I have read that there are 3 types of pricing models: local volatility, stochastic volatility and stochastic-local volatility models (LSV).
I am now looking at interest rates exotics pricing models and I see that LIBOR market model (LMM) is the market standard for simple exotics. But given this model cannot fit the smile since you are just simulating all the forward rates under the same measure, via a series of drift corrections, the solution is adding stochastic volatility to LMM to price more complex structures.
But how would you classify this model given that we can either have Local or Stochastic vol models (or mix of the two, as in LSV)? Does LMM with stochastic volatility fall under the LSV category?