Computational Economics

, Volume 53, Issue 1, pp 191–205 | Cite as

A Stable and Convergent Finite Difference Method for Fractional Black–Scholes Model of American Put Option Pricing

  • R. Kalantari
  • S. ShahmoradEmail author


We introduce the mathematical modeling of American put option under the fractional Black–Scholes model, which leads to a free boundary problem. Then the free boundary (optimal exercise boundary) that is unknown, is found by the quasi-stationary method that cause American put option problem to be solvable. In continuation we use a finite difference method for derivatives with respect to stock price, Grünwal Letnikov approximation for derivatives with respect to time and reach a fractional finite difference problem. We show that the set up fractional finite difference problem is stable and convergent. We also show that the numerical results satisfy the physical conditions of American put option pricing under the FBS model.


Fractional differential equation American option pricing Quasi-stationary Finite difference method Newton interpolation method 


  1. Amin, k, & Khanna, A. (1994). Convergence of American option values from discrete to continuous-time financial models. Mathematical Finance, 4, 289–304.CrossRefGoogle Scholar
  2. Baleanu, D., Diethelm, K., Scalar, E., & Trujillo, J. J. (2012). Fractionak caculus, model and numerical methods (Vol. 3). Singapore: World Scientific.CrossRefGoogle Scholar
  3. Barraquand, J., & Pudet, T. (1994). Pricing of American path-dependent contingent claims. Paris: Digital Research Laboratory.Google Scholar
  4. Black, F., & Scholes, M. S. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637654.CrossRefGoogle Scholar
  5. Broadie, M., & Detemple, J. (1996). American option valuation: New bounds, approximations, and a comparison of existing methods. Review of Financial Studies, 9(4), 121–150.CrossRefGoogle Scholar
  6. Hull, J. C. (1997). Options futures and other derivatives. Upper Saddle River: Prentice Hall.Google Scholar
  7. Jumarie, G. (2008). Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise application to fractional Black–Scholes equations. Insurance Mathematics and Economics, 42, 271–287.CrossRefGoogle Scholar
  8. Kalanatri, R., & Shahmorad, S., & Ahmadian, D. , (2015). The stability analysis of predictor-corrector method in solving American option pricing model. Computer Econonics. doi: 10.1007/s10614-015-9483-x.
  9. Kwok, Y. K. (1998). Mathematical models of financial derivatives. Heidelberg: Springer.Google Scholar
  10. Kwok, Y. K. (2009). Mathematical models of financial derivatives (Vol. 2). Berlin: Springer.Google Scholar
  11. Meng, Wu, Nanjing, H., & Huiqiang, M. (2014). American option pricing with time-varying parameters. Computer Economics, 241, 439–450.Google Scholar
  12. Merton, R. C. (1973). The theory of rational option pricing. The Bell Journal of Economics and Management of Science, 4, 141–183.CrossRefGoogle Scholar
  13. Podlubny, I. (1999). Fractional differential equations. Cambridge: Academic press.Google Scholar
  14. Ross, S. H. (1999). An Introduction to mathematical finance. Cambridge: Cambridge University Press.Google Scholar
  15. San-Lin, C. (2000). American option valuation under stochastic interest rates. Computer Economics, 3, 283–307.Google Scholar
  16. Smith, G. D. (1985). Numerical solution of partial differential equations: Finite difference methods. Oxford: Clarendon Press.Google Scholar
  17. Wilmott, P. (1998). The theory and practice of financial engineering. New York: Wiley.Google Scholar
  18. Wilmott, P., Dewynne, J., & Howison, S. (1993). Option pricing, mathematical models and computation. Oxford: Oxford Financial Press.Google Scholar
  19. Yu, D., & Tan, H. (2003). Numerical methods of differential equations. Beijing: Science Publisher.Google Scholar
  20. Zhuang, P., Liu, F., Anh, V., & Turner, I. (2009). Numerical methods for the variable-order fractional advection–diffusion equation with a nonlinear source term. SIAM Jouranl of Numerical Analysis, 47, 1760–1781.CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Faculty of Mathematical SciencesUniversity of TabrizTabrizIran

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