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Vietnam Journal of Mathematics

, Volume 44, Issue 3, pp 513–530 | Cite as

Regularization for the Inverse Problem of Finding the Purely Time-Dependent Volatility

  • Dang Duc Trong
  • Dinh Ngoc Thanh
  • Nguyen Nhu LanEmail author
Article
  • 87 Downloads

Abstract

We consider the inverse problem of finding the volatility σL ρ (0, T) such that \(U_{BS}(X,K,r,t,{{\int }_{0}^{t}}\sigma ^{2}(\tau )d\tau )=u(t)\), 0≤tT, where U B S is the Black–Scholes formula and u(t) is the observable fair price of an European call option. The problem is ill-posed. Using the residual method, we shall regularize the problem. An explicit error estimate is given.

Keywords

Calibration Volatility Ill-posed Regularization 

Mathematics Subject Classification (2010)

35R30 65J20 91B24 

Notes

Acknowledgments

The authors are grateful to three anonymous referees for their precious suggestions leading to the improvement version of our paper.

References

  1. 1.
    Baumeister, J.: Stable solution of inverse problems. Friedr. Vieweg & Son (1987)Google Scholar
  2. 2.
    Bouchouev, I., Isakov, V.: The inverse problem of option pricing. Inverse Probl. 13, L11–L17 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bouchouev, I., Isakov, V.: Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets. Inverse Probl. 15, R95–R116 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bouchouev, I., Isakov, V., Valdivia, N.: Recovery of volatility coefficient by linearization. Quant. Financ. 2, 257–263 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chargory-Corona, J., Ibarra-Valdez, C.: A note on Black–Scholes implied volatility. Phys. A 370, 681–688 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Crépey, S.: Calibration of the local volatility in a generalized Black–Scholes model using Tikhonov regularization. SIAM J. Math. Anal. 34, 1183–1206 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    De Cezaro, A., Scherzer, O., Zubelli, J.P.: Convex regularization of local volatility models from option prices: convergence analysis and rates. Nonlinear Anal. 75, 2398–2415 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Deng, Z.-C., Yu, J.-N., Yang, L.: An inverse problem of determining the implied volatility in option pricing. J. Math. Anal. Appl. 340, 16–31 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Egger, H., Engl, H.W.: Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates. Inverse Probl. 21, 1027–1045 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Egger, H., Hein, T., Hofmann, B.: On decoupling of volatility smile and term structure in inverse option pricing. Inverse Probl. 22, 1247–1259 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Engl, H.W., Zou, J.: A new approach to convergence rate analysis of Tikhonov regularization for parameter identification in heat conduction. Inverse Probl. 16, 1907–1923 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Guo, L.: The mollification analysis of Stochastic volatility. Actuar. Res. Clear. House 1, 409–419 (1998)Google Scholar
  13. 13.
    Hein, T.: Some analysis of Tikhonov regularization for the inverse problem of option pricing in the price-dependent case. Z. Anal. Anwend. 24, 593–609 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hein, T., Hofmann, B.: On the nature of ill-posedness of an inverse problem arising in option pricing. Inverse Probl. 19, 1319–1338 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hofmann, B., Krämer, R.: On maximum entropy regularization for a specific inverse problem of option pricing. J. Inverse Ill-Posed Probl. 13, 41–63 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Krämer, R., Mathé, P.: Modulus of continuity of Nemytskii operators with application to the problem of option pricing. J. Inverse Ill-Posed Probl. 16, 435–461 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kwok, Y.-K.: Mathematical models of financial derivatives. Springer, Berlin–Heidelberg (1998)zbMATHGoogle Scholar
  18. 18.
    Lishang, J., Youshan, T.: Identifying the volatility of underlying assets from option prices. Inverse Probl. 17, 137–155 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lu, L., Yi, L.: Recovery implied volatility of underlying asset from European option price. J. Inverse Ill-Posed Probl. 17, 499–509 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    McDonald, R.L.: Derivatives markets. Addison Wesley (2006)Google Scholar
  21. 21.
    Roberts, A.J.: Elementary calculus of financial mathematics. SIAM, Philadelphia (2009)CrossRefzbMATHGoogle Scholar
  22. 22.
    Tikhonov, A., Arsénine, V.: Méthodes de résolution de problèmes mal posés. Édition Mir (1974)Google Scholar
  23. 23.
    Trong, D.D., Thanh, D.N., Lan, N.N., Uyen, P.H.: Calibration of the purely T-independent Black–Scholes implied volatility. Appl. Anal. 93, 859–874 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zang, K., Wang, S.: A computational scheme for uncertain volatility model in option pricing. Appl. Numer. Math. 59, 1754–1767 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

Authors and Affiliations

  • Dang Duc Trong
    • 1
  • Dinh Ngoc Thanh
    • 1
  • Nguyen Nhu Lan
    • 2
    Email author
  1. 1.Department of Mathematics and Computer ScienceHo Chi Minh City University of Science, Vietnam National UniversityHo Chi Minh CityVietnam
  2. 2.Tay Do UniversityCan ThoVietnam

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