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

• Published in: Nonlinear analysis Vol. 75; no. 4; pp. 2398 - 2415
• Main Authors: De Cezaro, A., Scherzer, O., Zubelli, J.P.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.03.2012
• Subjects: Convergence; Convergence rates; Convex regularization; Convex risk measures; Functionals; Local volatility surface identification; Noise levels; Nonlinearity; Regularization; Risk; Source condition interpretation; Spreads; Volatility; Convex regularization; Convex risk measures; Local volatility surface identification; Convergence rates; primary; Source condition interpretation
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2011.10.037

We study a convex regularization of the local volatility surface identification problem for the Black–Scholes 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 bid–ask 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.