  Consistently fitting vanilla option surfaces is an important issue when it
comes to modelling in finance. Local volatility models introduced by Dupire in
1994 are widely used to price and manage the risks of structured products.
However, the inconsistencies observed between the dynamics of the smile in
those models and in real markets motivate researches for stochastic volatility
modelling. Combining both those ideas to form Local and Stochastic Volatility
models is of interest for practitioners. In this paper, we study the
calibration of the vanillas in those models. This problem can be written as a
nonlinear and nonlocal partial differential equation, for which we prove
short-time existence of solutions.
