Skip to main content
SpringerLink
Log in

Numerically Pricing Nonlinear Time-Fractional Black–Scholes Equation with Time-Dependent Parameters Under Transaction Costs

  • Published:
Computational Economics Aims and scope Submit manuscript

Abstract

One of the assumptions of the classical Black–Scholes (B–S) is that the market is frictionless. Also, the classical B–S model cannot show the memory effect of the stock price in the financial markets. Previously, Ankudinova and Ehrhardt (Comput Math Appl 56:799–812, 2008) priced a European option under the classical B–S model with transaction costs when dividends are paid on assets during that period. But due to the importance of the trend memory effect in financial pricing, we extend Ankudinova’s and Ehrhardt’s study under the fractional B–S model when the price change of the underlying asset with time follows a fractal transmission system. The option price is governed by a time-fractional B–S equation of order \( 0<\alpha <1 \). The main objective of this study is to obtain a numerical solution to determine the European option price with transaction costs based on the implicit difference scheme. This difference scheme is unconditionally stable and convergent and is shown stability and convergence by Fourier analysis. Numerical results and comparisons demonstrate that the introduced difference scheme has high accuracy and efficiency.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  • Ankudinova, J., & Ehrhardt, M. (2008). On the numerical solution of nonlinear Black–Scholes equations. Computers and Mathematics with Applications, 56, 799–812.

    Article  Google Scholar 

  • Barles, G., & Soner, H. M. (1998). Option pricing with transaction costs and a nonlinear Black–Scholes equation. Finance and Stochastics, 2, 369–397.

    Article  Google Scholar 

  • Bensaid, B., Lesne, J., Pages, H., & Scheinkman, J. (1992). Derivative asset pricing with transaction costs. Mathematical Finance, 2, 63–82.

    Article  Google Scholar 

  • Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of political economy, 81, 637–654.

    Article  Google Scholar 

  • Boyle, P. P., & Vorst, T. (1992). Option replication in discrete time with transaction costs. The Journal of Finance, 47, 271–293.

    Article  Google Scholar 

  • Cen, Z., & Le, A. (2011). A robust and accurate finite difference method for a generalized Black–Scholes equation. Journal of Computational and Applied Mathematics, 235, 3728–3733.

    Article  Google Scholar 

  • Chen, W., Xu, X., & Zhu, S. P. (2015). Analytically pricing double barrier options based on a time-fractional Black–Scholes equation. Computers and Mathematics with Applications, 69, 1407–1419.

    Article  Google Scholar 

  • Company, R., Jódar, L., & Pintos, J. R. (2009). A numerical method for European Option Pricing with transaction costs nonlinear equation. Mathematical and Computer Modelling, 50, 910–920.

    Article  Google Scholar 

  • Company, R., Navarro, E., Pintos, J. R., & Ponsoda, E. (2008). Numerical solution of linear and nonlinear Black–Scholes option pricing equations. Computers and Mathematics with Applications, 56, 813–821.

    Article  Google Scholar 

  • De Staelen, R. H., & Hendy, A. S. (2017). Numerically pricing double barrier options in a time-fractional Black–Scholes model. Computers and Mathematics with Applications, 74, 1166–1175.

    Article  Google Scholar 

  • Düring, B., Fournié, M., & Jüngel, A. (2003). High order compact finite difference schemes for a nonlinear Black–Scholes equation. International Journal of Theoretical and Applied Finance, 6, 767–789.

    Article  Google Scholar 

  • Giona, M., & Roman, H. E. (1992a). Fractional difusion equation for transport phenomena in random media. Physica A, 185, 87–97.

    Article  Google Scholar 

  • Giona, M., & Roman, H. E. (1992b). Fractional difusion equation on fractals: one-dimensional case and asymptotic behaviour. Journal of Physics A: Mathematical and General, 25, 2093–2105.

    Article  Google Scholar 

  • Golbabai, A., & Nikan, O. (2020). A computational method based on the moving least-squares approach for pricing double barrier options in a time-fractional Black–Scholes model. Computational Economics, 55, 119–141.

    Article  Google Scholar 

  • 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, 38, 173.

    Article  Google Scholar 

  • Gu, H., Liang, J. R., & Zhang, Y. X. (2012). Time-changed geometric fractional Brownian motion and option pricing with transaction costs. Physica A, 391, 3971–3977.

    Article  Google Scholar 

  • Haq, S., Hussain, M., & Ghafoor, A. (2019). A computational study of variable coefficients fractional advection-diffusion-reaction equations via implicit meshless spectral algorithm. Engineering with Computers, 36, 1243–1263.

    Article  Google Scholar 

  • Hull, C., & White, A. D. (1987). The pricing of options on assets with stochastic volatilities. The Journal of Finance, 42, 281–300.

    Article  Google Scholar 

  • Jandačka, M., & Ševčovič, D. (2005). On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile. Journal of Applied Mathematics, 2005, 235–258.

  • Kangro, R., & Nicolaides, R. (2000). Far field boundary conditions for Black–Scholes equations. SIAM Journal on Numerical Analysis, 38, 1357–1368.

    Article  Google Scholar 

  • Kou, S. G. (2002). A jump-diffusion model for option. Management science, 48, 1086–1101.

    Article  Google Scholar 

  • Kratka, M. (1998). No mystery behind the smile. Risk, 11, 67–71.

    Google Scholar 

  • Le Mehaute, A. (1984). Transfer processes in fractal media. Journal of Statistical Physics, 36, 665–676.

    Article  Google Scholar 

  • Leland, H. E. (1985). Option pricing and replication with transactions costs. The Journal of Finance, 40, 1283–1301.

    Article  Google Scholar 

  • Liu, H. K., & Chang, J. J. (2013). A closed-form approximation for the fractional Black–Scholes model with transaction costs. Computers and Mathematics with Applications, 65, 1719–1726.

    Article  Google Scholar 

  • Louis, O. S. (1997). Pricing stock options in a jump-diffusion model with stochastic volatility and interest rates: applications of Fourier inversion methods. Mathematical Finance, 7, 413–426.

    Article  Google Scholar 

  • Marco, A., & Antonio, P. (1994). Dynamic hedging portfolios for derivative securities in the presence of large transaction costs. Applied Mathematical Finance, 1, 165–194.

    Article  Google Scholar 

  • Merton, R. C. (1973). Theory of rational option pricing. The Bell Journal of economics and management science (The RAND Corporation), 4, 141–183.

    Google Scholar 

  • Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of financial economics, 3, 125–144.

    Article  Google Scholar 

  • Mohammadi, R. (2015). Quintic B-spline collocation approach for solving generalized Black–Scholes equation governing option pricing. Computers and Mathematics with Applications, 6, 777–797.

    Article  Google Scholar 

  • Murio, D. A. (2008). Implicit finite difference approximation for time fractional diffusion equations. Computers and Mathematics with Applications, 56, 1138–1145.

    Article  Google Scholar 

  • Nikan, O., Golbabai, A., Machado, J. T., & Nikazad, T. (2020a). Numerical approximation of the time-fractional cable model arising in neuronal dynamics. Engineering with Computers. https://doi.org/10.1007/s00366-020-01033-8.

    Article  Google Scholar 

  • Nikan, O., Jafari, H., & Golbabai, A. (2020b). Numerical analysis of the fractional evolution model for heat flow in materials with memory. Alexandria Engineering Journal, 59, 2627–2637.

    Article  Google Scholar 

  • Nikan, O., Machado, J. T., Avazzadeh, Z., & Jafari, H. (2020c). Numerical evaluation of fractional Tricomi-type model arising from physical problems of gas dynamics. Journal of Advanced Research. https://doi.org/10.1016/j.jare.2020.06.018.

    Article  Google Scholar 

  • Podlubny, I. (1999). Fractional differential equations. Cambridge: Academic Press.

    Google Scholar 

  • Rezaei, M., & Yazdanian, A. R. (2019). Numerical solution of the time-fractional Black–Scholes equation for European double barrier option with time-dependent parameters under the CEV model. Financial Engineering and Portfolio Management, 10, 339–369.

    Google Scholar 

  • Rezaei, M., Yazdanian, A. R., Ashrafi, A., & Mahmoudi, S. M. (2021). Numerical pricing based on fractional Black–Scholes equation with time-dependent parameters under the CEV model: Double barrier options. Computers and Mathematics with Applications, 90, 104–111.

    Article  Google Scholar 

  • Rezaei, M., Yazdanian, A. R., Mahmoudi, S. M., & Ashrafi, A. (2021). A compact difference scheme for time-fractional Black–Scholes equation with time-dependent parameters under the CEV model: American options. Computational Methods for Differential Equations, 9, 523–552.

    Google Scholar 

  • \(\breve{\mathrm{S}}\text{ev}\breve{\mathrm{c}}\text{ovi}\breve{\mathrm{c}}\), D. (2007). An iterative algorithm for evaluating approximations to the optimal exercise boundary for a nonlinear Black–Scholes equation. Canadian Applied Mathematics Quarterly, 15, 77–97.

  • Smith, G. D. (1985). Numerical solution of partial differential equations: Finite difference methods (3rd ed.). Oxford: Clarendon Press.

    Google Scholar 

  • Song, L. (2017). A space-time fractional derivative model for European option pricing with transaction costs in fractal market. Chaos, Solitons and Fractals, 103, 123–130.

    Article  Google Scholar 

  • Strikwerda, J. C. (2004). Finite difference schemes and partial differential equations. New Delhi: Siam.

    Google Scholar 

  • Tavella, D., & Randall, C. (2000). Pricing financial instruments. The finite difference method. New york: John Wiley & Sons Inc.

    Google Scholar 

  • Thomas, J. W. (1995). Numerical partial differential equations: Finite difference methods. Berlin: Springer.

    Book  Google Scholar 

  • Wang, X. T., Li, Z., & Zhuang, L. (2017). Risk preference, option pricing and portfolio hedging with proportional transaction costs. Chaos, Solitons and Fractals, 95, 111–130.

    Article  Google Scholar 

  • Wang, J., Liang, J. R., Lv, L. J., Qiu, W. Y., & Ren, F. Y. (2012). Continuous time Black–Scholes equation with transaction costs in subdiffusive fractional Brownian motion regime. Physica A, 391, 750–759.

    Article  Google Scholar 

  • Wang, X. T. (2010a). Scaling and long-range dependence in option pricing I: Pricing European option with transaction costs under the fractional Black–Scholes model. Physica A, 389, 438–444.

    Article  Google Scholar 

  • Wang, X. T. (2010b). Scaling and long-range dependence in option pricing, IV: pricing European options with transaction costs under the multifractional Black–Scholes model. Physica A, 389, 789–796.

    Article  Google Scholar 

  • Wang, X. T., Wu, M., Zhou, Z. M., & Jing, W. S. (2012). Pricing European option with transaction costs under the fractional long memory stochastic volatility model. Physica A, 391, 1469–1480.

    Article  Google Scholar 

  • Wang, X. T., Yan, H. G., Tang, M. M., & Zhu, E. H. (2010a). Scaling and long-range dependence in option pricing III: a fractional version of the Merton model with transaction costs. Physica A, 389, 452–458.

    Article  Google Scholar 

  • Wang, X. T., Zhu, E. H., Tang, M. M., & Yan, H. G. (2010b). Scaling and long-range dependence in option pricing II: Pricing European option with transaction costs under the mixed Brownian-fractional Brownian model. Physica A, 389, 445–451.

    Article  Google Scholar 

  • Yang, R., Qin, X., & Xia, B. (2008). Pricing barrier options with time-dependent parameters and curved boundaries. ISECS International Colloquium on Computing, Communication, Control, and Management, 3, 299–303.

    Article  Google Scholar 

  • Yousuf, M., Khaliq, A. Q. M., & Kleefeld, B. (2012). The numerical approximation of nonlinear Black–Scholes model for exotic path-dependent American options with transaction cost. International Journal of Computer Mathematics, 89, 1239–1254.

    Article  Google Scholar 

  • 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, 1772–1783.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematical Sciences, Semnan University, Semnan, Iran

    M. Rezaei, A. Ashrafi & S. M. Mahmoudi

  2. Faculty of Finance Sciences, Kharazmi University, Tehran, Iran

    A. R. Yazdanian

Authors
  1. M. Rezaei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. R. Yazdanian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Ashrafi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. S. M. Mahmoudi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. R. Yazdanian.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rezaei, M., Yazdanian, A.R., Ashrafi, A. et al. Numerically Pricing Nonlinear Time-Fractional Black–Scholes Equation with Time-Dependent Parameters Under Transaction Costs. Comput Econ 60, 243–280 (2022). https://doi.org/10.1007/s10614-021-10148-z

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10614-021-10148-z

Keywords

Search

Navigation