Something has been bugging me for a while, and I can't really find an answer to it in papers. Maybe somebody can help me out.
In addition to modelling the instantaneous vol, or modelling forward variance swap prices, in order to explain the implied volatility surface some have tried to model implied volatility directly a la HJM.
One of the first to do this is afaik Ledoit and Santa Clara and they derive a complicated PDE that must be satisfied by all the implied vols. Other and subsequent market models for implied volatility, such as Schonbucher's, are in the spirit of Ledoit and Santa-Clara.
Now here is my question:
Since the PDE for the implied volatilities are derived from the option prices PDE, and as the latter is under an arbitrage free framework, does this mean that the IVs which are solutions to the PDE for the IVs arbitrage free as well, specifically will they satisfy for instance Lee's no arbitrage conditions not only today but also in the future?
I would think so, but is this really the case?