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A New Approach for the Black–Scholes Model with Linear and Nonlinear Volatilities

Author

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  • Seda Gulen

    (Department of Mathematics, Tekirdag Namik Kemal University, Degirmenalti, Tekirdag 59030, Turkey)

  • Catalin Popescu

    (Department of Business Administration, Petroleum-Gas University, Blvd. Bucuresti, no.39, 100680 Ploiesti, Romania)

  • Murat Sari

    (Department of Mathematics, Yildiz Technical University, Istanbul 34220, Turkey)

Abstract

Since financial engineering problems are of great importance in the academic community, effective methods are still needed to analyze these models. Therefore, this article focuses mainly on capturing the discrete behavior of linear and nonlinear Black–Scholes European option pricing models. To achieve this, this article presents a combined method; a sixth order finite difference (FD6) scheme in space and a third–order strong stability preserving Runge–Kutta (SSPRK3) over time. The computed results are compared with available literature and the exact solution. The computed results revealed that the current method seems to be quite strong both quantitatively and qualitatively with minimal computational effort. Therefore, this method appears to be a very reliable alternative and flexible to implement in solving the problem while preserving the physical properties of such realistic processes.

Suggested Citation

  • Seda Gulen & Catalin Popescu & Murat Sari, 2019. "A New Approach for the Black–Scholes Model with Linear and Nonlinear Volatilities," Mathematics, MDPI, vol. 7(8), pages 1-14, August.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:8:p:760-:d:258983
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    References listed on IDEAS

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    Cited by:

    1. Tianbao Zhou & Xinghao Li & Peng Wang, 2021. "Statistics and Practice on the Trend’s Reversal and Turning Points of Chinese Stock Indices Based on Gann’s Time Theory and Solar Terms Effect," Mathematics, MDPI, vol. 9(15), pages 1-24, July.
    2. Claudiu Tiberiu Albulescu & Aviral Kumar Tiwari & Phouphet Kyophilavong, 2021. "Nonlinearities and Chaos: A New Analysis of CEE Stock Markets," Mathematics, MDPI, vol. 9(7), pages 1-13, March.
    3. Chaeyoung Lee & Jisang Lyu & Eunchae Park & Wonjin Lee & Sangkwon Kim & Darae Jeong & Junseok Kim, 2020. "Super-Fast Computation for the Three-Asset Equity-Linked Securities Using the Finite Difference Method," Mathematics, MDPI, vol. 8(3), pages 1-13, February.
    4. Sangkwon Kim & Darae Jeong & Chaeyoung Lee & Junseok Kim, 2020. "Finite Difference Method for the Multi-Asset Black–Scholes Equations," Mathematics, MDPI, vol. 8(3), pages 1-17, March.

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