghiorghes, E.J., Rustem, B., Siokos, S. (eds.) Computational Methods in Decision-Making, Economics and Finance, pp. 325–338. Springer, New York (2002)Google Scholar
  10. 10.
    Boyle, P., Draviam, T.: Pricing exotic options under regime switching. Insur. Math. Econ. 40(2), 267–282 (2007)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Buffington, J., Elliott, R.J.: American options with regime switching. Int. J. Theor. Appl. Finance 05(05), 497–514 (2002)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Chen, S., Insley, M.: Regime switching in stochastic models of commodity prices: an application to an optimal tree harvesting problem. J. Econ. Dyn. Control 36(2), 201–219 (2012)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Chen, Z., Forsyth, P.A.: Implications of a regime-switching model on natural gas storage valuation and optimal operation. Quant. Finance 10(2), 159–176 (2010)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Dai, M., Zhang, Q., Zhu, Q.J.: Trend following trading under a regime switching model. SIAM J. Financ. Math. 1(1), 780–810 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Driscoll, T.A., Hale, N., Trefethen, L.N.: Chebfun Guide. Pafnuty Publications, Oxford (2014)Google Scholar
  16. 16.
    Egorova, V.N., Company, R., Jódar, L.: A new efficient numerical method for solving American option under regime switching model. Comput. Math. Appl. 71(1), 224–237 (2016)MathSciNetGoogle Scholar
  17. 17.
    Fasshauer, G.E., Khaliq, A.Q.M., Voss, D.A.: Using meshfree approximation for multiasset American options. J. Chin. Inst. Eng. 27(4), 563–571 (2004)Google Scholar
  18. 18.
    Fornberg, B., Flyer, N.: A Primer on Radial Basis Functions with Applications to the Geosciences. SIAM, Philadelphia (2015)zbMATHGoogle Scholar
  19. 19.
    Fornberg, B., Flyer, N.: Solving PDEs with radial basis functions. Acta Numer. 24, 215–258 (2015)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Golbabai, A., Mohebianfar, E.: A new method for evaluating options based on multiquadric RBF-FD method. Appl. Math. Comput. 308(Supplement C), 130–141 (2017)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Guo, X.: Information and option pricings. Quant. Finance 1(1), 38–44 (2001)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Haldrup, N., Nielsen, M.O.: A regime switching long memory model for electricity prices. J. Econ. 135(12), 349–376 (2006)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Hardy, M.: A regime-switching model of long-term stock returns. N. Am. Actuar. J. 5(2), 41–53 (2001)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Holmes, A.D., Yang, H., Zhang, S.: A front-fixing finite element method for the valuation of American options with regime switching. Int. J. Comput. Math. 89(9), 1094–1111 (2012)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Hon, Y.C., Mao, X.Z.: A radial basis function method for solving options pricing model. Financ. Eng. 8, 31–49 (1999)Google Scholar
  26. 26.
    Huang, Y., Forsyth, P.A., Labahn, G.: Methods for pricing American options under regime switching. SIAM J. Sci. Comput. 33(5), 2144–2168 (2011)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Howison, S., Wilmott, P., Dewynne, J.: Option Pricing: Mathematical Models and Computation. Oxford Financial Press, Oxford (1993)zbMATHGoogle Scholar
  28. 28.
    Ikonen, S., Toivanen, J.: Operator splitting methods for American option pricing. Appl. Math. Lett. 17(7), 809–814 (2004)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Kanas, A.: On real interest rate dynamics and regime switching. J. Bank. Finance 32(10), 2089–2098 (2008)Google Scholar
  30. 30.
    Khaliq, A.Q.M., Kleefeld, B., Liu, R.H.: Solving complex PDE systems for pricing American options with regime-switching by efficient exponential time differencing schemes. Numer. Methods Partial Differ. Equ. 29(1), 320–336 (2013)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Khaliq, A.Q.M., Liu, R.H.: New numerical scheme for pricing American option with regime-switching. Int. J. Theor. Appl. Finance 12(03), 319–340 (2009)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Le, H., Wang, C.: A finite time horizon optimal stopping problem with regime switching. SIAM J. Control Optim. 48(8), 5193–5213 (2010)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Liu, R.H.: Regime-switching recombining tree for option pricing. Int. J. Theor. Appl. Finance 13(03), 479–499 (2010)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Liu, R.H., Nguyen, D.: A tree approach to options pricing under regime-switching jump diffusion models. Int. J. Comput. Math. 92(12), 2575–2595 (2015)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Mollapourasl, R., Fereshtian, A., Vanmaele, M.: Radial basis functions with partition of unity method for American options with stochastic volatility. Comput. Econ. (2017).
  36. 36.
    Pettersson, U., Larsson, E., Marcusson, G., Persson, J.: Improved radial basis function methods for multi-dimensional option pricing. J. Comput. Appl. Math. 222(1), 82–93 (2008). (Special issue: numerical PDE methods in finance) MathSciNetzbMATHGoogle Scholar
  37. 37.
    Pooley, D.M., Vetzal, K.R., Forsyth, P.A.: Convergence remedies for non-smooth payoffs in option pricing. J. Comput. Finance 6(4), 25–40 (2003)Google Scholar
  38. 38.
    Rad, J.A., Parand, K., Ballestra, L.V.: Pricing European and American options by radial basis point interpolation. Appl. Math. Comput. 251(Supplement C), 363–377 (2015)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Safdari-Vaighani, A., Heryudono, A., Larsson, E.: A radial basis function partition of unity collocation method for convection–diffusion equations arising in financial applications. J. Sci. Comput. 64(2), 341–367 (2015)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Salmi, S., Toivanen, J.: An iterative method for pricing American options under jump-diffusion models. Appl. Numer. Math. 61(7), 821–831 (2011)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Seydel, R.: Tools for Computational Finance, 4th edn. Springer, Berlin (2009)zbMATHGoogle Scholar
  42. 42.
    Shcherbakov, V., Larsson, E.: Radial basis function partition of unity methods for pricing vanilla basket options. Comput. Math. Appl. 71(1), 185–200 (2016)MathSciNetGoogle Scholar
  43. 43.
    Varga, R.S.: Matrix Iterative Analysis. Springer, Berlin (2010)Google Scholar
  44. 44.
    Wahab, M.I.M., Yin, Z., Edirisinghe, N.C.P.: Pricing swing options in the electricity markets under regime-switching uncertainty. Quant. Finance 10(9), 975–994 (2010)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Wendland, H.: Scattered Data Approximation. Number 17 in Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, New York (2005)Google Scholar
  46. 46.
    Wu, Z., Hon, Y.C.: Convergence error estimate in solving free boundary diffusion problem by radial basis functions method. Eng. Anal. Bound. Elem. 27(1), 73–79 (2003)zbMATHGoogle Scholar
  47. 47.
    Yang, H.: A numerical analysis of American options with regime switching. J. Sci. Comput. 44(1), 69–91 (2010)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Yin, G., Zhang, Q.: Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach. Springer, Berlin (1998)zbMATHGoogle Scholar
  49. 49.
    Young, D.M.: Iterative Solution of Large Linear Systems. Academic Press, New York (1971)zbMATHGoogle Scholar
  50. 50.
    Yousuf, M., Khaliq, A.Q.M., Liu, R.H.: Pricing American options under multi-state regime switching with an efficient L-stable method. Int. J. Comput. Math. 92(12), 2530–2550 (2015)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Zhang, Q., Zhou, X.Y.: Valuation of stock loans with regime switching. SIAM J. Control Optim. 48(3), 1229–1250 (2009)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Zvan, R., Forsyth, P.A., Vetzal, K.R.: Penalty methods for American options with stochastic volatility. J. Comput. Appl. Math. 91(2), 199–218 (1998)MathSciNetzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of MathematicsWayne State UniversityDetroitUSA
  2. 2.School of MathematicsShahid Rajaee Teacher Training UniversityLavizan, TehranIran
  3. 3.Department of MathematicsOregon State UniversityCorvallisUSA

Personalised recommendations