For the so-called local volatility model, Bruno Dupire derived a closed form solution, which, when naively applied, delivers cliffy volatility surfaces. A robust and fast parameter calibration scheme has to be found.
Call and/or put options on liquid assets or equity indices are traded for different expiries and for a range of strike prices. It turns out that the traded option prices do not fit into the constant volatility world of Black-Scholes but exhibit so-called “volatility smiles” or “volatility skewnesses”. A model for which such a behavior can be obtained without the need of stochastic volatility is the local volatility model. Bruno Dupire showed in 1994: If these call prices were available as a function, then the volatility must satisfy a complex function whose naive inversion yield extremly unstable results.
Mathematically, two conflicting targets should be achieved. On the one hand, the fit for the traded option data should be as good as possible (“model prices close to market prices”), on the other hand, the local volatility surface should be as smooth as possible. Nonlinear Tikhonov regularization with an appropriate regularization parameter choice fulfills the requirement of a fast and robust identification procedure.
Further Reading: Egger, Engl: Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates, Inverse Problems 21, 1027–1045, 2005.