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

Science China Mathematics 2009年第6期52卷 1157-1168页

作者： POUZO Demian Department of Economics New York University New York NY 10012 USA

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.

关键词： nonlinear ill-posed inverse problems Hilbert Scales optimal convergence rates pricing of defaultable bonds option prices 15A29 62G20 91B02

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Recovering implied risk-neutral probability density function using SVR

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Journal of Southeast University(English Edition) 2010年第3期26卷 489-493页

作者：胡小平崔海蓉朱丽华王新燕东南大学经济管理学院南京211189 南京农业大学工学院南京210032

Using support vector regression （SVR）, a novel non-parametric method for recovering implied risk-neutral probability density function （IRNPDF） is investigated by solving linear operator equations. First, the SVR principle for function approximation is introduced, and an SVR method for solving linear operator equations with knowing some values of the right-hand function and without knowing its form is depicted. Then, the principle for solving the IRNPDF based on SVR and the method for constructing cross-kernel functions are proposed. Finally, an empirical example is given to verify the validity of the method. The results show that the proposed method can overcome the shortcomings of the traditional parametric methods, which have strict restrictions on the option exercise price; meanwhile, it requires less data than other non-parametric methods, and it is a promising method for the recover of IRNPDF.

关键词： support vector regression option prices implied risk-neutral probability linear operator equation non-parametric method

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