The inverse problem of option pricing

The Black-Scholes formula [6] provides with an elegant and simple method to price financial derivatives under the assumption that the stock price is log-normally distributed. However, the actual distribution of most assets is rarely log-normal, and theoretical prices of options with different strikes generated by the Black-Scholes formula differ from observed market prices. One way to reconcile the differences is to replace the log-normal process with constant volatility by a more general diffusion model. While in the log-normal it is sufficient to calculate a single volatility number from each option quote to calibrate the model, in the general diffusion framework the whole volatility function must be restored from collection of simulteneous option quotes with different strikes. Several useful numerical algorithms [1], [5], [9] to recover time dependent volatility from time-space data have been proposed, these algorithms are mostly based on regularized least squares fitting and as for typically for nonconvex minimization convergence properties have not been satisfactory for practitioners. Recently, in the case of time-space data an interesting relation between implied and local volatilities was discovered [2] and