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The local vol model has exactly enough freedom to match the individual densities $X_t.$ There is no additional freedom in the local vol model to match even a joint density for a pair of times $(X_t,X_s).$ When you ask about the joint density across the continuum of times $t \in [0,T]$ it is pretty easy to show that any local vol model differs from any ...


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Look at Gilli & Schumann's paper. They provide a Bates' model estimates set, the way to improve such estimates calibrating those ones using an Heuristic model and, lastly, the relative codes in matlab, in order to be able to replicate the model. Unfortunately, there are not available the relative call prices estimated time series; I think that noone ...



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