Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates

This paper investigates the stable identification of local volatility surfaces sigma (S, t) in the Black-Scholes/Dupire equation from market prices of European Vanilla options. Based on the properties of the parameter-to-solution mapping, which assigns option prices to given volatilities, we show stability and convergence of approximations gained by Tikhonov regularization. In the case of a known term-structure of the volatility surface, in particular, if the volatility is assumed to be constant in time, we prove convergence rates under simple smoothness and decay conditions on the true volatility. The convergence rate analysis sheds light onto the importance of an appropriate a priori guess for the unknown volatility and the nature of the ill-posedness of the inverse problem, caused by smoothing properties and the nonlinearity of the direct problem. Finally, the theoretical results are illustrated by numerical experiments.