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A Unified Numerical Approach for a Large Class of Nonlinear Black-Scholes Models

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Large-Scale Scientific Computing (LSSC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10665))

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Abstract

In this paper, we consider a class of non-linear models in mathematical finance, where the volatility depends on the second spatial derivative of the option value. We study the convergence and realization of the constructed, on a fitted non-uniform meshes, implicit difference schemes. We implement various Picard and Newton iterative processes. Numerical experiments are discussed.

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References

  1. Barles, G.: Convergence of numerical schemes for degenerate parabolic equations arising in finance. In: Rogers, L.C.G., Talay, D. (eds.) Numerical Methods in Finance. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  2. Ehrhardt, M.: Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing. Nova Science Publishers, New York (2008)

    MATH  Google Scholar 

  3. Feng, J., Liu, S.: An improved generalized Newton method for absolute value equations. SpringerPlus 5(1), 10–42 (2016)

    Article  MathSciNet  Google Scholar 

  4. Haentjens, T., In’t Hout, K.J.: Alternating direction implicit finite difference schemes for the Heston-Hull-White PDE. J. Comput. Finan. 16(1), 83–110 (2012)

    Article  Google Scholar 

  5. Haghani, F.K.: On generalized Traub’s method for absolute value equations. J. Optim. Theory Appl. 166(2), 619–625 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Koleva, M.N., Vulkov, L.G.: On splitting-based numerical methods for nonlinear models of European options. Int. J. Comput. Math. 3(5), 781–796 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lesmana, D.C., Wang, S.: An upwind finite difference method for a nonlinear Black-Scholes equation governing European option valuation under transaction costs. Appl. Math. Comput. 219(16), 8811–8828 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Mangasarian, O.L.: A generalized Newton method for absolute value equations. Optim. Lett. 3(1), 101–108 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Pooley, D., Forsythy, P., Vetzalz, K.: Numerical convergence properties of option pricing PDEs with uncertain volatility. IMA J. Numer. Anal. 23(2), 241–267 (2003)

    Article  MathSciNet  Google Scholar 

  10. Ševčovič, D.: Nonlinear parabolic equations arising in mathematical finance. In: Ehrhardt, M., Günther, M., ter Maten, E.J.W. (eds.) Novel Methods in Computational Finance. MI, vol. 25, pp. 3–15. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-61282-9_1

    Chapter  Google Scholar 

  11. Varga, R.S.: Matrix Iterative Analysis. Prentice-Hall, Engelwood Cliffs (1962)

    Google Scholar 

  12. Yong, L.: An iterative method for absolute value equations problem. Information 16(1), 7–12 (2013). International Information Institute (Tokyo)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This research was supported by the Bulgarian National Fund of Science under Project “Advanced Analytical and Numerical Methods for Nonlinear Differential Equations with Applications in Finance and Environmental Pollution”-2017.

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Authors and Affiliations

  1. University of Ruse, 8 Studentska str., 7017, Ruse, Bulgaria

    Miglena N. Koleva & Lubin G. Vulkov

Authors
  1. Miglena N. Koleva
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  2. Lubin G. Vulkov
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Corresponding author

Correspondence to Miglena N. Koleva .

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Editors and Affiliations

  1. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Lirkov

  2. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Svetozar Margenov

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Koleva, M.N., Vulkov, L.G. (2018). A Unified Numerical Approach for a Large Class of Nonlinear Black-Scholes Models. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computing. LSSC 2017. Lecture Notes in Computer Science(), vol 10665. Springer, Cham. https://doi.org/10.1007/978-3-319-73441-5_64

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  • DOI: https://doi.org/10.1007/978-3-319-73441-5_64

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-73440-8

  • Online ISBN: 978-3-319-73441-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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