"A Linearization-Based Solution to the Ill-Posed Local Volatility Estimation Problem" 
June 1997, last revised December 2000.

Abstract

Given a set of option quotes at a point in time, we estimate a local volatility surface.  The heterogeneous time- and state-dependent parameters in this linearized model specification are identified by the Tikhonov regularization method.  Specifically, the optimal parameter estimates satisfy a least squares error criterion and are minimum sum of squared deviations from the Black-Scholes constant variance specification.  Applying Wahba’s (1983) generalized degrees of freedom (GDF) measure, we specify a test of time-dependent, state-dependent and maturity time- and state-dependent restrictions of the proposed time- and state-dependent local volatility specification.  As an application, we evaluate these restrictions for the November 25, 1991 set of Philadelphia Stock Exchange European Deutschemark option trades.  The restricted specifications either fail to meet the least squares error criterion or require more generalized degrees-of-freedom than the time- sand state-dependent local volatility specification which we propose..