AIMS Mathematics, 2020, 5(3): 2226-2243. doi: 10.3934/math.2020147

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

An accurate solution for the generalized Black-Scholes equations governing option pricing

Department of Mathematics, National Institute of Technology Calicut, Calicut - 673601, India

Today industries related to finance are essentially implementing advanced mathematical tools. In 1973, Fisher Black and Myron Scholes developed an eminent stochastic model which later coined as Black-Scholes differential equations for option pricing. This paper illustrates a convenient time integration scheme based on the generalized trapezoidal formulas (GTF $[\alpha=\frac{1}{3}]$) introduced by Chawla et al. in 1996. GTF is applied for the temporal discretization along with the classical finite difference schemes in space direction. The proposed scheme yields the (uniform) stability employing the uniform bound of the inverse operator, as well as second-order spatial accuracy and third-order temporal accuracy under reasonable conditions. Finally, the numerical illustrations and comparison with existing schemes demonstrate the stability and accuracy of the method.
  Article Metrics


1. B. Fischer, S. Myron, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637–654.

2. R. C. Merton, Theory of rational option pricing, Bell J. Econ. Manage. Sci., 4 (1973), 141–183.

3. P. Wilmott, S. Howson, J. Dewynne, The mathematics of financial derivatives: A student introduction, Cambridge University Press, 1995.

4. J. C. Zhao, M. Davison, R. M. Corless, Compact finite difference method for American option pricing, J. Comput. Appl. Math., 206 (2007), 306–321.

5. M. K. Kadalbajoo, L. P. Tripathi, P. Arora, A robust nonuniform B-spline collocation method for solving the generalized Black–Scholes equation, IMA J. Numer. Anal., 34 (2014), 252–278.

6. M. K. Kadalbajoo, L. P. Tripathi, A. Kumar, A cubic B-spline collocation method for a numerical solution of the generalized Black–Scholes equation, Mathe. Comput. Modell., 55 (2012), 1483–1505.

7. R. Company, E. Navarro, J. R. Pintos, Numerical solution of linear and nonlinear Black–Scholes option pricing equations, Comput. Math. Appl., 56 (2008), 813–821.

8. M. K. Salahuddin, M. Ahmed S. K. Bhowmilk, A note on numerical solution of a linear BlackScholes model, GANIT: J. Bangladesh Math. Soc., 33 (2013), 103–115.

9. R. Mohammadi, Quintic B-spline collocation approach for solving generalized Black–Scholes equation governing option pricing, Comput. Math. Appl., 69 (2015), 777–797.

10. R. Valkov, Fitted finite volume method for a generalized Black–Scholes equation transformed on finite interval, Numer. Algorithms, 65 (2014), 195–220.

11. S. Wang, A novel fitted finite volume method for the Black–Scholes equation governing option pricing, IMA J. Numer. Anal., 24 (2004), 699–720.

12. M. Ehrhardt, R. E. Mickens, A fast, stable and accurate numerical method for the Black–Scholes equation of American options, Inter. J. Theor. Appl. Finance, 11 (2008), 471–501.

13. C. S. Rao, Numerical solution of generalized Black–Scholes model, Appl. Math. Comput., 321 (2018), 401–421.

14. J. Huang, Z. D. Cen, Cubic spline method for a generalized Black-Scholes equation, Math. Probl. Eng., 2014 (2014).

15. R. E. Lynch, J. R. Rice, A high-order difference method for differential equations, Math. Comput., 34 (1980), 333–372.

16. R. D. Skeel, The second-order backward differentiation formula is unconditionally zero-stable, Appl. Numer. Math., 5 (1989), 145–149.

17. J. R. Cash, On the integration of stiff systems of ODEs using extended backward differentiation formulae, Numer. Math., 34 (1980), 235–246.

18. C. F. Curtiss, J. O. Hirschfelder, Integration of stiff equations, P. Natl. Acad. Sci., 38 (1952), 235–243.

19. W. Wyss, The fractional black-scholes equation, Fract. Calc. Appl. Anal., 3 (2000), 51–66.

20. J. R. Liang, J. Wang, W. J. Zhang, et al, Option pricing of a bi-fractional Black–Merton–Scholes model with the Hurst exponent H in [12, 1], Appl. Math. Lett., 23 (2010), 859–863.

21. M. L. Zheng, F. W. Liu, I. Turner, et al, A novel high order space-time spectral method for the time fractional Fokker–Planck equation, SIAM J. Sci. Comput., 37 (2015), A701–A724.

22. R. H. De Staelen, A. S. Hendy, Numerically pricing double barrier options in a time-fractional Black–Scholes model, Comput. Math. Appl., 74 (2017), 1166–1175.

23. M. M. Chawla, M. A. Al-Zanaidi, D. J. Evans, A class of generalized trapezoidal formulas for the numerical integration of, Inter. J. Comput. Math., 62 (1996), 131–142.

24. M. M. Chawla, M. A. Al-Zanaidi, D. J. Evans, Generalized trapezoidal formulas for parabolic equations, Inter. J. Comput. Math., 70 (1999), 429–443.

25. M. M. Chawla, M. A. Al-Zanaidi, D. J. Evans, Generalized trapezoidal formulas for convectiondiffusion equations, Inter. J. Comput. Math., 72 (1999), 141–154.

26. M. M. Chawla, M. A. Al-Zanaidi, D. J. Evans, Generalized trapezoidal formulas for the Black– Scholes equation of option pricing, Inter. J. Comput. Math., 80 (2003), 1521–1526.

27. O. A. Ladyzhenskaia, V. A Solonnikov, N. N. Ural'ceva, Linear and quasi-linear equations of parabolic type, Am. Math.Soc., 1968.

28. G. M. Lieberman, Second order parabolic differential equations, World Scientific, 1996.

29. D. Wei, Existence, uniqueness, and numerical analysis of solutions of a quasilinear parabolic problem, SIAM J. Numer. Anal., 29 (1992), 484–497.

30. A. Friedman, Partial differential equations of parabolic type, Courier Dover Publications, 2008.

31. V. Isakov, Inverse problems for partial differential equations, Springer International Publishing, 2017.

32. C. Vázquez, An upwind numerical approach for an American and European option pricing model, Appl. Math. Comput., 97 (1998), 273–286.

33. R. Kangro, R. Nicolaides, Far field boundary conditions for Black-Scholes equations, SIAM J. Numer. Anal., 38 (2000), 1357–1368.

34. C. K. Cho, T. Kim, Y. H. Kwon, Estimation of local volatilities in a generalized Black–Scholes model, Appl. Math. Comput., 162 (2005), 1135–1149.

35. Z. Cen, A. Le, A robust and accurate finite difference method for a generalized Black–Scholes equation, J. Comput. Appl. Math., 235 (2011), 3728–3733.

36. J. M. Varah, A lower bound for the smallest singular value of a matrix, Linear Algebra Appl., 11 (1975), 3–5.

37. R. S. Varga, Matrix iterative analysis, Second Revised and Expanded Edition, 2009.

38. R. Teman, Numerical analysis, D. Reidel Publishing Company, Dordrecht, Holland, 2012.

39. K. Atkinson, W. Han, Theoretical numerical analysis: A functional analysis framework, Springer New York, 2001.

40. J. W. Thomas, Numerical partial differential equations: Finite difference methods, Springer New York, 2013.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved