Non-parametric calibration of the local volatility surface for European options using a second-order Tikhonov regularization

Abstract

We calibrate the local volatility surface for European options across all strikes and maturities of the same underlying. There is no interpolation or extrapolation of either the option prices or the volatility surface. We do not make any assumption regarding the shape of the volatility surface except to assume that it is smooth. Due to the smoothness assumption, we apply a second-order Tikhonov regularization. We choose the Tikhonov regularization parameter as one of the singular values of the Jacobian matrix of the Dupire model. Finally we perform extensive numerical tests to assess and verify the aforementioned techniques for both volatility models with known analytical solutions of European option prices and real market option data.

Keywords:

• Local volatility surface
• Second-order Tikhonov regularization
• SVD
• Large scale non-linear inverse problem

JEL Classification:

• C61
• C

View correction statement:

Erratum

Acknowledgments

The authors would like to thank Dr. Cristian Homescu from Wells Fargo for his helpful comments. The authors also owe their sincere gratitude to the two reviewers for patiently pointing out both some schematic deficiencies that we were able to make up during the revision process and also some typographical errors that make this paper much stronger. Prof. I.M. Navon would like to acknowledge support from NSF grant ATM-0931198. Dr. Xiao Chen would like to acknowledge support from the US Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

Notes

There are several free automatic differentiation tools available, whose details are to be found on the website www.autodif.org Automatic differentiation can help speed up the process of developing the numerical code of an adjoint model especially for complicated models. However, some debugging and verification is usually necessary for checking the validity of the code generated by the free automatic differentiation tools. For a method to verity the correctness of the adjoint code, please see the gradient test in Navon et al. (1992).

This article was originally publishedwith errors. This version has been corrected. Please see Erratum (http://dx.doi.org/10.1080/14697688.2013.844894).