On nonlinear ill-posed inverse problems with applications to pricing of defaultable bonds and option pricing Authors XiaoHong Chen Department of Economics Yale University The Guanghua School of Management Peking University School of Economics Shanghai University of Finance and Economics Demian Pouzo Department of Economics New York University Article

First Online: 23 July 2009 Received: 30 November 2008 Accepted: 14 January 2009 DOI :
10.1007/s11425-009-0058-y

Cite this article as: Chen, X. & Pouzo, D. Sci. China Ser. A-Math. (2009) 52: 1157. doi:10.1007/s11425-009-0058-y
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.

Keywords nonlinear ill-posed inverse problems Hilbert Scales optimal convergence rates pricing of defaultable bonds option prices The first anther was supported by US National Science Foundation (Grant No. SES-0631613) and the Cowles Foundation for Research in Economics

References 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 CrossRef MathSciNet 2.

Horowitz J, Lee S. Nonparametric instrumental variables estimation of a quantile regression model.

Econometrica ,

75 : 1191–1208 (2007)

MATH CrossRef MathSciNet 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 CrossRef MathSciNet 4.

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

5.

Neubauer A. Tikhonov regularization of Nonlinear ill-posed Problems in Hilbert Scales.

Appl Anal ,

46 : 59–72 (1992)

MATH CrossRef MathSciNet 6.

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

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

8.

Hall P, J Horowitz Nonparametric Methods for Inference in the Presence of Instrumental Variables.

Ann Statist ,

33 : 2904–2929 (2005)

MATH CrossRef MathSciNet 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 CrossRef MathSciNet 10.

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

11.

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

12.

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

13.

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

14.

Yang Y, Barron A. Information-theoretic determination of minimax rates of convergence.

Ann Statist ,

27 : 1564–1599 (1999)

MATH CrossRef MathSciNet 15.

Arellano C. Default risk and income fluctuations in emerging economies.

Amer Econ Rev ,

98 : 690–713 (2008)

CrossRef 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 CrossRef MathSciNet 17.

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

18.

Kalthenbacher B, Neubauer A. Convergence of projected iterative regularization methods for nonlinear problems with smooth solultions.

Inverse Problems ,

22 : 1105–1119 (2006)

CrossRef MathSciNet 19.

Efromovch S, Koltchinskii V. On inverse problems with unknown operators.

IEEE Trans Inform Theory ,

47 : 2876–2894 (2001)

CrossRef MathSciNet 20.

Hofmann M, Reiss M. Nonlinear estimation for linear inverse problems with error in the operator.

Ann Statist ,

36 : 310–336 (2007)

CrossRef 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 CrossRef MathSciNet 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 23.

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

24.

Cavalier L, Raimondo M. Wavelet deconvolution with noisy eigenvalues.

IEEE Trans Signal Process ,

55 (3): 2414–2424 (2007)

CrossRef 25.

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

26.

Huang J. Projection estimation in multiple regression with application to functional ANOVA models.

Ann Statist ,

26 : 242–272 (1998)

MATH CrossRef MathSciNet 27.

Blundell R, Chen X, Kristensen D. Semi-nonparametric IV estimation of shape-invariant Engel curves.

Econometrica ,

75 : 1613–1669 (2007)

MATH CrossRef MathSciNet © Science in China Press and Springer-Verlag GmbH 2009