Advertisement

A Computational Method Based on the Moving Least-Squares Approach for Pricing Double Barrier Options in a Time-Fractional Black–Scholes Model

  • Ahmad GolbabaiEmail author
  • Omid Nikan
Article
  • 12 Downloads

Abstract

The mathematical modeling in trade and finance issues is the key purpose in the computation of the value and considering option during preferences in contract. This paper investigates the pricing of double barrier options when the price change of the underlying is considered as a fractal transmission system. Due to the outstanding memory effect present in the fractional derivatives, approximating financial options with regards to their hereditary characteristics can be well interpreted and stated. Motivated by the reason mentioned, relatively reliable and also efficient numerical approaches have to be found while facing with fractional differential equations. The main objective of the current paper is to obtain the approximation solution of the time fractional Black–Scholes model of order \(0<\alpha \le 1\) governing European options based on the moving least-squares (MLS) method. In proposed method, firstly, the mentioned equation is discretized in the time sense based on finite difference scheme of order \({\mathcal {O}}(\delta t^{2-\alpha })\) and then approximated by using MLS approach in the space variable. Furthermore, the stability and convergence of the proposed method are discussed in detail throughout the paper. Numerical evidences and comparisons demonstrate that the proposed method is very accurate and efficient.

Keywords

Time fractional Black–Scholes model Double barrier option MLS method Stability Convergence 

Mathematics Subject Classification

34K37 97N50 91G80 

Notes

Acknowledgements

The authors are thankful to the referees for their valuable comments and constructive suggestions towards the improvement of the original paper. The authors are also very grateful to the Editor-in-Chief, Professor Hans Amman.

References

  1. Armentano, M. G. (2001). Error estimates in sobolev spaces for moving least square approximations. SIAM Journal on Numerical Analysis, 39(1), 38–51.Google Scholar
  2. Armentano, M. G., & Durán, R. G. (2001). Error estimates for moving least square approximations. Applied Numerical Mathematics, 37(3), 397–416.Google Scholar
  3. Belytschko, T., Lu, Y. Y., & Gu, L. (1994). Element-free galerkin methods. International Journal for Numerical Methods in Engineering, 37(2), 229–256.Google Scholar
  4. Bhowmik, S. K. (2014). Fast and efficient numerical methods for an extended Black–Scholes model. Computers and Mathematics with Applications, 67(3), 636–654.Google Scholar
  5. Björk, T., & Hult, H. (2005). A note on wick products and the fractional Black–Scholes model. Finance and Stochastics, 9(2), 197–209.Google Scholar
  6. Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654.Google Scholar
  7. Cartea, Á. (2013). Derivatives pricing with marked point processes using tick-by-tick data. Quantitative Finance, 13(1), 111–123.Google Scholar
  8. Cartea, A., & del Castillo-Negrete, D. (2007). Fractional diffusion models of option prices in markets with jumps. Physica A: Statistical Mechanics and Its Applications, 374(2), 749–763.Google Scholar
  9. Chen, W. (2014). Numerical methods for fractional Black–Scholes equations and variational inequalities governing option pricing. Ph.D. thesis, University of Western Australia.Google Scholar
  10. Chen, W., Xu, X., & Zhu, S.-P. (2014). Analytically pricing European-style options under the modified Black–Scholes equation with a spatial-fractional derivative. Quarterly of Applied Mathematics, 72(3), 597–611.Google Scholar
  11. Chen, W., Xu, X., & Zhu, S.-P. (2015a). Analytically pricing double barrier options based on a time-fractional Black–Scholes equation. Computers and Mathematics with Applications, 69(12), 1407–1419.Google Scholar
  12. Chen, W., Xu, X., & Zhu, S.-P. (2015b). A predictor–corrector approach for pricing american options under the finite moment log-stable model. Applied Numerical Mathematics, 97, 15–29.Google Scholar
  13. De Staelen, R., & Hendy, A. S. (2017). Numerically pricing double barrier options in a time-fractional Black–Scholes model. Computers and Mathematics with Applications, 74(6), 1166–1175.Google Scholar
  14. Duan, J.-S., Lu, L., Chen, L., & An, Y.-L. (2018). Fractional model and solution for the Black–Scholes equation. Mathematical Methods in the Applied Sciences, 41(2), 697–704.Google Scholar
  15. Elbeleze, A. A., Kılıçman, A., & Taib, B. M. (2013). Homotopy perturbation method for fractional Black–Scholes European option pricing equations using Sumudu transform. Mathematical Problems in Engineering, 2013, 7.Google Scholar
  16. Farnoosh, R., Rezazadeh, H., Sobhani, A., & Beheshti, M. H. (2016). A numerical method for discrete single barrier option pricing with time-dependent parameters. Computational Economics, 48(1), 131–145.Google Scholar
  17. Farnoosh, R., Sobhani, A., & Beheshti, M. H. (2017). Efficient and fast numerical method for pricing discrete double barrier option by projection method. Computers and Mathematics with Applications, 73(7), 1539–1545.Google Scholar
  18. Farnoosh, R., Sobhani, A., Rezazadeh, H., & Beheshti, M. H. (2015). Numerical method for discrete double barrier option pricing with time-dependent parameters. Computers and Mathematics with Applications, 70(8), 2006–2013.Google Scholar
  19. Franke, R., & Nielson, G. (1980). Smooth interpolation of large sets of scattered data. International Journal for Numerical Methods in Engineering, 15(11), 1691–1704.Google Scholar
  20. Fries, T.-P., & Matthies, H. G. (2004). Classification and overview of meshfree methods. Department of Mathematics and Computer Science, Technical University of Braunschweig.Google Scholar
  21. Golbabai, A., Ahmadian, D., & Milev, M. (2012). Radial basis functions with application to finance: American put option under jump diffusion. Mathematical and Computer Modelling, 55(3–4), 1354–1362.Google Scholar
  22. Golbabai, A., Ballestra, L., & Ahmadian, D. (2014). A highly accurate finite element method to price discrete double barrier options. Computational Economics, 44(2), 153–173.Google Scholar
  23. Golbabai, A., & Mohebianfar, E. (2017a). A new method for evaluating options based on multiquadric RBF-FD method. Applied Mathematics and Computation, 308, 130–141.Google Scholar
  24. Golbabai, A., & Mohebianfar, E. (2017b). A new stable local radial basis function approach for option pricing. Computational Economics, 49(2), 271–288.Google Scholar
  25. Golbabai, A., Nikan, O., & Nikazad, T. (2019). Numerical analysis of time fractional Black–Scholes European option pricing model arising in financial market. Computational and Applied Mathematics, 1, 177–183.Google Scholar
  26. Hariharan, G., Padma, S., & Pirabaharan, P. (2013). An efficient wavelet based approximation method to time fractional Black–Scholes European option pricing problem arising in financial market. Applied Mathematical Sciences, 7(69), 3445–3456.Google Scholar
  27. Jumarie, G. (2010). Derivation and solutions of some fractional Black–Scholes equations in coarse-grained space and time. application to merton’s optimal portfolio. Computers and Mathematics with Applications, 59(3), 1142–1164.Google Scholar
  28. Koleva, M. N., & Vulkov, L. G. (2017). Numerical solution of time-fractional Black–Scholes equation. Computational and Applied Mathematics, 36(4), 1699–1715.Google Scholar
  29. Kumar, D., Singh, J., & Baleanu, D. (2016). Numerical computation of a fractional model of differential–difference equation. Journal of Computational and Nonlinear Dynamics, 11(6), 061004.Google Scholar
  30. Kumar, S., Yildirim, A., Khan, Y., Jafari, H., Sayevand, K., & Wei, L. (2012). Analytical solution of fractional Black–Scholes European option pricing equation by using Laplace transform. Journal of Fractional Calculus and Applications, 2(8), 1–9.Google Scholar
  31. Lancaster, P., & Salkauskas, K. (1981). Surfaces generated by moving least squares methods. Mathematics of Computation, 37(155), 141–158.Google Scholar
  32. Leonenko, N. N., Meerschaert, M. M., & Sikorskii, A. (2013). Fractional pearson diffusions. Journal of Mathematical Analysis and Applications, 403(2), 532–546.Google Scholar
  33. Li, G., Jin, X., & Alum, N. (2005). Meshless methods for numerical solution of partial differential equations. In Handbook of materials modeling (pp. 2447–2474). Berlin: Springer.Google Scholar
  34. Liang, J.-R., Wang, J., Zhang, W.-J., Qiu, W.-Y., & Ren, F.-Y. (2010). The solution to a bifractional Black–Scholes–Merton differential equation. International Journal of Pure and Applied Mathematics, 58(1), 99–112.Google Scholar
  35. Liu, G., & Gu, Y. (2004). Boundary meshfree methods based on the boundary point interpolation methods. Engineering Analysis with Boundary Elements, 28(5), 475–487.Google Scholar
  36. Mardani, A., Hooshmandasl, M., Heydari, M., & Cattani, C. (2017). A meshless method for solving the time fractional advection–diffusion equation with variable coefficients. Computers and Mathematics with Applications, 75, 122–133.Google Scholar
  37. Marom, O., & Momoniat, E. (2009). A comparison of numerical solutions of fractional diffusion models in finance. Nonlinear Analysis: Real World Applications, 10(6), 3435–3442.Google Scholar
  38. McLain, D. H. (1976). Two dimensional interpolation from random data. The Computer Journal, 19(2), 178–181.Google Scholar
  39. Meerschaert, M. M., & Sikorskii, A. (2012). Stochastic models for fractional calculus (Vol. 43). Berlin: Walter de Gruyter.Google Scholar
  40. Merton, R. C. (1973). Theory of rational option pricing. The Bell Journal of Economics and Management Science, 4, 141–183.Google Scholar
  41. Podlubny, I. (1999). Mathematics in science and engineering. Fractional Differential Equations, 198, 1–340.Google Scholar
  42. Rashidinia, J., & Jamalzadeh, S. (2017a). Collocation method based on modified cubic b-spline for option pricing models. Mathematical Communications, 22(1), 89–102.Google Scholar
  43. Rashidinia, J., & Jamalzadeh, S. (2017b). Modified b-spline collocation approach for pricing american style asian options. Mediterranean Journal of Mathematics, 14(3), 111.Google Scholar
  44. Rogers, L. C. G. (1997). Arbitrage with fractional brownian motion. Mathematical Finance, 7(1), 95–105.Google Scholar
  45. Shepard, D. (1968). A two-dimensional interpolation function for irregularly-spaced data. In Proceedings of the 1968 23rd ACM national conference (pp. 517–524). New York: ACM.Google Scholar
  46. Sobhani, A., & Milev, M. (2018). A numerical method for pricing discrete double barrier option by legendre multiwavelet. Journal of Computational and Applied Mathematics, 328, 355–364.Google Scholar
  47. Song, L., & Wang, W. (2013). Solution of the fractional Black–Scholes option pricing model by finite difference method. In Abstract and applied analysis (Vol. 2013). Cairo: Hindawi.Google Scholar
  48. Tayebi, A., Shekari, Y., & Heydari, M. (2017). A meshless method for solving two-dimensional variable-order time fractional advection–diffusion equation. Journal of Computational Physics, 340, 655–669.Google Scholar
  49. Uddin, M., & Haq, S. (2011). RBFs approximation method for time fractional partial differential equations. Communications in Nonlinear Science and Numerical Simulation, 16(11), 4208–4214.Google Scholar
  50. Wyss, W. (2017). The fractional Black–Scholes equation. Fractional Calculus and Applied Analysis, 3, 51–62.Google Scholar
  51. Zhang, H., Liu, F., Turner, I., & Yang, Q. (2016). Numerical solution of the time fractional Black–Scholes model governing European options. Computers and Mathematics with Applications, 71(9), 1772–1783.Google Scholar
  52. Zhang, X., Shuzhen, S., Lifei, W., & Xiaozhong, Y. (2014). \(\theta \)-difference numerical method for solving time-fractional Black–Scholes equation. Highlights of Science Paper Online, China Science and Technology Papers, 7(13), 1287–1295.Google Scholar
  53. Zhuang, P., Gu, Y., Liu, F., Turner, I., & Yarlagadda, P. (2011). Time-dependent fractional advection–diffusion equations by an implicit MLS meshless method. International Journal for Numerical Methods in Engineering, 88(13), 1346–1362.Google Scholar
  54. Zuppa, C. (2003a). Error estimates for moving least square approximations. Bulletin of the Brazilian Mathematical Society, 34(2), 231–249.Google Scholar
  55. Zuppa, C. (2003b). Good quality point sets and error estimates for moving least square approximations. Applied Numerical Mathematics, 47(3–4), 575–585.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsIran University of Science and TechnologyNarmak, TehranIran

Personalised recommendations