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Computational Mathematics and Mathematical Physics

, Volume 59, Issue 10, pp 1753–1758 | Cite as

Algorithm for Determining the Volatility Function in the Black–Scholes Model

  • V. M. IsakovEmail author
  • S. I. KabanikhinEmail author
  • A. A. ShananinEmail author
  • M. A. ShishleninEmail author
  • S. ZhangEmail author
Article
  • 55 Downloads

Abstract

An algorithm for reconstructing the volatility function in the modified Black–Scholes model is developed. Results of numerical computations are presented. It is shown that adding information about the prices of similar options with different issue dates makes it possible to improve the accuracy and increase the interval in which the volatility function can be reconstructed.

Keywords:

Black–Scholes equation coefficient inverse problem optimization local volatility 

Notes

FUNDING

This work was supported by the Russian Foundation for Basic Research (project nos. 19-01-00694, 17-07-00507, 17-51-150001, 17-51-540004, 16-29-15120, and 16-01-00437) and by NSF (project no. DMS 15-08902).

REFERENCES

  1. 1.
    F. Black and M. Scholes, “The pricing of options and corporate liabilities,” J. Political Econ. 81, 637–654 (1973).MathSciNetCrossRefGoogle Scholar
  2. 2.
    R. Merton, “Theory of rational option pricing,” Bell J. Econom. Manag. Sci. 4, 141–183 (1973).MathSciNetCrossRefGoogle Scholar
  3. 3.
    J. C. Cox, S. A. Ross, and M. Rubinstein, “Option pricing: A simplified approach,” J. Financial Econ. 7, 229–263 (1979).MathSciNetCrossRefGoogle Scholar
  4. 4.
    A. N. Shiryaev, Foundations of Stochastic Financial Mathemtics (Fazis, Moscow, 1998) [in Russian].Google Scholar
  5. 5.
    T. F. Coleman, Y. Li, and A. Verma, “Reconstructing the unknown local volatility function,” J. Comput. Finance 2 (3), 77–102 (1999).CrossRefGoogle Scholar
  6. 6.
    V. Isakov, I. Bouchouev, and N. Valdivia, “Recovery of volatility coefficient by linearization,” Quantitative Finance 2, 257–263 (2002).MathSciNetCrossRefGoogle Scholar
  7. 7.
    S. I. Kabanikhin, O. Scherzer, and M. A. Shishlenin, “Iteration methods for solving a two dimensional inverse problem for a hyperbolic equation,” J. Inverse Ill-Posed Probl. 11 (1), 87–109 (2003).MathSciNetCrossRefGoogle Scholar
  8. 8.
    H. Egger and H. W. Engl, “Tikhonov regularization applied to the inverse problem of option pricing: Convergence analysis and rates,” Inverse Probl. 21, 1027–1045 (2005).MathSciNetCrossRefGoogle Scholar
  9. 9.
    H. Egger, T. Hein, and B. Hofmann, “On decoupling of volatility smile and term structure in inverse option pricing,” Inverse Probl. 22, 1247–1259 (2006).MathSciNetCrossRefGoogle Scholar
  10. 10.
    H. Föllmer and A. Schied, Stochastic Finance: An Introduction in Discrete Time (Walter de Gruyter, New York, 2004).CrossRefGoogle Scholar
  11. 11.
    S. I. Kabanikhin, “Definitions and examples of inverse and ill-posed problems,” J. Inverse Ill-Posed Probl. 16 (4), 317–357 (2008).MathSciNetzbMATHGoogle Scholar
  12. 12.
    S. I. Kabanikhin and M. A. Shishlenin, “Quasi-solution in inverse coefficient problems,” J. Inverse Ill-Posed Probl. 16 (7), 87–109 (2008).MathSciNetzbMATHGoogle Scholar
  13. 13.
    A. Lakhal, M. M. Lakhal, and A. K. Louis, “Calibrating local volatility in inverse option pricing using the Levenberg–Marquardt method,” J. Inverse Ill-Posed Probl. 18 (5), 493–514 (2010).MathSciNetCrossRefGoogle Scholar
  14. 14.
    T. Björk, Arbitrage Theory in Continuous Time (Oxford Univ. Press, Oxford, 2004).CrossRefGoogle Scholar
  15. 15.
    S. I. Kabanikhin and M. A. Shishlenin, “On using a priori information in coefficient inverse problems for hyperbolic equations,” Trudy Inst. Mat. Mekh Ur. Otd. Ross. Akad. Nauk 18 (1), 147–164 (2012).Google Scholar
  16. 16.
    S. I. Kabanikhin, D. B. Nurseitov, M. A. Shishlenin, and B. B. Sholpanbaev, “Inverse problems for the ground penetrating radar,” J. Inverse Ill-Posed Probl. 21, 885–892 (2013).MathSciNetzbMATHGoogle Scholar
  17. 17.
    V. Isakov, “Recovery of time dependent volatility coefficient by linearization,” Evolution Equations Control Theory 3, 119–134 (2014).MathSciNetCrossRefGoogle Scholar
  18. 18.
    V. Isakov, Z.-C. Deng, and B. Hon, “Recovery of time dependent volatility in option pricing model,” Inverse Probl. 32 (11), 115010 (2016).MathSciNetCrossRefGoogle Scholar
  19. 19.
    R. Y. Zhang, F. F. Xu, and J. C. Huang, “Reconstructing local volatility using total variation,” Acta Math. Sinica— English Ser. 33 (2), 263–277 (2017).MathSciNetCrossRefGoogle Scholar
  20. 20.
    S. I. Kabanikhin and M. A. Shishlenin, “Recovering time-dependent diffusion coefficient from nonlocal data,” Sib. Zh. Vychisl. Mat. 21 (1), 55–63 (2018).zbMATHGoogle Scholar
  21. 21.
    S. I. Kabanikhin and M. A. Shishlenin, “Recovering a time-dependent diffusion coefficient from nonlocal data,” Numer. Anal. Appl. 11 (1), 38–44 (2018).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Wichita State UniversityWichitaUSA
  2. 2.Sobolev Institute of Mathematics, Russian Academy of Sciences, Novosibirsk State UniversityNovosibirskRussia
  3. 3.Moscow Institute of Physics and TechnologyDolgoprudnyiRussia
  4. 4.Tianjin University of Finance and EconomicsBeijingChina

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