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Journal of Applied Mathematics
Volume 2014, Article ID 510819, 9 pages
http://dx.doi.org/10.1155/2014/510819
Research Article

Calibration of the Volatility in Option Pricing Using the Total Variation Regularization

1College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
2Department of Mathematics, Hunan First Normal University, Changsha 410205, China
3Department of Information and Computing Science, Changsha University, Changsha 410003, China

Received 6 November 2013; Revised 27 January 2014; Accepted 24 February 2014; Published 25 March 2014

Academic Editor: Roberto Renò

Copyright © 2014 Yu-Hua Zeng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In market transactions, volatility, which is a very important risk measurement in financial economics, has significantly intimate connection with the future risk of the underlying assets. Identifying the implied volatility is a typical PDE inverse problem. In this paper, based on the total variation regularization strategy, a bivariate total variation regularization model is proposed to estimate the implied volatility. We not only prove the existence of the solution, but also provide the necessary condition of the optimal control problem—Euler-Lagrange equation. The stability and convergence analyses for the proposed approach are also given. Finally, numerical experiments have been carried out to show the effectiveness of the method.