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Numerical Identification of Time-Dependent Volatility in European Options with Two-Stage Regime-Switching

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Book cover Advances in High Performance Computing (HPC 2019)

Part of the book series: Studies in Computational Intelligence ((SCI,volume 902))

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Abstract

We develop numerical algorithms to solve inverse problems of determining time-dependent volatility according to point measurements inside of a truncated domain for regime-switching models of European options. An average linearization in time of the diffusion terms of the initial-boundary problems is used. Difference schemes on Tavella-Randall grids are derived. The numerical method is based on a decomposition of the difference solution with respect to the volatility for which the transition to the new time layer is carried out by solving two discrete elliptic system problems. Numerical experiments are performed to verify the effectiveness and robustness of the new algorithms.

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References

  1. Achdou, Y., Pironneau, O.: Computational methods for option pricing. In: SIAM Frontiers in Applied Mathematics (2005)

    Google Scholar 

  2. Avellaneda, M., Friedman, C., Holmes, R., Samperi, D.: Calibrating volatility surfaces via relative entropy minimization. Appl. Math. Finance 4, 37–64 (1997)

    Article  MATH  Google Scholar 

  3. Baumeister, J.: Inverse problems in finance. In: Gestner, T., Kloeden, P. (eds.) Recent Developments in Computational Finance, Interdisciplinary Mathematical Sciences, vol. 14, pp. 81-159 (2013)

    Google Scholar 

  4. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 61, 637–654 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bouchouev, I., Isakov, V.: The inverse problem of option pricing. Inverse Prob. 13, 7–11 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Buffington, J., Elliott, R.J.: Regime switching and European options. In: Pasik-Duncan, B. (ed.) Stochastic Theory and Control. Proceedings of a Workshop, pp. 73–82. Springer, Heidelberg (2002)

    Google Scholar 

  7. Coleman, T.F., Li, Y., Verma, A.: Reconstructing the unknown local volatility function. World Sci. Book Chapters 2, 77–102 (2015)

    MATH  Google Scholar 

  8. Crépey, S.: Calibration of the local volatility in a generalized Black-Scholes models using Tikhonov regularization. SIAM J. Math. Anal. 34, 1183–1206 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dai, M., Zhang, Q., Zhu, J.Q.: Trend following trading under regime switching models. SIAM J. Finance Math. 1(1), 780–810 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Deng, Z.-C., Hon, Y.C., Isakov, V.: Recovery of the time-dependent volatility in optionpricing model. Inverse Prob. 32, 115010 (2016)

    Article  MATH  Google Scholar 

  11. Dupire, B.: Pricing with a smile. Risk 7(2), 16–20 (1994)

    Google Scholar 

  12. Ehrhardt, M., Günther, M., ter Maten, E.J.W. (eds.) Novel Methods in Computational Finance. Springer (2017)

    Google Scholar 

  13. Georgiev, S.G., Vulkov, L.G.: Computational recovery of time-dependent volatility from integral observations in option pricing. J. Comput. Sci. 39, 101054 (2019)

    Article  MathSciNet  Google Scholar 

  14. Georgiev, S.G., Vulkov, L.G.: Numerical determination of the right boundary condition for regime-switching models of European options from point observations. In: AIP Conference Proceedings, vol. 2048, p. 030003 (2018)

    Google Scholar 

  15. Georgiev, S.G.: Numerical and analytical computation of the implied volatility from option price measurements under regime–switching. In: AIP Conference Proceedings, vol. 2172, p. 070007 (2019)

    Google Scholar 

  16. He, X.-J., Zhu, S.-P.: On full calibration of hybrid local volatility and regime-switching models. J. Future Mark. 38(5), 586–606 (2018)

    Article  Google Scholar 

  17. Isakov, V.: Recovery of time-dependent volatility coefficient by linearization. Evol. Equ. Control Theory 3(1), 119–134 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Naik, V.: Option valuation and hedging strategies with jumps in the volatility of asset returns. J. Finance 48(5), 1969–1984 (1993)

    Article  Google Scholar 

  19. Orlando, G., Taglilatela, G.: A review on implied volatility calculation. J. Comput Appl. Math. 320, 202–220 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Samarskii, A.A., Vabishchevich, P.: Numerical Methods for Solving Inverse Problems of Mathematical Physics. Walter de Gruter, Berlin (2007)

    Book  MATH  Google Scholar 

  21. Ševčovič, D., Stehlíková, B., Mikula, K.: Analytical and Numerical Methods for Pricing Financial Derivatives. Nova Science, Hauppauge (2011)

    Google Scholar 

  22. Siu, T.K., Yang, H., Liu, J.W.: Pricing currency option under two-factor Markov-modulated stochastic volatility models. Insur. Math. Econ. 43(3), 295–302 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tavella, D., Randall, C.: Pricing Financial Instruments: The Finite Difference Method. Wiley, Hoboken (2000)

    Google Scholar 

  24. Vabishchevich, P.N., Klibanov, M.V.: Numerical identification of the leading coefficient of a parabolic equation. Differ. Equ. 52(7), 896–953 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Valdez, A.R.L., Vargiolu, T.: Optimal portfolio in the regime-switching model, In: Dalang, R.C., Dozzi, M., Russo, F. (eds.) Proceedings of the Ascona’11 Seminar on Stochastic Analysis, Random Fields and Applications, pp. 435-449 (2013)

    Google Scholar 

  26. Valkov, R.: Fitted finite volume method for generalized Black-Scholes equation transformed on finite interval. Numer. Algorithm 65, 195–220 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Windcliff, H., Forsyth, P.A., Vetzal, K.R.: Analysis of the stability of linear boundary condition for the Black-Sholes equation. J. Comp. Finance 8, 65–92 (2004)

    Article  Google Scholar 

  28. Xi, X., Rodrigo, M.R., Mamon, R.S.: Parameter estimation of a regime-switching model using an inverse Stieltjes moment approach. In: Cohen, S.N., Madan, D., Siu, T.K., Yang, H. (eds.) Stochastic Processes, Finance And Control: A Festschrift in Honor of Robert J. Elliott, pp. 549-569 World scientific, Singapore (2012)

    Google Scholar 

  29. Zhang, F.: Matrix Theory - Basic Results and Techniques, 2nd edn. Springer, Heidelberg (2011)

    Book  MATH  Google Scholar 

  30. Zhang, K., Teo, K.L., Swarts, M.: A robust numerical scheme to pricing American options under regime-switching based on penalty method. Comput. Econ. 43, 463–483 (2013)

    Article  Google Scholar 

  31. Zhu, S.-P., Badzan, A., Lu, X.: A new exact solution for pricing European options in a two-state regime-switching economy. Comp. Math. Appl. 64(8), 2744–2755 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zubelli, J.P.: Inverse problems in finance, a short survey of calibration techniques. In: Kolev, N., Morettin, P. (eds.) Proceedings of the Second Brazilian Conference on Statistical Modelling in Insurance and Finance, pp. 64-76 (2005)

    Google Scholar 

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Acknowledgements

The authors are thankful to the reviewers for their constructive comments and suggestions, which significantly improved the quality of the paper.

This paper contains results of the work on project No 2019 - FNSE - 05, financed by “Scientific Research” Fund of Ruse University. The research is also supported by the Bulgarian National Science Fund under Project DN 12/4 “Advanced analytical and numerical methods for nonlinear differential equations with application in finance and environmental pollution” from 2017, and project No 2019 - FNSE - 03.

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Authors and Affiliations

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

    Slavi G. Georgiev & Lubin G. Vulkov

Authors
  1. Slavi G. Georgiev
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  2. Lubin G. Vulkov
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Correspondence to Slavi G. Georgiev .

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Editors and Affiliations

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

    Ivan Dimov

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

    Stefka Fidanova

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Georgiev, S.G., Vulkov, L.G. (2021). Numerical Identification of Time-Dependent Volatility in European Options with Two-Stage Regime-Switching. In: Dimov, I., Fidanova, S. (eds) Advances in High Performance Computing. HPC 2019. Studies in Computational Intelligence, vol 902. Springer, Cham. https://doi.org/10.1007/978-3-030-55347-0_21

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