Skip to main content

On nonlinear ill-posed inverse problems with applications to pricing of defaultable bonds and option pricing

Abstract

This paper considers the estimation of an unknown function h that can be characterized as a solution to a nonlinear operator equation mapping between two infinite dimensional Hilbert spaces. The nonlinear operator is unknown but can be consistently estimated, and its inverse is discontinuous, rendering the problem ill-posed. We establish the consistency for the class of estimators that are regularized using general lower semicompact penalty functions. We derive the optimal convergence rates of the estimators under the Hilbert scale norms. We apply our results to two important problems in economics and finance: (1) estimating the parameters of the pricing kernel of defaultable bonds; (2) recovering the volatility surface implied by option prices allowing for measurement error in the option prices and numerical error in the computation of the operator.

This is a preview of subscription content, access via your institution.

References

  1. 1

    Bissantz N, Hohage T, Munk A. Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise. Inverse Problems, 20: 1773–1789 (2004)

    MATH  Article  MathSciNet  Google Scholar 

  2. 2

    Horowitz J, Lee S. Nonparametric instrumental variables estimation of a quantile regression model. Econometrica, 75: 1191–1208 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  3. 3

    Lu S, Pereverzev S, Ramlau R. An Analysis of Tikhonov regularization for nonlinear ill-posed problems under a general smoothness assumption. Inverse Problems, 23: 217–230 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  4. 4

    Engl H W, Hanke M, Neubauer A. Regularization in Inverse Problems. Dordrecht: Kluwer, 1996

    Google Scholar 

  5. 5

    Neubauer A. Tikhonov regularization of Nonlinear ill-posed Problems in Hilbert Scales. Appl Anal, 46: 59–72 (1992)

    MATH  Article  MathSciNet  Google Scholar 

  6. 6

    Chen X, Pouzo D. Estimation of nonparametric conditional moment models with possibly nonsmooth moments. Cowles Foundation Discussion Paper No. 1640, 2008

  7. 7

    Chernozhukov V, Gagliardini P, Scaillet O. Nonparametric Instrumental Variable Estimation of Quantile Structural Effects. Mimeo, MIT, University of Lugano and Swiss Finance Institute, 2008

    Google Scholar 

  8. 8

    Hall P, J Horowitz Nonparametric Methods for Inference in the Presence of Instrumental Variables. Ann Statist, 33: 2904–2929 (2005)

    MATH  Article  MathSciNet  Google Scholar 

  9. 9

    Nair M T, Pereverzev S V, Tautenhahn U. Regularization in Hilbert scales under general smoothing conditions. Inverse Problems, 21(6): 1851–1869 (2005)

    MATH  Article  MathSciNet  Google Scholar 

  10. 10

    Darolles S, Florens J P, Renault E. Nonparametric Instrumental Regression. Mimeo, Toulouse School of Economics, 2006

    Google Scholar 

  11. 11

    Chen X, Reiss M. On rate optimality for ill-posed problems in econometrics. Cowles Foundation Discussion Paper No. 1626, 2008

  12. 12

    Florens J P, Johannes J, Van Bellegem S. Identification and estimation by penalization in Nonparametric Instrumental Regression. Mimeo, IDEI, University Toulouse I, 2008

    Google Scholar 

  13. 13

    Florens J P, Van Bellegem S. Slides Presentation in CIRANO and CIREQ Econometrics Conference on GMM, Montreal, Canada, November, 2007

  14. 14

    Yang Y, Barron A. Information-theoretic determination of minimax rates of convergence. Ann Statist, 27: 1564–1599 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  15. 15

    Arellano C. Default risk and income fluctuations in emerging economies. Amer Econ Rev, 98: 690–713 (2008)

    Article  Google Scholar 

  16. 16

    Chatterjee S, Corbae D, Nakajima M, et al. A Quantitative Theory of Unsecured Consumer Credit with Risk of Default. Econometrica, 75: 1525–1589 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  17. 17

    Haroske D. Entropy Numbers and Approximation Numbers in weighted Function spaces of type B s p,q and F s p,q , Eigenvalue Distributions of Some degenerate Pseudodifferential Operators. PhD Dissertation, Friedrich Schiller University, Jena, 1995

  18. 18

    Kalthenbacher B, Neubauer A. Convergence of projected iterative regularization methods for nonlinear problems with smooth solultions. Inverse Problems, 22: 1105–1119 (2006)

    Article  MathSciNet  Google Scholar 

  19. 19

    Efromovch S, Koltchinskii V. On inverse problems with unknown operators. IEEE Trans Inform Theory, 47: 2876–2894 (2001)

    Article  MathSciNet  Google Scholar 

  20. 20

    Hofmann M, Reiss M. Nonlinear estimation for linear inverse problems with error in the operator. Ann Statist, 36: 310–336 (2007)

    Article  Google Scholar 

  21. 21

    Egger H, Engl H W. Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates. Inverse Problems, 21: 1027–1045 (2005)

    MATH  Article  MathSciNet  Google Scholar 

  22. 22

    Yi B, Ye Y. Recovering the local volatility in Black-Scholes model by numerical differentiation. Appl Anal, 85(6–7): 681–692 (2006)

    MathSciNet  Google Scholar 

  23. 23

    Cavalier L. Nonparametric statistical inverse problems. Inverse Problems, 24(3), DOI: 10.108810266-5611/24 /3/2034004, 2008

  24. 24

    Cavalier L, Raimondo M. Wavelet deconvolution with noisy eigenvalues. IEEE Trans Signal Process, 55(3): 2414–2424 (2007)

    Article  Google Scholar 

  25. 25

    Gu C. Smoothing noisy data via regularization: statistical perspectives. Inverse Problems, 24: DOI: 10.108810266-5611/24/3/034002, 2008

  26. 26

    Huang J. Projection estimation in multiple regression with application to functional ANOVA models. Ann Statist, 26: 242–272 (1998)

    MATH  Article  MathSciNet  Google Scholar 

  27. 27

    Blundell R, Chen X, Kristensen D. Semi-nonparametric IV estimation of shape-invariant Engel curves. Econometrica, 75: 1613–1669 (2007)

    MATH  Article  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to XiaoHong Chen.

Additional information

The first anther was supported by US National Science Foundation (Grant No. SES-0631613) and the Cowles Foundation for Research in Economics

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chen, X., Pouzo, D. On nonlinear ill-posed inverse problems with applications to pricing of defaultable bonds and option pricing. Sci. China Ser. A-Math. 52, 1157–1168 (2009). https://doi.org/10.1007/s11425-009-0058-y

Download citation

Keywords

  • nonlinear ill-posed inverse problems
  • Hilbert Scales
  • optimal convergence rates
  • pricing of defaultable bonds
  • option prices

MSC(2000)

  • 15A29
  • 62G20
  • 91B02