Recovering Volatility in the Black-Scholes Model

In a Black-Scholes world prices of financial derivatives depend, for known inerest rate and dividend yield, solely on the local volatility of the underlying. Market prices of liquidly traded options, such as Vanilla Calls, are directly observable on financial markets and can be used for estimating local volatilities and thus calibrating the model to the market. The problem of volatility estimation is, like many parameter estimation problems for partial differential equations, in general illposed and has to be regularized. We formulate this inverse problem by means of the dual (Dupire) equation, use Tikhonov regularization and establish the basic stability and convergence results for a wide class of parameters, i.e. bounded volatilities with gradient in L_2, and various observation spaces (continuous, discrete in time, discrete in time and state). Additionally we show convergence rates for the case of time independent volatility (smile structure).