Online Local Volatility Calibration by Convex Regularization

Applicable Analysis and Discrete Mathematics, Vol. 8 (2014) Doi:10.2298/AADM140811012A

23 Pages Posted: 3 Nov 2012 Last revised: 27 Aug 2014

See all articles by Vinicius Albani

Vinicius Albani

Federal University of Santa Catarina

Jorge P. Zubelli

Instituto de Matematica Pura e Aplicada (IMPA)

Date Written: August 26, 2014

Abstract

We address the inverse problem of local volatility surface calibration from market given option prices. We integrate the ever-increasing ow of option price information into the well-accepted local volatility model of Dupire. This leads to considering both the local volatility surfaces and their corresponding prices as indexed by the observed underlying stock price as time goes by in appropriate function spaces. The resulting parameter to data map is defined in appropriate Bochner-Sobolev spaces. Under this framework, we prove key regularity properties. This enable us to build a calibration technique that combines online methods with convex Tikhonov regularization tools. Such procedure is used to solve the inverse problem of local volatility identification. As a result, we prove convergence rates with respect to noise and a corresponding discrepancy-based choice for the regularization parameter. We conclude by illustrating the theoretical results by means of numerical tests.

Keywords: Local Volatility Calibration, Convex Regularization, Online Estimation, Morozov's Principle, Convergence Rates

JEL Classification: C00, C60, C63, C80, C81

Suggested Citation

Albani, Vinicius and Zubelli, Jorge P., Online Local Volatility Calibration by Convex Regularization (August 26, 2014). Applicable Analysis and Discrete Mathematics, Vol. 8 (2014) Doi:10.2298/AADM140811012A, Available at SSRN: https://ssrn.com/abstract=2169927 or http://dx.doi.org/10.2139/ssrn.2169927

Vinicius Albani (Contact Author)

Federal University of Santa Catarina ( email )

Campus Trindade
Florianopolis, Santa Catarina 88040-900
Brazil
+554837216560 (Phone)
+554837214612 (Fax)

HOME PAGE: http://mtm.ufsc.br/~v.albani/

Jorge P. Zubelli

Instituto de Matematica Pura e Aplicada (IMPA) ( email )

Estrada Dona Castorina 110
Jardim Botanico
Rio de Janeiro, 22460
Brazil

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