On the calibration of local jump-diffusion asset price models

We consider the inverse problem of calibrating a localized jump-diffusion process to given option price data. It is shown that applying Tikhonov regularization to the originally ill-posed problem yields a well-posed optimization problem. For the solution of the latter, i.e., the calibrated (infinite-dimensional) parameter of the process, we prove the stability and furthermore obtain convergence results. The work-horse for these proofs is the forward partial integro-differential equation associated to the European call price. Moreover, by providing a precise link between the parameters and the corresponding asset price models, we are able to carry over the stability and convergence results to the associated asset price models and hence to the model prices of exotic derivatives. Finally we indicate some possible applications.