Convex regularization of local volatility models from option prices: Convergence analysis and rates

A. De Cezaro, O. Scherzer, J. P. Zubelli

Research output: Contribution to journalArticlepeer-review

19 Scopus citations

Abstract

We study a convex regularization of the local volatility surface identification problem for the BlackScholes partial differential equation from prices of European call options. This is a highly nonlinear ill-posed problem which in practice is subject to different noise levels associated to bidask spreads and sampling errors. We analyze, in appropriate function spaces, different properties of the parameter-to-solution map that assigns to a given volatility surface the corresponding option prices. Using such properties, we show stability and convergence of the regularized solutions in terms of the Bregman distance with respect to a class of convex regularization functionals when the noise level goes to zero. We improve convergence rates available in the literature for the volatility identification problem. Furthermore, in the present context, we relate convex regularization with the notion of exponential families in Statistics. Finally, we connect convex regularization functionals with convex risk measures through Fenchel conjugation. We do this by showing that if the source condition for the regularization functional is satisfied, then convex risk measures can be constructed.

Original languageBritish English
Pages (from-to)2398-2415
Number of pages18
JournalNonlinear Analysis, Theory, Methods and Applications
Volume75
Issue number4
DOIs
StatePublished - Mar 2012

Keywords

  • Convergence rates
  • Convex regularization
  • Convex risk measures
  • Local volatility surface identification
  • Source condition interpretation

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