A convex-regularization framework for local-Volatility calibration in derivative markets: the connection with convex-risk measures and exponential families

Abstract

We present a unified framework for the calibration of local volatility models that makes use of recent tools of convex regularization of ill-posed Inverse Problems. The unique aspect of the present approach is that it address in a general and rigorous way the key issue of convergence and sensitivity of the regularized solution when the noise level of the observed prices goes to zero. In particular, we present convergence results that include convergence rates with respect to noise level in fairly general contexts and go well beyond the classical quadratic regularization. Our approach directly relates to many of the different techniques that have been used in volatility surface estimation. In particular, it directly connects with the Statistical concept of exponentia families and entropy-based estimation. Finally, we also show that our framework connects with the Financial concept of Convex Risk Measures.