Science in China Series A: Mathematics

, Volume 52, Issue 6, pp 1157–1168

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

Article

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 

MSC(2000)

15A29 62G20 91B02 

Copyright information

© Science in China Press and Springer-Verlag GmbH 2009

Authors and Affiliations

  1. 1.Department of EconomicsYale UniversityNew HavenUSA
  2. 2.The Guanghua School of ManagementPeking UniversityBeijingChina
  3. 3.School of EconomicsShanghai University of Finance and EconomicsShanghaiChina
  4. 4.Department of EconomicsNew York UniversityNew YorkUSA