Finance and Stochastics

, Volume 10, Issue 4, pp 449–474 | Cite as

Spectral calibration of exponential Lévy models

Article

Abstract

We investigate the problem of calibrating an exponential Lévy model based on market prices of vanilla options. We show that this inverse problem is in general severely ill-posed and we derive exact minimax rates of convergence. The estimation procedure we propose is based on the explicit inversion of the option price formula in the spectral domain and a cut-off scheme for high frequencies as regularisation.

Keywords

European option Jump diffusion Minimax rates Severely ill-posed Nonlinear inverse problem Spectral cut-off 

Mathematics Subject Classification (2000)

60G51 62G20 91B28 

JEL Classification

G13 C14 

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References

  1. 1.
    Aït-Sahalia Y., Duarte J. Nonparametric option pricing under shape restrictions. J. Econom. 116(1–2), 9–47 (2003)Google Scholar
  2. 2.
    Aït-Sahalia Y., Jacod, J.: Volatility estimators for discretely sampled Lévy processes. Ann. Stat. (in press) (2006)Google Scholar
  3. 3.
    Belomestny, D., Reiß, M.: Optimal calibration of exponential Lévy models, Preprint 1017, Weierstraß Institute, Berlin. http://www.wias-berlin.de (2005)Google Scholar
  4. 4.
    Belomestny, D., Reiß, M.: Spectral calibration of exponential Lévy models [2]. Discussion Paper 35, Collaborative Research Center 649 Economic Risk, Berlin. http://sfb649.wiwi.hu-berlin.de (2006)Google Scholar
  5. 5.
    Breeden D., Litzenberger R. (1978) Prices of state-contingent claims implicit in options prices. J. Business 51, 621–651CrossRefGoogle Scholar
  6. 6.
    Brown L.D., Low M.G. (1996) Asymptotic equivalence of nonparametric regression and white noise. Ann. Stat. 24, 2384–2398MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Butucea C., Matias C. (2005) Minimax estimation of the noise level and of the deconvolution density in a semiparametric convolution model. Bernoulli 11, 309–340MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Carr P., Geman H., Madan D.B., Yor M. (2002) The fine structure of asset returns: An empirical investigation. J. Business 75, 305–332CrossRefGoogle Scholar
  9. 9.
    Carr P., Madan D. (1999) Option valuation using the fast Fourier transform. J. Comput. Financ. 2, 61–73Google Scholar
  10. 10.
    Cont R., Tankov P. Financial Modelling With Jump Processes. In: Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton (2004)Google Scholar
  11. 11.
    Cont R., Tankov P. (2004) Nonparametric calibration of jump-diffusion option pricing models. J. Comput. Financ. 7(3): 1–49Google Scholar
  12. 12.
    Cont R., Tankov P. Retrieving Lévy processes from option prices: regularization of an ill-posed inverse problem. SIAM J. Numer. Opt. Control (in press) (2005)Google Scholar
  13. 13.
    Cont R., Voltchkova E. (2005) Integro-differential equations for option prices in exponential Lévy models. Financ. Stoch. 9, 299–325MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Crépey S. (2003) Calibration of the local volatility in a generalized Black–Scholes model using Tikhonov regularization. SIAM J. Math. Anal. 34: 1183–1206MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Duffie D., Filipović D., Schachermayer W. (2003) Affine processes and applications in finance. Ann. Appl. Probab. 13, 984–1053MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Eberlein E., Keller U., Prause K. (1998) New insights into smile, mispricing, and value at risk: the hyperbolic model. J. Business 71, 371–405CrossRefGoogle Scholar
  17. 17.
    Emmer S., Klüppelberg C. (2004) Optimal portfolios when stock prices follow an exponential Lévy process. Financ. Stoch. 8, 17–44MATHCrossRefGoogle Scholar
  18. 18.
    Fengler, M.: Semiparametric modeling of implied volatility. In: Springer Finance Series (2005)Google Scholar
  19. 19.
    Goldenshluger A., Tsybakov A., Zeevi A. (2006) Optimal change-point estimation from indirect observations. Ann. Stat. 34, 350–372MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Jackson N., Süli E., Howison S. (1999) Computation of deterministic volatility surfaces. J. Comput. Financ. 2(2): 5–32Google Scholar
  21. 21.
    Kallsen J. (2000) Optimal portfolios for exponential Lévy processes. Math. Meth. Oper. Res. 51, 357–374MathSciNetCrossRefGoogle Scholar
  22. 22.
    Korostelev, A., Tsybakov, A.: Minimax Theory of Image Reconstruction. Lecture Notes in Statistics vol. 82. Springer, Berlin Heidelberg New York (1993)Google Scholar
  23. 23.
    Kou S. (2002) A jump diffusion model for option pricing. Manag. Sci. 48: 1086–1101CrossRefGoogle Scholar
  24. 24.
    Merton R. (1976) Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3, 125–144CrossRefMATHGoogle Scholar
  25. 25.
    Mordecki E. (2002) Optimal stopping and perpetual options for Lévy processes. Financ. Stoch. 6, 473–493MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Weierstraß Institute for Applied Analysis and Stochastics (WIAS)BerlinGermany
  2. 2.Institute of Applied MathematicsUniversity of HeidelbergHeidelbergGermany

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