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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

6 Issues per year


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Convergence rates results for recovering the volatility term structure including at-the-money options

T. Hein1

1Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany. Email:

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 17, Issue 4, Pages 359–373, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: 10.1515/JIIP.2009.024, June 2009

Publication History

Received:
2007-01-20
Published Online:
2009-06-16

Abstract

Determining the term structure of local volatilities with at-the-money options represents a singular situation. On the one hand, prices of options with strikes close to the current asset price are known as most sensitive to variations in the volatility. Therefore such options should be preferred for reconstruction problems. On the other hand, the analysis of corresponding inverse problems seems to be much easier, if at-the-money options are excluded. In particular, convergence rate results for regularization approaches were formulated preferably for in-the-money and out-of-the-money options. This paper is contribution to bridge the gap. By application of a generalized Tikhonov regularization approach we present convergence rate results by formulating the underling inverse problem in appropriate spaces.

Key words.: inverse problem of option pricing; identification of local volatilities; Black–Scholes model; parabolic equations; ill-posed problem; regularization

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