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We present an Hilbert space formulation for a set of implied volatilitymodels introduced in \cite{BraceGoldys01} in which the authors studiedconditions for a family of European call options, varying the maturing time andthe strike price $T$ an $K$, to be arbitrage free. The arbitrage freeconditions give a system of stochastic PDEs for the evolution of the impliedvolatility surface ${\hat\sigma}_t(T,K)$. We will focus on the family obtainedfixing a strike $K$ and varying $T$. In order to give conditions to prove anexistence-and-uniqueness result for the solution of the system it is hereexpressed in terms of the square root of the forward implied volatility andrewritten in an Hilbert space setting. The existence and the uniqueness for the(arbitrage free) evolution of the forward implied volatility, and then of thethe implied volatility, among a class of models, are proved. Specific examplesare also given.

An Hilbert space approach for a class of arbitrage free implied volatilities models