We present an Hilbert space formulation for a set of implied volatility
models introduced in \cite{BraceGoldys01} in which the authors studied
conditions for a family of European call options, varying the maturing time and
the strike price T an K, to be arbitrage free. The arbitrage free
conditions give a system of stochastic PDEs for the evolution of the implied
volatility surface σ^t(T,K). We will focus on the family obtained
fixing a strike K and varying T. In order to give conditions to prove an
existence-and-uniqueness result for the solution of the system it is here
expressed in terms of the square root of the forward implied volatility and
rewritten in an Hilbert space setting. The existence and the uniqueness for the
(arbitrage free) evolution of the forward implied volatility, and then of the
the implied volatility, among a class of models, are proved. Specific examples
are also given.Comment: 21 page